Patent Document

[0001]    This application claims priority to U.S. Application Nos. 60/878,055, filed 3 Jan. 2007; 60/880,061, filed 12 Jan. 2007; 60/898,415, filed 31 Jan. 2007; 60/904,164, filed 1 Mar. 2007; 60/907,433, filed 2 Apr. 2007; 60/907,722, filed 13 Apr. 2007; 60/913,556, filed 24 Apr. 2007; 60/986,675, filed 9 Nov. 2007; 60/986,750, filed 9 Nov. 2007; and 60/988,537, filed 16 Nov. 2007, the complete disclosures of which are incorporated herein by reference. 
     
    
     FIELD OF THE INVENTION 
       [0002]    This invention relates to a system and method of physically solving the charge, mass, and current density functions of molecules, molecular ions, compounds and materials, and at least one part thereof, comprising at least one from the group of pharmaceuticals, allotropes of carbon, metals, silicon molecules, semiconductors, boron molecules, aluminum molecules, coordinate compounds, and organometallic molecules, and tin molecules, or any portion of these species, and computing and rendering the nature of these species using the solutions. The results can be displayed on visual or graphical media. The displayed information provides insight into the nature of these species and is useful to anticipate their reactivity, physical properties, and spectral absorption and emission, and permits the solution and display of other species. 
         [0003]    Rather than using postulated unverifiable theories that treat atomic particles as if they were not real, physical laws are now applied to atoms and ions. In an attempt to provide some physical insight into atomic problems and starting with the same essential physics as Bohr of the e −  moving in the Coulombic field of the proton with a true wave equation, as opposed to the diffusion equation of Schrödinger, a classical approach is explored which yields a model that is remarkably accurate and provides insight into physics on the atomic level. The proverbial view deeply seated in the wave-particle duality notion that there is no large-scale physical counterpart to the nature of the electron is shown not to be correct. Physical laws and intuition may be restored when dealing with the wave equation and quantum atomic problems. 
         [0004]    Specifically, a theory of classical quantum mechanics (CQM) was derived from first principles as reported previously [reference Nos. 1-8] that successfully applies physical laws to the solution of atomic problems that has its basis in a breakthrough in the understanding of the stability of the bound electron to radiation. Rather than using the postulated Schrödinger boundary condition: “Ψ→0 as r→∞”, which leads to a purely mathematical model of the electron, the constraint is based on experimental observation. Using Maxwell&#39;s equations, the classical wave equation is solved with the constraint that the bound n=1-state electron cannot radiate energy. Although it is well known that an accelerated point particle radiates, an extended distribution modeled as a superposition of accelerating charges does not have to radiate. A simple invariant physical model arises naturally wherein the predicted results are extremely straightforward and internally consistent requiring minimal math, as in the case of the most famous equations of Newton, Maxwell, Poincare, de Broglie, and Planck on which the model is based. No new physics is needed; only the known physical laws based on direct observation are used. 
         [0005]    Applicant&#39;s previously filed WO2005/067678 discloses a method and system of physically solving the charge, mass, and current density functions of atoms and atomic ions and computing and rendering the nature of these species using the solutions. The complete disclosure of this published PCT application is incorporated herein by reference. 
         [0006]    Applicant&#39;s previously filed WO2005/116630 discloses a method and system of physically solving the charge, mass, and current density functions of excited states of atoms and atomic ions and computing and rendering the nature of these species using the solutions. The complete disclosure of this published PCT application is incorporated herein by reference. 
         [0007]    Applicant&#39;s previously filed U.S. Published Patent Application No. 20050209788A1, relates to a method and system of physically solving the charge, mass, and current density functions of hydrogen-type molecules and molecular ions and computing and rendering the nature of the chemical bond using the solutions. The complete disclosure of this published application is incorporated herein by reference. 
         [0008]    Applicant&#39;s previously filed WO2007/051078 discloses a method and system of physically solving the charge, mass, and current density functions of polyatomic molecules and polyatomic molecular ions and computing and rendering the nature of these species using the solutions. The complete disclosure of this published PCT application is incorporated herein by reference. This incorporated application discloses complete flow charts and written description of a computer program that can be modified using the novel equations and description below to physically solve the charge, mass, and current density functions of the specific groups of molecules, molecular ions, compounds and materials disclosed herein and computing and rendering the nature of these specific groups. 
       BACKGROUND OF THE INVENTION 
       [0009]    The old view that the electron is a zero or one-dimensional point in an all-space probability wave function Ψ(x) is not taken for granted. The theory of classical quantum mechanics (CQM), derived from first principles, must successfully and consistently apply physical laws on all scales [1-8]. Stability to radiation was ignored by all past atomic models. Historically, the point at which QM broke with classical laws can be traced to the issue of nonradiation of the one electron atom. Bohr just postulated orbits stable to radiation with the further postulate that the bound electron of the hydrogen atom does not obey Maxwell&#39;s equations—rather it obeys different physics [1-12]. Later physics was replaced by “pure mathematics” based on the notion of the inexplicable wave-particle duality nature of electrons which lead to the Schrödinger equation wherein the consequences of radiation predicted by Maxwell&#39;s equations were ignored. Ironically, Bohr, Schrödinger, and Dirac used the Coulomb potential, and Dirac used the vector potential of Maxwell&#39;s equations. But, all ignored electrodynamics and the corresponding radiative consequences. Dirac originally attempted to solve the bound electron physically with stability with respect to radiation according to Maxwell&#39;s equations with the further constraints that it was relativistically invariant and gave rise to electron spin [13]. He and many founders of QM such as Sommerfeld, Bohm, and Weinstein wrongly pursued a planetary model, were unsuccessful, and resorted to the current mathematical-probability-wave model that has many problems [9-16]. Consequently, Feynman for example, attempted to use first principles including Maxwell&#39;s equations to discover new physics to replace quantum mechanics [17]. 
         [0010]    Physical laws may indeed be the root of the observations thought to be “purely quantum mechanical”, and it was a mistake to make the assumption that Maxwell&#39;s electrodynamic equations must be rejected at the atomic level. Thus, in the present approach, the classical wave equation is solved with the constraint that a bound n=1-state electron cannot radiate energy. 
         [0011]    Herein, derivations consider the electrodynamic effects of moving charges as well as the Coulomb potential, and the search is for a solution representative of the electron wherein there is acceleration of charge motion without radiation. The mathematical formulation for zero radiation based on Maxwell&#39;s equations follows from a derivation by Haus [18]. The function that describes the motion of the electron must not possess spacetime Fourier components that are synchronous with waves traveling at the speed of light. Similarly, nonradiation is demonstrated based on the electron&#39;s electromagnetic fields and the Poynting power vector. 
         [0012]    It was shown previously [1-8] that CQM gives closed form solutions for the atom, including the stability of the n=1 state and the instability of the excited states, the equation of the photon and electron in excited states, and the equation of the free electron and photon, which predict the wave-particle duality behavior of particles and light. The current and charge density functions of the electron may be directly physically interpreted. For example, spin angular momentum results from the motion of negatively charged mass moving systematically, and the equation for angular momentum, r×p, can be applied directly to the wave function (a current density function) that describes the electron. The magnetic moment of a Bohr magneton, Stern Gerlach experiment, g factor, Lamb shift, resonant line width and shape, selection rules, correspondence principle, wave-particle duality, excited states, reduced mass, rotational energies, and momenta, orbital and spin splitting, spin-orbital coupling, Knight shift, and spin-nuclear coupling, and elastic electron scattering from helium atoms, are derived in closed-form equations based on Maxwell&#39;s equations. The calculations agree with experimental observations. 
         [0013]    The Schrödinger equation gives a vague and fluid model of the electron. Schrödinger interpreted eΨ*(x)Ψ(x) as the charge-density or the amount of charge between x and x+dx (Ψ* is the complex conjugate of Ψ). Presumably, then, he pictured the electron to be spread over large regions of space. After Schrödinger&#39;s interpretation, Max Born, who was working with scattering theory, found that this interpretation led to inconsistencies, and he replaced the Schrödinger interpretation with the probability of finding the electron between x and x+dx as 
         [0000]      ∫Ψ(x)Ψ*(x)dx  (1)
 
         [0000]    Born&#39;s interpretation is generally accepted. Nonetheless, interpretation of the wave function is a never-ending source of confusion and conflict. Many scientists have solved this problem by conveniently adopting the Schrödinger interpretation for some problems and the Born interpretation for others. This duality allows the electron to be everywhere at one time—yet have no volume. Alternatively, the electron can be viewed as a discrete particle that moves here and there (from r=0 to r=∞), and ΨΨ* gives the time average of this motion. 
         [0014]    In contrast to the failure of the Bohr theory and the nonphysical, adjustable-parameter approach of quantum mechanics, multielectron atoms [1, 4] and the nature of the chemical bond [1, 5] are given by exact closed-form solutions containing fundamental constants only. Using the nonradiative wave equation solutions that describe the bound electron having conserved momentum and energy, the radii are determined from the force balance of the electric, magnetic, and centrifugal forces that corresponds to the minimum of energy of the system. The ionization energies are then given by the electric and magnetic energies at these radii. The spreadsheets to calculate the energies from exact solutions of one through twenty-electron atoms are given in &#39;06 Mills GUT [1] and are available from the interne [19]. For 400 atoms and ions, as well as hundreds of molecules, the agreement between the predicted and experimental results is remarkable. 
         [0015]    The background theory of classical quantum mechanics (CQM) for the physical solutions of atoms and atomic ions is disclosed in R. L. Mills, The Grand Unified Theory of Classical Quantum Mechanics, January 2000 Edition, BlackLight Power, Inc., Cranbury, N.J., ISBN 0963517147, Library of Congress Control Number 200091384, (“&#39;00 GUT”), provided by BlackLight Power, Inc. 493 Old Trenton Road, Cranbury, N.J. 08512; R. L. Mills, The Grand Unified Theory of Classical Quantum Mechanics, September 2001 Edition, BlackLight Power, Inc., Cranbury, N.J., ISBN 0963517155, Library of Congress Control Number 2001097371, (“&#39;01 GUT”), provided by BlackLight Power, Inc. 493 Old Trenton Road, Cranbury, N.J. 08512; R. L. Mills, The Grand Unified Theory of Classical Quantum Mechanics, May 2005 Edition, BlackLight Power, Inc., Cranbury, N.J., ISBN 0963517163, Library of Congress Control Number 2004101976, (“&#39;05 GUT”), provided by BlackLight Power, Inc. 493 Old Trenton Road, Cranbury, N.J. 08512; R. L. Mills, The Grand Unified Theory of Classical Quantum Mechanics, June 2006 Edition, Cadmus Professional Communications—Science Press Division, Ephrata, Pa., ISBN 0963517171, Library of Congress Control Number 2005936834, (“&#39;06 GUT”), provided by BlackLight Power, Inc. 493 Old Trenton Road, Cranbury, N.J. 08512; R. L. Mills, The Grand Unified Theory of Classical Quantum Mechanics, October 2007 Edition, Cadmus Professional Communications—A Conveo Company, Richmond, Va., ISBN 096351718X, Library of Congress Control Number 2007938695, (“&#39;07 GUT”), provided by BlackLight Power, Inc. 493 Old Trenton Road, Cranbury, N.J. 08512 posted at http://www.blacklightpower.com/theory/bookdownload.shtml., and in prior published PCT applications WO90/13126; WO92/10838; WO94/29873; WO96/42085; WO99/05735; WO99/26078; WO99/34322; WO99/35698; WO00/07931; WO00/07932; WO01/095944; WO01/18948; WO01/21300; WO01/22472; WO01/70627; WO02/087291; WO02/088020; WO02/16956; WO03/093173; WO03/066516; WO04/092058; WO05/041368; WO05/067678; WO2005/116630; WO2007/051078; and WO2007/053486, and U.S. Pat. Nos. 6,024,935 and 7,188,033; the entire disclosures of which are all incorporated herein by reference (hereinafter “Mills Prior Publications”). 
         [0016]    The following list of references, which are also incorporated herein by reference in their entirety, are referred to in the above sections using [brackets]:
   1. R. L. Mills, “The Grand Unified Theory of Classical Quantum Mechanics”, October 2007 Edition, Cadmus Professional Communications—A Conveo Company, Richmond, Va., ISBN 096351718X, Library of Congress Control Number 2007938695, at www.blacklightpower.com.   2. R. L. Mills, “Classical Quantum Mechanics”, Physics Essays, Vol. 16, No. 4, December, (2003), pp. 433-498; posted with spreadsheets at www.blacklightpower.com/techpapers.shtml.   3. R. Mills, “Physical Solutions of the Nature of the Atom, Photon, and Their Interactions to Form Excited and Predicted Hydrino States”, in press, http://www.blacklightpower.comAechpapers.shtml.   4. R. L. Mills, “Exact Classical Quantum Mechanical Solutions for One-Through Twenty-Electron Atoms”, Phys. Essays, Vol. 18, (2005), 321-361, posted with spreadsheets at http://www.blacklightpower.com/techpapers.shtml.   5. R. L. Mills, “The Nature of the Chemical Bond Revisited and an Alternative Maxwellian Approach”, Physics Essays, Vol. 17, (2004), pp. 342-389, posted with spreadsheets at http://www.blacklightpower.com/techpapers.shtml.   6. R. L. Mills, “Maxwell&#39;s Equations and QED: Which is Fact and Which is Fiction”, in press, posted with spreadsheets at http://www.blacklightpower.com/techpapers.shtml.   7. R. L. Mills, “Exact Classical Quantum Mechanical Solution for Atomic Helium Which Predicts Conjugate Parameters from a Unique Solution for the First Time”, submitted, posted with spreadsheets at http://www.blacklightpower.com/theory/theory.shtml.   8. R. Mills, “The Grand Unified Theory of Classical Quantum Mechanics”, Int. J. Hydrogen Energy, Vol. 27, No. 5, (2002), pp. 565-590.   9. R. L. Mills, “The Fallacy of Feynman&#39;s Argument on the Stability of the Hydrogen Atom According to Quantum Mechanics”, Annales de la Fondation Louis de Broglie, Vol. 30, No. 2, (2005), pp. 129-151, posted at http://www.blacklightpower.com/techpapers.shtml.   10. R. Mills, The Nature of Free Electrons in Superfluid Helium—a Test of Quantum Mechanics and a Basis to Review its Foundations and Make a Comparison to Classical Theory, Int. J. Hydrogen Energy, Vol. 26, No. 10, (2001), pp. 1059-1096.   11. R. Mills, “The Hydrogen Atom Revisited”, Int. J. of Hydrogen Energy, Vol. 25, Issue 12, December, (2000), pp. 1171-1183.   12. F. Laloë, Do we really understand quantum mechanics? Strange correlations, paradoxes, and theorems, Am. J. Phys. 69 (6), June 2001, 655-701.   13. P. Pearle, Foundations of Physics, “Absence of radiationless motions of relativistically rigid classical electron”, Vol. 7, Nos. 11/12, (1977), pp. 931-945.   14. V. F. Weisskopf, Reviews of Modern Physics, Vol. 21, No. 2, (1949), pp. 305-315.   15. H. Wergeland, “The Klein Paradox Revisited”,  Old and New Questions in Physics, Cosmology, Philosophy, and Theoretical Biology , A. van der Merwe, Editor, Plenum Press, New York, (1983), pp. 503-515.   16. A. Einstein, B. Podolsky, N. Rosen, Phys. Rev., Vol. 47, (1935), p. 777.   17. F. Dyson, “Feynman&#39;s proof of Maxwell equations”, Am. J. Phys., Vol. 58, (1990), pp. 209-211.   18. Haus, H. A., “On the radiation from point charges”, American Journal of Physics, 54, (1986), pp. 1126-1129.   19. http://www.blacklightpower.com/new.shtml.   
 
       SUMMARY OF THE INVENTION 
       [0036]    The present invention, an exemplary embodiment of which is also referred to as Millsian software, stems from a new fundamental insight into the nature of the atom. Applicant&#39;s new theory of Classical Quantum Mechanics (CQM) reveals the nature of atoms, molecules, molecular ions, compounds and materials using classical physical laws for the first time. As discussed above, traditional quantum mechanics can solve neither multi-electron atoms nor molecules exactly. By contrast, CQM produces exact, closed-form solutions containing physical constants only for even the most complex atoms, molecules, molecular ions, compounds and materials. 
         [0037]    The present invention is the first and only molecular modeling program ever built on the CQM framework. For example, all the major functional groups that make up most organic molecules have been solved exactly in closed-form solutions with CQM. By using these functional groups as building blocks, or independent units, a potentially infinite number of organic molecules can be solved. As a result, the present invention can be used to visualize the exact 3D structure and calculate the heat of formation of any organic molecule. 
         [0038]    For the first time, the significant building-block molecules of chemistry have been successfully solved using classical physical laws in exact closed-form equations having fundamental constants only. The major functional groups have been solved from which molecules of infinite length can be solved almost instantly with a computer program. The predictions are accurate within experimental error for over 375 exemplary molecules. 
         [0039]    Applicant&#39;s CQM is the theory that physical laws (Maxwell&#39;s Equations, Newton&#39;s Laws, Special and General Relativity) must hold on all scales. The theory is based on an often overlooked result of Maxwell&#39;s Equations, that an extended distribution of charge may, under certain conditions, accelerate without radiating. This “condition of no radiation” is invoked to solve the physical structure of subatomic particles, atoms, and molecules. 
         [0040]    In exact closed-form equations with physical constants only, solutions to thousands of known experimental values arise that were beyond the reach of previous outdated theories. These include the electron spin, g-factor, multi-electron atoms, excited states, polyatomic molecules, wave-particle duality and the nature of the photon, the masses and families of fundamental particles, and the relationships between fundamental laws of the universe that reveal why the universe is accelerating as it expands. CQM is successful to over 85 orders of magnitude, from the level of quarks to the cosmos. Applicant now has over 65 peer-reviewed journal articles and also books discussing the CQM and supporting experimental evidence. 
         [0041]    The molecular modeling market was estimated to be a two-billion-dollar per year industry in 2002, with hundreds of millions of government and industry dollars invested in computer algorithms and supercomputer centers. This makes it the largest effort of computational chemistry and physics. 
         [0042]    The present invention&#39;s advantages over other models includes: Rendering true molecular structures; Providing precisely all characteristics, spatial and temporal charge distributions and energies of every electron in every bond, and of every bonding atom; Facilitating the identification of biologically active sites in drugs; and Facilitating drug design. 
         [0043]    An objective of the present invention is to solve the charge (mass) and current-density functions of specific groups of molecules, molecular ions, compounds and materials disclosed herein or any portion of these species from first principles. In an embodiment, the solution for the molecules, molecular ions, compounds and materials, or any portion of these species is derived from Maxwell&#39;s equations invoking the constraint that the bound electron before excitation does not radiate even though it undergoes acceleration. 
         [0044]    Another objective of the present invention is to generate a readout, display, or image of the solutions so that the nature of the molecules, molecular ions, compounds and materials, or any portion of these species be better understood and potentially applied to predict reactivity and physical and optical properties. 
         [0045]    Another objective of the present invention is to apply the methods and systems of solving the nature of the molecules, molecular ions, compounds and materials, or any portion of these species and their rendering to numerical or graphical. 
         [0046]    These objectives and other objectives are obtained by a system of computing and rendering the nature of at least one specie selected from the groups of molecules, molecular ions, compounds and materials disclosed herein, comprising physical, Maxwellian solutions of charge, mass, and current density functions of said specie, said system comprising a processor for processing physical, Maxwellian equations representing charge, mass, and current density functions of said specie; and an output device in communication with the processor for displaying said physical, Maxwellian solutions of charge, mass, and current density functions of said specie. 
         [0047]    Also provided is a composition of matter comprising a plurality of atoms, the improvement comprising a novel property or use discovered by calculation of at least one of a bond distance between two of the atoms, a bond angle between three of the atoms, and a bond energy between two of the atoms, orbital intercept distances and angles, charge-density functions of atomic, hybridized, and molecular orbitals, the bond distance, bond angle, and bond energy being calculated from physical solutions of the charge, mass, and current density functions of atoms and atomic ions, which solutions are derived from Maxwell&#39;s equations using a constraint that a bound electron(s) does not radiate under acceleration. 
         [0048]    The presented exact physical solutions for known species of the groups of molecules, molecular ions, compounds and materials disclosed herein can be applied to other unknown species. These solutions can be used to predict the properties of presently unknown species and engineer compositions of matter in a manner which is not possible using past quantum mechanical techniques. The molecular solutions can be used to design synthetic pathways and predict product yields based on equilibrium constants calculated from the heats of formation. Not only can new stable compositions of matter be predicted, but now the structures of combinatorial chemistry reactions can be predicted. 
         [0049]    Pharmaceutical applications include the ability to graphically or computationally render the structures of drugs that permit the identification of the biologically active parts of the specie to be identified from the common spatial charge-density functions of a series of active species. Novel drugs can now be designed according to geometrical parameters and bonding interactions with the data of the structure of the active site of the drug. 
         [0050]    The molecular solutions can be used to design synthetic pathways and predict product yields based on equilibrium constants calculated from the heats of formation. New stable compositions of matter can be predicted as well as the structures of combinatorial chemistry reactions. Further important pharmaceutical applications include the ability to graphically or computationally render the structures of drugs that permit the identification of the biologically active parts of the molecules to be identified from the common spatial charge-density functions of a series of active molecules. Drugs can be designed according to geometrical parameters and bonding interactions with the data of the structure of the active site of the drug. 
         [0051]    To calculate conformations, folding, and physical properties, the exact solutions of the charge distributions in any given molecule are used to calculate the fields, and from the fields, the interactions between groups of the same molecule or between groups on different molecules are calculated wherein the interactions are distance and relative orientation dependent. The fields and interactions can be determined using a finite-element-analysis approach of Maxwell&#39;s equations. 
         [0052]    The system can be used to calculate conformations, folding, and physical properties, and the exact solutions of the charge distributions in any given specie are used to calculate the fields. From the fields, the interactions between groups of the same specie or between groups on different species are calculated wherein the interactions are distance and relative orientation dependent. The fields and interactions can be determined using a finite-element-analysis approach of Maxwell&#39;s equations. 
         [0053]    In another embodiment of the system, metabolites or inhibitors that bind to a target enzyme are rendered and based on the topography of the electron density revealed by these renderings, nonmetabolizable analogues with the same or similar electron topography that bind to this enzyme to provide inhibition are rendered by the system. Thus, the system provides candidate drug agents based on charge density and geometry without direct knowledge of the structure of the enzyme. For example, metabolites or inhibitors that bind to 3-hydroxy-3-methylglutaryl-CoA reductase which catalyzes the rate-limiting and irreversible step of cholesterol synthesis are modeled. Then, based on the topography of the electron density revealed by these renderings, nonmetabolizable analogues with the same or similar electron topography that bind to this enzyme to provide inhibition at this step are rendered by the system. Thus, the system provides candidate anticholesterol agents based on charge density and geometry without direct knowledge of the structure of the enzyme. In an embodiment, the metabolites or inhibitors are at least one from the list of 3-hydroxy-3-methylglutarate, 3-hydroxybutyrate, 3-hydroxy-3-methylpentanoate, 4-bromocrotonyl-CoA, but-3-ynoyl-CoA, pent-3-ynoyl-CoA, dec-3-ynoyl-CoA, ML-236A, ML-236B (compactin), ML-236C, mevinolin, mevinolinic acid, or a mevalonic acid analogue. Further metabolites and inhibitors of corresponding enzymes that are rendered by system which then outputs renderings of analogues as candidate new drugs based on similarities of geometry and charge density are disclosed in my previous U.S. Pat. No. 5,773,592, Randell L. Mills, Jun. 30, 1998, entitled, “Prodrugs for Selective Drug Delivery” and U.S. Pat. No. 5,428,163, Randell L. Mills, Jun. 27, 1995 entitled “Prodrugs for Selective Drug Delivery” which are herein incorporated in their entirety by reference. 
         [0054]    Embodiments of the system for performing computing and rendering of the nature of the groups of molecules and molecular ions, or any portion of these species using the physical solutions may comprise a general purpose computer. Such a general purpose computer may have any number of basic configurations. For example, such a general purpose computer may comprise a central processing unit (CPU), one or more specialized processors, system memory, a mass storage device such as a magnetic disk, an optical disk, or other storage device, an input means, such as a keyboard or mouse, a display device, and a printer or other output device. A system implementing the present invention can also comprise a special purpose computer or other hardware system and all should be included within its scope. A complete description and drawing of a flow chart of how a computer can be used is disclosed in Applicant&#39;s prior incorporated WO2007/051078 application. 
         [0055]    Although not preferred, any of the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to +10%, if desired. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0056]      FIG. 1  illustrates Aspirin (acetylsalicylic acid). 
           [0057]      FIG. 2  illustrates grey scale, translucent view of the charge-density of aspirin showing the orbitals of the atoms at their radii, the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atom(s) participating in each bond, and the hydrogen nuclei. 
           [0058]      FIG. 3  illustrates the structure of diamond. (A) Twenty six C—C-bond MOs. (B). Fifty one C—C-bond MOs. 
           [0059]      FIG. 4  illustrates C 60  MO comprising a hollow cage of sixty carbon atoms bound with the linear combination of sixty sets of C—C-bond MOs bridged by 30 sets of C═C-bond MOs. A C═C group is bound to two C—C groups at each vertex carbon atom of C 60 . Color scale, translucent pentagonal view of the charge-density of the C 60 -bond MO with each C2sp 3  HO shown transparently. For each C—C and C═C bond, the ellipsoidal surface of the H 2 -type ellipsoidal MO that transitions to the C2sp 3  HO, the C2sp 3  HO shell, inner most C1s shell, and the nuclei, are shown. 
           [0060]      FIG. 5  illustrates an opaque pentagonal view of the charge-density of the C 60  MO high-lighting the twenty hexagonal and twelve pentagonal units joined together such that no two pentagons share an edge. The six-six ring edges are C═C bonds and the five-five ring edges are C—C-bonds such that each hexagon is comprised of alternating C═C-bond MOs and C—C-bond MOs and each pentagon is comprised of only C—C-bond MOs. 
           [0061]      FIG. 6  illustrates a hexagonal translucent view. 
           [0062]      FIG. 7  illustrates a hexagonal opaque view. 
           [0063]      FIG. 8  illustrates the structure of graphite. (A). Single plane of macromolecule of indefinite size. (B). Layers of graphitic planes. 
           [0064]      FIG. 9  illustrates a point charge above an infinite planar conductor. 
           [0065]      FIG. 10  illustrates a point charge above an infinite planar conductor and the image charge to meet the boundary condition Φ=0 at z=0. 
           [0066]      FIG. 11  illustrates electric field lines from a positive point charge near an infinite planar conductor. 
           [0067]      FIG. 12  illustrates the surface charge density distribution on the surface of the conduction planar conductor induced by the point charge at the position +. (A) The surface charge density −σ(ρ) (shown in color-scale relief). (B) The cross-sectional view of the surface charge density. 
           [0068]      FIG. 13  illustrates a point charge located between two infinite planar conductors. 
           [0069]      FIG. 14  illustrates the surface charge density −σ(ρ) of a planar electron shown in color scale. 
           [0070]      FIG. 15  illustrates the body-centered cubic lithium metal lattice showing the electrons of as planar two-dimensional membranes of zero thickness that are each an equipotential energy surface comprised of the superposition of multiple electrons. (A) and (B) The unit-cell component of the surface charge density of a planar electron having an electric field equivalent to that of image point charge for each corresponding positive ion of the lattice. (C) Opaque view of the ions and electrons of a unit cell. (D) Transparent view of the ions and electrons of a unit cell. 
           [0071]      FIG. 16  illustrates the body-centered cubic metal lattice of lithium showing the unit cell of electrons and ions. (A) Diagonal view. (B) Top view. 
           [0072]      FIG. 17  illustrates a portion of the crystalline lattice of Li metal comprising 3 3  body-centered cubic unit cells of electrons and ions. (A) Rotated diagonal opaque view. (B) Rotated diagonal transparent view. (C) Side transparent view. 
           [0073]      FIG. 18  illustrates the crystalline unit cells of the alkali metals showing each lattice of ions and electrons to the same scale. (A) The crystal structure of Li. (B) The crystal structure of Na. (C) The crystal structure of K. (D) The crystal structure of Rb. (E) The crystal structure of Cs. 
           [0074]      FIG. 19A-D  illustrates grey scale, translucent view of the charge-densities of the series SiH n=1,2,3,4 , showing the orbitals of each member Si atom at their radii, the ellipsoidal surface of each H 2 -type ellipsoidal MO of H that transitions to the outer shell of the Si atom participating in each Si—H bond, and the hydrogen nuclei. 
           [0075]      FIG. 20  illustrates Disilane. Color scale, translucent view of the charge-density of H 3 SiSiH 3  comprising the linear combination of two sets of three Si—H-bond MOs and a Si—Si-bond MO with the Si silane 3sp 3  HOs of the Si—Si-bond MO shown transparently. The Si—Si-bond MO comprises a H 2 -type ellipsoidal MO bridging two Si silane 3sp3 HOs. For each Si—H and the Si—Si bond, the ellipsoidal surface of the H 2 -type ellipsoidal MO that transitions to the Si silane 3sp3 HO, the Si silane 3SP3 HO shell with radius 0.97295a 0  (Eq. (20.21)), inner Si1s, Si2s, and Si2p shells with radii of Si1s=0.07216a 0  (Eq. (10.51)), Si2s=0.31274a 0  (Eq. (10.62)), and Si2p=0.40978a 0  (Eq. (10.212)), respectively, and the nuclei, are shown. 
           [0076]      FIG. 21  illustrates Dimethylsilane. Grey scale, translucent view of the charge-density of (H 3 C) 2  SiH 2  showing the orbitals of the Si and C atoms at their radii, the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the hydrogen nuclei. 
           [0077]      FIG. 22  illustrates Hexamethyldisilane. Grey scale, opaque view of the charge-density of (CH 3 ) 3  SiSi (CH 3 ) 3  showing the orbitals of the Si and C atoms at their radii, the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the hydrogen nuclei. 
           [0078]      FIG. 23  illustrates grey scale, translucent view of the charge-density of ((CH 3 ) 2 SiO) 3  showing the orbitals of the Si, O, and C atoms at their radii, the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the nuclei. 
           [0079]      FIG. 24  illustrates grey scale, translucent view of the charge-density of (CH 3 ) 3  SiOSi (CH 3 ) 3  showing the orbitals of the Si, O, and C atoms at their radii, the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the nuclei. 
           [0080]      FIG. 25A-B  illustrates the diamond structure of silicon in the insulator state. Axes indicate positions of additional bonds of the repeating structure. (A) Twenty six C—C-bond MOs. (B) Fifty one C—C-bond MOs. 
           [0081]      FIG. 26A-B  illustrates STM topographs of the clean Si(111)-(7×7) surface. Reprinted with permission from Ref [1]. Copyright 1995 American Chemical Society. 
           [0082]      FIG. 27  (A), (B), and (C) illustrate the conducting state of crystalline silicon showing the covalent diamond-structure network of the unit cell with two electrons ionized from σ MO shown as a planar two-dimensional membrane of zero thickness that is the perpendicular bisector of the former Si—Si bond axis. The corresponding two Si +  ions (smaller radii) are centered at the positions of the atoms that contributed the ionized Si3sp 3 -HO electrons. The electron equipotential energy surface may superimpose with multiple planar electron membranes. The surface charge density of each electron gives rise to an electric field equivalent to that of image point charge for each corresponding positive ion of the lattice. 
           [0083]      FIG. 28  illustrates Diborane. Grey scale, opague view of the charge-density of B 2 H 6  comprising the linear combination of two sets of two B—H-bond MOs and two B—H—B-bond MOs. For each B—H and B—H—B bond, the ellipsoidal surface of the H 2 -type ellipsoidal MO transitions to the B2sp 3  HO shell with radius 0.89047a 0  (Eq. (22.17)). The inner B1s radius is 0.20670a 0  (Eq. (10.51)). 
           [0084]      FIG. 29  illustrates Trimethylborane. Grey scale, translucent view of the charge-density of (H 3 C) 3  B showing the orbitals of the B and C atoms at their radii, the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the hydrogen nuclei. 
           [0085]      FIG. 30  illustrates Tetramethyldiborane. Grey scale, translucent view of the charge-density of (CH 3 ) 2  BH 2 B(CH 3 ) 2  showing the orbitals of the B and C atoms at their radii, the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the hydrogen nuclei. 
           [0086]      FIG. 31  illustrates Trimethoxyborane. Grey scale, translucent view of the charge-density of (H 3 CO) 3  B showing the orbitals of the B, O, and C atoms at their radii, the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the hydrogen nuclei. 
           [0087]      FIG. 32  illustrates Boric Acid. Grey scale, translucent view of the charge-density of (HO) 3  B showing the orbitals of the B and O atoms at their radii, the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the hydrogen nuclei. 
           [0088]      FIG. 33  illustrates Phenylborinic Anhydride. Grey scale, translucent view of the charge-density of phenylborinic anhydride showing the orbitals of the B and O atoms at their radii, the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the hydrogen nuclei. 
           [0089]      FIG. 34  illustrates Trisdimethylaminoborane. Grey scale, translucent view of the charge-density of ((H 3 C) 2  N) 3  B showing the orbitals of the B, N, and C atoms at their radii, the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the hydrogen nuclei. 
           [0090]      FIG. 35  illustrates Trimethylaminotrimethylborane. Grey scale, translucent view of the charge-density of (CH 3 ) 3  BN(CH 3 ) 3  showing the orbitals of the B, N, and C atoms at their radii, the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the hydrogen nuclei. 
           [0091]      FIG. 36  illustrates Boron Trifluoride. Grey scale, translucent view of the charge-density of BF 3  showing the orbitals of the B and F atoms at their radii, and the ellipsoidal surface of each H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond. 
           [0092]      FIG. 37  illustrates Boron Trichloride. Grey scale, translucent view of the charge-density of BCl 3  showing the orbitals of the B and Cl atoms at their radii, and the ellipsoidal surface of each H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond. 
           [0093]      FIG. 38  illustrates Trimethylaluminum. Grey scale, translucent view of the charge-density of (H 3 C) 3  Al comprising the linear combination of three sets of three C—H-bond MOs and three C—Al-bond MOs with the Al ogranoAl 3sp 3  HOs and C2sp 3  HOs shown transparently. Each C—Al-bond MO comprises a H 2 -type ellipsoidal MO bridging C2sp 3  and Al3sp 3  HOs. For each C—H and C—Al bond, the ellipsoidal surface of the H 2 -type ellipsoidal MO that transitions to the C2sp 3  HO shell with radius 0.89582; (Eq. (15.32)) or Al3sp 3  HO, the Al3sp 3  HO shell with radius 0.85503; (Eq. (15.32)), inner Al1s, Al2s, and Al2p shells with radii of Al1s=0.07778; (Eq. (10.51)), Al2s=0.33923; (Eq. (10.62)), and Al2p=0.45620; (Eq. (10.212)), respectively, and the nuclei (red, not to scale), are shown. 
           [0094]      FIG. 39  illustrates Scandium Trifluoride. Grey scale, translucent view of the charge-density of ScF 3  showing the orbitals of the Sc and F atoms at their radii, the ellipsoidal surface of each H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the nuclei. 
           [0095]      FIG. 40  illustrates Titanium Tetrafluoride. Grey scale, translucent view of the charge-density of TiF 4  showing the orbitals of the Ti and F atoms at their radii, the ellipsoidal surface of each H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the nuclei. 
           [0096]      FIG. 41  illustrates Vanadium Hexacarbonyl. Grey scale, translucent view of the charge-density of V (CO) 6  showing the orbitals of the V, C, and O atoms at their radii, the ellipsoidal surface of each H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the nuclei. 
           [0097]      FIG. 42  illustrates Dibenzene Vanadium. Grey scale, translucent view of the charge-density of V(C 6 H 6 ) 2  showing the orbitals of the V and C atoms at their radii, the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the hydrogen nuclei. 
           [0098]      FIG. 43  illustrates Toluene. 
           [0099]      FIG. 44  illustrates Chromium Hexacarbonyl. Grey scale, translucent view of the charge-density of Cr (CO) 6  showing the orbitals of the Cr, C, and O atoms at their radii, the ellipsoidal surface of each H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the nuclei. 
           [0100]      FIG. 45  illustrates Di-(1,2,4-trimethylbenzene) Chromium. Grey scale, opaque view of the charge-density of Cr((CH 3 ) 3 C 6 H 3 ) 2  showing the orbitals of the Cr and C atoms at their radii and the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond. 
           [0101]      FIG. 46  illustrates Diamanganese decacarbonyl. Grey scale, opaque view of the charge-density of Mn 2  (CO) 10  showing the orbitals of the Mn, C, and O atoms at their radii and the ellipsoidal surface of each H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond. 
           [0102]      FIG. 47  illustrates Iron Pentacarbonyl. Grey scale, translucent view of the charge-density of Fe (CO) 5  showing the orbitals of the Fe, C, and O atoms at their radii, the ellipsoidal surface of each H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the nuclei. 
           [0103]      FIG. 48  illustrates Bis-cylopentadienyl Iron. Grey scale, opaque view of the charge-density of Fe (C 5 H 5 ) 2  showing the orbitals of the Fe and C atoms at their radii and the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond. 
           [0104]      FIG. 49  illustrates Cobalt Tetracarbonyl Hydride. Color scale, translucent view of the charge-density of CoH(CO) 4  showing the orbitals of the Co, C, and O atoms at their radii, the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the nuclei. 
           [0105]      FIG. 50  illustrates Nickel Tetracarbonyl. Grey scale, translucent view of the charge-density of Ni (CO) 4  showing the orbitals of the Ni, C, and O atoms at their radii, the ellipsoidal surface of each H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the nuclei. 
           [0106]      FIG. 51  illustrates Nickelocene. Grey scale, opaque view of the charge-density of Ni(C 5 H 5 ) 2  showing the orbitals of the Ni and C atoms at their radii and the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond. 
           [0107]      FIG. 52  illustrates Copper Chloride. Grey scale, translucent view of the charge-density of CuCl showing the orbitals of the Cu and Cl atoms at their radii, the ellipsoidal surface of each H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the nuclei. 
           [0108]      FIG. 53  illustrates Copper Dichloride. Grey scale, translucent view of the charge-density of CuCl 2  showing the orbitals of the Cu and Cl atoms at their radii, the ellipsoidal surface of each H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the nuclei. 
           [0109]      FIG. 54  illustrates Zinc Chloride. Grey scale, translucent view of the charge-density of ZnCl showing the orbitals of the Zn and Cl atoms at their radii, the ellipsoidal surface of each H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the nuclei. 
           [0110]      FIG. 55  illustrates Di-n-butylzinc. Grey scale, translucent view of the charge-density of Zn(C 4 H 9 ) 2  showing the orbitals of the Zn and C atoms at their radii, the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the nuclei. 
           [0111]      FIG. 56  illustrates Tin Tetrachloride. Grey scale, translucent view of the charge-density of SnCl 4  showing the orbitals of the Sn and Cl atoms at their radii, the ellipsoidal surface of each H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond, and the nuclei. 
           [0112]      FIG. 57  illustrates Hexaphenyldistannane. Grey scale, opaque view of the charge-density of (C 6 H 5 ) 3 SnSn(C 6 H 5 ) 3  showing the orbitals of the Sn and C atoms at their radii and the ellipsoidal surface of each H or H 2 -type ellipsoidal MO that transitions to the corresponding outer shell of the atoms participating in each bond. 
       
    
    
     DETAILED DESCRIPTION 
       [0113]    The inventions disclosed herein will now be described with reference to the attached non-limiting Figures. 
       Organic Molecular Functional Groups and Molecules 
     Derivation of the General Geometrical and Energy Equations of Organic Chemistry 
       [0114]    Organic molecules comprising an arbitrary number of atoms can be solved using similar principles and procedures as those used to solve alkanes of arbitrary length. Alkanes can be considered to be comprised of the functional groups of CH 3 , CH 2 , and C—C. These groups with the corresponding geometrical parameters and energies can be added as a linear sum to give the solution of any straight chain alkane as shown in the Continuous-Chain Alkanes section. Similarly, the geometrical parameters and energies of all functional groups such as alkanes, branched alkanes, alkenes, branched alkenes, alkynes, alkyl fluorides, alkyl chlorides, alkyl bromides, alkyl iodides, alkene halides, primary alcohols, secondary alcohols, tertiary alcohols, ethers, primary amines, secondary amines, tertiary amines, aldehydes, ketones, carboxylic acids, carboxylic esters, amides, N-alkyl amides, N,N-dialkyl amides, urea, acid halides, acid anhydrides, nitriles, thiols, sulfides, disulfides, sulfoxides, sulfones, sulfites, sulfates, nitro alkanes, nitrites, nitrates, conjugated polyenes, aromatics, heterocyclic aromatics, substituted aromatics, and others can be solved. The functional-group solutions can be made into a linear superposition and sum, respectively, to give the solution of any organic molecule. The solutions of the functional groups can be conveniently obtained by using generalized forms of the geometrical and energy equations. The total bond energies of exemplary organic molecules calculated using the functional group composition and the corresponding energies derived in the following sections compared to the experimental values are given in Tables 15.333.1-15.333.36. 
         [0115]    Consider the case wherein at least two atomic orbital hybridize as a linear combination of electrons at the same energy in order to achieve a bond at an energy minimum, and the sharing of electrons between two or more such orbitals to form σ MO permits the participating hybridized orbitals to decrease in energy through a decrease in the radius of one or more of the participating orbitals. The force generalized constant k′ of a H 2 -type ellipsoidal MO due to the equivalent of two point charges of at the foci is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                     ′ 
                   
                   = 
                   
                     
                       
                         C 
                         1 
                       
                        
                       
                         C 
                         2 
                       
                        
                       2 
                        
                       
                          
                         2 
                       
                     
                     
                       4 
                        
                       
                         πɛ 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where C 1  is the fraction of the H 2 -type ellipsoidal MO basis function of a chemical bond of the molecule or molecular ion which is 0.75 (Eq. (13.59)) in the case of H bonding to a central atom and 0.5 (Eq. (14.152)) otherwise, and C 2  is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of the chemical bond. From Eqs. (13.58-13.63), the distance from the origin of the MO to each focus c′ is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     c 
                     ′ 
                   
                   = 
                   
                     
                       a 
                        
                       
                         
                           
                             
                               ℏ 
                               2 
                             
                              
                             4 
                              
                             
                               πɛ 
                               0 
                             
                           
                           
                             
                               m 
                               e 
                             
                              
                             
                                
                               2 
                             
                              
                             2 
                              
                             
                               C 
                               1 
                             
                              
                             
                               C 
                               2 
                             
                              
                             a 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           aa 
                           0 
                         
                         
                           2 
                            
                           
                             C 
                             1 
                           
                            
                           
                             C 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The internuclear distance is 
         [0000]    
       
         
           
             
               
                 
                   
                     2 
                      
                     
                       c 
                       ′ 
                     
                   
                   = 
                   
                     2 
                      
                     
                       
                         
                           aa 
                           0 
                         
                         
                           2 
                            
                           
                             C 
                             1 
                           
                            
                           
                             C 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The length of the semiminor axis of the prolate spheroidal MO b=c is given by 
         [0000]        b =√{square root over ( a   2   −c′   2 )}  (15.4)
 
         [0000]    And, the eccentricity, e, is 
         [0000]    
       
         
           
             
               
                 
                    
                   = 
                   
                     
                       c 
                       ′ 
                     
                     a 
                   
                 
               
               
                 
                   ( 
                   15.5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    From Eqs. (11.207-11.212), the potential energy of the two electrons in the central field of the nuclei at the foci is 
         [0000]    
       
         
           
             
               
                 
                   
                     V 
                     e 
                   
                   = 
                   
                     
                       n 
                       1 
                     
                      
                     
                       c 
                       1 
                     
                      
                     
                       c 
                       2 
                     
                      
                     
                       
                         
                           - 
                           2 
                         
                          
                         
                            
                           2 
                         
                       
                       
                         8 
                          
                         
                           πɛ 
                           0 
                         
                          
                         
                           
                             
                               a 
                               2 
                             
                             - 
                             
                               b 
                               2 
                             
                           
                         
                       
                     
                      
                     ln 
                      
                     
                       
                         a 
                         + 
                         
                           
                             
                               a 
                               2 
                             
                             - 
                             
                               b 
                               2 
                             
                           
                         
                       
                       
                         a 
                         - 
                         
                           
                             
                               a 
                               2 
                             
                             - 
                             
                               b 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The potential energy of the two nuclei is 
         [0000]    
       
         
           
             
               
                 
                   
                     V 
                     p 
                   
                   = 
                   
                     
                       n 
                       1 
                     
                      
                     
                       
                          
                         2 
                       
                       
                         8 
                          
                         
                           πɛ 
                           0 
                         
                          
                         
                           
                             
                               a 
                               2 
                             
                             - 
                             
                               b 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The kinetic energy of the electrons is 
         [0000]    
       
         
           
             
               
                 
                   T 
                   = 
                   
                     
                       n 
                       1 
                     
                      
                     
                       c 
                       1 
                     
                      
                     
                       c 
                       2 
                     
                      
                     
                       
                         ℏ 
                         2 
                       
                       
                         2 
                          
                         
                           m 
                           e 
                         
                          
                         a 
                          
                         
                           
                             
                               a 
                               2 
                             
                             - 
                             
                               b 
                               2 
                             
                           
                         
                       
                     
                      
                     ln 
                      
                     
                       
                         a 
                         + 
                         
                           
                             
                               a 
                               2 
                             
                             - 
                             
                               b 
                               2 
                             
                           
                         
                       
                       
                         a 
                         - 
                         
                           
                             
                               a 
                               2 
                             
                             - 
                             
                               b 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    And, the energy, V m , of the magnetic force between the electrons is 
         [0000]    
       
         
           
             
               
                 
                   
                     V 
                     m 
                   
                   = 
                   
                     
                       n 
                       1 
                     
                      
                     
                       c 
                       1 
                     
                      
                     
                       c 
                       2 
                     
                      
                     
                       
                         - 
                         
                           ℏ 
                           2 
                         
                       
                       
                         4 
                          
                         
                           m 
                           e 
                         
                          
                         a 
                          
                         
                           
                             
                               a 
                               2 
                             
                             - 
                             
                               b 
                               2 
                             
                           
                         
                       
                     
                      
                     ln 
                      
                     
                       
                         a 
                         + 
                         
                           
                             
                               a 
                               2 
                             
                             - 
                             
                               b 
                               2 
                             
                           
                         
                       
                       
                         a 
                         - 
                         
                           
                             
                               a 
                               2 
                             
                             - 
                             
                               b 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The total energy of the H 2 -type prolate spheroidal MO, E T (H 2 MO), is given by the sum of the energy terms: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       T 
                     
                      
                     
                       ( 
                       
                         
                           H 
                           2 
                         
                          
                         MO 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       V 
                       e 
                     
                     + 
                     T 
                     + 
                     
                       V 
                       m 
                     
                     + 
                     
                       V 
                       p 
                     
                   
                 
               
               
                 
                   ( 
                   15.10 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 H 
                                 2 
                               
                                
                               MO 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           - 
                           
                             
                               
                                 n 
                                 1 
                               
                                
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 
                                   
                                     a 
                                     2 
                                   
                                   - 
                                   
                                     b 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           [ 
                           
                             
                               
                                 c 
                                 1 
                               
                                
                               
                                 
                                   c 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     2 
                                     - 
                                     
                                       
                                         a 
                                         0 
                                       
                                       a 
                                     
                                   
                                   ) 
                                 
                               
                                
                               ln 
                                
                               
                                 
                                   a 
                                   + 
                                   
                                     
                                       
                                         a 
                                         2 
                                       
                                       - 
                                       
                                         b 
                                         2 
                                       
                                     
                                   
                                 
                                 
                                   a 
                                   - 
                                   
                                     
                                       
                                         a 
                                         2 
                                       
                                       - 
                                       
                                         b 
                                         2 
                                       
                                     
                                   
                                 
                               
                             
                             - 
                             1 
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           - 
                           
                             
                               
                                 n 
                                 1 
                               
                                
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 c 
                                 ′ 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           [ 
                           
                             
                               
                                 c 
                                 1 
                               
                                
                               
                                 
                                   c 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     2 
                                     - 
                                     
                                       
                                         a 
                                         0 
                                       
                                       a 
                                     
                                   
                                   ) 
                                 
                               
                                
                               ln 
                                
                               
                                 
                                   a 
                                   + 
                                   
                                     c 
                                     ′ 
                                   
                                 
                                 
                                   a 
                                   - 
                                   
                                     c 
                                     ′ 
                                   
                                 
                               
                             
                             - 
                             1 
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where n 1  is the number of equivalent bonds of the MO and applies in the case of functional groups. In the case of independent MOs not in contact with the bonding atoms, the terms based on charge are multiplied by c BO , the bond-order factor. It is 1 for a single bond, 4 for an independent double bond as in the case of the CO 2  and NO 2  molecules, and 9 for an independent triplet bond. Then, the kinetic energy term is multiplied by c′ BO  which is 1 for a single bond, 2 for a double bond, and 9/2 for a triple bond. c 1  is the fraction of the H 2 -type ellipsoidal MO basis function of an MO which is 0.75 (Eqs. (13.67-13.73)) in the case of H bonding to an unhybridized central atom and 1 otherwise, and c 2  is the factor that results in an equipotential energy match of the participating the MO and the at least two atomic orbitals of the chemical bond. Specifically, to meet the equipotential condition and energy matching conditions for the union of the H 2 -type-ellipsoidal-MO and the HOs or AOs of the bonding atoms, the factor c 2  of a H 2 -type ellipsoidal MO may given by (i) one, (ii) the ratio of the Coulombic or valence energy of the AO or HO of at least one atom of the bond and 13.605804 eV, the Coulombic energy between the electron and proton of H, (iii) the ratio of the valence energy of the AO or HO of one atom and the Coulombic energy of another, (iv) the ratio of the valence energies of the AOs or HOs of two atoms, (v) the ratio of two c 2  factors corresponding to any of cases (ii)-(iv), and (vi) the product of two different c 2  factors corresponding to any of the cases (i)-(v). Specific examples of the factor c 2  of a H 2 -type ellipsoidal MO given in previous sections are 0.936127, the ratio of the ionization energy of N 14.53414 eV and 13.605804 eV, the Coulombic energy between the electron and proton of H,
 
0.91771, the ratio of 14.82575 eV, −E Coulomb (C,2sp 3 ), and 13.605804 eV;
 
0.87495, the ratio of 15.55033 eV, −E Coulomb (C ethane ,2sp 3 ), and 13.605804 eV;
 
0.85252, the ratio of 15.95955 eV, −E Coulomb (C ethylene ,2sp 3 ), and 13.605804 eV;
 
0.85252, the ratio of 15.95955 eV, −E Coulomb (C benzene ,2sp 3 ), and 13.605804 eV, and
 
0.86359, the ratio of 15.55033 eV, −E Coulomb (C alkane ,2sp 3 ), and 13.605804 eV.
 
         [0116]    In the generalization of the hybridization of at least two atomic-orbital shells to form a shell of hybrid orbitals, the hybridized shell comprises a linear combination of the electrons of the atomic-orbital shells. The radius of the hybridized shell is calculated from the total Coulombic energy equation by considering that the central field decreases by an integer for each successive electron of the shell and that the total energy of the shell is equal to the total Coulombic energy of the initial AO electrons. The total energy E T (atom,msp 3 ) (m is the integer of the valence shell) of the AO electrons and the hybridized shell is given by the sum of energies of successive ions of the atom over the n electrons comprising total electrons of the at least one AO shell. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       T 
                     
                      
                     
                       ( 
                       
                         atom 
                         , 
                         
                           msp 
                           3 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     - 
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                           
                       
                        
                       
                         IP 
                         m 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where IP m  is the m th ionization energy (positive) of the atom. The radius r msp     3   , of the hybridized shell is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     r 
                     
                       msp 
                       3 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         q 
                         = 
                         
                           Z 
                           - 
                           n 
                         
                       
                       
                         Z 
                         - 
                         1 
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           - 
                           
                             ( 
                             
                               Z 
                               - 
                               q 
                             
                             ) 
                           
                         
                          
                         
                            
                           2 
                         
                       
                       
                         8 
                          
                         
                           πɛ 
                           0 
                         
                          
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               atom 
                               , 
                               
                                 msp 
                                 3 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Then, the Coulombic energy E Coulomb (atom,msp 3 ) of the outer electron of the atom msp 3  shell is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       Coulomb 
                     
                      
                     
                       ( 
                       
                         atom 
                         , 
                         
                           msp 
                           3 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       - 
                       
                          
                         2 
                       
                     
                     
                       8 
                        
                       
                         πɛ 
                         0 
                       
                        
                       
                         r 
                         
                           msp 
                           3 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In the case that during hybridization at least one of the spin-paired AO electrons is unpaired in the hybridized orbital (HO), the energy change for the promotion to the unpaired state is the magnetic energy E(magnetic) at the initial radius r of the AO electron: 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       ( 
                       magnetic 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         2 
                          
                         
                           πμ 
                           0 
                         
                          
                         
                            
                           2 
                         
                          
                         
                           ℏ 
                           2 
                         
                       
                       
                         
                           m 
                           e 
                           2 
                         
                          
                         
                           r 
                           3 
                         
                       
                     
                     = 
                     
                       
                         8 
                          
                         
                           πμ 
                           0 
                         
                          
                         
                           μ 
                           B 
                           2 
                         
                       
                       
                         r 
                         3 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Then, the energy E(atom,msp 3 ) of the outer electron of the atom msp 3  shell is given by the sum of E Coulomb (atom,msp 3 ) and E(magnetic): 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       ( 
                       
                         atom 
                         , 
                         
                           msp 
                           3 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                            
                           2 
                         
                       
                       
                         8 
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         
                           ɛ 
                           0 
                         
                          
                         
                           r 
                           
                             msp 
                             3 
                           
                         
                       
                     
                     + 
                     
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         
                           μ 
                           0 
                         
                          
                         
                            
                           2 
                         
                          
                         
                           ℏ 
                           2 
                         
                       
                       
                         
                           m 
                           e 
                           2 
                         
                          
                         
                           r 
                           3 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.16 
                   ) 
                 
               
             
           
         
       
     
         [0117]    Consider next that the at least two atomic orbitals hybridize as a linear combination of electrons at the same energy in order to achieve a bond at an energy minimum with another atomic orbital or hybridized orbital. As a further generalization of the basis of the stability of the MO, the sharing of electrons between two or more such hybridized orbitals to form σ MO permits the participating hybridized orbitals to decrease in energy through a decrease in the radius of one or more of the participating orbitals. In this case, the total energy of the hybridized orbitals is given by the sum of E(atom,msp 3 ) and the next energies of successive ions of the atom over the n electrons comprising the total electrons of the at least two initial AO shells. Here, E(atom,msp 3 ) is the sum of the first ionization energy of the atom and the hybridization energy. An example of E(atom,msp 3 ) for E(C, 2sp 3 ) is given in Eq. (14.503) where the sum of the negative of the first ionization energy of C, −11.27671 eV, plus the hybridization energy to form the C2sp 3  shell given by Eq. (14.146) is E(C,2sp 3 )=−14.63489 eV. 
         [0118]    Thus, the sharing of electrons between two atom msp 3  HOs to form an atom-atom-bond MO permits each participating hybridized orbital to decrease in radius and energy. In order to further satisfy the potential, kinetic, and orbital energy relationships, each atom msp 3  HO donates an excess of 25% per bond of its electron density to the atom-atom-bond MO to form an energy minimum wherein the atom-atom bond comprises one of a single, double, or triple bond. In each case, the radius of the hybridized shell is calculated from the Coulombic energy equation by considering that the central field decreases by an integer for each successive electron of the shell and the total energy of the shell is equal to the total Coulombic energy of the initial AO electrons plus the hybridization energy. The total energy E T (mol.atom,msp 3 ) (m is the integer of the valence shell) of the HO electrons is given by the sum of energies of successive ions of the atom over the n electrons comprising total electrons of the at least one initial AO shell and the hybridization energy: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       T 
                     
                      
                     
                       ( 
                       
                         
                           mol 
                           · 
                           atom 
                         
                         , 
                         
                           msp 
                           3 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       E 
                        
                       
                         ( 
                         
                           atom 
                           , 
                           
                             msp 
                             3 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           2 
                         
                         n 
                       
                        
                       
                           
                       
                        
                       
                         IP 
                         m 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.17 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where IP m  is the m th ionization energy (positive) of the atom and the sum of plus the hybridization energy is E(atom,msp 3 ). Thus, the radius r msp     3    of the hybridized shell due to its donation of a total charge −Qe to the corresponding MO is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             msp 
                             3 
                           
                         
                         = 
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     q 
                                     = 
                                     
                                       Z 
                                       - 
                                       n 
                                     
                                   
                                   
                                     Z 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     q 
                                   
                                   ) 
                                 
                               
                               - 
                               Q 
                             
                             ) 
                           
                            
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               π 
                                
                               
                                   
                               
                                
                               
                                 ɛ 
                                 0 
                               
                                
                               
                                 
                                   E 
                                   T 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       mol 
                                       · 
                                       atom 
                                     
                                     , 
                                     
                                       msp 
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     q 
                                     = 
                                     
                                       Z 
                                       - 
                                       n 
                                     
                                   
                                   
                                     Z 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     q 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 s 
                                  
                                 
                                   ( 
                                   0.25 
                                   ) 
                                 
                               
                             
                             ) 
                           
                            
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               π 
                                
                               
                                   
                               
                                
                               
                                 ɛ 
                                 0 
                               
                                
                               
                                 
                                   E 
                                   T 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       mol 
                                       · 
                                       atom 
                                     
                                     , 
                                     
                                       msp 
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.18 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where −e is the fundamental electron charge and s=1,2,3 for a single, double, and triple bond, respectively. The Coulombic energy E Coulomb (mol.atom,msp 3 ) of the outer electron of the atom msp 3  shell is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       Coulomb 
                     
                      
                     
                       ( 
                       
                         
                           mol 
                           · 
                           atom 
                         
                         , 
                         
                           msp 
                           3 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       - 
                       
                          
                         2 
                       
                     
                     
                       8 
                        
                       
                           
                       
                        
                       π 
                        
                       
                           
                       
                        
                       
                         ɛ 
                         0 
                       
                        
                       
                         r 
                         
                           msp 
                           3 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.19 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In the case that during hybridization at least one of the spin-paired AO electrons is unpaired in the hybridized orbital (HO), the energy change for the promotion to the unpaired state is the magnetic energy E(magnetic) at the initial radius r of the AO electron given by Eq. (15.15). Then, the energy E(mol.atom,msp 3 ) of the outer electron of the atom msp 3  shell is given by the sum of E Coulomb (mol.atom,msp 3 ) and E(magnetic): 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       ( 
                       
                         
                           mol 
                           · 
                           atom 
                         
                         , 
                         
                           msp 
                           3 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                            
                           2 
                         
                       
                       
                         8 
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         
                           ɛ 
                           0 
                         
                          
                         
                           r 
                           
                             msp 
                             3 
                           
                         
                       
                     
                     + 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         
                           μ 
                           0 
                         
                          
                         
                            
                           2 
                         
                          
                         
                           ℏ 
                           2 
                         
                       
                       
                         
                           m 
                           e 
                           2 
                         
                          
                         
                           r 
                           3 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.20 
                   ) 
                 
               
             
           
         
       
     
         [0000]    E T (atom−atom,msp 3 ), the energy change of each atom msp 3  shell with the formation of the atom-atom-bond MO is given by the difference between E(mol.atom,msp 3 ) and E(atom,msp 3 ): 
         [0000]        E   T (atom−atom,msp 3 )= E (mol.atom,msp 3 )− E (atom,msp 3 )  (15.21)
 
         [0119]    As examples from prior sections, E Coulomb (mol.atom,msp 3 ) is one of: 
         [0120]    E Coulomb (C ethylene ,2sp 3 ) E Coulomb (C ethane ,2sp 3 ), E Coulomb (C acetylene ,2sp 3  and E Coulomb (C alkane ,2sp 3 ); 
         [0121]    E Coulomb  and E Coulomb (atom,msp 3 ) is one of E Coulomb (C,2sp 3 ) and E Coulomb (Cl,3sp 3 ); 
         [0122]    E(mol.atom,msp 3 ) is one of E(C ethylene ,2sp 3 ), E(C ethane ,2sp 3 ), 
         [0123]    E(C acetylene ,2sp 3 )E(C alkane ,2sp 3 ); 
         [0124]    E(atom,msp 3 ) is one of and E(C,2sp 3 ) and E(Cl,3sp 3 ); 
         [0125]    E T (atom−atom,msp 3 ) is one of E(C—C,2sp 3 ), E(C═C,2sp 3 ), and. E(C≡C,2sp 3 ); 
         [0126]    atom msp 3  is one of C2sp 3 , Cl3sp 3    
         [0127]    E T (atom−atom(s 1 ),msp 3 ) is E T (C—C,2sp 3 ) and E T (atom−atom(s 2 ),msp 3 ) is E T (C═C,2sp 3 ), and 
         [0128]    r msp     3    is one of r C3sp     3   , r ethane2sp     3   r acetylene2sp     3   , r alkane2sp     3   , and r Cl3sp     3   . 
         [0129]    In the case of the C2sp 3  HO, the initial parameters (Eqs. (14.142-14.146)) are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               sp 
                               3 
                             
                           
                         
                         = 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               2 
                             
                             5 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   Z 
                                   - 
                                   n 
                                 
                                 ) 
                               
                                
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               π 
                                
                               
                                   
                               
                                
                               
                                 
                                   ɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     148.25751 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             10 
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               
                                 ɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   148.25751 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           0.91771 
                            
                           
                               
                           
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.22 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               C 
                               , 
                               
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               ɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               ɛ 
                               0 
                             
                              
                             
                                 
                             
                              
                             0.91771 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             14.82575 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.23 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               μ 
                               0 
                             
                              
                             
                                
                               2 
                             
                              
                             
                               ℏ 
                               2 
                             
                           
                           
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   3 
                                 
                                 ) 
                               
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             8 
                              
                             
                                 
                             
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               μ 
                               0 
                             
                              
                             
                               μ 
                               B 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 0.84317 
                                  
                                 
                                     
                                 
                                  
                                 
                                   a 
                                   0 
                                 
                               
                               ) 
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           0.19086 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.24 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               C 
                               , 
                               
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               π 
                                
                               
                                   
                               
                                
                               
                                 ɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               π 
                                
                               
                                   
                               
                                
                               
                                 μ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     3 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               14.82575 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.19086 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             14.63489 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.25 
                   ) 
                 
               
             
           
         
       
     
       In Eq. (15.18), 
       [0130]    
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         q 
                         = 
                         
                           Z 
                           - 
                           n 
                         
                       
                       
                         Z 
                         - 
                         1 
                       
                     
                      
                     
                         
                     
                      
                     
                       ( 
                       
                         Z 
                         - 
                         q 
                       
                       ) 
                     
                   
                   = 
                   10 
                 
               
               
                 
                   ( 
                   15.26 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Eqs. (14.147) and (15.17) give 
         [0000]      E T (mol.atom,msp 3 )=E T (C ethane ,2 sp   3 )=−151.61569 eV  (15.27)
 
         [0000]    Using Eqs. (15.18-15.28), the final values of r c2sp     3   , E Coulomb (C2sp 3 ), and E(C2sp 3 ), and the resulting 
         [0000]    
       
         
           
             
               E 
               T 
             
              
             
               ( 
               
                 
                   C 
                    
                   
                     - 
                     BO 
                   
                    
                   C 
                 
                 , 
                 
                   C 
                    
                   
                       
                   
                    
                   2 
                    
                   
                     sp 
                     3 
                   
                 
               
               ) 
             
           
         
       
     
         [0000]    of the MO due to charge donation from the HO to the MO where 
         [0000]    
       
         
           
             C 
              
             
               - 
               BO 
             
              
             C 
           
         
       
     
         [0000]    refers to the bond order of the carbon-carbon bond for different values of the parameter s are given in Table 15.1. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 15.1 
               
               
                   
               
             
             
               
                 The final values of r C2sp     3   , E Coulomb (C2sp 3 ), and E(C2sp 3 )  
                   and                 the                 resulting                     E   T          (       C        —   BO        C     ,     C                 2                   sp   3         )                     of                 the                 MO                 due                 to               charge                 donation                 from                 the                 HO                 to                 the                 MO                 where                 C        —   BO        C               
 refers to the bond order of the carbon-carbon bond. 
               
             
          
           
               
                 MO Bond Order (BO) 
                       s 1 
                       s 2 
                   r C2sp     3    (a 0 ) Final 
                 E Coulomb  (C2sp 3 ) (eV) Final 
                   E(C2sp 3 ) (eV) Final 
                   
         E   T          (       C        —   BO        C     ,     C                 2                   sp   3         )         
 (eV) 
               
               
                   
               
               
                 I 
                 1 
                 0 
                 0.87495 
                 −15.55033 
                 −15.35946 
                 −0.72457 
               
               
                 II 
                 2 
                 0 
                 0.85252 
                 −15.95955 
                 −15.76868 
                 −1.13379 
               
               
                 III 
                 3 
                 0 
                 0.83008 
                 −16.39089 
                 −16.20002 
                 −1.56513 
               
               
                 IV 
                 4 
                 0 
                 0.80765 
                 −16.84619 
                 −16.65532 
                 −2.02043 
               
               
                   
               
             
          
         
       
     
         [0131]    In another generalized case of the basis of forming a minimum-energy bond with the constraint that it must meet the energy matching condition for all MOs at all HOs or AOs, the energy E(mol.atom,msp 3 ) of the outer electron of the atom msp 3  shell of each bonding atom must be the average of E(mol.atom,msp 3 ) for two different values of s: 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       ( 
                       
                         
                           mol 
                           · 
                           atom 
                         
                         , 
                         
                           msp 
                           3 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               E 
                                
                               
                                 ( 
                                 
                                   
                                     mol 
                                     · 
                                     
                                       atom 
                                        
                                       
                                         ( 
                                         
                                           s 
                                           1 
                                         
                                         ) 
                                       
                                     
                                   
                                   , 
                                   
                                     msp 
                                     3 
                                   
                                 
                                 ) 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   mol 
                                   · 
                                   
                                     atom 
                                      
                                     
                                       ( 
                                       
                                         s 
                                         2 
                                       
                                       ) 
                                     
                                   
                                 
                                 , 
                                 
                                   msp 
                                   3 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   15.28 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In this case, E T (atom−atom,msp 3 ), the energy change of each atom msp 3  shell with the formation of each atom-atom-bond MO, is average for two different values of s: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       T 
                     
                      
                     
                       ( 
                       
                         
                           atom 
                           - 
                           atom 
                         
                         , 
                         
                           msp 
                           3 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               
                                 E 
                                 T 
                               
                                
                               
                                 ( 
                                 
                                   
                                     atom 
                                     - 
                                     
                                       atom 
                                        
                                       
                                         ( 
                                         
                                           s 
                                           1 
                                         
                                         ) 
                                       
                                     
                                   
                                   , 
                                   
                                     msp 
                                     3 
                                   
                                 
                                 ) 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               E 
                               T 
                             
                              
                             
                               ( 
                               
                                 
                                   atom 
                                   - 
                                   
                                     atom 
                                      
                                     
                                       ( 
                                       
                                         s 
                                         2 
                                       
                                       ) 
                                     
                                   
                                 
                                 , 
                                 
                                   msp 
                                   3 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   15.29 
                   ) 
                 
               
             
           
         
       
     
         [0132]    Consider an aromatic molecule such as benzene given in the Benzene Molecule section. Each C═C double bond comprises a linear combination of a factor of 0.75 of four paired electrons (three electrons) from two sets of two C2sp 3  HOs of the participating carbon atoms. Each C—H bond of CH having two spin-paired electrons, one from an initially unpaired electron of the carbon atom and the other from the hydrogen atom, comprises the linear combination of 75% H 2 -type ellipsoidal MO and 25% C2sp 3  HO as given by Eq. (13.439). However, E T (atom−atom,msp 3 ) of the C—H-bond MO is given by 0.5E T (C═C,2sp 3 ) (Eq. (14.247)) corresponding to one half of a double bond that matches the condition for a single-bond order for C—H that is lowered in energy due to the aromatic character of the bond. 
         [0133]    A further general possibility is that a minimum-energy bond is achieved with satisfaction of the potential, kinetic, and orbital energy relationships by the formation of an MO comprising an allowed multiple of a linear combination of H 2 -type ellipsoidal MOs and corresponding HOs or AOs that contribute a corresponding allowed multiple (e.g. 0.5, 0.75, 1) of the bond order given in Table 15.1. For example, the alkane MO given in the Continuous-Chain Alkanes section comprises a linear combination of factors of 0.5 of a single bond and 0.5 of a double bond. 
         [0134]    Consider a first MO and its HOs comprising a linear combination of bond orders and a second MO that shares a HO with the first. In addition to the mutual HO, the second MO comprises another AO or HO having a single bond order or a mixed bond order. Then, in order for the two MOs to be energy matched, the bond order of the second MO and its HOs or its HO and AO is a linear combination of the terms corresponding to the bond order of the mutual HO and the bond order of the independent HO or AO. Then, in general, E T (atom−atom,msp 3 ), the energy change of each atom msp 3  shell with the formation of each atom-atom-bond MO, is a weighted linear sum for different values of s that matches the energy of the bonded MOs, HOs, and AOs: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       T 
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               atom 
                               - 
                             
                           
                         
                         
                           
                             
                               atom 
                               , 
                               
                                 msp 
                                 3 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         1 
                       
                       N 
                     
                      
                     
                         
                     
                      
                     
                       
                         c 
                         
                           s 
                           n 
                         
                       
                        
                       
                         
                           E 
                           T 
                         
                          
                         
                           ( 
                           
                             
                               
                                 
                                   atom 
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   
                                     atom 
                                      
                                     
                                       ( 
                                       
                                         s 
                                         n 
                                       
                                       ) 
                                     
                                   
                                   , 
                                   
                                     msp 
                                     3 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.30 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where c s     n    is the multiple of the BO of s n . The radius r msp     3    of the atom msp 3  shell of each bonding atom is given by the Coulombic energy using the initial energy E Coulomb (atom,msp 3 ) and E T (atom−atom,msp 3 ), the energy change of each atom msp 3  shell with the formation of each atom-atom-bond MO: 
         [0000]    
       
         
           
             
               
                 
                   
                     r 
                     
                       msp 
                       3 
                     
                   
                   = 
                   
                     
                       - 
                       
                          
                         2 
                       
                     
                     
                       8 
                        
                       
                           
                       
                        
                       π 
                        
                       
                           
                       
                        
                       
                         ɛ 
                         0 
                       
                        
                       
                         
                           a 
                           0 
                         
                          
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     
                                       E 
                                       Coulomb 
                                     
                                      
                                     
                                       ( 
                                       
                                         atom 
                                         , 
                                         
                                           msp 
                                           3 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     E 
                                     T 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         atom 
                                         - 
                                         atom 
                                       
                                       , 
                                       
                                         msp 
                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.31 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E Coulomb (C2sp 3 )=−14.825751 eV. The Coulombic energy E Coulomb (mol.atom,msp 3 ) of the outer electron of the atom msp 3  shell is given by Eq. (15.19). In the case that during hybridization, at least one of the spin-paired AO electrons is unpaired in the hybridized orbital (HO), the energy change for the promotion to the unpaired state is the magnetic energy E(magnetic) (Eq. (15.15)) at the initial radius r of the AO electron. Then, the energy E(mol.atom,msp 3 ) of the outer electron of the atom msp 3  shell is given by the sum of E Coulomb (mol.atom,msp 3 ) and E(magnetic) (Eq. (15.20)). E T (atom−atom,msp 3 ), the energy change of each atom msp 3  shell with the formation of the atom-atom-bond MO is given by the difference between E(mol.atom,msp 3 ) and E(atom,msp 3 ) given by Eq. (15.21). Using Eq. (15.23) for E Coulomb (C,2sp 3 ) in Eq. (15.31), the single bond order energies given by Eqs. (15.18-15.27) and shown in Table 15.1, and the linear combination energies (Eqs. (15.28-15.30)), the parameters of linear combinations of bond orders and linear combinations of mixed bond orders are given in Table 15.2. 
         [0000]                                                                  TABLE 15.2                   The final values of r C2sp     3   , E Colomb (C2sp 3 ) and E(C2sp 3 ) and the 
       resulting                     E   T          (       C        —   BO        C     ,     C                 2                   sp   3         )                     of                 the                 MO                 comprising                 a                 linear                 combination       
 of H 2 -type ellipsoidal MOs and corresponding HOs of single or mixed bond order where c s     n    is the multiple of the bond order parameter E T (atom—atom (s n ), msp 3 ) given in Table 15.1.            MO Bond Order (BO)   s 1   c s     1       s 2   c s     2       s 3   c s     3       r C2sp     3   (a 0 ) Final   E Coulomb (C2sp 3 ) (eV) Final   E(C2sp 3 ) (ev) Final                         E   T          (             C        —   BO        C     ,               C                 2                   sp   3             )                 (   eV   )                               1/2I   1   0.5    0   0     0   0     0.89582   −15.18804   −14.99717   −0.36228       1/2II   2   0.5    0   0     0   0     0.88392   −15.39265   −15.20178   −0.56689       I + 1/2II   1   0.5    2   0.25   0   0     0.87941   −15.47149   −15.28062   −0.64573       1/2II + (I + II)   2   0.25   1   0.25   2   0.25   0.87363   −15.57379   −15.38293   −0.74804       3/4II   2   0.75   0   0     0   0     0.86793   −15.67610   −15.48523   −0.85034       I + II   1   0.5    2   0.5    0   0     0.86359   −15.75493   −15.56407   −0.92918       I + III   1   0.5    3   0.5    0   0     0.85193   −15.97060   −15.77974   −1.14485       I + IV   1   0.5    4   0.5    0   0     0.83995   −16.19826   −16.00739   −1.37250       II + III   2   0.5    3   0.5    0   0     0.84115   −16.17521   −15.98435   −1.34946       II + IV   2   0.5    4   0.5    0   0     0.82948   −16.40286   −16.21200   −1.57711       III + IV   3   0.5    4   0.5    0   0     0.81871   −16.61853   −16.42767   −1.79278       IV + IV   4   0.5    4   0.5    0   0     0.80765   −16.84619   −16.65532   −2.02043                    
Consider next the radius of the AO or HO due to the contribution of charge to more than one bond. The energy contribution due to the charge donation at each atom such as carbon superimposes linearly. In general, the radius r mol2sp     3    of the C2sp 3  HO of a carbon atom of a given molecule is calculated using Eq. (14.514) by considering ΣE T     mol   (MO,2sp 3 ), the total energy donation to all bonds with which it participates in bonding. The general equation for the radius is given by
 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             
                               mol 
                                
                               2 
                             
                              
                             
                                 
                             
                              
                             
                               sp 
                               3 
                             
                           
                         
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               
                                 ɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       E 
                                       Coulomb 
                                     
                                      
                                     
                                       ( 
                                       
                                         C 
                                         , 
                                         
                                           2 
                                            
                                           
                                             sp 
                                             3 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     ∑ 
                                     
                                       
                                         E 
                                         
                                           T 
                                           mol 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           MO 
                                           , 
                                           
                                             2 
                                              
                                             
                                               sp 
                                               3 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                              
                             2 
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               
                                 ɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     14.825751 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   + 
                                   
                                     ∑ 
                                     
                                        
                                       
                                         
                                           E 
                                           
                                             T 
                                             mol 
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             MO 
                                             , 
                                             
                                               2 
                                                
                                               
                                                 sp 
                                                 3 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.32 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The Coulombic energy E Coulomb (mol.atom,msp 3 ) of the outer electron of the atom msp 3  shell is given by Eq. (15.19). In the case that during hybridization, at least one of the spin-paired AO electrons is unpaired in the hybridized orbital (HO), the energy change for the promotion to the unpaired state is the magnetic energy E(magnetic) (Eq. (15.15)) at the initial radius r of the AO electron. Then, the energy E(mol.atom,msp 3 ) of the outer electron of the atom msp 3  shell is given by the sum of E Coulomb (mol.atom,msp 3 ) and E(magnetic) (Eq. (15.20)). 
         [0135]    For example, the C2sp 3  HO of each methyl group of an alkane contributes −0.92918 eV (Eq. (14.513)) to the corresponding single C—C bond; thus, the corresponding C2sp 3  HO radius is given by Eq. (14.514). The C2sp 3  HO of each methylene group of C n H 2+2  contributes −0.92918 eV to each of the two corresponding C—C bond MOs. Thus, the radius (Eq. (15.32)), the Coulombic energy (Eq. (15.19)), and the energy (Eq. (15.20)) of each alkane methylene group are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             
                               alkaneC 
                               methylene 
                             
                              
                             2 
                              
                             
                               sp 
                               3 
                             
                           
                         
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         
                                           
                                             E 
                                             Coulomb 
                                           
                                            
                                           
                                             ( 
                                             
                                               C 
                                               , 
                                               
                                                 2 
                                                  
                                                 
                                                   sp 
                                                   3 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         ∑ 
                                         
                                             
                                         
                                          
                                         
                                           
                                             E 
                                             
                                               T 
                                               alkane 
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   methylene 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   C 
                                                 
                                                 - 
                                                 C 
                                               
                                               , 
                                               
                                                 2 
                                                  
                                                 
                                                   sp 
                                                   3 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                              
                             2 
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         
                                            
                                            
                                           
                                               
                                           
                                            
                                           14.825751 
                                            
                                           
                                               
                                           
                                            
                                           eV 
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                            
                                            
                                           
                                               
                                           
                                            
                                           0.92918 
                                            
                                           
                                               
                                           
                                            
                                           eV 
                                         
                                         + 
                                         
                                           0 
                                            
                                           .92918 
                                            
                                           
                                               
                                           
                                            
                                           eV 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           0.81549 
                            
                           
                               
                           
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.33 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 C 
                                 methylene 
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   0.81549 
                                    
                                   
                                       
                                   
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             16.68412 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.34 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 C 
                                 methylene 
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     0.81549 
                                      
                                     
                                         
                                     
                                      
                                     
                                       a 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     0.84317 
                                      
                                     
                                         
                                     
                                      
                                     
                                       a 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             16.49325 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.35 
                   ) 
                 
               
             
           
         
       
     
         [0136]    In the determination of the parameters of functional groups, heteroatoms bonding to C2sp 3  HOs to form MOs are energy matched to the C2sp 3  HOs. Thus, the radius and the energy parameters of a bonding heteroatom are given by the same equations as those for C2sp 3  HOs. Using Eqs. (15.15), (15.19-15.20), (15.24), and (15.32) in a generalized fashion, the final values of the radius of the HO or AO, r Atom.HO.AO , E Coulomb (mol.atom,msp3), and E(C mol 2sp 3 ) are calculated using ΣE T     group   (MO,2sp 3 ), the total energy donation to each group bond with which an atom participates in bonding corresponding to the values of E T (C BO —C,C2sp 3 ) of the MO due to charge donation from the AO or HO to the MO given in Tables 15.1 and 15.2. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 15.3.A 
               
             
             
               
                   
               
               
                 The final values of r Atom.HO.AO , E Coulomb (mol.atom.msp 3 ), and E(C mol C2sp 3 ) calculated 
       using                 the                 values                 of                     E   T          (       C        —   BO        C     ,     C                 2                   sp   3         )                     given                 in                 Tables                 15.1                 and                   15.2   .         
 
               
             
          
           
               
                 Atom Hybrid- ization Desig- nation 
                 
                   
                     
                       
                         
                           E 
                           T 
                         
                          
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     C 
                                      
                                     
                                       — 
                                       BO 
                                     
                                      
                                     C 
                                   
                                   , 
                                 
                               
                             
                             
                               
                                 
                                   C 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           E 
                           T 
                         
                          
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     C 
                                      
                                     
                                       — 
                                       BO 
                                     
                                      
                                     C 
                                   
                                   , 
                                 
                               
                             
                             
                               
                                 
                                   C 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           E 
                           T 
                         
                          
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     C 
                                      
                                     
                                       — 
                                       BO 
                                     
                                      
                                     C 
                                   
                                   , 
                                 
                               
                             
                             
                               
                                 
                                   C 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           E 
                           T 
                         
                          
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     C 
                                      
                                     
                                       — 
                                       BO 
                                     
                                      
                                     C 
                                   
                                   , 
                                 
                               
                             
                             
                               
                                 
                                   C 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           E 
                           T 
                         
                          
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     C 
                                      
                                     
                                       — 
                                       BO 
                                     
                                      
                                     C 
                                   
                                   , 
                                 
                               
                             
                             
                               
                                 
                                   C 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 r Atom.HO.AO  Final 
                 E Coulomb  (mol.atom,msp 3 ) (eV) Final 
                 E(C mol 2sp 3 ) (eV) Final 
               
               
                   
               
               
                  1 
                 0         
                 0         
                 0         
                 0         
                 0 
                 0.91771 
                 −14.83575 
                 −14.63489 
               
               
                  2 
                 −0.36229 
                 0         
                 0         
                 0         
                 0 
                 0.89582 
                 −15.18804 
                 −14.99717 
               
               
                  3 
                 −0.46459 
                 0         
                 0         
                 0         
                 0 
                 0.88983 
                 −15.29034 
                 −15.09948 
               
               
                  4 
                 −0.56689 
                 0         
                 0         
                 0         
                 0 
                 0.88392 
                 −15.39265 
                 −15.20178 
               
               
                  5 
                 −0.72457 
                 0         
                 0         
                 0         
                 0 
                 0.87495 
                 −15.55033 
                 −15.35946 
               
               
                  6 
                 −0.85034 
                 0         
                 0         
                 0         
                 0 
                 0.86793 
                 −15.6761  
                 −15.48523 
               
               
                  7 
                 −0.92918 
                 0         
                 0         
                 0         
                 0 
                 0.86359 
                 −15.75493 
                 −15.56407 
               
               
                  8 
                 −0.54343 
                 −0.54343 
                 0         
                 0         
                 0 
                 0.85503 
                 −15.91261 
                 −15.72175 
               
               
                  9 
                 −1.13379 
                 0         
                 0         
                 0         
                 0 
                 0.85252 
                 −15.95955 
                 −15.76868 
               
               
                 10 
                 −1.14485 
                 0         
                 0         
                 0         
                 0 
                 0.85193 
                 −15.9706  
                 −15.77974 
               
               
                 11 
                 −0.46459 
                 −0.82688 
                 0         
                 0         
                 0 
                 0.84418 
                 −16.11722 
                 −15.92636 
               
               
                 12 
                 −1.34946 
                 0         
                 0         
                 0         
                 0 
                 0.84115 
                 −16.17521 
                 −15.98435 
               
               
                 13 
                 −1.3725  
                 0         
                 0         
                 0         
                 0 
                 0.83995 
                 −16.19826 
                 −16.00739 
               
               
                 14 
                 −0.46459 
                 −0.92918 
                 0         
                 0         
                 0 
                 0.83885 
                 −16.21952 
                 −16.02866 
               
               
                 15 
                 −0.72457 
                 −0.72457 
                 0         
                 0         
                 0 
                 0.836  
                 −16.2749  
                 −16.08404 
               
               
                 16 
                 −0.5669  
                 −0.92918 
                 0         
                 0         
                 0 
                 0.8336  
                 −16.32183 
                 −16.13097 
               
               
                 17 
                 −0.82688 
                 −0.72457 
                 0         
                 0         
                 0 
                 0.83078 
                 −16.37721 
                 −16.18634 
               
               
                 18 
                 −1.56513 
                 0         
                 0         
                 0         
                 0 
                 0.83008 
                 −16.39089 
                 −16.20002 
               
               
                 19 
                 −0.64574 
                 −0.92918 
                 0         
                 0         
                 0 
                 0.82959 
                 −16.40067 
                 −16.20981 
               
               
                 20 
                 −1.57711 
                 0         
                 0         
                 0         
                 0 
                 0.82948 
                 −16.40286 
                 −16.212  
               
               
                 21 
                 −0.72457 
                 −0.92918 
                 0         
                 0         
                 0 
                 0.82562 
                 −16.47951 
                 −16.28865 
               
               
                 22 
                 −0.85035 
                 −0.85035 
                 0         
                 0         
                 0 
                 0.82327 
                 −16.52645 
                 −16.33559 
               
               
                 23 
                 −1.79278 
                 0         
                 0         
                 0         
                 0 
                 0.81871 
                 −16.61853 
                 −16.42767 
               
               
                 24 
                 −1.13379 
                 −0.72457 
                 0         
                 0         
                 0 
                 0.81549 
                 −16.68411 
                 −16.49325 
               
               
                 25 
                 −0.92918 
                 −0.92918 
                 0         
                 0         
                 0 
                 0.81549 
                 −16.68412 
                 −16.49325 
               
               
                 26 
                 −2.02043 
                 0         
                 0         
                 0         
                 0 
                 0.80765 
                 −16.84619 
                 −16.65532 
               
               
                 27 
                 −1.13379 
                 −0.92918 
                 0         
                 0         
                 0 
                 0.80561 
                 −16.88872 
                 −16.69786 
               
               
                 28 
                 −0.56690 
                 −0.56690 
                 −0.92918 
                 0         
                 0 
                 0.80561 
                 −16.88873 
                 −16.69786 
               
               
                 29 
                 −0.85035 
                 −0.85035 
                 −0.46459 
                 0         
                 0 
                 0.80076 
                 −16.99104 
                 −16.80018 
               
               
                 30 
                 −0.85035 
                 −0.42517 
                 −0.92918 
                 0         
                 0 
                 0.79891 
                 −17.03045 
                 −16.83959 
               
               
                 31 
                 −0.5669  
                 −0.72457 
                 −0.92918 
                 0         
                 0 
                 0.78916 
                 −17.04641 
                 −16.85554 
               
               
                 32 
                 −1.13379 
                 −1.13379 
                 0         
                 0         
                 0 
                 0.79597 
                 −17.09334 
                 −16.90248 
               
               
                 33 
                 −1.34946 
                 −0.92918 
                 0         
                 0         
                 0 
                 0.79546 
                 −17.1044  
                 −18.91353 
               
               
                 34 
                 −0.46459 
                 −0.92918 
                 −0.92918 
                 0         
                 0 
                 0.79340 
                 −17.14871 
                 −16.95784 
               
               
                 35 
                 −0.64574 
                 −0.85034 
                 −0.85034 
                 0         
                 0 
                 0.79232 
                 −17.17217 
                 −16.98131 
               
               
                 36 
                 −0.85035 
                 −0.5669  
                 −0.92918 
                 0         
                 0 
                 0.79232 
                 −17.17218 
                 −16.98132 
               
               
                 37 
                 −0.72457 
                 −0.72457 
                 −0.92918 
                 0         
                 0 
                 0.79085 
                 −17.20408 
                 −17.01322 
               
               
                 38 
                 −0.75586 
                 −0.75586 
                 −0.92918 
                 0         
                 0 
                 0.78798 
                   17.26666 
                   17.07580 
               
               
                 39 
                 −0.74804 
                 −0.85034 
                 −0.85034 
                 0         
                 0 
                 0.78762 
                   17.27448 
                   17.08362 
               
               
                 40 
                 −0.82688 
                 −0.72457 
                 −0.92918 
                 0         
                 0 
                 0.78617 
                 −17.30638 
                 −17.11552 
               
               
                 41 
                 −0.72457 
                 −0.92918 
                 −0.92918 
                 0         
                 0 
                 0.78155 
                 −17.40868 
                 −17.21782 
               
               
                 42 
                 −0.92918 
                 −0.72457 
                 −0.92918 
                 0         
                 0 
                 0.78155 
                 −17.40869 
                 −17.21783 
               
               
                 43 
                 −0.54343 
                 −0.54343 
                 −0.5669  
                 −0.92918 
                 0 
                 0.78155 
                 −17.40869 
                 −17.21783 
               
               
                 44 
                 −0.92918 
                 −0.85034 
                 −0.85034 
                 0         
                 0 
                 0.77945 
                 −17.45561 
                 −17.26475 
               
               
                 45 
                 −0.42517 
                 −0.42517 
                 −0.85035 
                 −0.92918 
                 0 
                 0.77945 
                 −17.45563 
                 −17.24676 
               
               
                 46 
                 −0.82688 
                 −0.92918 
                 −0.92918 
                 0         
                 0 
                 0.77699 
                 −17.51099 
                 −17.32013 
               
               
                 47 
                 −0.92918 
                 −0.92918 
                 −0.92918 
                 0         
                 0 
                 0.77247 
                 −17.6133  
                 −17.42244 
               
               
                 48 
                 −0.85035 
                 −0.54343 
                 −0.5669  
                 −0.92918 
                 0 
                 0.76801 
                 −17.71561 
                 −17.52475 
               
               
                 49 
                 −1.34946 
                 −0.64574 
                 −0.92918 
                 0         
                 0 
                 0.76652 
                 −17.75013 
                 −17.55927 
               
               
                 50 
                 −0.85034 
                 −0.54343 
                 −0.60631 
                 −0.92918 
                 0 
                 0.76631 
                 −17.75502 
                 −17.56415 
               
               
                 51 
                 −1.1338  
                 −0.92918 
                 −0.92918 
                 0         
                 0 
                 0.7636  
                 −17.81791 
                 −17.62705 
               
               
                 52 
                 −0.46459 
                 −0.85035 
                 −0.85035 
                 −0.92918 
                 0 
                 0.75924 
                 −17.92022 
                 −17.72936 
               
               
                 53 
                 −0.82688 
                 −1.34946 
                 −0.92918 
                 0         
                 0 
                 0.75877 
                 −17.93128 
                 −17.74041 
               
               
                 54 
                 −0.92918 
                 −1.34946 
                 −0.92918 
                 0         
                 0 
                 0.75447 
                 −18.03358 
                 −17.84272 
               
               
                 55 
                 −1.13379 
                 −1.13379 
                 −1.13379 
                 0         
                 0 
                 0.74646 
                 −18.22712 
                 −18.03626 
               
               
                 56 
                 −1.79278 
                 −0.92918 
                 −0.92918 
                 0         
                 0 
                 0.73637 
                 −18.47690 
                 −18.28604 
               
               
                   
               
             
          
         
       
     
         [0000]                                                          TABLE 15.3.B                   The final values of r Atom.HO.AO , E Coulomb (mol.atom.msp 3 ), and E(C mol C2sp 3 ) calculated for 
       heterocyclic                 groups                 using                 the                 values                 of                     E   T          (       C        —   BO        C     ,     C                 2                   sp   3         )                     given                 in                 Tables                 15.1                 and                   15.2   .         
            Atom Hybrid- ization Desig- nation             E   T          (             C        —   BO        C     ,               C                 2                   sp   3             )                       E   T          (             C        —   BO        C     ,               C                 2                   sp   3             )                       E   T          (             C        —   BO        C     ,               C                 2                   sp   3             )                       E   T          (             C        —   BO        C     ,               C                 2                   sp   3             )                       E   T          (             C        —   BO        C     ,               C                 2                   sp   3             )             r Atom.HO.AO  Final   E Coulomb  (mol.atom,msp 3 ) (eV) Final   E(C mol 2sp 3 ) (eV) Final                1   0           0           0           0           0   0.91771   −14.82575   −14.63489        2   −0.56690   0           0           0           0   0.88392   −15.39265   −15.20178        3   −0.72457   0           0           0           0   0.87495   −15.55033   −15.35946        4   −0.92918   0           0           0           0   0.86359   −15.75493   −15.56407        5   −0.54343   −0.54343   0           0           0   0.85503   −15.91261   −15.72175        6   −1.13379   0           0           0           0   0.85252   −15.95954   −15.76868        7   −0.60631   −0.60631   0           0           0   0.84833   −16.03838   −15.84752        8   −0.46459   −0.92918   0           0           0   0.83885   −16.21953   −16.02866        9   −0.72457   −0.72457   0           0           0   0.83600   −16.27490   −16.08404       10   −0.92918   −0.60631   0           0           0   0.83159   −16.36125   −16.17038       11   −0.92918   −0.72457   0           0           0   0.82562   −16.47951   −16.28864       12   −0.85035   −0.85035   0           0           0   0.82327   −16.52644   −16.33558       13   −0.92918   −0.92918   0           0           0   0.81549   −16.68411   −16.49325       14   −1.13379   −0.72457   0           0           0   0.81549   −16.68412   −16.49325       15   −1.13379   −0.92918   0           0           0   0.80561   −16.88873   −16.69786       16   −0.85035   −0.85035   −0.46459   0           0   0.80076   −16.99103   −16.80017       17   −0.85034   −0.85034   −0.56690   0           0   0.79595   −17.09334   −16.90247       18   −1.13379   −1.13380   0           0           0   0.79597   −17.09334   −16.90248       19   −0.85035   −0.54343     0.00000   −0.92918   0   0.79340   −17.14871   −16.95785       20   −0.85035   −0.56690   −0.92918   0           0   0.79232   −17.17218   −16.98132       21   −0.54343   −0.54343   −0.56690   −0.92918   0   0.78155   −17.40869   −17.21783       22   −0.85034   −0.28345   −0.54343   −0.92918   0   0.78050   −17.43216   −17.24130       23   −0.92918   −0.92918   −0.92918   0           0   0.77247   −17.61330   −17.42243       24   −0.85034   −0.54343   −0.56690   −0.92918   0   0.76801   −17.71560   −17.52474       25   −0.85034   −0.54343   −0.60631   −0.92918   0   0.76631   −17.75502   −17.56416       26   −1.13379   −0.92918   −0.92918   0           0   0.76360   −17.81791   −17.62704       27   −1.13379   −1.13380   −0.72457   0           0   0.76360   −17.81791   −17.62705       28   −0.46459   −0.85035   −0.85035   −0.92918   0   0.75924   −17.92022   −17.72935       29   −1.13380   −1.13379   −0.92918   0           0   0.75493   −18.02252   −17.83166       30   −1.13379   −1.13379   −1.13379   0           0   0.74646   −18.22713   −18.03627                    
From Eq. (15.18), the general equation for the radius due to a total charge −Qe of an AO or a HO that participates in bonding to form σ MO is given by
 
         [0000]    
       
         
           
             
               
                 
                   
                     r 
                     
                       msp 
                       3 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             ∑ 
                             
                               q 
                               = 
                               
                                 Z 
                                 - 
                                 n 
                               
                             
                             
                               Z 
                               - 
                               1 
                             
                           
                            
                           
                               
                           
                            
                           
                             ( 
                             
                               Z 
                               - 
                               q 
                             
                             ) 
                           
                         
                         - 
                         Q 
                       
                       ) 
                     
                      
                     
                       
                         - 
                         
                            
                           2 
                         
                       
                       
                         8 
                          
                         
                           πɛ 
                           0 
                         
                          
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 mol 
                                 · 
                                 atom 
                               
                               , 
                               
                                 msp 
                                 3 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.36 
                   ) 
                 
               
             
           
         
       
     
         [0000]    By equating the radii of Eqs. (15.36) and (15.32), the total charge parameter Q of the AO or HO can be calculated wherein the excess charge is on the MO: 
         [0000]    
       
         
           
             
               
                 
                   Q 
                   = 
                   
                     
                       ( 
                       
                         
                           ∑ 
                           
                             q 
                             = 
                             
                               Z 
                               - 
                               n 
                             
                           
                           
                             Z 
                             - 
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           ( 
                           
                             Z 
                             - 
                             q 
                           
                           ) 
                         
                       
                       ) 
                     
                     - 
                     
                       
                          
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 mol 
                                 · 
                                 atom 
                               
                               , 
                               
                                 msp 
                                 3 
                               
                             
                             ) 
                           
                         
                          
                       
                       
                         ( 
                         
                           
                             
                               
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   14.825751 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 Σ 
                                  
                                 
                                    
                                   
                                     
                                       E 
                                       
                                         T 
                                         mol 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         MO 
                                         , 
                                         
                                           2 
                                            
                                           
                                             sp 
                                             3 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.37 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The modulation of the constant function by the time and spherically harmonic functions as given in Eq. (1.65) time-averages to zero such that the charge density of any HO or AO is determined by the constant function. The charge density a is then given by the fundamental charge −e times the number of electrons n divided by the area of the spherical shell of radius r mol2sp     3    given by Eq. (15.32): 
         [0000]    
       
         
           
             
               
                 
                   σ 
                   = 
                   
                     
                       
                         ( 
                         
                           n 
                           - 
                           Q 
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           - 
                           e 
                         
                         ) 
                       
                     
                     
                       
                         4 
                         3 
                       
                        
                       π 
                        
                       
                           
                       
                        
                       
                         r 
                         
                           mol 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             sp 
                             3 
                           
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.38 
                   ) 
                 
               
             
           
         
       
     
         [0137]    The charge density of an ellipsoidal MO in rectangular coordinates (Eqs. 11.42-11.45)) is 
         [0000]    
       
         
           
             
               
                 
                   σ 
                   = 
                   
                     
                       
                         q 
                         
                           4 
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           abc 
                         
                       
                        
                       
                         1 
                         
                           
                             
                               
                                 x 
                                 2 
                               
                               
                                 a 
                                 4 
                               
                             
                             + 
                             
                               
                                 y 
                                 2 
                               
                               
                                 b 
                                 4 
                               
                             
                             + 
                             
                               
                                 z 
                                 2 
                               
                               
                                 c 
                                 4 
                               
                             
                           
                         
                       
                     
                     = 
                     
                       
                         q 
                         
                           4 
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           abc 
                         
                       
                        
                       D 
                     
                   
                 
               
               
                 
                   ( 
                   15.39 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where D is the distance from the origin to the tangent plane. The charge q is given by the fundamental electron charge −e times the sum of parameter n 1  of Eqs. (15.51) and (15.61) and the charge donation parameter Q (Eq. (15.37)) of each AO or HO to the MO. Thus, the charge density of the MO is given by 
         [0000]    
       
         
           
             
               
                 
                   σ 
                   = 
                   
                     
                       
                         - 
                         
                           e 
                            
                           
                             ( 
                             
                               
                                 n 
                                 1 
                               
                               + 
                               Q 
                             
                             ) 
                           
                         
                       
                       
                         4 
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         abc 
                       
                     
                      
                     D 
                   
                 
               
               
                 
                   ( 
                   15.40 
                   ) 
                 
               
             
           
         
       
     
         [0138]    The charge density of the MO that is continuous with the surface of the AO or HO and any radial bisector current resulting from the intersection of two or more MOs as given in the Methane Molecule (CH 4 ) section is determined by the current continuity condition. Consider the continuity of the current due to the intersection of an MO with a corresponding AO or HO. The parameters of each point of intersection of each H 2 -type ellipsoidal MO and the corresponding atom AO or HO determined from the polar equation of the ellipse are given by Eqs. (15.80-15.87). The overlap charge Δq is given by the total charge of the prolate-spheroidal MO minus the integral of the charge density of the MO over the area between curves of intersection with the AOs or HOs that forms the MO: 
         [0000]    
       
         
           
             
               
                 
                   
                     Δ 
                      
                     
                         
                     
                      
                     q 
                   
                   = 
                   
                     
                       
                         - 
                         
                           e 
                            
                           
                             ( 
                             
                               
                                 n 
                                 1 
                               
                               + 
                               Q 
                             
                             ) 
                           
                         
                       
                       - 
                       
                         ∫ 
                         
                           σ 
                            
                           
                              
                             A 
                           
                         
                       
                     
                     = 
                     
                       
                         - 
                         
                           e 
                            
                           
                             ( 
                             
                               
                                 n 
                                 1 
                               
                               + 
                               Q 
                             
                             ) 
                           
                         
                       
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                             ∫ 
                             
                               
                                 D 
                                 
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   abc 
                                 
                               
                                
                               
                                  
                                 A 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.41 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The overlap charge of the prolate-spheroidal MO Δq is uniformly distributed on the external spherical surface of the AO or HO of radius r mol2sp     3     3  such that the charge density σ from Eq. (15.41) is 
         [0000]    
       
         
           
             
               
                 
                   σ 
                   = 
                   
                     
                       Δ 
                        
                       
                           
                       
                        
                       q 
                     
                     A 
                   
                 
               
               
                 
                   ( 
                   15.42 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where A is the external surface area of the AO or HO between the curves of intersection with the MO surface. 
         [0139]    At the curves of intersection of two or more MOs where they occur, the current between the AO or HO shell and curves of mutual contact is projected onto and flows in the direction of the radial vector to the surface of the AO or HO shell. This current designated the bisector current (BC) meets the AO or HO surface and does not travel to distances shorter than its radius. Due to symmetry, a radial axis through the AO or HO exists such that current travels from the MOs to the AO or HO along the radial vector in one direction and returns to the MO along the radial vector in the opposite direction from the AO or HO surface to conserve current flow. Since the continuation of the MO charge density on the bisector current and the external surface of the AO or HO is an equipotential, the charge density on these surfaces must be uniform. Thus, σ on these surfaces is given by Eq. (15.42) where Δq is given by Eq. (15.41) with the integral over the MO area between curves of intersection of the MOs, and A is the sum of the surface areas of the bisector current and the external surface of the AO or HO between the curves of intersection of the bisector current with the AO or HO surface. 
         [0140]    The angles at which any two prolate spheroidal A-C and B—C-bond MOs intersect can be determined using Eq. (13.85) by equating the radii of the elliptic cross sections of the MOs: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           a 
                           1 
                         
                         - 
                         
                           c 
                           1 
                           ′ 
                         
                       
                       ) 
                     
                      
                     
                       
                         1 
                         + 
                         
                           
                             c 
                             1 
                             ′ 
                           
                           
                             a 
                             1 
                           
                         
                       
                       
                         1 
                         + 
                         
                           
                             
                               c 
                               1 
                               ′ 
                             
                             
                               a 
                               1 
                             
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             1 
                             ′ 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           a 
                           2 
                         
                         - 
                         
                           c 
                           2 
                           ′ 
                         
                       
                       ) 
                     
                      
                     
                       
                         1 
                         + 
                         
                           
                             c 
                             2 
                             ′ 
                           
                           
                             a 
                             2 
                           
                         
                       
                       
                         1 
                         + 
                         
                           
                             
                               c 
                               2 
                               ′ 
                             
                             
                               a 
                               2 
                             
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             2 
                             ′ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.43 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and by using the following relationship between the polar angles θ 1 ′ and θ 2 ′: 
         [0000]      θ ∠ACB =θ 1 ′+θ 2 ′−360°  (15.44)
 
         [0000]    where θ ∠ACB  is the bond angle of atoms A and B with central atom C. From either angle, the polar radius of intersection can be determined using Eq. (13.85). An example for methane is shown in Eqs. (13.597-13.600). Using these coordinates and the radius of the AO or HO, the limits of the integrals for the determination of the charge densities as well as the regions of each charge density are determined. 
         [0141]    The energy of the MO is matched to each of the participating outermost atomic or hybridized orbitals of the bonding atoms wherein the energy match includes the energy contribution due to the AO or HO&#39;s donation of charge to the MO. The force constant k′ (Eq. (15.1)) is used to determine the ellipsoidal parameter c′ (Eq. (15.2)) of the each H 2 -type-ellipsoidal-MO in terms of the central force of the foci. Then, c′ is substituted into the energy equation (from Eq. (15.11))) which is set equal to n 1  times the total energy of H 2  where n 1  is the number of equivalent bonds of the MO and the energy of H 2 , −31.63536831 eV, Eq. (11.212) is the minimum energy possible for a prolate spheroidal MO. From the energy equation and the relationship between the axes, the dimensions of the MO are solved. The energy equation has the semimajor axis a as it only parameter. The solution of the semimajor axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of each MO (Eqs. (15.3-15.5)). The parameter solutions then allow for the component and total energies of the MO to be determined. 
         [0142]    The total energy, E T (H 2 MO), is given by the sum of the energy terms (Eqs. (15.6-15.11)) plus E T (AO/HO): 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         E 
                         T 
                       
                        
                       
                         ( 
                         
                             
                           
                             
                               H 
                               2 
                             
                              
                             MO 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         V 
                         e 
                       
                       + 
                       T 
                       + 
                       
                         V 
                         m 
                       
                       + 
                       
                         V 
                         p 
                       
                       + 
                       
                         
                           E 
                           T 
                         
                          
                         
                           ( 
                           
                             AO 
                             / 
                             HO 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.45 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                                 
                               
                                 
                                   H 
                                   2 
                                 
                                  
                                 MO 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                               
                                 
                                   
                                     n 
                                     1 
                                   
                                    
                                   
                                     e 
                                     2 
                                   
                                 
                                 
                                   8 
                                    
                                   
                                       
                                   
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ɛ 
                                     0 
                                   
                                    
                                   
                                     
                                       
                                         a 
                                         2 
                                       
                                       - 
                                       
                                         b 
                                         2 
                                       
                                     
                                   
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       c 
                                       1 
                                     
                                      
                                     
                                       
                                         c 
                                         2 
                                       
                                        
                                       
                                         ( 
                                         
                                           2 
                                           - 
                                           
                                             
                                               a 
                                               0 
                                             
                                             a 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     ln 
                                      
                                     
                                       
                                         a 
                                         + 
                                         
                                           
                                             
                                               a 
                                               2 
                                             
                                             - 
                                             
                                               b 
                                               2 
                                             
                                           
                                         
                                       
                                       
                                         a 
                                         - 
                                         
                                           
                                             
                                               a 
                                               2 
                                             
                                             - 
                                             
                                               b 
                                               2 
                                             
                                           
                                         
                                       
                                     
                                   
                                   - 
                                   1 
                                 
                                 ] 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               AO 
                               / 
                               HO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                               
                                 
                                   
                                     n 
                                     1 
                                   
                                    
                                   
                                     e 
                                     2 
                                   
                                 
                                 
                                   8 
                                    
                                   
                                       
                                   
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ɛ 
                                     0 
                                   
                                    
                                   
                                     c 
                                     ′ 
                                   
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       c 
                                       1 
                                     
                                      
                                     
                                       
                                         c 
                                         2 
                                       
                                        
                                       
                                         ( 
                                         
                                           2 
                                           - 
                                           
                                             
                                               a 
                                               0 
                                             
                                             a 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     ln 
                                      
                                     
                                       
                                         a 
                                         + 
                                         
                                           c 
                                           ′ 
                                         
                                       
                                       
                                         a 
                                         - 
                                         
                                           c 
                                           ′ 
                                         
                                       
                                     
                                   
                                   - 
                                   1 
                                 
                                 ] 
                               
                             
                           
                           + 
                           
                             
                               E 
                               T 
                             
                              
                             
                               ( 
                               
                                 AO 
                                 / 
                                 HO 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.46 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where n 1  is the number of equivalent bonds of the MO, c 1  is the fraction of the H 2 -type ellipsoidal MO basis function of a chemical bond of the group, c 2  is the factor that results in an equipotential energy match of the participating at least two atomic orbitals of each chemical bond, and E T (AO/HO) is the total energy comprising the difference of the energy E(AO/HO) of at least one atomic or hybrid orbital to which the MO is energy matched and any energy component ΔE H     2     MO (AO/HO) due to the AO or HO&#39;s charge donation to the MO. 
         [0000]      E T (AO/HO)= E (AO/HO)−Δ E   H     2     MO (AO/HO)  (15.47)
 
         [0000]    As specific examples given in previous sections, E T (AO/HO) is one from the group of 
         [0143]    E T (AO/HO)=E(O2p shell)=−E(ionization; O)=−13.6181 eV; 
         [0144]    E T (AO/HO)=E(N2p shell)=−E(ionization; N)=−14.53414 eV; 
         [0145]    E T (AO/HO)=E(C,2sp 3 )=−14.63489 eV; 
         [0146]    E T (AO/HO)=E Coulomb (C1,3sp 3 )=−14.60295 eV; 
         [0147]    E T (AO/HO)=E(ionization; C)+E(ionization; C + ); 
         [0148]    E T (AO/HO)=E(C ethane ,2sp 3 )=−15.35946 eV; 
         [0149]    E T (AO/HO=+E(C ethylene ,2sp 3 )−E(C ethylene ,2sp 3 ); 
         [0150]    E T (AO/HO)=E(C,2sp 3 )−2E T (C═C,2sp 3 )=−14.63489 eV+2.26758 eV); 
         [0000]    E T (AO/HO)=E(C acetylene ,2sp 3 )−E(C acetylene ,2sp 3 )−E(C acetylene ,2sp 3 )=16.20002 eV; 
         [0151]    E T (AO/HO)=E(C,2sp 3 )−2E T (C≡C,2sp 3 )=−14.63489 eV −(−3.13026 eV); 
         [0152]    E T (AO/HO)=E(C benzene ,2sp 3 )−E(C benzene ,2sp 3 ); 
         [0153]    E T (AO/HO=E(C,2sp 3 )−E T (C═C,2sp 3 )=−14.63489 eV−(−1.13379 eV), and 
         [0154]    E T (AO/HO)=E(C benzene ,2sp 3 )=−15.56407 eV. 
         [0155]    To solve the bond parameters and energies, 
         [0000]    
       
         
           
             
               c 
               ′ 
             
             = 
             
               
                 a 
                  
                 
                   
                     
                       
                         ℏ 
                         2 
                       
                        
                       4 
                        
                       
                           
                       
                        
                       π 
                        
                       
                           
                       
                        
                       
                         ɛ 
                         0 
                       
                     
                     
                       
                         m 
                         e 
                       
                        
                       
                         e 
                         2 
                       
                        
                       2 
                        
                       
                           
                       
                        
                       
                         C 
                         1 
                       
                        
                       
                         C 
                         2 
                       
                        
                       a 
                     
                   
                 
               
               = 
               
                 
                   
                     aa 
                     0 
                   
                   
                     2 
                      
                     
                         
                     
                      
                     
                       C 
                       1 
                     
                      
                     
                       C 
                       2 
                     
                   
                 
               
             
           
         
       
     
         [0000]    (Eq. (15.2)) is substituted into E T (H 2 MO) to give 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                                 
                               
                                 
                                   H 
                                   2 
                                 
                                  
                                 MO 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                               
                                 
                                   
                                     n 
                                     1 
                                   
                                    
                                   
                                     e 
                                     2 
                                   
                                 
                                 
                                   8 
                                    
                                   
                                       
                                   
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ɛ 
                                     0 
                                   
                                    
                                   
                                     
                                       
                                         a 
                                         2 
                                       
                                       - 
                                       
                                         b 
                                         2 
                                       
                                     
                                   
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       c 
                                       1 
                                     
                                      
                                     
                                       
                                         c 
                                         2 
                                       
                                        
                                       
                                         ( 
                                         
                                           2 
                                           - 
                                           
                                             
                                               a 
                                               0 
                                             
                                             a 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     ln 
                                      
                                     
                                       
                                         a 
                                         + 
                                         
                                           
                                             
                                               a 
                                               2 
                                             
                                             - 
                                             
                                               b 
                                               2 
                                             
                                           
                                         
                                       
                                       
                                         a 
                                         - 
                                         
                                           
                                             
                                               a 
                                               2 
                                             
                                             - 
                                             
                                               b 
                                               2 
                                             
                                           
                                         
                                       
                                     
                                   
                                   - 
                                   1 
                                 
                                 ] 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               AO 
                               / 
                               HO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                               
                                 
                                   
                                     n 
                                     1 
                                   
                                    
                                   
                                     e 
                                     2 
                                   
                                 
                                 
                                   8 
                                    
                                   
                                       
                                   
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ɛ 
                                     0 
                                   
                                    
                                   
                                     c 
                                     ′ 
                                   
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       c 
                                       1 
                                     
                                      
                                     
                                       
                                         c 
                                         2 
                                       
                                        
                                       
                                         ( 
                                         
                                           2 
                                           - 
                                           
                                             
                                               a 
                                               0 
                                             
                                             a 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     ln 
                                      
                                     
                                       
                                         a 
                                         + 
                                         
                                           c 
                                           ′ 
                                         
                                       
                                       
                                         a 
                                         - 
                                         
                                           c 
                                           ′ 
                                         
                                       
                                     
                                   
                                   - 
                                   1 
                                 
                                 ] 
                               
                             
                           
                           + 
                           
                             
                               E 
                               T 
                             
                              
                             
                               ( 
                               
                                 AO 
                                 / 
                                 HO 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                               
                                 
                                   
                                     n 
                                     1 
                                   
                                    
                                   
                                     e 
                                     2 
                                   
                                 
                                 
                                   8 
                                    
                                   
                                       
                                   
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ɛ 
                                     0 
                                   
                                    
                                   
                                     
                                       
                                         aa 
                                         0 
                                       
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         
                                           C 
                                           1 
                                         
                                          
                                         
                                           C 
                                           2 
                                         
                                       
                                     
                                   
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       c 
                                       1 
                                     
                                      
                                     
                                       
                                         c 
                                         2 
                                       
                                        
                                       
                                         ( 
                                         
                                           2 
                                           - 
                                           
                                             
                                               a 
                                               0 
                                             
                                             a 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     ln 
                                      
                                     
                                       
                                         
                                           
                                             
                                               a 
                                               + 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 
                                                   aa 
                                                   0 
                                                 
                                                 
                                                   2 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     C 
                                                     1 
                                                   
                                                    
                                                   
                                                     C 
                                                     2 
                                                   
                                                 
                                               
                                             
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               a 
                                               - 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 
                                                   aa 
                                                   0 
                                                 
                                                 
                                                   2 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     C 
                                                     1 
                                                   
                                                    
                                                   
                                                     C 
                                                     2 
                                                   
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                   - 
                                   1 
                                 
                                 ] 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               AO 
                               / 
                               HO 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.48 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The total energy is set equal to E(basis energies) which in the most general case is given by the sum of a first integer n 1  times the total energy of H 2  minus a second integer n 2  times the total energy of H, minus a third integer n 3  times the valence energy of E(A0) (e.g. E(N)=−14.53414 eV) where the first integer can be 1,2,3 . . . , and each of the second and third integers can be 0, 1, 2, 3 . . . . 
         [0000]        E (basis energies)= n   1 (−31.63536831 eV)− n   2 (−13.605804 eV)− n   3   E ( AO )  (15.49)
 
         [0000]    In the case that the MO bonds two atoms other than hydrogen, E(basis energies) is n 1  times the total energy of H 2  where n 1  is the number of equivalent bonds of the MO and the energy of H 2 , −31.63536831 eV, Eq. (11.212) is the minimum energy possible for a prolate spheroidal MO: 
         [0000]        E (basis energies)= n   1 (−31.63536831 eV)  (15.50)
 
         [0156]    E T (H 2 MO), is set equal to E(basis energies), and the semimajor axis a is solved. Thus, the semimajor axis a is solved from the equation of the form: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       - 
                       
                         
                           
                             
                               n 
                               1 
                             
                              
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               ɛ 
                               0 
                             
                              
                             
                               
                                 
                                   aa 
                                   0 
                                 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     C 
                                     1 
                                   
                                    
                                   
                                     C 
                                     2 
                                   
                                 
                               
                             
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 c 
                                 1 
                               
                                
                               
                                 
                                   c 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     2 
                                     - 
                                     
                                       
                                         a 
                                         0 
                                       
                                       a 
                                     
                                   
                                   ) 
                                 
                               
                                
                               ln 
                                
                               
                                 
                                   a 
                                   + 
                                   
                                     
                                       
                                         aa 
                                         0 
                                       
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         
                                           C 
                                           1 
                                         
                                          
                                         
                                           C 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 
                                   a 
                                   - 
                                   
                                     
                                       
                                         aa 
                                         0 
                                       
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         
                                           C 
                                           1 
                                         
                                          
                                         
                                           C 
                                           2 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             - 
                             1 
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         E 
                         T 
                       
                        
                       
                         ( 
                         
                           AO 
                           / 
                           HO 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     E 
                      
                     
                       ( 
                       
                         basis 
                          
                         
                             
                         
                          
                         energies 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   15.51 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The distance from the origin of the H 2 -type-ellipsoidal-MO to each focus c′, the internuclear distance 2c′, and the length of the semiminor axis of the prolate spheroidal H 2 -type MO b=c are solved from the semimajor axis a using Eqs. (15.2-15.4). Then, the component energies are given by Eqs. (15.6-15.9) and (15.48). 
         [0157]    The total energy of the MO of the functional group, E T (MO), is the sum of the total energy of the components comprising the energy contribution of the MO formed between the participating atoms and E T (atom−atom,msp 3 .AO), the change in the energy of the AOs or HOs upon forming the bond. From Eqs. (15.48-15.49), E T (Mo) is 
         [0000]      E T (MO)=E(basis energies)+E T (atom−atom,msp 3 .AO)  (15.52)
 
         [0158]    During bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Ē osc  is the sum of the Doppler, Ē D , and average vibrational kinetic energies, Ē Kvib : 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       _ 
                     
                     osc 
                   
                   = 
                   
                     
                       
                         n 
                         1 
                       
                        
                       
                         ( 
                         
                           
                             
                               E 
                               _ 
                             
                             D 
                           
                           + 
                           
                             
                               E 
                               _ 
                             
                             Kvib 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         n 
                         1 
                       
                        
                       
                         ( 
                         
                           
                             
                               E 
                               hv 
                             
                              
                             
                               
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       E 
                                       _ 
                                     
                                     K 
                                   
                                 
                                 
                                   
                                     m 
                                     e 
                                   
                                    
                                   
                                     c 
                                     2 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               1 
                               2 
                             
                              
                             ℏ 
                              
                             
                               
                                 k 
                                 μ 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.53 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where n 1  is the number of equivalent bonds of the MO, k is the spring constant of the equivalent harmonic oscillator, and μ is the reduced mass. The angular frequency of the reentrant oscillation in the transition state corresponding to Ē D  is determined by the force between the central field and the electrons in the transition state. The force and its derivative are given by 
         [0000]    
       
         
           
             
               
                 
                   
                     f 
                      
                     
                       ( 
                       R 
                       ) 
                     
                   
                   = 
                   
                     
                       - 
                       
                         c 
                         BO 
                       
                     
                      
                     
                       
                         
                           C 
                           
                             1 
                              
                             
                                 
                             
                              
                             o 
                           
                         
                          
                         
                           C 
                           
                             2 
                              
                             
                                 
                             
                              
                             o 
                           
                         
                          
                         
                           e 
                           2 
                         
                       
                       
                         4 
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         
                           ɛ 
                           0 
                         
                          
                         
                           R 
                           3 
                         
                       
                     
                      
                     
                         
                     
                      
                     and 
                   
                 
               
               
                 
                   ( 
                   15.54 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       f 
                       ′ 
                     
                      
                     
                       ( 
                       a 
                       ) 
                     
                   
                   = 
                   
                     2 
                      
                     
                         
                     
                      
                     
                       c 
                       BO 
                     
                      
                     
                       
                         
                           C 
                           
                             1 
                              
                             
                                 
                             
                              
                             o 
                           
                         
                          
                         
                           C 
                           
                             2 
                              
                             
                                 
                             
                              
                             o 
                           
                         
                          
                         
                           e 
                           2 
                         
                       
                       
                         4 
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         
                           ɛ 
                           0 
                         
                          
                         
                           R 
                           3 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.55 
                   ) 
                 
               
             
           
         
       
     
         [0000]    such that the angular frequency of the oscillation in the transition state is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ω 
                         = 
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       - 
                                       3 
                                     
                                     a 
                                   
                                    
                                   
                                     f 
                                      
                                     
                                       ( 
                                       a 
                                       ) 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     f 
                                     ′ 
                                   
                                    
                                   
                                     ( 
                                     a 
                                     ) 
                                   
                                 
                               
                               ] 
                             
                             
                               m 
                               e 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             k 
                             
                               m 
                               e 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 c 
                                 BO 
                               
                                
                               
                                 
                                   
                                     C 
                                     
                                       1 
                                        
                                       
                                           
                                       
                                        
                                       o 
                                     
                                   
                                    
                                   
                                     C 
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       o 
                                     
                                   
                                    
                                   
                                     e 
                                     2 
                                   
                                 
                                 
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ɛ 
                                     0 
                                   
                                    
                                   
                                     R 
                                     3 
                                   
                                 
                               
                             
                             
                               m 
                               e 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.56 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where R is the semimajor axis a or the semiminor axis b depending on the eccentricity of the bond that is most representative of the oscillation in the transition state, c BO , is the bond-order factor which is 1 for a single bond and when the MO comprises n 1  equivalent single bonds as in the case of functional groups. c BO  is 4 for an independent double bond as in the case of the CO 2  and NO 2  molecules and 9 for an independent triplet bond. C 1o  is the fraction of the H 2 -type ellipsoidal MO basis function of the oscillatory transition state of a chemical bond of the group, and C 2o  is the factor that results in an equipotential energy match of the participating at least two atomic orbitals of the transition state of the chemical bond. Typically, C 1o ═C 1  and C 2o ═C 2 . The kinetic energy, E K , corresponding to Ē D  is given by Planck&#39;s equation for functional groups: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       _ 
                     
                     K 
                   
                   = 
                   
                     
                       ℏ 
                        
                       
                           
                       
                        
                       ω 
                     
                     = 
                     
                       ℏ 
                        
                       
                         
                           
                             
                               
                                 C 
                                 
                                   1 
                                    
                                   
                                       
                                   
                                    
                                   o 
                                 
                               
                                
                               
                                 C 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   o 
                                 
                               
                                
                               
                                 e 
                                 2 
                               
                             
                             
                               4 
                                
                               
                                   
                               
                                
                               π 
                                
                               
                                   
                               
                                
                               
                                 ɛ 
                                 0 
                               
                                
                               
                                 R 
                                 3 
                               
                             
                           
                           
                             m 
                             e 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.57 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The Doppler energy of the electrons of the reentrant orbit is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         E 
                         _ 
                       
                       D 
                     
                     ≅ 
                     
                       
                         E 
                         hv 
                       
                        
                       
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               
                                 E 
                                 _ 
                               
                               K 
                             
                           
                           
                             
                               m 
                               e 
                             
                              
                             
                               c 
                               2 
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       E 
                       hv 
                     
                      
                     
                       
                         
                           2 
                            
                           
                               
                           
                            
                           ℏ 
                            
                           
                             
                               
                                 
                                   
                                     C 
                                     
                                       1 
                                        
                                       
                                           
                                       
                                        
                                       o 
                                     
                                   
                                    
                                   
                                     C 
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       o 
                                     
                                   
                                    
                                   
                                     e 
                                     2 
                                   
                                 
                                 
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ɛ 
                                     0 
                                   
                                    
                                   
                                     R 
                                     3 
                                   
                                 
                               
                               
                                 m 
                                 e 
                               
                             
                           
                         
                         
                           
                             m 
                             e 
                           
                            
                           
                             c 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.58 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Ē osc  given by the sum of Ē D  and Ē Kvib  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               E 
                               _ 
                             
                             osc 
                           
                            
                           
                             ( 
                             group 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             n 
                             1 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   E 
                                   _ 
                                 
                                 D 
                               
                               + 
                               
                                 
                                   E 
                                   _ 
                                 
                                 Kvib 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             n 
                             1 
                           
                           ( 
                           
                             
                               
                                 E 
                                 hv 
                               
                                
                               
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     ℏ 
                                      
                                     
                                       
                                         
                                           
                                             
                                               C 
                                               
                                                 1 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 o 
                                               
                                             
                                              
                                             
                                               C 
                                               
                                                 2 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 o 
                                               
                                             
                                              
                                             
                                               e 
                                               2 
                                             
                                           
                                           
                                             4 
                                              
                                             
                                                 
                                             
                                              
                                             π 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               ɛ 
                                               0 
                                             
                                              
                                             
                                               R 
                                               3 
                                             
                                           
                                         
                                         
                                           m 
                                           e 
                                         
                                       
                                     
                                   
                                   
                                     
                                       m 
                                       e 
                                     
                                      
                                     
                                       c 
                                       2 
                                     
                                   
                                 
                               
                             
                             + 
                             
                               E 
                               vib 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.59 
                   ) 
                 
               
             
           
         
       
     
         [0000]    E hv  of a group having n 1  bonds is given by E T (Mo)/n 1  such that 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             _ 
                           
                           osc 
                         
                         = 
                           
                          
                         
                           
                             n 
                             1 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   E 
                                   _ 
                                 
                                 D 
                               
                               + 
                               
                                 
                                   E 
                                   _ 
                                 
                                 Kvib 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             n 
                             1 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       E 
                                       T 
                                     
                                      
                                     
                                       ( 
                                       MO 
                                       ) 
                                     
                                   
                                   / 
                                   
                                     n 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           E 
                                           _ 
                                         
                                         K 
                                       
                                     
                                     
                                       M 
                                        
                                       
                                           
                                       
                                        
                                       
                                         c 
                                         2 
                                       
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   1 
                                   2 
                                 
                                  
                                 ℏ 
                                  
                                 
                                   
                                     k 
                                     μ 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.60 
                   ) 
                 
               
             
           
         
       
     
         [0000]    E T+osc  (Group) is given by the sum of E T (Mo) (Eq. (15.51)) and Ē osc  (Eq. (15.60)): 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     ( 
                     15.61 
                     ) 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       E 
                       
                         T 
                         + 
                         OSC 
                       
                     
                      
                     
                       ( 
                       
                           
                         Group 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           E 
                           T 
                         
                          
                         
                           ( 
                           
                               
                             MO 
                           
                           ) 
                         
                       
                       + 
                       
                         
                           E 
                           _ 
                         
                         osc 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         
                                           - 
                                           
                                             
                                               
                                                 
                                                   n 
                                                   1 
                                                 
                                                  
                                                 
                                                   e 
                                                   2 
                                                 
                                               
                                               
                                                 8 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   ɛ 
                                                   0 
                                                 
                                                  
                                                 
                                                   
                                                     
                                                       aa 
                                                       0 
                                                     
                                                     
                                                       2 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       
                                                         C 
                                                         1 
                                                       
                                                        
                                                       
                                                         C 
                                                         2 
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                              
                                             
                                               [ 
                                               
                                                 
                                                   
                                                     c 
                                                     1 
                                                   
                                                    
                                                   
                                                     
                                                       c 
                                                       2 
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         2 
                                                         - 
                                                         
                                                           
                                                             a 
                                                             0 
                                                           
                                                           a 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                    
                                                   ln 
                                                    
                                                   
                                                     
                                                       a 
                                                       + 
                                                       
                                                         
                                                           
                                                             aa 
                                                             0 
                                                           
                                                           
                                                             2 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             C 
                                                             1 
                                                             
                                                              
                                                             
                                                             C 
                                                             2 
                                                             
                                                           
                                                         
                                                       
                                                     
                                                     
                                                       a 
                                                       - 
                                                       
                                                         
                                                           
                                                             aa 
                                                             0 
                                                           
                                                           
                                                             2 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             C 
                                                             1 
                                                             
                                                              
                                                             
                                                             C 
                                                             2 
                                                             
                                                           
                                                         
                                                       
                                                     
                                                   
                                                 
                                                 - 
                                                 1 
                                               
                                               ] 
                                             
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           
                                             E 
                                             T 
                                           
                                            
                                           
                                             ( 
                                             
                                               AO 
                                               / 
                                               HO 
                                             
                                             ) 
                                           
                                         
                                         + 
                                         
                                           
                                             E 
                                             T 
                                           
                                            
                                           
                                             ( 
                                             
                                               atom—atom 
                                               , 
                                               
                                                 
                                                   msp 
                                                   3 
                                                 
                                                 . 
                                                 AO 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 
                                   [ 
                                   
                                     1 
                                     + 
                                     
                                       
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           ℏ 
                                            
                                           
                                             
                                               
                                                 
                                                   
                                                     C 
                                                     
                                                       1 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       o 
                                                     
                                                   
                                                    
                                                   
                                                     C 
                                                     
                                                       2 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       o 
                                                     
                                                   
                                                    
                                                   
                                                     e 
                                                     2 
                                                   
                                                 
                                                 
                                                   4 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   π 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     ɛ 
                                                     0 
                                                   
                                                    
                                                   
                                                     R 
                                                     3 
                                                   
                                                 
                                               
                                               
                                                 m 
                                                 e 
                                               
                                             
                                           
                                         
                                         
                                           
                                             m 
                                             e 
                                           
                                            
                                           
                                             c 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 + 
                                 
                                   
                                     n 
                                     1 
                                   
                                    
                                   
                                     1 
                                     2 
                                   
                                    
                                   ℏ 
                                    
                                   
                                     
                                       k 
                                       μ 
                                     
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                       = 
                       
                         
                           ( 
                           
                             
                               E 
                                
                               
                                 ( 
                                 
                                   basis 
                                    
                                   
                                       
                                   
                                    
                                   energies 
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 E 
                                 T 
                               
                                
                               
                                 ( 
                                 
                                   atom—atom 
                                   , 
                                   
                                     
                                       msp 
                                       3 
                                     
                                     . 
                                     AO 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                          
                         
                             
                           
                             
                               [ 
                               
                                   
                               
                                
                               
                                 1 
                                 + 
                                 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       ℏ 
                                        
                                       
                                         
                                           
                                             
                                               
                                                 C 
                                                 
                                                   1 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   o 
                                                 
                                               
                                                
                                               
                                                 C 
                                                 
                                                   2 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   o 
                                                 
                                               
                                                
                                               
                                                 e 
                                                 2 
                                               
                                             
                                             
                                               4 
                                                
                                               
                                                   
                                               
                                                
                                               π 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 ɛ 
                                                 0 
                                               
                                                
                                               
                                                 R 
                                                 3 
                                               
                                             
                                           
                                           
                                             m 
                                             e 
                                           
                                         
                                       
                                     
                                     
                                       
                                         m 
                                         e 
                                       
                                        
                                       
                                         c 
                                         2 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             + 
                             
                               
                                 n 
                                 1 
                               
                                
                               
                                 1 
                                 2 
                               
                                
                               ℏ 
                                
                               
                                 
                                   k 
                                   μ 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
         [0159]    The total energy of the functional group E T (group) is the sum of the total energy of the components comprising the energy contribution of the MO formed between the participating atoms, E(basis energies), the change in the energy of the AOs or HOs upon forming the bond (E T (atom−atom,msp 3 .AO)), the energy of oscillation in the transition state, and the change in magnetic energy with bond formation, E mag . From Eq. (15.61), the total energy of the group E T (Group) is 
         [0000]    
       
         
           
             
                 
             
              
             
               ( 
               15.62 
               ) 
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       E 
                       T 
                     
                      
                     
                       ( 
                       
                           
                         Group 
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         
                                           E 
                                            
                                           
                                             ( 
                                             
                                               basis 
                                                
                                               
                                                   
                                               
                                                
                                               energies 
                                             
                                             ) 
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           E 
                                           T 
                                         
                                          
                                         
                                           ( 
                                           
                                             atom—atom 
                                             , 
                                             
                                               
                                                 msp 
                                                 3 
                                               
                                               . 
                                               AO 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                               [ 
                               
                                 1 
                                 + 
                                 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       ℏ 
                                        
                                       
                                         
                                           
                                             
                                               
                                                 C 
                                                 
                                                   1 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   o 
                                                 
                                               
                                                
                                               
                                                 C 
                                                 
                                                   2 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   o 
                                                 
                                               
                                                
                                               
                                                 e 
                                                 2 
                                               
                                             
                                             
                                               4 
                                                
                                               
                                                   
                                               
                                                
                                               π 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 ɛ 
                                                 0 
                                               
                                                
                                               
                                                 R 
                                                 3 
                                               
                                             
                                           
                                           
                                             m 
                                             e 
                                           
                                         
                                       
                                     
                                     
                                       
                                         m 
                                         e 
                                       
                                        
                                       
                                         c 
                                         2 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   n 
                                   1 
                                 
                                  
                                 
                                   
                                     E 
                                     _ 
                                   
                                   Kvib 
                                 
                               
                               + 
                               
                                 E 
                                 mag 
                               
                             
                           
                         
                       
                       + 
                     
                     ) 
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
         [0000]    The change in magnetic energy E mag  which arises due to the formation of unpaired electrons in the corresponding fragments relative to the bonded group is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                     mag 
                   
                   = 
                   
                     
                       
                         c 
                         3 
                       
                        
                       
                         
                           2 
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           
                             μ 
                             0 
                           
                            
                           
                             e 
                             2 
                           
                            
                           
                             ℏ 
                             2 
                           
                         
                         
                           
                             m 
                             e 
                             2 
                           
                            
                           
                             r 
                             3 
                           
                         
                       
                     
                     = 
                     
                       
                         c 
                         3 
                       
                        
                       
                         
                           8 
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           
                             μ 
                             0 
                           
                            
                           
                             μ 
                             B 
                             2 
                           
                         
                         
                           r 
                           3 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.63 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where r 3  is the radius of the atom that reacts to form the bond and c 3  is the number of electron pairs. 
         [0000]    
       
         
           
             
                 
             
              
             
               ( 
               15.64 
               ) 
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       E 
                       T 
                     
                      
                     
                       ( 
                       
                           
                         Group 
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         
                                           E 
                                            
                                           
                                             ( 
                                             
                                               basis 
                                                
                                               
                                                   
                                               
                                                
                                               energies 
                                             
                                             ) 
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           E 
                                           T 
                                         
                                          
                                         
                                           ( 
                                           
                                             atom—atom 
                                             , 
                                             
                                               
                                                 msp 
                                                 3 
                                               
                                               . 
                                               AO 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                               [ 
                               
                                 1 
                                 + 
                                 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       ℏ 
                                        
                                       
                                         
                                           
                                             
                                               
                                                 C 
                                                 
                                                   1 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   o 
                                                 
                                               
                                                
                                               
                                                 C 
                                                 
                                                   2 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   o 
                                                 
                                               
                                                
                                               
                                                 e 
                                                 2 
                                               
                                             
                                             
                                               4 
                                                
                                               
                                                   
                                               
                                                
                                               π 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 ɛ 
                                                 0 
                                               
                                                
                                               
                                                 R 
                                                 3 
                                               
                                             
                                           
                                           
                                             m 
                                             e 
                                           
                                         
                                       
                                     
                                     
                                       
                                         m 
                                         e 
                                       
                                        
                                       
                                         c 
                                         2 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   n 
                                   1 
                                 
                                  
                                 
                                   
                                     E 
                                     _ 
                                   
                                   Kvib 
                                 
                               
                               + 
                               
                                 
                                   c 
                                   3 
                                 
                                  
                                 
                                   
                                     8 
                                      
                                     
                                         
                                     
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     
                                       μ 
                                       0 
                                     
                                      
                                     
                                       μ 
                                       B 
                                       2 
                                     
                                   
                                   
                                     r 
                                     3 
                                   
                                 
                               
                             
                           
                         
                       
                       + 
                     
                     ) 
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
         [0000]    The total bond energy of the group E D (Group) is the negative difference of the total energy of the group (Eq. (15.64)) and the total energy of the starting species given by the sum of C 4 E initial (c 4  AO/HO) and c 5 E initial  (c 5  AO/HO): 
         [0000]    
       
         
           
             
                 
             
              
             
               ( 
               15.65 
               ) 
             
           
         
       
       
         
           
             
               
                 E 
                 D 
               
                
               
                 ( 
                 
                     
                   Group 
                 
                 ) 
               
             
             = 
             
               - 
               
                 ( 
                 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       E 
                                        
                                       
                                         ( 
                                         
                                           basis 
                                            
                                           
                                               
                                           
                                            
                                           energies 
                                         
                                         ) 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       E 
                                       T 
                                     
                                      
                                     
                                       ( 
                                       
                                         atom—atom 
                                         , 
                                         
                                           
                                             msp 
                                             3 
                                           
                                           . 
                                           AO 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           [ 
                           
                             1 
                             + 
                             
                               
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   ℏ 
                                    
                                   
                                     
                                       
                                         
                                           
                                             C 
                                             
                                               1 
                                                
                                               
                                                   
                                               
                                                
                                               o 
                                             
                                           
                                            
                                           
                                             C 
                                             
                                               2 
                                                
                                               
                                                   
                                               
                                                
                                               o 
                                             
                                           
                                            
                                           
                                             e 
                                             2 
                                           
                                         
                                         
                                           4 
                                            
                                           
                                               
                                           
                                            
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           
                                             ɛ 
                                             0 
                                           
                                            
                                           
                                             R 
                                             3 
                                           
                                         
                                       
                                       
                                         m 
                                         e 
                                       
                                     
                                   
                                 
                                 
                                   
                                     m 
                                     e 
                                   
                                    
                                   
                                     c 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                     
                       
                         
                           
                             
                               n 
                               1 
                             
                              
                             
                               
                                 E 
                                 _ 
                               
                               Kvib 
                             
                           
                           + 
                           
                             
                               c 
                               3 
                             
                              
                             
                               
                                 8 
                                  
                                 
                                     
                                 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 
                                   μ 
                                   0 
                                 
                                  
                                 
                                   μ 
                                   B 
                                   2 
                                 
                               
                               
                                 r 
                                 n 
                                 3 
                               
                             
                           
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       
                                         c 
                                         4 
                                       
                                        
                                       
                                         
                                           E 
                                           initial 
                                         
                                          
                                         
                                           ( 
                                           
                                             AO 
                                             / 
                                             HO 
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       c 
                                       5 
                                     
                                      
                                     
                                       
                                         E 
                                         initial 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             c 
                                             5 
                                           
                                            
                                           
                                             AO 
                                             / 
                                             HO 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   + 
                 
                 ) 
               
             
           
         
       
     
         [0000]    In the case of organic molecules, the atoms of the functional groups are energy matched to the C2sp 3  HO such that 
         [0000]      E(AO/HO=−14.63489 eV  (15.66)
 
         [0000]    For examples of E mag  from previous sections: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             mag 
                           
                            
                           
                             ( 
                             
                               C 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               c 
                               3 
                             
                              
                             
                               
                                 8 
                                  
                                 
                                     
                                 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 
                                   μ 
                                   0 
                                 
                                  
                                 
                                   μ 
                                   B 
                                   2 
                                 
                               
                               
                                 r 
                                 3 
                               
                             
                           
                           = 
                           
                             
                               c 
                               3 
                             
                              
                             
                               
                                 8 
                                  
                                 
                                     
                                 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 
                                   μ 
                                   0 
                                 
                                  
                                 
                                   μ 
                                   B 
                                   2 
                                 
                               
                               
                                 
                                   ( 
                                   
                                     0.91771 
                                      
                                     
                                         
                                     
                                      
                                     
                                       a 
                                       0 
                                     
                                   
                                   ) 
                                 
                                 3 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             c 
                             3 
                           
                            
                           0.14803 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.67 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       E 
                       mag 
                     
                      
                     
                       ( 
                       
                         O 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         p 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         c 
                         3 
                       
                        
                       
                         
                           8 
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           
                             μ 
                             0 
                           
                            
                           
                             μ 
                             B 
                             2 
                           
                         
                         
                           r 
                           3 
                         
                       
                     
                     = 
                     
                       
                         
                           c 
                           3 
                         
                          
                         
                           
                             8 
                              
                             
                                 
                             
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               μ 
                               0 
                             
                              
                             
                               μ 
                               B 
                               2 
                             
                           
                           
                             a 
                             0 
                             3 
                           
                         
                       
                       = 
                       
                         
                           c 
                           3 
                         
                          
                         0.11441 
                          
                         
                             
                         
                          
                         eV 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.68 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             mag 
                           
                            
                           
                             ( 
                             
                               N 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               p 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               c 
                               3 
                             
                              
                             
                               
                                 8 
                                  
                                 
                                     
                                 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 
                                   μ 
                                   0 
                                 
                                  
                                 
                                   μ 
                                   B 
                                   2 
                                 
                               
                               
                                 r 
                                 3 
                               
                             
                           
                           = 
                           
                             
                               c 
                               3 
                             
                              
                             
                               
                                 8 
                                  
                                 
                                     
                                 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 
                                   μ 
                                   0 
                                 
                                  
                                 
                                   μ 
                                   B 
                                   2 
                                 
                               
                               
                                 
                                   ( 
                                   
                                     0.93084 
                                      
                                     
                                         
                                     
                                      
                                     
                                       a 
                                       0 
                                     
                                   
                                   ) 
                                 
                                 3 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             c 
                             3 
                           
                            
                           0.14185 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.69 
                   ) 
                 
               
             
           
         
       
     
         [0160]    In the general case of the solution of an organic functional group, the geometric bond parameters are solved from the semimajor axis and the relationships between the parameters by first using Eq. (15.51) to arrive at a. Then, the remaining parameters are determined using Eqs. (15.1-15.5). Next, the energies are given by Eqs. (15.61-15.68). To meet the equipotential condition for the union of the H 2 -type-ellipsoidal-MO and the HO or AO of the atom of a functional group, the factor c 2  of a H 2 -type ellipsoidal MO in principal Eqs. (15.51) and (15.61) may given by 
         [0161]    (i) one: 
         [0000]      c 2 =1  (15.70)
 
         [0162]    (ii) the ratio that is less than one of 13.605804 eV, the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243), and the magnitude of the Coulombic energy of the participating AO or HO of the atom, E Coulomb (MO.atom,msp 3 ) given by Eqs. (15.19) and (15.31-15.32). For |E Coulomb (MO.atom,msp 3 )|&gt;13.605804 eV: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       c 
                       2 
                     
                     = 
                     
                       
                         
                           
                             e 
                             2 
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               ɛ 
                               0 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                         
                           
                             e 
                             2 
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               ɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 A 
                                 - 
                                 
                                   B 
                                    
                                   
                                       
                                   
                                    
                                   A 
                                    
                                   
                                       
                                   
                                    
                                   or 
                                    
                                   
                                       
                                   
                                    
                                   
                                     Bsp 
                                     3 
                                   
                                 
                               
                             
                           
                         
                       
                       = 
                       
                         
                           13.605804 
                            
                           
                               
                           
                            
                           eV 
                         
                         
                            
                           
                             
                               E 
                               Coulomb 
                             
                              
                             
                               ( 
                               
                                 
                                   MO 
                                   . 
                                   atom 
                                 
                                 , 
                                 
                                   msp 
                                   3 
                                 
                               
                               ) 
                             
                           
                            
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       For 
                        
                       
                           
                       
                        
                       
                          
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 MO 
                                 . 
                                 atom 
                               
                               , 
                               
                                 msp 
                                 3 
                               
                             
                             ) 
                           
                         
                          
                       
                     
                     &lt; 
                     
                       13.605804 
                        
                       
                           
                       
                        
                       eV 
                        
                       
                         : 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.71 
                   ) 
                 
               
             
             
               
                 
                   
                     c 
                     2 
                   
                   = 
                   
                     
                       
                         
                           e 
                           2 
                         
                         
                           8 
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           
                             ɛ 
                             0 
                           
                            
                           
                             r 
                             
                               A 
                               - 
                               
                                 B 
                                  
                                 
                                     
                                 
                                  
                                 A 
                                  
                                 
                                     
                                 
                                  
                                 or 
                                  
                                 
                                     
                                 
                                  
                                 
                                   Bsp 
                                   3 
                                 
                               
                             
                           
                         
                       
                       
                         
                           e 
                           2 
                         
                         
                           8 
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           
                             ɛ 
                             0 
                           
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                     = 
                     
                       
                          
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 MO 
                                 . 
                                 atom 
                               
                               , 
                               
                                 msp 
                                 3 
                               
                             
                             ) 
                           
                         
                          
                       
                       
                         13.605804 
                          
                         
                             
                         
                          
                         eV 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.72 
                   ) 
                 
               
             
           
         
       
     
         [0163]    (iii) the ratio that is less than one of 13.605804 eV, the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243), and the magnitude of the valence energy, E(valence), of the participating AO or HO of the atom where E(valence) is the ionization energy or E(MO.atom,msp 3 ) given by Eqs. (15.20) and (15.31-15.32). For 1E(valence)|&gt;13.605804 eV: 
         [0000]    
       
         
           
             
               
                 
                   
                     c 
                     2 
                   
                   = 
                   
                     
                       
                         
                           e 
                           2 
                         
                         
                           8 
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           
                             ɛ 
                             0 
                           
                            
                           
                             a 
                             0 
                           
                         
                       
                       
                         
                           e 
                           2 
                         
                         
                           8 
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           
                             ɛ 
                             0 
                           
                            
                           
                             r 
                             
                               A 
                               - 
                               
                                 B 
                                  
                                 
                                     
                                 
                                  
                                 A 
                                  
                                 
                                     
                                 
                                  
                                 or 
                                  
                                 
                                     
                                 
                                  
                                 
                                   Bsp 
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                     = 
                     
                       
                         13.605804 
                          
                         
                             
                         
                          
                         eV 
                       
                       
                          
                         
                           E 
                            
                           
                             ( 
                             valence 
                             ) 
                           
                         
                          
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.73 
                   ) 
                 
               
             
           
         
       
     
         [0164]    For |E(valence)|&lt;13.605804 eV: 
         [0000]    
       
         
           
             
               
                 
                   
                     c 
                     2 
                   
                   = 
                   
                     
                       
                         
                           e 
                           2 
                         
                         
                           8 
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           
                             ɛ 
                             0 
                           
                            
                           
                             r 
                             
                               A 
                               - 
                               
                                 B 
                                  
                                 
                                     
                                 
                                  
                                 A 
                                  
                                 
                                     
                                 
                                  
                                 or 
                                  
                                 
                                     
                                 
                                  
                                 
                                   Bsp 
                                   3 
                                 
                               
                             
                           
                         
                       
                       
                         
                           e 
                           2 
                         
                         
                           8 
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           
                             ɛ 
                             0 
                           
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                     = 
                     
                       
                          
                         
                           E 
                            
                           
                             ( 
                             valence 
                             ) 
                           
                         
                          
                       
                       
                         13.605804 
                          
                         
                             
                         
                          
                         eV 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.74 
                   ) 
                 
               
             
           
         
       
     
         [0165]    (iv) the ratio that is less than one of the magnitude of the Coulombic energy of the participating AO or HO of a first atom, E Coulomb (MO.atom,msp 3 ) given by Eqs. (15.19) and (15.31-15.32), and the magnitude of the valence energy, E(valence), of the participating AO or HO of a second atom to which the first is energy matched where E(valence) is the ionization energy or E(MO.atom,msp 3 ) given by Eqs. (15.20) and (15.31-15.32). For |E Coulomb (MO.atom,msp 3 )|&gt;E(valence): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       c 
                       2 
                     
                     = 
                     
                       
                          
                         
                           E 
                            
                           
                             ( 
                             valence 
                             ) 
                           
                         
                          
                       
                       
                          
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 MO 
                                 . 
                                 atom 
                               
                               , 
                               
                                 msp 
                                 3 
                               
                             
                             ) 
                           
                         
                          
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       For 
                        
                       
                           
                       
                        
                       
                          
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 MO 
                                 . 
                                 atom 
                               
                               , 
                               
                                 msp 
                                 3 
                               
                             
                             ) 
                           
                         
                          
                       
                     
                     &lt; 
                     
                       
                         E 
                          
                         
                           ( 
                           valence 
                           ) 
                         
                       
                        
                       
                         : 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.75 
                   ) 
                 
               
             
             
               
                 
                   
                     c 
                     2 
                   
                   = 
                   
                     
                        
                       
                         
                           E 
                           Coulomb 
                         
                          
                         
                           ( 
                           
                             
                               MO 
                               . 
                               atom 
                             
                             , 
                             
                               msp 
                               3 
                             
                           
                           ) 
                         
                       
                        
                     
                     
                        
                       
                         E 
                          
                         
                           ( 
                           valence 
                           ) 
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   15.76 
                   ) 
                 
               
             
           
         
       
     
         [0166]    (v) the ratio of the magnitude of the valence-level energies, E n  (valence), of the AO or HO of the nth participating atom of two that are energy matched where E(valence) is the ionization energy or E(MO.atom,msp 3 ) given by Eqs. (15.20) and (15.31-15.32): 
         [0000]    
       
         
           
             
               
                 
                   
                     c 
                     2 
                   
                   = 
                   
                     
                       
                         E 
                         1 
                       
                        
                       
                         ( 
                         valence 
                         ) 
                       
                     
                     
                       
                         E 
                         2 
                       
                        
                       
                         ( 
                         valence 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.77 
                   ) 
                 
               
             
           
         
       
     
         [0167]    (vi) the factor that is the ratio of the hybridization factor c 2  (1) of the valence AO or HO of a first atom and the hybridization factor c 2  (2) of the valence AO or HO of a second atom to which the first is energy matched where c 2  (n) is given by Eqs. (15.71-15.77); alternatively c 2  is the hybridization factor c 2  (1) of the valence AOs or HOs a first pair of atoms and the hybridization factor c 2  (2) of the valence AO or HO a third atom or second pair to which the first two are energy matched: 
         [0000]    
       
         
           
             
               
                 
                   
                     c 
                     2 
                   
                   = 
                   
                     
                       
                         c 
                         2 
                       
                        
                       
                         ( 
                         1 
                         ) 
                       
                     
                     
                       
                         c 
                         2 
                       
                        
                       
                         ( 
                         2 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.78 
                   ) 
                 
               
             
           
         
       
     
         [0168]    (vii) the factor that is the product of the hybridization factor c 2  (1) of the valence AO or HO of a first atom and the hybridization factor c 2  (2) of the valence AO or HO of a second atom to which the first is energy matched where c 2  (n) is given by Eqs. (15.71-15.78); alternatively c 2  is the hybridization factor c 2  (1) of the valence AOs or HOs a first pair of atoms and the hybridization factor c 2  (2) of the valence AO or HO a third atom or second pair to which the first two are energy matched: 
         [0000]        c   2   =c   2 (1) c   2 (2)  (15.79)
 
         [0000]    The hybridization factor c 2  corresponds to the force constant k (Eqs. (11.65) and (13.58)). In the case that the valence or Coulombic energy of the AO or HO is less than 13.605804 eV, the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243), then C 2  corresponding to k′ (Eq. (15.1)) is given by Eqs. (15.71-15.79). 
         [0169]    Specific examples of the factors c 2  and C 2  of a H 2 -type ellipsoidal MO of Eq. (15.60) given in following sections are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       c 
                       2 
                     
                      
                     
                       ( 
                       
                         C 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         
                             
                         
                          
                         HO 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         F 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         E 
                          
                         
                           ( 
                           
                             C 
                             , 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                             
                           
                           ) 
                         
                       
                       
                         E 
                          
                         
                           ( 
                           F 
                           ) 
                         
                       
                     
                      
                     
                       
                         c 
                         2 
                       
                        
                       
                         ( 
                         
                           C 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             sp 
                             3 
                           
                            
                           HO 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           - 
                           14.63489 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                       
                         
                           - 
                           17.42282 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                     
                      
                     
                       ( 
                       0.91771 
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     0.77087 
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       C 
                       2 
                     
                      
                     
                       ( 
                       
                         C 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         Cl 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         E 
                          
                         
                           ( 
                           Cl 
                           ) 
                         
                       
                       
                         E 
                          
                         
                           ( 
                           
                             C 
                             , 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                             
                           
                           ) 
                         
                       
                     
                      
                     
                       
                         c 
                         2 
                       
                        
                       
                         ( 
                         
                           C 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             sp 
                             3 
                           
                            
                           HO 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           - 
                           12.96764 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                       
                         
                           - 
                           14.63489 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                     
                      
                     
                       ( 
                       0.91771 
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     0.81317 
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       C 
                       2 
                     
                      
                     
                       ( 
                       
                         C 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         Br 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         E 
                          
                         
                           ( 
                           Br 
                           ) 
                         
                       
                       
                         E 
                          
                         
                           ( 
                           
                             C 
                             , 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                             
                           
                           ) 
                         
                       
                     
                      
                     
                       
                         c 
                         2 
                       
                        
                       
                         ( 
                         
                           C 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             sp 
                             3 
                           
                            
                           HO 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           - 
                           11.81381 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                       
                         
                           - 
                           14.63489 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                     
                      
                     
                       ( 
                       0.91771 
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     0.74081 
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       C 
                       2 
                     
                      
                     
                       ( 
                       
                         C 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         I 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         E 
                          
                         
                           ( 
                           I 
                           ) 
                         
                       
                       
                         E 
                          
                         
                           ( 
                           
                             C 
                             , 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                             
                           
                           ) 
                         
                       
                     
                      
                     
                       
                         c 
                         2 
                       
                        
                       
                         ( 
                         
                           C 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             sp 
                             3 
                           
                            
                           HO 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           - 
                           10.45126 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                       
                         
                           - 
                           14.63489 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                     
                      
                     
                       ( 
                       0.91771 
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     0.65537 
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       c 
                       2 
                     
                      
                     
                       ( 
                       
                         C 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         O 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         E 
                          
                         
                           ( 
                           O 
                           ) 
                         
                       
                       
                         E 
                          
                         
                           ( 
                           
                             C 
                             , 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                             
                           
                           ) 
                         
                       
                     
                      
                     
                       
                         c 
                         2 
                       
                        
                       
                         ( 
                         
                           C 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             sp 
                             3 
                           
                            
                           HO 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           - 
                           13.61806 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                       
                         
                           - 
                           14.63489 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                     
                      
                     
                       ( 
                       0.91771 
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     0.85395 
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   c 
                   2 
                 
                  
                 
                   ( 
                   
                     H 
                      
                     
                         
                     
                      
                     to 
                      
                     
                         
                     
                      
                     1 
                      
                     ° 
                      
                     
                         
                     
                      
                     N 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     E 
                      
                     
                       ( 
                       N 
                       ) 
                     
                   
                   
                     E 
                      
                     
                       ( 
                       
                         C 
                         , 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             sp 
                             3 
                           
                         
                       
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       
                         - 
                         14.53414 
                       
                        
                       
                           
                       
                        
                       eV 
                     
                     
                       
                         - 
                         15.35946 
                       
                        
                       
                           
                       
                        
                       eV 
                     
                   
                   = 
                   0.94627 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   
                     
                       c 
                       2 
                     
                      
                     
                       ( 
                       
                         C 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         N 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         E 
                          
                         
                           ( 
                           N 
                           ) 
                         
                       
                       
                         E 
                          
                         
                           ( 
                           
                             C 
                             , 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                             
                           
                           ) 
                         
                       
                     
                      
                     
                       
                         c 
                         2 
                       
                        
                       
                         ( 
                         
                           C 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             sp 
                             3 
                           
                            
                           HO 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           - 
                           14.53414 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                       
                         
                           - 
                           14.63489 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                     
                      
                     
                       ( 
                       0.91771 
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     0.91140 
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   c 
                   2 
                 
                  
                 
                   ( 
                   
                     H 
                      
                     
                         
                     
                      
                     to 
                      
                     
                         
                     
                      
                     2 
                      
                     ° 
                      
                     
                         
                     
                      
                     N 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     E 
                      
                     
                       ( 
                       N 
                       ) 
                     
                   
                   
                     E 
                      
                     
                       ( 
                       
                         C 
                         , 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             sp 
                             3 
                           
                         
                       
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       
                         - 
                         14.53414 
                       
                        
                       
                           
                       
                        
                       eV 
                     
                     
                       
                         - 
                         15.56407 
                       
                        
                       
                           
                       
                        
                       eV 
                     
                   
                   = 
                   0.93383 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   C 
                   2 
                 
                  
                 
                   ( 
                   
                     S 
                      
                     
                         
                     
                      
                     3 
                      
                     
                         
                     
                      
                     p 
                      
                     
                         
                     
                      
                     to 
                      
                     
                         
                     
                      
                     H 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     E 
                      
                     
                       ( 
                       
                         S 
                         , 
                         
                           3 
                            
                           
                               
                           
                            
                           p 
                         
                       
                       ) 
                     
                   
                   
                     E 
                      
                     
                       ( 
                       H 
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       
                         - 
                         10.36001 
                       
                        
                       
                           
                       
                        
                       eV 
                     
                     
                       
                         - 
                         13.60580 
                       
                        
                       
                           
                       
                        
                       eV 
                     
                   
                   = 
                   0.76144 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   
                     
                       C 
                       2 
                     
                      
                     
                       ( 
                       
                         C 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         S 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         E 
                          
                         
                           ( 
                           S 
                           ) 
                         
                       
                       
                         E 
                          
                         
                           ( 
                           
                             C 
                             , 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                             
                           
                           ) 
                         
                       
                     
                      
                     
                       
                         c 
                         2 
                       
                        
                       
                         ( 
                         
                           C 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             sp 
                             3 
                           
                            
                           HO 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           - 
                           10.36001 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                       
                         
                           - 
                           14.63489 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                     
                      
                     
                       ( 
                       0.91771 
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     0.64965 
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       c 
                       2 
                     
                      
                     
                       ( 
                       
                         O 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         S 
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         C 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         E 
                          
                         
                           ( 
                           O 
                           ) 
                         
                       
                       
                         E 
                          
                         
                           ( 
                           S 
                           ) 
                         
                       
                     
                      
                     
                       
                         c 
                         2 
                       
                        
                       
                         ( 
                         
                           C 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             sp 
                             3 
                           
                            
                           HO 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           - 
                           13.61806 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                       
                         
                           - 
                           10.36001 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                     
                      
                     
                       ( 
                       0.91771 
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     1.20632 
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   c 
                   2 
                 
                  
                 
                   ( 
                   
                     S 
                      
                     
                         
                     
                      
                     3 
                      
                     
                         
                     
                      
                     
                       sp 
                       3 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     
                       E 
                       Coulomb 
                     
                      
                     
                       ( 
                       
                         S 
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                       
                       ) 
                     
                   
                   
                     E 
                      
                     
                       ( 
                       H 
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       
                         - 
                         11.57099 
                       
                        
                       
                           
                       
                        
                       eV 
                     
                     
                       
                         - 
                         13.60580 
                       
                        
                       
                           
                       
                        
                       eV 
                     
                   
                   = 
                   0.85045 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   
                     
                       C 
                       2 
                     
                      
                     
                       ( 
                       
                         C 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         S 
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         E 
                          
                         
                           ( 
                           
                             S 
                              
                             
                                 
                             
                              
                             3 
                              
                             
                                 
                             
                              
                             
                               sp 
                               3 
                             
                           
                           ) 
                         
                       
                       
                         E 
                          
                         
                           ( 
                           
                             C 
                             , 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                             
                           
                           ) 
                         
                       
                     
                      
                     
                       
                         c 
                         2 
                       
                        
                       
                         ( 
                         
                           S 
                            
                           
                               
                           
                            
                           3 
                            
                           
                               
                           
                            
                           
                             sp 
                             3 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           - 
                           11.52126 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                       
                         
                           - 
                           14.63489 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                     
                      
                     
                       ( 
                       0.85045 
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     0.66951 
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       C 
                       2 
                     
                      
                     
                       ( 
                       
                         S 
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         O 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         C2 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         E 
                          
                         
                           ( 
                           
                             S 
                              
                             
                                 
                             
                              
                             3 
                              
                             
                                 
                             
                              
                             
                               sp 
                               3 
                             
                           
                           ) 
                         
                       
                       
                         E 
                          
                         
                           ( 
                           
                             O 
                             , 
                             
                               2 
                                
                               
                                   
                               
                                
                               p 
                             
                           
                           ) 
                         
                       
                     
                      
                     
                       
                         c 
                         2 
                       
                        
                       
                         ( 
                         
                           C 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             sp 
                             3 
                           
                            
                           HO 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           - 
                           11.52126 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                       
                         
                           - 
                           13.61806 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                     
                      
                     
                       ( 
                       0.91771 
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     0.77641 
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       c 
                       2 
                     
                      
                     
                       ( 
                       
                         O 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         N 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         p 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         C 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         E 
                          
                         
                           ( 
                           O 
                           ) 
                         
                       
                       
                         E 
                          
                         
                           ( 
                           N 
                           ) 
                         
                       
                     
                      
                     
                       
                         c 
                         2 
                       
                        
                       
                         ( 
                         
                           C 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             sp 
                             3 
                           
                            
                           HO 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           - 
                           13.61806 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                       
                         
                           - 
                           14.53414 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                     
                      
                     
                       ( 
                       0.91771 
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     0.85987 
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   c 
                   2 
                 
                  
                 
                   ( 
                   
                     N 
                      
                     
                         
                     
                      
                     2 
                      
                     
                         
                     
                      
                     p 
                      
                     
                         
                     
                      
                     to 
                      
                     
                         
                     
                      
                     O 
                      
                     
                         
                     
                      
                     2 
                      
                     
                         
                     
                      
                     p 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     
                       c 
                       2 
                     
                      
                     
                       ( 
                       
                         C 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         N 
                       
                       ) 
                     
                   
                   
                     
                       c 
                       2 
                     
                      
                     
                       ( 
                       
                         C 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         O 
                       
                       ) 
                     
                   
                 
                 = 
                 
                   
                     0.91140 
                     0.85395 
                   
                   = 
                   1.06727 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   
                     
                       C 
                       2 
                     
                      
                     
                       ( 
                       
                         benzene 
                          
                         
                             
                         
                          
                         C 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       c 
                       2 
                     
                      
                     
                       ( 
                       
                         benzene 
                          
                         
                             
                         
                          
                         C 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       13.605804 
                        
                       
                           
                       
                        
                       eV 
                     
                     
                       15.95955 
                        
                       
                           
                       
                        
                       eV 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     0.85252 
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       c 
                       2 
                     
                      
                     
                       ( 
                       
                         aryl 
                          
                         
                             
                         
                          
                         C 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         O 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         E 
                          
                         
                           ( 
                           O 
                           ) 
                         
                       
                       
                         E 
                          
                         
                           ( 
                           
                             C 
                             , 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                             
                           
                           ) 
                         
                       
                     
                      
                     
                       
                         c 
                         2 
                       
                        
                       
                         ( 
                         
                           aryl 
                            
                           
                               
                           
                            
                           C 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             sp 
                             3 
                           
                            
                           HO 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           - 
                           13.61806 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                       
                         
                           - 
                           14.63489 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                     
                      
                     
                       ( 
                       0.85252 
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     0.79329 
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   c 
                   2 
                 
                  
                 
                   ( 
                   
                     H 
                      
                     
                         
                     
                      
                     to 
                      
                     
                         
                     
                      
                     anline 
                      
                     
                         
                     
                      
                     N 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     E 
                      
                     
                       ( 
                       N 
                       ) 
                     
                   
                   
                     E 
                      
                     
                       ( 
                       
                         C 
                         , 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             sp 
                             3 
                           
                         
                       
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       
                         - 
                         14.53414 
                       
                        
                       
                           
                       
                        
                       eV 
                     
                     
                       
                         - 
                         15.76868 
                       
                        
                       
                           
                       
                        
                       eV 
                     
                   
                   = 
                   0.92171 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   
                     
                       c 
                       2 
                     
                      
                     
                       ( 
                       
                         aryl 
                          
                         
                             
                         
                          
                         C 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         N 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         E 
                          
                         
                           ( 
                           N 
                           ) 
                         
                       
                       
                         E 
                          
                         
                           ( 
                           
                             C 
                             , 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                             
                           
                           ) 
                         
                       
                     
                      
                     
                       
                         c 
                         2 
                       
                        
                       
                         ( 
                         
                           aryl 
                            
                           
                               
                           
                            
                           C 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             sp 
                             3 
                           
                            
                           HO 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           - 
                           14.53414 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                       
                         
                           - 
                           14.63489 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                     
                      
                     
                       ( 
                       0.85252 
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     0.84665 
                   
                   , 
                 
               
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               
                 
                   
                     
                       C 
                       2 
                     
                      
                     
                       ( 
                       
                         S 
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         p 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         aryl 
                          
                         
                           - 
                         
                          
                         type 
                          
                         
                             
                         
                          
                         C 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       E 
                        
                       
                         ( 
                         
                           S 
                           , 
                           
                             3 
                              
                             
                                 
                             
                              
                             p 
                           
                         
                         ) 
                       
                     
                     
                       E 
                        
                       
                         ( 
                         
                           C 
                           , 
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               sp 
                               3 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         - 
                         10.36001 
                       
                        
                       
                           
                       
                        
                       eV 
                     
                     
                       
                         - 
                         15.76868 
                       
                        
                       
                           
                       
                        
                       eV 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     0.65700 
                     . 
                   
                 
               
             
           
         
       
     
       Mo Intercept Angles and Distances 
       [0170]    Consider the general case of Eqs. (13.84-13.95), wherein the nucleus of a B atom and the nucleus of a A atom comprise the foci of each H 2 -type ellipsoidal MO of an A-B bond. The parameters of the point of intersection of each H 2 -type ellipsoidal MO and the A-atom AO are determined from the polar equation of the ellipse: 
         [0000]    
       
         
           
             
               
                 
                   r 
                   = 
                   
                     
                       r 
                       0 
                     
                      
                     
                       
                         1 
                         + 
                         e 
                       
                       
                         1 
                         + 
                         
                           e 
                            
                           
                               
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             ′ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.80 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The radius of the A shell is r A , and the polar radial coordinate of the ellipse and the radius of the A shell are equal at the point of intersection such that 
         [0000]    
       
         
           
             
               
                 
                   
                     r 
                     A 
                   
                   = 
                   
                     
                       ( 
                       
                         a 
                         - 
                         
                           c 
                           ′ 
                         
                       
                       ) 
                     
                      
                     
                       
                         1 
                         + 
                         
                           
                             c 
                             ′ 
                           
                           a 
                         
                       
                       
                         1 
                         + 
                         
                           
                             
                               c 
                               ′ 
                             
                             a 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             ′ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.81 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The polar angle θ′ at the intersection point is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     θ 
                     ′ 
                   
                   = 
                   
                     
                       cos 
                       
                         - 
                         1 
                       
                     
                     ( 
                     
                       
                         a 
                         
                           c 
                           ′ 
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               ( 
                               
                                 a 
                                 - 
                                 
                                   c 
                                   ′ 
                                 
                               
                               ) 
                             
                              
                             
                               
                                 1 
                                 + 
                                 
                                   
                                     c 
                                     ′ 
                                   
                                   a 
                                 
                               
                               
                                 r 
                                 A 
                               
                             
                           
                           - 
                           1 
                         
                         ) 
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   15.82 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Then, the angle θ A AO  the radial vector of the A AO makes with the internuclear axis is 
         [0000]      θ A AO =180°−θ′  (15.83)
 
         [0000]    The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals such that the angle ωt=θ H     2     MO  between the internuclear axis and the point of intersection of each H 2 -type ellipsoidal MO with the A radial vector obeys the following relationship: 
         [0000]      r A  sin θ A AO =b sin θ H     2     MO   (15.84)
 
         [0000]    such that 
         [0000]    
       
         
           
             
               
                 
                   
                     θ 
                     
                       
                         H 
                         2 
                       
                        
                       MO 
                     
                   
                   = 
                   
                     
                       sin 
                       
                         - 
                         1 
                       
                     
                      
                     
                       
                         
                           r 
                           a 
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         
                           θ 
                           AAO 
                         
                       
                       b 
                     
                   
                 
               
               
                 
                   ( 
                   15.85 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The distance d H     2     MO  along the internuclear axis from the origin of H 2 -type ellipsoidal MO to the point of intersection of the orbitals is given by 
         [0000]      d H     2     MO =a cos θ H     2     MO   (15.86)
 
         [0000]    The distance d A AO  along the internuclear axis from the origin of the A atom to the point of intersection of the orbitals is given by 
         [0000]        d   A AO   =c′−d   H     2     MO   (15.87)
 
       Bond Angles 
       [0171]    Further consider an ACB MO comprising a linear combination of C-A-bond and C—B-bond MOs where C is the general central atom. A bond is also possible between the A and B atoms of the C-A and C—B bonds. Such A-B bonding would decrease the C-A and C—B bond strengths since electron density would be shifted from the latter bonds to the former bond. Thus, the ∠ACB bond angle is determined by the condition that the total energy of the H 2 -type ellipsoidal MO between the terminal A and B atoms is zero. The force constant k′ of a H 2 -type ellipsoidal MO due to the equivalent of two point charges of at the foci is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                     ′ 
                   
                   = 
                   
                     
                       
                         C 
                         1 
                       
                        
                       
                         C 
                         2 
                       
                        
                       2 
                        
                       
                           
                       
                        
                       
                         e 
                         2 
                       
                     
                     
                       4 
                        
                       
                         πɛ 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.88 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where C 1  is the fraction of the H 2 -type ellipsoidal MO basis function of a chemical bond of the molecule which is 0.75 (Eq. (13.59)) for a terminal A-H (A is H or other atom) and 1 otherwise and C 2  is the factor that results in an equipotential energy match of the participating at least two atomic orbitals of the chemical bond and is equal to the corresponding factor of Eqs. (15.51) and (15.61). The distance from the origin of the MO to each focus c′ of the A-B ellipsoidal MO is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     c 
                     ′ 
                   
                   = 
                   
                     
                       a 
                        
                       
                         
                           
                             
                               ℏ 
                               2 
                             
                              
                             4 
                              
                             
                               πɛ 
                               0 
                             
                           
                           
                             
                               m 
                               e 
                             
                              
                             
                               e 
                               2 
                             
                              
                             2 
                              
                             
                                 
                             
                              
                             
                               C 
                               1 
                             
                              
                             
                               C 
                               2 
                             
                              
                             a 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           aa 
                           0 
                         
                         
                           2 
                            
                           
                               
                           
                            
                           
                             C 
                             1 
                           
                            
                           
                             C 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.89 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The internuclear distance is 
         [0000]    
       
         
           
             
               
                 
                   
                     2 
                      
                     
                         
                     
                      
                     
                       c 
                       ′ 
                     
                   
                   = 
                   
                     2 
                      
                     
                       
                         
                           aa 
                           0 
                         
                         
                           2 
                            
                           
                               
                           
                            
                           
                             C 
                             1 
                           
                            
                           
                             C 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.90 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The length of the semiminor axis of the prolate spheroidal A-B MO b=c is given by Eq. (15.4). 
         [0172]    The component energies and the total energy, E T (H 2 MO), of the A-B bond are given by the energy equations (Eqs. (11.207-11.212), (11.213-11.217), and (11.239)) of H 2  except that the terms based on charge are multiplied by c BO , the bond-order factor which is 1 for a single bond and when the MO comprises n 1  equivalent single bonds as in the case of functional groups. c BO  is 4 for an independent double bond as in the case of the CO 2  and NO 2  molecules. The kinetic energy term is multiplied by c′ 80  which is 1 for a single bond, 2 for a double bond, and 9/2 for a triple bond. The electron energy terms are multiplied by c 1 , the fraction of the H 2 -type ellipsoidal MO basis function of a terminal chemical bond which is 0.75 (Eq. (13.233)) for a terminal A-H (A is H or other atom) and 1 otherwise, The electron energy terms are further multiplied by c′ 2 , the hybridization or energy-matching factor that results in an equipotential energy match of the participating at least two atomic orbitals of each terminal bond. Furthermore, when A-B comprises atoms other than H, E T (atom−atom,msp 3 .AO), the energy component due to the AO or HO&#39;s charge donation to the terminal MO, is added to the other energy terms to give E T (H 2 MO): 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                     
                       T 
                        
                       
                         ( 
                         
                           
                             H 
                             2 
                           
                            
                           MO 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           - 
                           
                              
                             2 
                           
                         
                         
                           8 
                            
                           
                             πɛ 
                             0 
                           
                            
                           
                             c 
                             ′ 
                           
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                              
                             
                               
                                 c 
                                 2 
                                 ′ 
                               
                                
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       c 
                                       BO 
                                     
                                   
                                   - 
                                   
                                     
                                       c 
                                       BO 
                                       ′ 
                                     
                                      
                                     
                                       
                                         a 
                                         0 
                                       
                                       a 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             ln 
                              
                             
                               
                                 a 
                                 + 
                                 
                                   c 
                                   ′ 
                                 
                               
                               
                                 a 
                                 - 
                                 
                                   c 
                                   ′ 
                                 
                               
                             
                           
                           - 
                           1 
                         
                         ] 
                       
                     
                     + 
                     
                       
                         E 
                         T 
                       
                        
                       
                         ( 
                         
                           
                             atom 
                             - 
                             atom 
                           
                           , 
                           
                             
                               msp 
                               3 
                             
                             . 
                             AO 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.91 
                   ) 
                 
               
             
           
         
       
     
         [0173]    The radiation reaction force in the case of the vibration of A-B in the transition state corresponds to the Doppler energy, E D , given by Eq. (11.181) that is dependent on the motion of the electrons and the nuclei. The total energy that includes the radiation reaction of the A-B MO is given by the sum of E T (H 2 MO) (Eq. (15.91)) and Ē osc  given Eqs. (11.213-11.220), (11.231-11.236), and (11.239-11.240). Thus, the total energy E T (A-B) of the A-B MO including the Doppler term is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       T 
                     
                      
                     
                       ( 
                       
                         A 
                         - 
                         B 
                       
                       ) 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           - 
                                           
                                             e 
                                             2 
                                           
                                         
                                         
                                           8 
                                            
                                           
                                             πɛ 
                                             0 
                                           
                                            
                                           
                                             c 
                                             ′ 
                                           
                                         
                                       
                                        
                                       
                                         [ 
                                         
                                           
                                             
                                               c 
                                               1 
                                             
                                              
                                             
                                               
                                                 c 
                                                 2 
                                                 ′ 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     2 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     
                                                       c 
                                                       BO 
                                                     
                                                   
                                                   - 
                                                   
                                                     
                                                       c 
                                                       BO 
                                                       ′ 
                                                     
                                                      
                                                     
                                                       
                                                         a 
                                                         0 
                                                       
                                                       a 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                              
                                             ln 
                                              
                                             
                                               
                                                 a 
                                                 + 
                                                 
                                                   c 
                                                   ′ 
                                                 
                                               
                                               
                                                 a 
                                                 - 
                                                 
                                                   c 
                                                   ′ 
                                                 
                                               
                                             
                                           
                                           - 
                                           1 
                                         
                                         ] 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       E 
                                       T 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           atom 
                                           - 
                                           atom 
                                         
                                         , 
                                         
                                           
                                             msp 
                                             3 
                                           
                                           . 
                                           AO 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     [ 
                                     
                                       1 
                                       + 
                                       
                                         
                                           
                                             2 
                                              
                                             ℏ 
                                              
                                             
                                               
                                                 
                                                   
                                                     c 
                                                     BO 
                                                   
                                                    
                                                   
                                                     
                                                       
                                                         C 
                                                         
                                                           1 
                                                            
                                                           
                                                               
                                                           
                                                            
                                                           o 
                                                         
                                                       
                                                        
                                                       
                                                         C 
                                                         
                                                           2 
                                                            
                                                           
                                                               
                                                           
                                                            
                                                           o 
                                                         
                                                       
                                                        
                                                       
                                                         e 
                                                         2 
                                                       
                                                     
                                                     
                                                       4 
                                                        
                                                       
                                                         πɛ 
                                                         0 
                                                       
                                                        
                                                       
                                                         a 
                                                         3 
                                                       
                                                     
                                                   
                                                 
                                                 
                                                   m 
                                                   e 
                                                 
                                               
                                             
                                           
                                           
                                             
                                               m 
                                               e 
                                             
                                              
                                             
                                               c 
                                               2 
                                             
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     1 
                                     2 
                                   
                                    
                                   ℏ 
                                    
                                   
                                     
                                       
                                         
                                           
                                             c 
                                             BO 
                                           
                                            
                                           
                                             
                                               
                                                 c 
                                                 1 
                                               
                                                
                                               
                                                 c 
                                                 2 
                                                 ′ 
                                               
                                                
                                               
                                                 e 
                                                 2 
                                               
                                             
                                             
                                               8 
                                                
                                               
                                                 πɛ 
                                                 0 
                                               
                                                
                                               
                                                 a 
                                                 3 
                                               
                                             
                                           
                                         
                                         - 
                                         
                                           
                                             
                                               c 
                                               BO 
                                             
                                              
                                             
                                               e 
                                               2 
                                             
                                           
                                           
                                             8 
                                              
                                             
                                               
                                                 
                                                   πɛ 
                                                   0 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     a 
                                                     + 
                                                     
                                                       c 
                                                       ′ 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               3 
                                             
                                           
                                         
                                       
                                       μ 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   15.92 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where C 1o  is the fraction of the H 2 -type ellipsoidal MO basis function of the oscillatory transition state of the A-B bond which is 0.75 (Eq. (13.233)) in the case of H bonding to a central atom and 1 otherwise, C 2o  is the factor that results in an equipotential energy match of the participating at least two atomic orbitals of the transition state of the chemical bond, and 
         [0000]    
       
         
           
             μ 
             = 
             
               
                 
                   m 
                   1 
                 
                  
                 
                   m 
                   2 
                 
               
               
                 
                   m 
                   1 
                 
                 + 
                 
                   m 
                   2 
                 
               
             
           
         
       
     
         [0000]    is the reduced mass of the nuclei given by Eq. (11.154). To match the boundary condition that the total energy of the A-B ellipsoidal MO is zero, E T (A-B) given by Eq. (15.92) is set equal to zero. Substitution of Eq. (15.90) into Eq. (15.92) gives 
         [0000]    
       
         
           
             
               
                 
                   0 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           - 
                                           
                                             e 
                                             2 
                                           
                                         
                                         
                                           8 
                                            
                                           
                                             πɛ 
                                             0 
                                           
                                            
                                           
                                             
                                               
                                                 aa 
                                                 0 
                                               
                                               
                                                 2 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   C 
                                                   1 
                                                 
                                                  
                                                 
                                                   C 
                                                   2 
                                                 
                                               
                                             
                                           
                                         
                                       
                                        
                                       
                                         [ 
                                         
                                           
                                             
                                               
                                                 
                                                   c 
                                                   1 
                                                 
                                                  
                                                 
                                                   
                                                     c 
                                                     2 
                                                     ′ 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         2 
                                                          
                                                         
                                                             
                                                         
                                                          
                                                         
                                                           c 
                                                           BO 
                                                         
                                                       
                                                       - 
                                                       
                                                         
                                                           c 
                                                           BO 
                                                           ′ 
                                                         
                                                          
                                                         
                                                           
                                                             a 
                                                             0 
                                                           
                                                           a 
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   ln 
                                                    
                                                   
                                                     
                                                       a 
                                                       + 
                                                       
                                                         
                                                           
                                                             aa 
                                                             0 
                                                           
                                                           
                                                             2 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             C 
                                                             1 
                                                             
                                                              
                                                             
                                                             C 
                                                             2 
                                                             
                                                           
                                                         
                                                       
                                                     
                                                     
                                                       a 
                                                       - 
                                                       
                                                         
                                                           
                                                             aa 
                                                             0 
                                                           
                                                           
                                                             2 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             C 
                                                             1 
                                                             
                                                              
                                                             
                                                             C 
                                                             2 
                                                             
                                                           
                                                         
                                                       
                                                     
                                                   
                                                 
                                                 - 
                                                 1 
                                               
                                             
                                           
                                         
                                         ] 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       E 
                                       T 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           atom 
                                           - 
                                           atom 
                                         
                                         , 
                                         
                                           
                                             msp 
                                             3 
                                           
                                           . 
                                           AO 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     [ 
                                     
                                       1 
                                       + 
                                       
                                         
                                           
                                             2 
                                              
                                             ℏ 
                                              
                                             
                                               
                                                 
                                                   
                                                     c 
                                                     BO 
                                                   
                                                    
                                                   
                                                     
                                                       
                                                         C 
                                                         
                                                           1 
                                                            
                                                           
                                                               
                                                           
                                                            
                                                           o 
                                                         
                                                       
                                                        
                                                       
                                                         C 
                                                         
                                                           2 
                                                            
                                                           
                                                               
                                                           
                                                            
                                                           o 
                                                         
                                                       
                                                        
                                                       
                                                         e 
                                                         2 
                                                       
                                                     
                                                     
                                                       4 
                                                        
                                                       
                                                         πɛ 
                                                         0 
                                                       
                                                        
                                                       
                                                         a 
                                                         3 
                                                       
                                                     
                                                   
                                                 
                                                 
                                                   m 
                                                   e 
                                                 
                                               
                                             
                                           
                                           
                                             
                                               m 
                                               e 
                                             
                                              
                                             
                                               c 
                                               2 
                                             
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     1 
                                     2 
                                   
                                    
                                   ℏ 
                                    
                                   
                                     
                                       
                                         
                                           
                                             c 
                                             BO 
                                           
                                            
                                           
                                             
                                               
                                                 c 
                                                 1 
                                               
                                                
                                               
                                                 c 
                                                 2 
                                                 ′ 
                                               
                                                
                                               
                                                 e 
                                                 2 
                                               
                                             
                                             
                                               8 
                                                
                                               
                                                 πɛ 
                                                 0 
                                               
                                                
                                               
                                                 a 
                                                 3 
                                               
                                             
                                           
                                         
                                         - 
                                         
                                           
                                             
                                               c 
                                               BO 
                                             
                                              
                                             
                                               e 
                                               2 
                                             
                                           
                                           
                                             8 
                                              
                                             
                                               
                                                 
                                                   πɛ 
                                                   0 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     a 
                                                     + 
                                                     
                                                       
                                                         
                                                           aa 
                                                           0 
                                                         
                                                         
                                                           2 
                                                            
                                                           
                                                               
                                                           
                                                            
                                                           
                                                             C 
                                                             1 
                                                           
                                                            
                                                           
                                                             C 
                                                             2 
                                                           
                                                         
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               3 
                                             
                                           
                                         
                                       
                                       μ 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   15.93 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The vibrational energy-term of Eq. (15.93) is determined by the forces between the central field and the electrons and those between the nuclei (Eqs. (11.141-11.145)). The electron-central-field force and its derivative are given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       f 
                        
                       
                         ( 
                         a 
                         ) 
                       
                     
                     = 
                     
                       
                         - 
                         
                           c 
                           BO 
                         
                       
                        
                       
                         
                           
                             c 
                             1 
                           
                            
                           
                             c 
                             2 
                             ′ 
                           
                            
                           
                             e 
                             2 
                           
                         
                         
                           4 
                            
                           
                             πɛ 
                             0 
                           
                            
                           
                             a 
                             3 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   15.94 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       f 
                       ′ 
                     
                      
                     
                       ( 
                       a 
                       ) 
                     
                   
                   = 
                   
                     
                       - 
                       2 
                     
                      
                     
                       c 
                       BO 
                     
                      
                     
                       
                         
                           c 
                           1 
                         
                          
                         
                           c 
                           2 
                           ′ 
                         
                          
                         
                           e 
                           2 
                         
                       
                       
                         4 
                          
                         
                           πɛ 
                           0 
                         
                          
                         
                           a 
                           3 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.95 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The nuclear repulsion force and its derivative are given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       f 
                        
                       
                         ( 
                         
                           a 
                           + 
                           
                             c 
                             ′ 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         e 
                         2 
                       
                       
                         8 
                          
                         
                           
                             
                               πɛ 
                               0 
                             
                              
                             
                               ( 
                               
                                 a 
                                 + 
                                 
                                   c 
                                   ′ 
                                 
                               
                               ) 
                             
                           
                           2 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   15.96 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       f 
                       ′ 
                     
                      
                     
                       ( 
                       
                         a 
                         + 
                         
                           c 
                           ′ 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     - 
                     
                       
                         e 
                         2 
                       
                       
                         4 
                          
                         
                           
                             
                               πɛ 
                               0 
                             
                              
                             
                               ( 
                               
                                 a 
                                 + 
                                 
                                   c 
                                   ′ 
                                 
                               
                               ) 
                             
                           
                           3 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.97 
                   ) 
                 
               
             
           
         
       
     
         [0000]    such that the angular frequency of the oscillation is given by 
         [0000]    
       
         
           
             
               
                 
                   ω 
                   = 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   - 
                                   3 
                                 
                                 a 
                               
                                
                               
                                 f 
                                  
                                 
                                   ( 
                                   a 
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 f 
                                 ′ 
                               
                                
                               
                                 ( 
                                 a 
                                 ) 
                               
                             
                           
                           ] 
                         
                         μ 
                       
                     
                     = 
                     
                       
                         
                           k 
                           
                             m 
                             e 
                           
                         
                       
                       = 
                       
                         
                           
                             
                               
                                 c 
                                 BO 
                               
                                
                               
                                 
                                   
                                     c 
                                     1 
                                   
                                    
                                   
                                     c 
                                     2 
                                     ′ 
                                   
                                    
                                   
                                     e 
                                     2 
                                   
                                 
                                 
                                   4 
                                    
                                   
                                     πɛ 
                                     0 
                                   
                                    
                                   
                                     a 
                                     3 
                                   
                                 
                               
                             
                             - 
                             
                               
                                 e 
                                 2 
                               
                               
                                 8 
                                  
                                 
                                   
                                     
                                       πɛ 
                                       0 
                                     
                                      
                                     
                                       ( 
                                       
                                         a 
                                         + 
                                         
                                           c 
                                           ′ 
                                         
                                       
                                       ) 
                                     
                                   
                                   2 
                                 
                               
                             
                           
                           μ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.98 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Since both terms of Ē osc =Ē D +Ē Kvib  are small due to the large values of a and c′, to very good approximation, a convenient form of Eq. (15.93) which is evaluated to determine the bond angles of functional groups is given by 
         [0000]    
       
         
           
             
               
                 
                   0 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           - 
                                           
                                             e 
                                             2 
                                           
                                         
                                         
                                           8 
                                            
                                           
                                             πɛ 
                                             0 
                                           
                                            
                                           
                                             
                                               
                                                 aa 
                                                 0 
                                               
                                               
                                                 2 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   C 
                                                   1 
                                                 
                                                  
                                                 
                                                   C 
                                                   2 
                                                 
                                               
                                             
                                           
                                         
                                       
                                        
                                       
                                         [ 
                                         
                                           
                                             
                                               c 
                                               1 
                                             
                                              
                                             
                                               
                                                 c 
                                                 2 
                                                 ′ 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   2 
                                                   - 
                                                   
                                                     
                                                       a 
                                                       0 
                                                     
                                                     a 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                              
                                             ln 
                                              
                                             
                                               
                                                 a 
                                                 + 
                                                 
                                                   
                                                     
                                                       aa 
                                                       0 
                                                     
                                                     
                                                       2 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       
                                                         C 
                                                         1 
                                                       
                                                        
                                                       
                                                         C 
                                                         2 
                                                       
                                                     
                                                   
                                                 
                                               
                                               
                                                 a 
                                                 - 
                                                 
                                                   
                                                     
                                                       aa 
                                                       0 
                                                     
                                                     
                                                       2 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       
                                                         C 
                                                         1 
                                                       
                                                        
                                                       
                                                         C 
                                                         2 
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                           - 
                                           1 
                                         
                                         ] 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       E 
                                       T 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           atom 
                                           - 
                                           atom 
                                         
                                         , 
                                         
                                           
                                             msp 
                                             3 
                                           
                                           . 
                                           AO 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     [ 
                                     
                                       1 
                                       + 
                                       
                                         
                                           
                                             2 
                                              
                                             ℏ 
                                              
                                             
                                               
                                                 
                                                   
                                                     
                                                       c 
                                                       1 
                                                     
                                                      
                                                     
                                                       e 
                                                       2 
                                                     
                                                   
                                                   
                                                     4 
                                                      
                                                     
                                                       πɛ 
                                                       0 
                                                     
                                                      
                                                     
                                                       a 
                                                       3 
                                                     
                                                   
                                                 
                                                 
                                                   m 
                                                   e 
                                                 
                                               
                                             
                                           
                                           
                                             
                                               m 
                                               e 
                                             
                                              
                                             
                                               c 
                                               2 
                                             
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     1 
                                     2 
                                   
                                    
                                   ℏ 
                                    
                                   
                                     
                                       
                                         
                                           
                                             
                                               c 
                                               1 
                                             
                                              
                                             
                                               e 
                                               2 
                                             
                                           
                                           
                                             8 
                                              
                                             
                                               πɛ 
                                               0 
                                             
                                              
                                             
                                               a 
                                               3 
                                             
                                           
                                         
                                         - 
                                         
                                           
                                             e 
                                             2 
                                           
                                           
                                             8 
                                              
                                             
                                               
                                                 
                                                   πɛ 
                                                   0 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     a 
                                                     + 
                                                     
                                                       
                                                         
                                                           aa 
                                                           0 
                                                         
                                                         
                                                           2 
                                                            
                                                           
                                                               
                                                           
                                                            
                                                           
                                                             C 
                                                             1 
                                                           
                                                            
                                                           
                                                             C 
                                                             2 
                                                           
                                                         
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               3 
                                             
                                           
                                         
                                       
                                       μ 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   15.99 
                   ) 
                 
               
             
           
         
       
     
         [0000]    From the energy relationship given by Eq. (15.99) and the relationship between the axes given by Eqs. (15.2-15.5), the dimensions of the A-B MO can be solved. The most convenient way to solve Eq. (15.99) is by the reiterative technique using a computer. 
         [0174]    A factor c 2  of a given atom in the determination of c; for calculating the zero of the total A-B bond energy is typically given by Eqs. (15.71-15.74). In the case of a H—H terminal bond of an alkyl or alkenyl group, c; is typically the ratio of c 2  of Eq. (15.71) for the H—H bond which is one and c 2  of the carbon of the corresponding C—H bond: 
         [0000]    
       
         
           
             
               
                 
                   
                     c 
                     2 
                     ′ 
                   
                   = 
                   
                     
                       1 
                       
                         
                           c 
                           2 
                         
                          
                         
                           ( 
                           
                             C 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                                 
                             
                              
                             
                               sp 
                               3 
                             
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         
                           E 
                           Coulomb 
                         
                          
                         
                           ( 
                           
                             C 
                              
                             
                               - 
                             
                              
                             H 
                              
                             
                                 
                             
                              
                             C 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                                 
                             
                              
                             
                               sp 
                               3 
                             
                           
                           ) 
                         
                       
                       
                         13.605804 
                          
                         
                             
                         
                          
                         eV 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.100 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In the case of the determination of the bond angle of the ACH MO comprising a linear combination of C-A-bond and C—H-bond MOs where A and C are general, C is the central atom, and c 2  for an atom is given by Eqs. (15.71-15.79), c; of the A-H terminal bond is typically the ratio of c 2  of the A atom for the A-H terminal bond and c 2  of the C atom of the corresponding C—H bond: 
         [0000]    
       
         
           
             
               
                 
                   
                     c 
                     2 
                     ′ 
                   
                   = 
                   
                     
                       
                         c 
                         2 
                       
                        
                       
                         ( 
                         
                           
                             A 
                             ( 
                             
                               A 
                                
                               
                                 - 
                               
                                
                               H 
                             
                             ) 
                           
                            
                           
                             msp 
                             3 
                           
                         
                         ) 
                       
                     
                     
                       
                         c 
                         2 
                       
                       ( 
                       
                         
                           C 
                            
                           
                             ( 
                             
                               C 
                                
                               
                                 - 
                               
                                
                               H 
                             
                             ) 
                           
                         
                          
                         
                           ( 
                           
                             msp 
                             3 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.101 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In the case of the determination of the bond angle of the COH MO of an alcohol comprising a linear combination of C—O-bond and O—H-bond MOs where C, O, and H are carbon, oxygen, and hydrogen, respectively, c; of the C—H terminal bond is typically 0.91771 since the oxygen and hydrogen atoms are at the Coulomb potential of a proton and an electron (Eqs. (1.236) and (10.162), respectively) that is energy matched to the C2sp 3  HO. 
         [0175]    In the determination of the hybridization factor c′ 2  of Eq. (15.99) from Eqs. (15.71-15.79), the Coulombic energy, E Coulomb (MO.atom,msp 3 ), or the energy, E(MO.atom,msp 3 ), the radius r A-B AorBsp     3    of the A or B AO or HO of the heteroatom of the A-B terminal bond MO such as the C2sp 3  HO of a terminal C—C bond is calculated using Eq. (15.32) by considering ΣE T     mol   (MO,2sp 3 ), the total energy donation to each bond with which it participates in bonding as it forms the terminal bond. The Coulombic energy E Coulomb (MO.atom,msp3) of the outer electron of the atom msp 3  shell is given by Eq. (15.19). In the case that during hybridization, at least one of the spin-paired AO electrons is unpaired in the hybridized orbital (HO), the energy change for the promotion to the unpaired state is the magnetic energy E(magnetic) (Eq. (15.15)) at the initial radius r of the AO electron. Then, the energy E(MO.atom,msp 3 ) of the outer electron of the atom msp 3  shell is given by the sum of E Coulomb (MO.atom,msp 3 ) and E(magnetic) (Eq. (15.20)). 
         [0176]    In the specific case of the terminal bonding of two carbon atoms, the c 2  factor of each carbon given by Eq. (15.71) is determined using the Coulombic energy E Coulomb (C—C C2sp 3 ) of the outer electron of the C2sp 3  shell given by Eq. (15.19) with the radius r C—C C 2sp   3    of each C2sp 3  HO of the terminal C—C bond calculated using Eq. (15.32) by considering ΣE T     mol   (MO,2sp 3 ), the total energy donation to each bond with which it participates in bonding as it forms the terminal bond including the contribution of the methylene energy, 0.92918 eV (Eq. (14.513)), corresponding to the terminal C—C bond. The corresponding E T (atom−atom,msp 3 .AO) in Eq. (15.99) is E T (C—C C2sp 3 )=−1.85836 eV. 
         [0177]    In the case that the terminal atoms are carbon or other heteroatoms, the terminal bond comprises a linear combination of the HOs or AOs; thus, c′ 2  is the average of the hybridization factors of the participating atoms corresponding to the normalized linear sum: 
         [0000]    
       
         
           
             
               
                 
                   
                     c 
                     2 
                     ′ 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       ( 
                       
                         
                           
                             c 
                             2 
                             ′ 
                           
                            
                           
                             ( 
                             
                               atom 
                                
                               
                                   
                               
                                
                               1 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             c 
                             2 
                             ′ 
                           
                            
                           
                             ( 
                             
                               atom 
                                
                               
                                   
                               
                                
                               2 
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   15.102 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In the exemplary cases of C—C, O—O, and N—N where C is carbon: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           c 
                           2 
                           ′ 
                         
                         = 
                           
                          
                         
                           
                             1 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     e 
                                     2 
                                   
                                   
                                     8 
                                      
                                     
                                         
                                     
                                      
                                     
                                       πɛ 
                                       0 
                                     
                                      
                                     
                                       a 
                                       0 
                                     
                                   
                                 
                                 
                                   
                                     e 
                                     2 
                                   
                                   
                                     8 
                                      
                                     
                                         
                                     
                                      
                                     
                                       πɛ 
                                       0 
                                     
                                      
                                     
                                       r 
                                       
                                         A 
                                          
                                         
                                           - 
                                         
                                          
                                         A 
                                          
                                         
                                             
                                         
                                          
                                         
                                           A 
                                           1 
                                         
                                          
                                         
                                           AO 
                                           / 
                                           HO 
                                         
                                       
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     e 
                                     2 
                                   
                                   
                                     8 
                                      
                                     
                                         
                                     
                                      
                                     
                                       πɛ 
                                       0 
                                     
                                      
                                     
                                       a 
                                       0 
                                     
                                   
                                 
                                 
                                   
                                     e 
                                     2 
                                   
                                   
                                     8 
                                      
                                     
                                         
                                     
                                      
                                     
                                       πɛ 
                                       0 
                                     
                                      
                                     
                                       r 
                                       
                                         A 
                                          
                                         
                                           
                                             - 
                                           
                                         
                                          
                                         A 
                                          
                                         
                                             
                                         
                                          
                                         
                                           A 
                                           2 
                                         
                                          
                                         
                                           AO 
                                           / 
                                           HO 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             1 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       
                                         13.605804 
                                          
                                         
                                             
                                         
                                          
                                         eV 
                                       
                                       
                                         
                                           E 
                                           Coulomb 
                                         
                                          
                                         
                                           ( 
                                           
                                             A 
                                              
                                             
                                               - 
                                             
                                              
                                             
                                               A 
                                               . 
                                               
                                                 A 
                                                 1 
                                               
                                             
                                              
                                             
                                               AO 
                                               / 
                                               HO 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       13.605804 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     
                                       
                                         E 
                                         Coulomb 
                                       
                                        
                                       
                                         ( 
                                         
                                           A 
                                            
                                           
                                             - 
                                           
                                            
                                           
                                             A 
                                             . 
                                             
                                               A 
                                               2 
                                             
                                           
                                            
                                           
                                             AO 
                                             / 
                                             HO 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.103 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In the exemplary cases of C—N, C—O, and C—S, 
         [0000]    
       
         
           
             
               
                 
                   
                     c 
                     2 
                     ′ 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       ( 
                       
                         
                           
                             13.605804 
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               E 
                               Coulomb 
                             
                              
                             
                               ( 
                               
                                 C 
                                  
                                 
                                   - 
                                 
                                  
                                 B 
                                  
                                 
                                     
                                 
                                  
                                 C 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               C 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               B 
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   15.104 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where C is carbon and c 2  (C to B) is the hybridization factor of Eqs. (15.61) and (15.93) that matches the energy of the atom B to that of the atom C in the group. For these cases, the corresponding E T (atom−atom,msp 3 .AO) term in Eq. (15.99) depends on the hybridization and bond order of the terminal atoms in the molecule, but typical values matching those used in the determination of the bond energies (Eq. (15.65)) are 
         [0178]    E T (C—O C2sp 3 .O2p)=−1.44915 eV; E T (C—O C2sp 3 .O2p)=−1.65376 eV; 
         [0179]    E T (C—N C2sp 3 .N2p)=−1.44915 eV; E T (C—S C2sp 3 .S2p)=−0.72457 eV; 
         [0180]    E T (O—O O2p. O2p)=−1.44915 eV; E T (O—O O2p. O2p)=−1.65376 eV; 
         [0181]    E T (N—N N2p.N2p)=−1.44915 eV; E T (N—O N2p. O2p)=−1.44915 eV; 
         [0182]    E T (F—F F2p.F2p)=−1.44915 eV; E T (Cl—Cl Cl3p.C13p)=−0.92918 eV; 
         [0183]    E T (Br—Br Br4p.Br4p)=−0.92918 eV; E T (I—I I5p.I5p)=−0.36229 eV; 
         [0184]    E T (C—F C2sp 3 .F2p)=−1.85836 eV; E T (C—Cl C2sp 3 .C13p)=−0.92918 eV; 
         [0185]    E T (C—Br C2sp 3 .Br4p)=−0.72457 eV; E T (C—I C2sp 3 .I5p)=−0.36228 eV, and 
         [0186]    E T (O—O O2p.C13p)=−0.92918 eV. 
         [0187]    In the case that the terminal bond is X—X where X is a halogen atom, c 1  is one, and c′ 2  is the average (Eq. (15.102)) of the hybridization factors of the participating halogen atoms given by Eqs. (15.71-15.72) where E Coulomb (MO.atom,msp 3 ) is determined using Eq. (15.32) and E Coulomb (MO.atom,msp 3 )=13.605804 eV for X═I. The factor C 1  of Eq. (15.99) is one for all halogen atoms. The factor C 2  of fluorine is one since it is the only halogen wherein the ionization energy is greater than that 13.605804 eV, the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243). For each of the other halogens, Cl, Br, and I, C 2  is the hybridization factor of Eq. (15.61) given by Eq. (15.79) with c 2  (1) being that of the halogen given by Eq. (15.77) that matches the valence energy of X (E 1  (valence)) to that of the C2sp 3  HO (E 2 (valence)=−14.63489 eV, Eq. (15.25)) and to the hybridization of C2sp 3  HO (c 2  (2)=0.91771, Eq. (13.430)). E T (atom−atom, msp 3 .AO) of Eq. (15.99) is the maximum for the participating atoms which is −1.44915 eV, −0.92918 eV, −0.92918 eV, and −0.33582 eV for F, Cl, Br, and I, respectively. 
         [0188]    Consider the case that the terminal bond is C—X where C is a carbon atom and X is a halogen atom. The factors c 1  and C, of Eq. (15.99) are one for all halogen atoms. For X═F, c′ 2  is the average (Eq. (15.104)) of the hybridization factors of the participating carbon and F atoms where c 2  for carbon is given by Eq. (15.71) and c 2  for fluorine matched to carbon is given by Eq. (15.79) with c 2  (1) for the fluorine atom given by Eq. (15.77) that matches the valence energy of F (E 1 (valence)=−17.42282 eV) to that of the C2sp 3  HO (E 2 (valence)=−14.63489 eV, Eq. (15.25)) and to the hybridization of C2sp 3  HO (c 2  (2)=0.91771, Eq. (13.430)). The factor C 2  of fluorine is one since it is the only halogen wherein the ionization energy is greater than that 13.605804 eV, the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243). For each of the other halogens, Cl, Br, and I, c′ 2  is the hybridization factor of the participating carbon atom since the halogen atom is energy matched to the carbon atom. C 2  of the terminal-atom bond matches that used to determine the energies of the corresponding C—X-bond MO. Then, C 2  is the hybridization factor of Eq. (15.61) given by Eq. (15.79) with c 2  (1) for the halogen atom given by Eq. (15.77) that matches the valence energy of X (E 1  (valence)) to that of the C2sp 3  HO (E 2 (valence)=−14.63489 eV, Eq. (15.25)) and to the hybridization of C2sp 3  HO (c 2 (2)=0.91771, Eq. (13.430)). E T (atom−atom,msp 3 .AO) of Eq. (15.99) is the maximum for the participating atoms which is −1.85836 eV, −0.92918 eV, −0.72457 eV, and −0.33582 eV for F, Cl, Br, and I, respectively. 
         [0189]    Consider the case that the terminal bond is H—X corresponding to the angle of the atoms HCX where C is a carbon atom and X is a halogen atom. The factors c 1  and C 1  of Eq. (15.99) are 0.75 for all halogen atoms. For X═F, c′ 2  is given by Eq. (15.78) with c 2  of the participating carbon and F atoms given by Eq. (15.71) and Eq. (15.74), respectively. The factor C 2  of fluorine is one since it is the only halogen wherein the ionization energy is greater than that 13.605804 eV, the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243). For each of the other halogens, Cl, Br, and I, c′ 2  is also given by Eq. (15.78) with c 2  of the participating carbon given by Eq. (15.71) and c 2  of the participating X atom given by c 2 =0.91771 (Eq. (13.430)) since the X atom is energy matched to the C2sp 3  HO. In these cases, C 2  is given by Eq. (15.74) for the corresponding atom X where C 2  matches the energy of the atom X to that of H. 
         [0190]    Using the distance between the two atoms A and B of the general molecular group ACB when the total energy of the corresponding A-B MO is zero, the corresponding bond angle can be determined from the law of cosines: 
         [0000]        s   1   2   +s   2   2 −2 s   1   s   2 cosine θ= s   3   2   (15.105)
 
         [0000]    With s 1 =2c′ C-A , the internuclear distance of the C-A bond, s 2 =2c′ C—B , the internuclear distance of each C—B bond, and s 3 =2c′ A-B  the internuclear distance of the two terminal atoms, the bond angle θ ∠ACB  between the C-A and C—B bonds is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             c 
                             
                               C 
                                
                               
                                 - 
                               
                                
                               A 
                             
                             ′ 
                           
                         
                         ) 
                       
                       2 
                     
                     + 
                     
                       ( 
                       
                         2 
                          
                         
                             
                         
                          
                         
                           c 
                           
                             C 
                              
                             
                               - 
                             
                              
                             B 
                           
                           ′ 
                         
                       
                       ) 
                     
                     - 
                     
                       2 
                        
                       
                         ( 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             c 
                             
                               C 
                                
                               
                                 - 
                               
                                
                               A 
                             
                             ′ 
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             c 
                             
                               C 
                                
                               
                                 - 
                               
                                
                               B 
                             
                             ′ 
                           
                         
                         ) 
                       
                        
                       cosine 
                        
                       
                           
                       
                        
                       θ 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         2 
                          
                         
                             
                         
                          
                         
                           c 
                           
                             A 
                              
                             
                               - 
                             
                              
                             B 
                           
                           ′ 
                         
                       
                       ) 
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   15.106 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       θ 
                       ∠ABC 
                     
                     = 
                     
                       
                         cos 
                         
                           - 
                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     c 
                                     
                                       C 
                                        
                                       
                                         - 
                                       
                                        
                                       A 
                                     
                                     ′ 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     c 
                                     
                                       C 
                                        
                                       
                                         - 
                                       
                                        
                                       B 
                                     
                                     ′ 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             - 
                             
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     c 
                                     
                                       A 
                                        
                                       
                                         - 
                                       
                                        
                                       B 
                                     
                                     ′ 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           
                             2 
                              
                             
                               ( 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   c 
                                   
                                     C 
                                      
                                     
                                       - 
                                     
                                      
                                     A 
                                   
                                   ′ 
                                 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   c 
                                   
                                     C 
                                      
                                     
                                       - 
                                     
                                      
                                     B 
                                   
                                   ′ 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.107 
                   ) 
                 
               
             
           
         
       
     
         [0191]    Consider the exemplary structure C b C a (O a )O b  wherein C a  is bound to C b , O a  and O b . In the general case that the three bonds are coplanar and two of the angles are known, say θ 1  and θ 2 , then the third θ 3  can be determined geometrically: 
         [0000]      θ 3 =360−θ 1 −θ 2   (15.108)
 
         [0000]    In the general case that two of the three coplanar bonds are equivalent and one of the angles is known, say θ 1 , then the second and third can be determined geometrically: 
         [0000]    
       
         
           
             
               
                 
                   
                     θ 
                     2 
                   
                   = 
                   
                     
                       θ 
                       3 
                     
                     = 
                     
                       
                         ( 
                         
                           360 
                           - 
                           
                             θ 
                             1 
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   15.109 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Angles and Distances for an Mo that Forms an Isosceles Triangle
 
In the general case where the group comprises three A-B bonds having B as the central atom at the apex of a pyramidal structure formed by the three bonds with the A atoms at the base in the xy-plane. The C 3v  axis centered on B is defined as the vertical or z-axis, and any two A-B bonds form an isosceles triangle. Then, the angle of the bonds and the distances from and along the z-axis are determined from the geometrical relationships given by Eqs. (13.412-13.416):
 
         [0192]    the distance d origin-B  from the origin to the nucleus of a terminal B atom is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     d 
                     origin—B 
                   
                   = 
                   
                     
                       2 
                        
                       
                           
                       
                        
                       
                         c 
                         
                           B 
                            
                           
                             - 
                           
                            
                           B 
                         
                         ′ 
                       
                     
                     
                       2 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       60 
                        
                       ° 
                     
                   
                 
               
               
                 
                   ( 
                   15.110 
                   ) 
                 
               
             
           
         
       
     
         [0193]    the height along the z-axis from the origin to the A nucleus d height  is given by 
         [0000]        d   height =√{square root over ((2 c′   A-B ) 2 −( d   origin-B ) 2 )}{square root over ((2 c′   A-B ) 2 −( d   origin-B ) 2 )}, and  (15.111)
 
         [0194]    the angle θ v  of each A-B bond from the z-axis is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     θ 
                     v 
                   
                   = 
                   
                     
                       tan 
                       
                         - 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           d 
                           
                             origin 
                             - 
                             B 
                           
                         
                         
                           d 
                           height 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   15.112 
                   ) 
                 
               
             
           
         
       
     
         [0195]    Consider the case where the central atom B is further bound to a fourth atom C and the B—C bond is along the z-axis. Then, the bond θ ∠ABC  given by Eq. (14.206) is 
         [0000]      θ ∠ABC =180−θ v   (15.113)
 
       Dihedral Angle 
       [0196]    Consider the plane defined by a general ACA MO comprising a linear combination of two C-A-bond MOs where C is the central atom. The dihedral angle θ ∠BCI ACA  between the ACA-plane and a line defined by a third bond with C, specifically that corresponding to a C—B-bond MO, is calculated from the bond angle θ ∠ACA  and the distances between the A, B, and C atoms. The distance d 1  along the bisector of θ ∠ACA  from C to the internuclear-distance line between A and A, 2c′ A-A  is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     d 
                     1 
                   
                   = 
                   
                     2 
                      
                     
                         
                     
                      
                     
                       c 
                       
                         C 
                          
                         
                           - 
                         
                          
                         A 
                       
                       ′ 
                     
                      
                     cos 
                      
                     
                         
                     
                      
                     
                       
                         θ 
                         
                           ∠ 
                            
                           ACA 
                         
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   15.114 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where 2c′ C-A  is the internuclear distance between A and C. The atoms A, A, and B define the base of a pyramid. Then, the pyramidal angle θ ∠ABA  can be solved from the internuclear distances between A and A, 2c′ A-A  and between A and B, 2c′ A-B  using the law of cosines (Eq. (15.107)): 
         [0000]    
       
         
           
             
               
                 
                   
                     θ 
                     
                       ∠ 
                        
                       ABA 
                     
                   
                   = 
                   
                     
                       cos 
                       
                         - 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               ( 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   c 
                                   
                                     A 
                                      
                                     
                                       - 
                                     
                                      
                                     B 
                                   
                                   ′ 
                                 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   c 
                                   
                                     A 
                                      
                                     
                                       - 
                                     
                                      
                                     B 
                                   
                                   ′ 
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             
                               ( 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   c 
                                   
                                     A 
                                      
                                     
                                       - 
                                     
                                      
                                     A 
                                   
                                   ′ 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         
                           2 
                            
                           
                             ( 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 c 
                                 
                                   A 
                                    
                                   
                                     - 
                                   
                                    
                                   B 
                                 
                                 ′ 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 c 
                                 
                                   A 
                                    
                                   
                                     - 
                                   
                                    
                                   B 
                                 
                                 ′ 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   15.115 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Then, the distance d 2  along the bisector of θ ∠ABA  from B to the internuclear-distance line 2c′ A-A  is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     d 
                     2 
                   
                   = 
                   
                     2 
                      
                     
                         
                     
                      
                     
                       c 
                       
                         A 
                          
                         
                           - 
                         
                          
                         B 
                       
                       ′ 
                     
                      
                     cos 
                      
                     
                         
                     
                      
                     
                       
                         θ 
                         
                           ∠ 
                            
                           ACA 
                         
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   15.116 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The lengths d 1 , d 2 , and 2c′ C—B  define a triangle wherein the angle between d 1  and the internuclear distance between B and C, 2c′ C—B  is the dihedral angle θ ∠BCI ACA  that can be solved using the law of cosines (Eq. (15.107)): 
         [0000]    
       
         
           
             
               
                 
                   
                     θ 
                     
                       
                         ∠ 
                          
                         BC 
                       
                       / 
                       ACA 
                     
                   
                   = 
                   
                     
                       cos 
                       
                         - 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           
                             d 
                             1 
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   c 
                                   
                                     C 
                                      
                                     
                                       - 
                                     
                                      
                                     B 
                                   
                                   ′ 
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             d 
                             2 
                             2 
                           
                         
                         
                           2 
                            
                           
                               
                           
                            
                           
                             
                               d 
                               1 
                             
                              
                             
                               ( 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   c 
                                   
                                     C 
                                      
                                     
                                       - 
                                     
                                      
                                     B 
                                   
                                   ′ 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   15.117 
                   ) 
                 
               
             
           
         
       
     
       General Dihedral Angle 
       [0197]    Consider the plane defined by a general ACB MO comprising a linear combination of C-A and C—B-bond MOs where C is the central atom. The dihedral angle θ ∠CDI ACB  between the ACB-plane and a line defined by a third bond of C with D, specifically that corresponding to a C-D-bond MO, is calculated from the bond angle θ ∠ACB  and the distances between the A, B, C, and D atoms. The distance d 1  from C to the bisector of the internuclear-distance line between A and B, 2c′ A-B  is given by two equations involving the law of cosines (Eq. (15.105)). One with s 1 =2c′ C-A , the internuclear distance of the C-A bond, s 2 =d 1 , 
         [0000]    
       
         
           
             
               
                 s 
                 3 
               
               = 
               
                 
                   2 
                    
                   
                       
                   
                    
                   
                     c 
                     
                       A 
                        
                       
                         - 
                       
                        
                       B 
                     
                     ′ 
                   
                 
                 2 
               
             
             , 
           
         
       
     
         [0000]    half the internuclear distance between A and B, and θ=θ ∠ACd     1   , the angle between d 1  and the C-A bond is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             c 
                             
                               C 
                                
                               
                                 - 
                               
                                
                               A 
                             
                             ′ 
                           
                         
                         ) 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           d 
                           1 
                         
                         ) 
                       
                       2 
                     
                     - 
                     
                       2 
                        
                       
                         ( 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             c 
                             
                               C 
                                
                               
                                 - 
                               
                                
                               A 
                             
                             ′ 
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           d 
                           1 
                         
                         ) 
                       
                        
                       cosine 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           
                             ∠ 
                              
                             ACd 
                           
                           1 
                         
                       
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           2 
                            
                           
                               
                           
                            
                           
                             c 
                             A—B 
                             ′ 
                           
                         
                         2 
                       
                       ) 
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   15.118 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The other with s 1 =2c′ C—B , the internuclear distance of the C—B bond, s 2 =d 1 , 
         [0000]    
       
         
           
             
               
                 s 
                 3 
               
               = 
               
                 
                   2 
                    
                   
                       
                   
                    
                   
                     c 
                     
                       A 
                       - 
                       B 
                     
                     ′ 
                   
                 
                 2 
               
             
             , 
           
         
       
     
         [0000]    and θ=θ ∠ACB −θ ∠ACd     1    where θ ∠ACB  is the bond angle between the C-A and C—B bonds is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             c 
                             
                               C 
                               - 
                               B 
                             
                             
                               ′ 
                                
                               
                                   
                               
                             
                           
                         
                         ) 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           d 
                           1 
                         
                         ) 
                       
                       2 
                     
                     - 
                     
                       2 
                        
                       
                         ( 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             c 
                             
                               C 
                               - 
                               B 
                             
                             ′ 
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           d 
                           1 
                         
                         ) 
                       
                        
                       
                         cosine 
                          
                         
                           ( 
                           
                             
                               θ 
                               
                                 ∠ 
                                  
                                 ACB 
                               
                             
                             - 
                             
                               θ 
                               
                                 
                                   ∠ 
                                    
                                   AC 
                                    
                                   d 
                                 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           2 
                            
                           
                               
                           
                            
                           
                             c 
                             
                               A 
                               - 
                               B 
                             
                             ′ 
                           
                         
                         2 
                       
                       ) 
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   15.119 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Subtraction of Eq. (15.119) from Eq. (15.118) gives 
         [0000]    
       
         
           
             
               
                 
                   
                     d 
                     1 
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               c 
                               
                                 C 
                                 - 
                                 A 
                               
                               ′ 
                             
                           
                           ) 
                         
                         2 
                       
                       - 
                       
                         
                           ( 
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               c 
                               
                                 C 
                                 - 
                                 B 
                               
                               ′ 
                             
                           
                           ) 
                         
                         2 
                       
                     
                     
                       2 
                        
                       
                         ( 
                         
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         c 
                                         
                                           C 
                                           - 
                                           A 
                                         
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                    
                                   cosine 
                                    
                                   
                                       
                                   
                                    
                                   
                                     θ 
                                     
                                       
                                         ∠ 
                                          
                                         AC 
                                          
                                         d 
                                       
                                       1 
                                     
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       c 
                                       
                                         C 
                                         - 
                                         B 
                                       
                                       ′ 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   cosine 
                                    
                                   
                                     ( 
                                     
                                       
                                         θ 
                                         
                                           ∠ 
                                            
                                           ACB 
                                         
                                       
                                       - 
                                       
                                         θ 
                                         
                                           
                                             ∠ 
                                              
                                             AC 
                                              
                                             d 
                                           
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.120 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Substitution of Eq. (15.120) into Eq. (15.118) gives 
         [0000]    
       
         
           
             
                 
             
              
             
               ( 
               15.121 
               ) 
             
           
         
       
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                               
                              
                             
                               
                                 
                                   ( 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       c 
                                       
                                         C 
                                         - 
                                         A 
                                       
                                       ′ 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               c 
                                               
                                                 C 
                                                 - 
                                                 A 
                                               
                                               ′ 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                       - 
                                       
                                         
                                           ( 
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               c 
                                               
                                                 C 
                                                 - 
                                                 B 
                                               
                                               ′ 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                     
                                       2 
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     ( 
                                                     
                                                       2 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       
                                                         c 
                                                         
                                                           C 
                                                           - 
                                                           A 
                                                         
                                                         ′ 
                                                       
                                                     
                                                     ) 
                                                   
                                                    
                                                   cosine 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     θ 
                                                     
                                                       
                                                         ∠ 
                                                          
                                                         AC 
                                                          
                                                         d 
                                                       
                                                       1 
                                                     
                                                   
                                                 
                                                 - 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   ( 
                                                   
                                                     2 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     
                                                       c 
                                                       
                                                         C 
                                                         - 
                                                         B 
                                                       
                                                       ′ 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 
                                                   cosine 
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         θ 
                                                         
                                                           ∠ 
                                                            
                                                           ACB 
                                                         
                                                       
                                                       - 
                                                       
                                                         θ 
                                                         
                                                           
                                                             ∠ 
                                                              
                                                             AC 
                                                              
                                                             d 
                                                           
                                                           1 
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                       
                         
                           
                               
                              
                             
                               
                                 - 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     c 
                                     
                                       C 
                                       - 
                                       A 
                                     
                                     ′ 
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           
                                             c 
                                             
                                               C 
                                               - 
                                               A 
                                             
                                             ′ 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     - 
                                     
                                       
                                         ( 
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           
                                             c 
                                             
                                               C 
                                               - 
                                               B 
                                             
                                             ′ 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                   
                                     2 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   ( 
                                                   
                                                     2 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     
                                                       c 
                                                       
                                                         C 
                                                         - 
                                                         A 
                                                       
                                                       ′ 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 cosine 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   θ 
                                                   
                                                     
                                                       ∠ 
                                                        
                                                       AC 
                                                        
                                                       d 
                                                     
                                                     1 
                                                   
                                                 
                                               
                                               - 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   2 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     c 
                                                     
                                                       C 
                                                       - 
                                                       B 
                                                     
                                                     ′ 
                                                   
                                                 
                                                 ) 
                                               
                                                
                                               
                                                 cosine 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       θ 
                                                       
                                                         ∠ 
                                                          
                                                         ACB 
                                                       
                                                     
                                                     - 
                                                     
                                                       θ 
                                                       
                                                         
                                                           ∠ 
                                                            
                                                           AC 
                                                            
                                                           d 
                                                         
                                                         1 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                                
                               cosine 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 
                                   
                                     ∠ 
                                      
                                     AC 
                                      
                                     d 
                                   
                                   1 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                               
                              
                             
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         c 
                                         
                                           A 
                                           - 
                                           B 
                                         
                                         ′ 
                                       
                                     
                                     2 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                     ) 
                   
                   = 
                   0 
                 
               
               
                 
                     
                 
               
             
           
         
       
     
         [0000]    The angle between d 1  and the C-A bond, θ ∠ACd     1   , can be solved reiteratively using Eq. (15.121), and the result can be substituted into Eq. (15.120) to give d 1 . 
         [0198]    The atoms A, B, and D define the base of a pyramid. Then, the pyramidal angle θ ∠ADB  can be solved from the internuclear distances between A and D, 2c′ A-D , between B and D, 2c′ B-D , and between A and B, 2c′ A-B , using the law of cosines (Eq. (15.107)): 
         [0000]    
       
         
           
             
               
                 
                   
                     θ 
                     
                       ∠ 
                        
                       
                           
                       
                        
                       ADB 
                     
                   
                   = 
                   
                     
                       cos 
                       
                         - 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               ( 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   c 
                                   
                                     A 
                                     - 
                                     D 
                                   
                                   ′ 
                                 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   c 
                                   
                                     B 
                                     - 
                                     D 
                                   
                                   ′ 
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             
                               ( 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   c 
                                   
                                     A 
                                     - 
                                     B 
                                   
                                   ′ 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         
                           2 
                            
                           
                             ( 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 c 
                                 
                                   A 
                                   - 
                                   D 
                                 
                                 ′ 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 c 
                                 
                                   B 
                                   - 
                                   D 
                                 
                                 ′ 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   15.122 
                   ) 
                 
               
             
           
         
       
     
         [0199]    Then, the distance d 2  from D to the bisector of the internuclear-distance line between A and B,2c′ A-B , is given by two equations involving the law of cosines (Eq. (15.105)). One with s 1 =2c A-D , the internuclear distance between A and D, s 2 =d 2 , 
         [0000]    
       
         
           
             
               
                 s 
                 3 
               
               = 
               
                 
                   2 
                    
                   
                       
                   
                    
                   
                     c 
                     
                       A 
                       - 
                       B 
                     
                     ′ 
                   
                 
                 2 
               
             
             , 
           
         
       
     
         [0000]    half the internuclear distance between A and B, and θ=θ ∠ADd     2   , the angle between d 2  and the A-D axis is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             c 
                             
                               A 
                               - 
                               D 
                             
                             ′ 
                           
                         
                         ) 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           d 
                           2 
                         
                         ) 
                       
                       2 
                     
                     - 
                     
                       2 
                        
                       
                         ( 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             c 
                             
                               A 
                               - 
                               D 
                             
                             ′ 
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           d 
                           2 
                         
                         ) 
                       
                        
                       cosine 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           ∠ 
                            
                           
                               
                           
                            
                           
                             ADd 
                             2 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           2 
                            
                           
                               
                           
                            
                           
                             c 
                             
                               A 
                               - 
                               B 
                             
                             ′ 
                           
                         
                         2 
                       
                       ) 
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   15.123 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The other with s 1 =2c′ B-D , the internuclear distance between B and D, s 2 =d 2 , and θ=θ ∠ADB −θ ∠ADd     2    where θ ∠ADB  is the bond angle between the A-D and B-D axes is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             c 
                             
                               B 
                               - 
                               D 
                             
                             ′ 
                           
                         
                         ) 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           d 
                           2 
                         
                         ) 
                       
                       2 
                     
                     - 
                     
                       2 
                        
                       
                         ( 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             c 
                             
                               B 
                               - 
                               D 
                             
                             ′ 
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           d 
                           2 
                         
                         ) 
                       
                        
                       
                         cosine 
                         ( 
                         
                           
                             θ 
                             
                               ∠ 
                                
                               
                                   
                               
                                
                               ADB 
                             
                           
                           - 
                           
                             θ 
                             
                               ∠ 
                                
                               
                                   
                               
                                
                               
                                 ADd 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           2 
                            
                           
                               
                           
                            
                           
                             c 
                             
                               A 
                               - 
                               B 
                             
                             ′ 
                           
                         
                         2 
                       
                       ) 
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   15.124 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Subtraction of Eq. (15.124) from Eq. (15.123) gives 
         [0000]    
       
         
           
             
               
                 
                   
                     d 
                     2 
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               c 
                               
                                 A 
                                 - 
                                 D 
                               
                               ′ 
                             
                           
                           ) 
                         
                         2 
                       
                       - 
                       
                         
                           ( 
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               c 
                               
                                 B 
                                 - 
                                 D 
                               
                               ′ 
                             
                           
                           ) 
                         
                         2 
                       
                     
                     
                       2 
                        
                       
                         ( 
                         
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         c 
                                         
                                           A 
                                           - 
                                           D 
                                         
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                    
                                   cosine 
                                    
                                   
                                       
                                   
                                    
                                   
                                     θ 
                                     
                                       
                                         ∠ 
                                          
                                         AD 
                                          
                                         d 
                                       
                                       2 
                                     
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       c 
                                       
                                         B 
                                         - 
                                         D 
                                       
                                       ′ 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   cosine 
                                    
                                   
                                     ( 
                                     
                                       
                                         θ 
                                         
                                           ∠ 
                                            
                                           ADB 
                                         
                                       
                                       - 
                                       
                                         θ 
                                         
                                           
                                             ∠ 
                                              
                                             AD 
                                              
                                             d 
                                           
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15.125 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Substitution of Eq. (15.125) into Eq. (15.123) gives 
         [0000]    
       
         
           
             
                 
             
              
             
               ( 
               15.126 
               ) 
             
           
         
       
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                               
                              
                             
                               
                                 
                                   ( 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       c 
                                       
                                         A 
                                         - 
                                         D 
                                       
                                       ′ 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               c 
                                               
                                                 A 
                                                 - 
                                                 D 
                                               
                                               ′ 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                       - 
                                       
                                         
                                           ( 
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               c 
                                               
                                                 B 
                                                 - 
                                                 D 
                                               
                                               ′ 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                     
                                       2 
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     ( 
                                                     
                                                       2 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       
                                                         c 
                                                         
                                                           A 
                                                           - 
                                                           D 
                                                         
                                                         ′ 
                                                       
                                                     
                                                     ) 
                                                   
                                                    
                                                   cosine 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     θ 
                                                     
                                                       
                                                         ∠ 
                                                          
                                                         AD 
                                                          
                                                         d 
                                                       
                                                       2 
                                                     
                                                   
                                                 
                                                 - 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   ( 
                                                   
                                                     2 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     
                                                       c 
                                                       
                                                         B 
                                                         - 
                                                         D 
                                                       
                                                       ′ 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 
                                                   cosine 
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         θ 
                                                         
                                                           ∠ 
                                                            
                                                           ADB 
                                                         
                                                       
                                                       - 
                                                       
                                                         θ 
                                                         
                                                           
                                                             ∠ 
                                                              
                                                             AD 
                                                              
                                                             d 
                                                           
                                                           2 
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                       
                         
                           
                               
                              
                             
                               
                                 - 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     c 
                                     
                                       A 
                                       - 
                                       D 
                                     
                                     ′ 
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           
                                             c 
                                             
                                               A 
                                               - 
                                               D 
                                             
                                             ′ 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     - 
                                     
                                       
                                         ( 
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           
                                             c 
                                             
                                               B 
                                               - 
                                               D 
                                             
                                             ′ 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                   
                                     2 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   ( 
                                                   
                                                     2 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     
                                                       c 
                                                       
                                                         A 
                                                         - 
                                                         D 
                                                       
                                                       ′ 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 cosine 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   θ 
                                                   
                                                     
                                                       ∠ 
                                                        
                                                       AD 
                                                        
                                                       d 
                                                     
                                                     2 
                                                   
                                                 
                                               
                                               - 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   2 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     c 
                                                     
                                                       B 
                                                       - 
                                                       D 
                                                     
                                                     ′ 
                                                   
                                                 
                                                 ) 
                                               
                                                
                                               
                                                 cosine 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       θ 
                                                       
                                                         ∠ 
                                                          
                                                         ADB 
                                                       
                                                     
                                                     - 
                                                     
                                                       θ 
                                                       
                                                         
                                                           ∠ 
                                                            
                                                           AD 
                                                            
                                                           d 
                                                         
                                                         2 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                                
                               cosine 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 
                                   
                                     ∠ 
                                      
                                     AD 
                                      
                                     d 
                                   
                                   2 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                               
                              
                             
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         c 
                                         
                                           A 
                                           - 
                                           B 
                                         
                                         ′ 
                                       
                                     
                                     2 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                     ) 
                   
                   = 
                   0 
                 
               
               
                 
                     
                 
               
             
           
         
       
     
         [0000]    The angle between d 2  and the A-D axis, θ ∠ADd     2   , can be solved reiteratively using Eq. (15.126), and the result can be substituted into Eq. (15.125) to give d 2 . 
         [0200]    The lengths d 1 , d 2 , and 2c′ C—B  define a triangle wherein the angle between d 1  and the internuclear distance between C and D, 2c′ C-D  is the dihedral angle θ ∠CDI ACB  that can be solved using the law of cosines (Eq. (15.107)): 
         [0000]    
       
         
           
             
               
                 
                   
                     θ 
                     
                       ∠ 
                        
                       
                           
                       
                        
                       
                         CD 
                         / 
                         ACB 
                       
                     
                   
                   = 
                   
                     
                       cos 
                       
                         - 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           
                             d 
                             1 
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   c 
                                   
                                     C 
                                     - 
                                     D 
                                   
                                   ′ 
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             d 
                             2 
                             2 
                           
                         
                         
                           2 
                            
                           
                               
                           
                            
                           
                             
                               d 
                               1 
                             
                              
                             
                               ( 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   c 
                                   
                                     C 
                                     - 
                                     D 
                                   
                                   ′ 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   15.127 
                   ) 
                 
               
             
           
         
       
     
       Solution of Geometrical and Energy Parameters of Major Functional Groups and Corresponding Organic Molecules 
       [0201]    The exemplary molecules given in the following sections were solved using the solutions of organic chemical functional groups as basis elements wherein the structures and energies where linearly added to achieve the molecular solutions. Each functional group can be treated as a building block to form any desired molecular solution from the corresponding linear combination. Each functional group element was solved using the atomic orbital and hybrid orbital spherical orbitsphere solutions bridged by molecular orbitals comprised of the H 2 -type prolate spheroidal solution given in the Nature of the Chemical Bond of Hydrogen-Type Molecules section. The energy of each MO was matched at the HO or AO by matching the hybridization and total energy of the MO to the AOs and HOs. The energy E mag  (e.g. given by Eq. (15.67)) for a C2sp 3  HO and Eq. (15.68) for an O2p AO) was subtracted for each set of unpaired electrons created by bond breakage. 
         [0202]    The bond energy is not equal to the component energy of each bond as it exists in the molecule; although, they are close. The total energy of each group is its contribution to the total energy of the molecule as a whole. The determination of the bond energies for the creation of the separate parts must take into account the energy of the formation of any radicals and any redistribution of charge density within the pieces and the corresponding energy change with bond cleavage. Also, the vibrational energy in the transition state is dependent on the other groups that are bound to a given functional group. This will effect the functional-group energy. But, because the variations in the energy based on the balance of the molecular composition are typically of the order of a few hundreds of electron volts at most, they were neglected. 
         [0203]    The energy of each functional-group MO bonding to a given carbon HO is independently matched to the HO by subtracting the contribution to the change in the energy of the HO from the total MO energy given by the sum of the MO contributions and E(C,2sp 3 )=−14.63489 eV (Eq. (13.428)). The intercept angles are determined from Eqs. (15.80-15.87) using the final radius of the HO of each atom. The final carbon-atom radius is determined using Eqs. (15.32) wherein the sum of the energy contributions of each atom to all the MOs in which it participates in bonding is determined. This final radius is used in Eqs. (15.19) and (15.20) to calculate the final valence energy of the HO of each atom at the corresponding final radius. The radius of any bonding heteroatom that contributes to σ MO is calculated in the same manner, and the energy of its outermost shell is matched to that of the MO by the hybridization factor between the carbon-HO energy and the energy of the heteroatomic shell. The donation of electron density to the AOs and HOs reduces the energy. The donation of the electron density to the MO&#39;s at each AO or HO is that which causes the resulting energy to be divided equally between the participating AOs or HOs to achieve energy matching. 
         [0204]    The molecular solutions can be used to design synthetic pathways and predict product yields based on equilibrium constants calculated from the heats of formation. New stable compositions of matter can be predicted as well as the structures of combinatorial chemistry reactions. Further important pharmaceutical applications include the ability to graphically or computationally render the structures of drugs that permit the identification of the biologically active parts of the molecules to be identified from the common spatial charge-density functions of a series of active molecules. Drugs can be designed according to geometrical parameters and bonding interactions with the data of the structure of the active site of the drug. 
         [0205]    To calculate conformations, folding, and physical properties, the exact solutions of the charge distributions in any given molecule are used to calculate the fields, and from the fields, the interactions between groups of the same molecule or between groups on different molecules are calculated wherein the interactions are distance and relative orientation dependent. The fields and interactions can be determined using a finite-element-analysis approach of Maxwell&#39;s equations. 
       Pharmaceutical Molecular Functional Groups and Molecules 
     General Considerations of the Bonding in Pharmaceuticals 
       [0206]    Pharmaceutical molecules comprising an arbitrary number of atoms can be solved using similar principles and procedures as those used to solve general organic molecules of arbitrary length and complexity. Pharmaceuticals can be considered to be comprised of functional groups such those of alkanes, branched alkanes, alkenes, branched alkenes, alkynes, alkyl fluorides, alkyl chlorides, alkyl bromides, alkyl iodides, alkene halides, primary alcohols, secondary alcohols, tertiary alcohols, ethers, primary amines, secondary amines, tertiary amines, aldehydes, ketones, carboxylic acids, carboxylic esters, amides, N-alkyl amides, N,N-dialkyl amides, ureas, acid halides, acid anhydrides, nitriles, thiols, sulfides, disulfides, sulfoxides, sulfones, sulfites, sulfates, nitro alkanes, nitrites, nitrates, conjugated polyenes, aromatics, heterocyclic aromatics, substituted aromatics, and others given in the Organic Molecular Functional Groups and Molecules section. The solutions of the functional groups can be conveniently obtained by using generalized forms of the geometrical and energy equations. The functional-group solutions can be made into a linear superposition and sum, respectively, to give the solution of any pharmaceutical molecule comprising these groups. The total bond energies of exemplary pharmaceutical molecules such as aspirin are calculated using the functional group composition and the corresponding energies derived in the previous sections. 
       Aspirin (Acetylsalicylic Acid) 
       [0207]    Aspirin comprises salicylic acid (ortho-hydroxybenzoic acid) with the H of the phenolic OH group replaced by an acetyl group. Thus, aspirin comprises the benzoic acid C—C(O)—OH moiety that comprises C═O and OH functional groups that are the same as those of carboxylic acids given in the corresponding section. The single bond of aryl carbon to the carbonyl carbon atom, C—C(O), is also a functional group given in the Benzoic Acid Compounds section. The aromatic 
         [0000]    
       
         
           
             C 
              
             
               = 
               
                 3 
                  
                 e 
               
             
              
             C 
           
         
       
     
         [0000]    and C—H functional groups are equivalent to those of benzene given in Aromatic and Heterocyclic Compounds section. The phenolic ester C—O functional group is equivalent to that given in the Phenol section. The acetyl O—C(O)—CH 3  moiety comprises (i) C═O and C—C functional groups that are the same as those of carboxylic acids and esters given in the corresponding sections, (ii) a CH 3  group that is equivalent to that of alkanes given in the corresponding sections, (iii) and a C—O bridging the carbonyl carbon and the phenolic ester which is equivalent to that of esters given in the corresponding section. 
         [0208]    The symbols of the functional groups of aspirin are given in Table 16.1. 
         [0209]    The corresponding designations of aspirin are shown in  FIG. 1 . The geometrical (Eqs. (15.1-15.5) and (15.51)), intercept (Eqs. (15.80-15.87)), and energy (Eqs. (15.6-15.11) and (15.17-15.65)) parameters of aspirin are given in Tables 16.2, 16.3, and 16.4, respectively (all as shown in the priority document). The total energy of aspirin given in Table 16.5 (as shown in the priority document) was calculated as the sum over the integer multiple of each E D  (Group) of Table 16.4 (as shown in the priority document) corresponding to functional-group composition of the molecule. The bond angle parameters of aspirin determined using Eqs. (15.88-15.117) are given in Table 16.6 (as shown in the priority document). The color scale, translucent view of the charge-density of aspirin comprising the concentric shells of atoms with the outer shell bridged by one or more H 2 -type ellipsoidal MOs or joined with one or more hydrogen MOs is shown in  FIG. 2 . 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 16.1 
               
             
             
               
                   
               
               
                 The symbols of functional groups of aspirin. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
                   
               
               
                   
                 CC (aromatic bond) 
                 
                   
                     
                       
                         C 
                          
                         
                           
                             — 
                             — 
                           
                           
                               
                           
                           
                             3 
                              
                             e 
                           
                         
                          
                         C 
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                 CH (aromatic) 
                 CH 
               
               
                   
                 Aryl C—C(O) 
                 C—C(O) (i) 
               
               
                   
                 Alkyl C—C(O) 
                 C—C(O) (ii) 
               
               
                   
                 C═O (aryl carboxylic acid) 
                 C═O 
               
               
                   
                 Aryl (O)C—O 
                 C—O (i) 
               
               
                   
                 Alkyl (O)C—O 
                 C—O (ii) 
               
               
                   
                 Aryl C—O 
                 C—O (iii) 
               
               
                   
                 OH group 
                 OH 
               
               
                   
                 CH 3  group 
                 CH 3   
               
               
                   
                   
               
             
          
         
       
     
       REFERENCES 
       [0000]    
       
         1. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), pp. 9-19 to 9-45. 
         2. G. A. Sim, J. M. Robertson, T. H. Goodwin, “The crystal and molecular structure of benzoic acid”, Acta Cryst., Vol. 8, (1955), pp. 157-164. 
         3. G. Herzberg,  Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules , Van Nostrand Reinhold Company, New York, N.Y., (1945), pp. 362-369. 
         4. acetic acid at http://webbook.nist.gov/. 
         5. G. Herzberg,  Molecular Spectra and Molecular Structure II Infrared and Raman Spectra of Polyatomic Molecules , Krieger Publishing Company, Malabar, Fla., (1991), p. 195. 
         6. D. Lin-Vien. N. B. Colthup, W. G. Fateley, J. G. Grasselli,  The Handbook of Infrared and Raman Frequencies of Organic Molecules , Academic Press, Inc., Harcourt Brace Jovanovich, Boston, (1991), p. 138. 
         7. methyl formate at http://webbook.nist.gov/. 
         8. methanol at http://webbook.nist.gov/. 
         9. K. P. Huber, G. Herzberg,  Molecular Spectra and Molecular Structure, IV Constants of Diatomic Molecules , Van Nostrand Reinhold Company, New York, (1979). 
         10. J. Crovisier, Molecular Database—Constants for molecules of astrophysical interest in the gas phase: photodissociation, microwave and infrared spectra, Ver. 4.2, Observatoire de Paris, Section de Meudon, Meudon, France, May 2002, pp. 34-37, available at http://wwwusr.obspm.fr/˜crovisie/. 
         11. W. I. F. David, R. M. Ibberson, G. A. Jeffrey, J. R. Ruble, “The structure analysis of deuterated benzene and deuterated nitromethane by pulsed-neutron powder diffraction: a comparison with single crystal neutron analysis”, Physica B (1992), 180 &amp; 181, pp. 597-600. 
         12. G. A. Jeffrey, J. R. Ruble, R. K. McMullan, J. A. Pople, “The crystal structure of deuterated benzene,” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 414, No. 1846, (Nov. 9, 1987), pp. 47-57. 
         13. H. B. Burgi, S. C. Capelli, “Getting more out of crystal-structure analyses,” Helvetica Chimica Acta, Vol. 86, (2003), pp. 1625-1640. 
       
     
       Nature of the Solid Molecular Bond of the Three Allotropes of Carbon 
     General Considerations of the Solid Molecular Bond 
       [0223]    The solid molecular bond of a material comprising an arbitrary number of atoms can be solved using similar principles and procedures as those used to solve organic molecules of arbitrary length. Molecular solids are also comprised of functional groups. Depending on the material, exemplary groups are C—C, C═C, C—O, C—N, C—S, and others given in the Organic Molecular Functional Groups and Molecules section. The solutions of these functional groups or any others corresponding to the particular solid can be conveniently obtained by using generalized forms of the geometrical and energy equations given in the Derivation of the General Geometrical and Energy Equations of Organic Chemistry section. The appropriate functional groups with the their geometrical parameters and energies can be added as a linear sum to give the solution of any molecular solid. 
       Diamond 
       [0224]    It is demonstrated in this Diamond section as well as the Fullerene (C 60 ) and Graphite sections, that very complex macromolecules can be simply solved from the groups at each vertex carbon atom of the structure. Specifically, for fullerene a C═C group is bound to two C—C bonds at each vertex carbon atom of C 60 . The solution of the macromolecule is given by superposition of the geometrical and energy parameters of the corresponding two groups. In graphite, each sheet of joined hexagons can be constructed with a C═C group bound to two C—C bonds at each vertex carbon atom that hybridize to an aromatic-like functional group, 
         [0000]    
       
         
           
             
               C 
                
               
                 = 
                 
                   
                     8 
                     / 
                     3 
                   
                    
                   e 
                 
               
                
               C 
             
             , 
           
         
       
     
         [0000]    with 8/3 electron-number per bond compared to the pure aromatic functional group, 
         [0000]    
       
         
           
             
               C 
                
               
                 = 
                 
                   3 
                    
                   e 
                 
               
                
               C 
             
             , 
           
         
       
     
         [0000]    with 3 electron-number per bond as given the Aromatics section. Similarly, diamond comprising, in principle, an infinite network of carbons can be solved using the functional group solutions where the task is also simple since diamond has only one functional group, the diamond C—C functional group. 
         [0225]    The diamond C—C bonds are all equivalent, and each C—C bond can be considered bound to a t-butyl group at the corresponding vertex carbon. Thus, the parameters of the diamond C—C functional group are equivalent to those of the t-butyl C—C group of branched alkanes given in the Branched Alkanes section. Based on symmetry, the parameter R in Eqs. (15.56) and (15.61) is the semimajor axis a, and the vibrational energy in the Ē acs  term is that of diamond. Also, the C2sp 3  HO magnetic energy E mag  given by Eq. (15.67) was subtracted for each t-butyl group of alkyl fluorides, alkyl chlorides, alkyl iodides, thiols, sulfides, disulfides, and nitroalkanes as given in the corresponding sections of Chapter 15 due to a set of unpaired electrons being created by bond breakage. Since each C—C group of diamond bonds with a t-butyl group at each vertex carbon, c 3  of Eq. (15.65) is one, and E mag  is given by Eq. (15.67). 
         [0226]    The symbol of the functional group of diamond is given in Table 17.1. The geometrical (Eqs. (15.1-15.5) and (15.51)) parameters of diamond are given in Table 17.2. The lattice parameter a l  was calculated from the bond distance using the law of cosines: 
         [0000]        s   1   2   +s   2   2 −2 s   1   s   2 cosine θ= s   3   2   (17.1)
 
         [0000]    With the bond angle θ ∠CCC =109.5° [1] and s 1 =s 2 =2c′ C—C , the internuclear distance of the C—C bond, s 3 =2c′ C     1     -C     1   , the internuclear distance of the two terminal C atoms is given by 
         [0000]      2 c′   C     1     -C     1   =√{square root over (2(2 c′   C     1     -C     1   ) 2 (1−cosine(109.5°)}{square root over (2(2 c′   C     1     -C     1   ) 2 (1−cosine(109.5°)}  (17.2)
 
         [0000]    Two times the distance 2c′ C     1     -C     1    is the hypotenuse of the isosceles triangle having equivalent sides of length equal to the lattice parameter a 1 . Using Eq. (17.2) and 2c′ C     1     -C     1   =1.53635 Å from Table 17.2, the lattice parameter a 1  for the cubic diamond structure is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           a 
                           l 
                         
                         = 
                           
                          
                         
                           
                             
                               2 
                                
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     c 
                                     
                                       
                                         C 
                                         t 
                                       
                                       - 
                                       
                                         C 
                                         t 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                             
                               2 
                             
                           
                           = 
                           
                             
                               2 
                             
                              
                             
                               
                                 2 
                                  
                                 
                                   
                                     ( 
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         c 
                                         
                                           C 
                                           - 
                                           C 
                                         
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       cosine 
                                        
                                       
                                         ( 
                                         
                                           109.5 
                                            
                                           ° 
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           3.54867 
                            
                           
                               
                           
                            
                           Å 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17.3 
                   ) 
                 
               
             
           
         
       
     
         [0227]    The intercept (Eqs. (15.80-15.87)) and energy (Eqs. (15.6-15.11) and (15.17-15.65)) parameters of diamond are given in Tables 17.2, 17.3 (as shown in priority document), and 17.4, respectively. The total energy of diamond given in Table 17.5 was calculated as the sum over the integer multiple of each E D  (Group) of Table 17.4 corresponding to functional-group composition of the molecular solid. The experimental C—C bond energy of diamond, E D     exp   (C—C) at 298 K, is given by the difference between the enthalpy of formation of gaseous carbon atoms from graphite (ΔH f (C graphite (gas))) and the heat of formation of diamond (ΔH f  (C (diamond))) wherein graphite has a defined heat of formation of zero (ΔH f  (C(graphite)=0): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       
                         D 
                         exp 
                       
                     
                      
                     
                       ( 
                       
                         C 
                         - 
                         C 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       [ 
                       
                         
                           Δ 
                            
                           
                               
                           
                            
                           
                             
                               H 
                               f 
                             
                              
                             
                               ( 
                               
                                 
                                   C 
                                   graphite 
                                 
                                  
                                 
                                   ( 
                                   gas 
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                         - 
                         
                           Δ 
                            
                           
                               
                           
                            
                           
                             
                               H 
                               f 
                             
                              
                             
                               ( 
                               
                                 C 
                                  
                                 
                                   ( 
                                   diamond 
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   17.4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the heats of formation of atomic carbon and diamond are [2]: 
         [0000]      Δ H   f ( C   graphite (gas))=716.68 kJ/mole(7.42774 eV/atom)  (17.5)
 
         [0000]      Δ H   f ( C (diamond))=1.9 kJ/mole(0.01969 eV/atom)  (17.6)
 
       Using Eqs. (17.4-17.6), E D     exp   (C—C) is 
       [0228]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               D 
                               exp 
                             
                           
                            
                           
                             ( 
                             
                               C 
                               - 
                               C 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             1 
                             2 
                           
                            
                           
                             [ 
                             
                               
                                 7.42774 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               - 
                               
                                 0.01969 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           3.704 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17.7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the factor of one half corresponds to the ratio of two electrons per bond and four electrons per carbon atom. The bond angle parameters of diamond determined using Eqs. (15.88-15.117) are given in Table 17.6 (as shown in priority document). The structure of diamond is shown in  FIG. 3 . 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 17.1 
               
             
             
               
                   
               
               
                 The symbols of the functional group of diamond. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
                   
               
               
                   
                 CC bond (diamond-C) 
                 C—C 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 17.2 
               
             
             
               
                   
               
               
                 The geometrical bond parameters of 
               
               
                 diamond and experimental values [1, 3]. 
               
             
          
           
               
                   
                   
                 C—C 
               
               
                   
                 Parameter 
                 Group 
               
               
                   
                   
               
             
          
           
               
                   
                 a (a 0 ) 
                 2.10725 
               
               
                   
                 c′ (a 0 ) 
                 1.45164 
               
               
                   
                 Bond Length 2c′ (Å) 
                 1.53635 
               
               
                   
                 Exp. Bond Length (Å) 
                 1.54428 
               
               
                   
                 b, c (a 0 ) 
                 1.52750 
               
               
                   
                 e 
                 0.68888 
               
               
                   
                 Lattice Parameter a l  (Å) 
                 3.54867 
               
               
                   
                 Exp. Lattice Parameter a l  (Å) 
                 3.5670 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 17.4 
               
             
             
               
                   
               
               
                 The energy parameters (eV) of the functional group of diamond. 
               
             
          
           
               
                   
                   
                 C—C 
               
               
                   
                 Parameters 
                 Group 
               
               
                   
                   
               
             
          
           
               
                   
                 n 1   
                 1 
               
               
                   
                 n 2   
                 0 
               
               
                   
                 n 3   
                 0 
               
               
                   
                 C 1   
                 0.5 
               
               
                   
                 C 2   
                 1 
               
               
                   
                 c 1   
                 1 
               
               
                   
                 c 2   
                 0.91771 
               
               
                   
                 c 3   
                 1 
               
               
                   
                 c 4   
                 2 
               
               
                   
                 c 5   
                 0 
               
               
                   
                 C 1o   
                 0.5 
               
               
                   
                 C 2o   
                 1 
               
               
                   
                 V e  (eV) 
                 −29.10112 
               
               
                   
                 V p  (eV) 
                 9.37273 
               
               
                   
                 T (eV) 
                 6.90500 
               
               
                   
                 V m  (eV) 
                 −3.45250 
               
               
                   
                 E (AO/HO) (eV) 
                 −15.35946 
               
               
                   
                 ΔE H     2     MO  (AO/HO) (eV) 
                 0 
               
               
                   
                 E T  (AO/HO) (eV) 
                 −15.35946 
               
               
                   
                 E T  (H 2 MO) (eV) 
                 −31.63535 
               
               
                   
                 E T  (atom-atom, msp 3  · AO) (eV) 
                 −1.44915 
               
               
                   
                 E T  (MO) (eV) 
                 −33.08452 
               
               
                   
                 ω (10 15  rad/s) 
                 9.55643 
               
               
                   
                 E K  (eV) 
                 6.29021 
               
               
                   
                 Ē D  (eV) 
                 −0.16416 
               
               
                   
                 Ē Kvib  (eV) 
                 0.16515 [4] 
               
               
                   
                 Ē osc  (eV) 
                 −0.08158 
               
               
                   
                 E mag  (eV) 
                 0.14803 
               
               
                   
                 E T  (Group) (eV) 
                 −33.16610 
               
               
                   
                 E initial  (c 4  AO/HO) (eV) 
                 −14.63489 
               
               
                   
                 E initial  (c 5  AO/HO) (eV) 
                 0 
               
               
                   
                 E D  (Group) (eV) 
                 3.74829 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 17.5 
               
             
             
               
                   
               
               
                 The total bond energy of diamond calculated using 
               
               
                 the functional group composition and the energy of 
               
               
                 Table 17.4 compared to the experimental value [1-2]. 
               
             
          
           
               
                   
                   
                   
                 Calculated 
                 Experimental 
                   
               
               
                   
                   
                   
                 Total Bond 
                 Total Bond 
                 Relative 
               
               
                 Formula 
                 Name 
                 C—C 
                 Energy (eV) 
                 Energy (eV) 
                 Error 
               
               
                   
               
               
                 C n   
                 Diamond 
                 1 
                 3.74829 
                 3.704 
                 −0.01 
               
               
                   
               
             
          
         
       
     
       Fullerene (C 60 ) 
       [0229]    C 60  comprises 60 equivalent carbon atoms that are bound as 60 single bonds and 30 double bonds in the geometric form of a truncated icosahedron: twelve pentagons and twenty hexagons joined such that no two pentagons share an edge. To achieve this minimum energy structure each equivalent carbon atom serves as a vertex incident with one double and two single bonds. Each type of bond serves as a functional group which has aromatic character. The aromatic bond is uniquely stable and requires the sharing of the electrons of multiple H 2 -type MOs. The results of the derivation of the parameters of the benzene molecule given in the Benzene Molecule (C 6 H 6 ) section was generalized to any aromatic functional group of aromatic and heterocyclic compounds in the Aromatic and Heterocyclic Compounds section. Ethylene serves as a basis element for the 
         [0000]    
       
         
           
             C 
              
             
               = 
               
                 3 
                  
                 e 
               
             
              
             C 
           
         
       
     
         [0000]    bonding of the aromatic bond wherein each of the 
         [0000]    
       
         
           
             C 
              
             
               = 
               
                 3 
                  
                 e 
               
             
              
             C 
           
         
       
     
         [0000]    aromatic bonds comprises (0.75)(4)=3 electrons according to Eq. (15.161) wherein C 2  of Eq. (15.51) for the aromatic C 3e ═C-bond MO given by Eq. (15.162) is C 2 (aromaticC2sp 3 HO)=c 2  (aromaticC2sp 3 HO)=0.85252 and E T (atom−atom,msp 3 .AO)=−2.26759 eV. In C 60 , the minimum energy structure with equivalent carbon atoms wherein each carbon forms bonds with three other such carbons requires a redistribution of charge within an aromatic system of bonds. The C═C functional group of C 60  comprises the aromatic bond with the exception that it comprises four electrons. Thus, E T (Group) and E D  (Group) are given by Eqs. (15.165) and (15.166), respectively, with f 1 =1, c 4 =4, and Ē Kvib  (eV) is that of C 60 . 
         [0230]    In addition to the C═C bond, each vertex carbon atom of C 60  is bound to two C—C bonds that substitute for the aromatic 
         [0000]    
       
         
           
             C 
              
             
               = 
               
                 3 
                  
                 e 
               
             
              
             C 
           
         
       
     
         [0000]    and C—H bonds. As in the case of the C—C-bond MO of naphthalene, to match energies within the MO that bridges single and double-bond MOs, E(A0/HO) and ΔE H     2     MO (AO/HO) in Eq. (15.51) are −14.63489 eV and −2.26759 eV, respectively. 
         [0231]    To meet the equipotential condition of the union of the C2sp 3  HOs of the C—C single bond bridging double bonds, the parameters c 1 , C 2 , and C 2o  of Eq. (15.51) are one for the C—C group, C 1o  and C 1  are 0.5, and c 2  given by Eq. (13.430) is c 2  (C2sp 3 HO=0.91771. To match the energies of the functional groups with the electron-density shift to the double bond, E T (atom−atom,msp 3 .AO) of each of the equivalent C—C-bond MOs in Eq. (15.61) due to the charge donation from the C atoms to the MO can be considered a linear combination of that of C—C-bond MO of toluene, −1.13379 eV and the that of the aromatic C—H-bond MO, 
         [0000]    
       
         
           
             
               
                 
                   - 
                   1.13379 
                 
                  
                 
                     
                 
                  
                 eV 
               
               2 
             
             . 
           
         
       
     
         [0000]    Thus, E T (atom−atom,msp 3 .AO) of each C—C-bond MO of C 60  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           - 
                           1.13379 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                       + 
                       
                         0.5 
                          
                         
                           ( 
                           
                             
                               - 
                               1.13379 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           ) 
                         
                       
                     
                     2 
                   
                   = 
                   
                     0.75 
                      
                     
                       ( 
                       
                         
                           - 
                           1.13379 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       - 
                       0.85034 
                     
                      
                     
                         
                     
                      
                     
                       eV 
                       . 
                     
                   
                 
               
             
           
         
       
     
         [0000]    As in the case of the aromatic C—H bond, c 3 =1 in Eq. (15.65) with E mag , given by Eq. (15.67), and Ē Kvib (eV) is that of C 60 . 
         [0232]    The symbols of the functional groups of C 60  are given in Table 17.7. The geometrical (Eqs. (15.1-15.5) and (15.51)), intercept (Eqs. (15.80-15.87)), and energy (Eqs. (15.6-15.11), (15.17-15.65), and (15.165-15.166)) parameters of C 60  are given in Tables 17.8, 17.9 (as shown in priority document), and 17.10, respectively. The total energy of C 60  given in Table 17.11 was calculated as the sum over the integer multiple of each E D  (Group) of Table 17.10 corresponding to functional-group composition of the molecule. The bond angle parameters of C 60  determined using Eqs. (15.87-15.117) are given in Table 17.12 (as shown in the priority document). The structure of C 60  is shown in  FIGS. 4 and 7 . The fullerene vertex-atom group comprising a double and two single bonds can serve as a basis element to form other higher-order fullerene-type macromolecules, hyperfullerenes, and complex hybrid conjugated carbon and aromatic structures comprising a mixture of elements from the group of fullerene, graphitic, and diamond carbon described in the corresponding sections. 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 17.7 
               
             
             
               
                   
               
               
                 The symbols of functional groups of C 60 . 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
                   
               
               
                   
                 C═C (aromatic-type) 
                 C═C 
               
               
                   
                 C—C (bound to C═C aromatic-type) 
                 C—C 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 17.8 
               
             
             
               
                   
               
               
                 The geometrical bond parameters of C 60  and 
               
               
                 experimental values [5]. 
               
             
          
           
               
                   
                   
                 C═C 
                 C—C 
               
               
                   
                 Parameter 
                 Group 
                 Group 
               
               
                   
                   
               
               
                   
                 a (a 0 ) 
                 1.47348 
                 1.88599 
               
               
                   
                 c′ (a 0 ) 
                 1.31468 
                 1.37331 
               
               
                   
                 Bond Length 2c′ (Å) 
                 1.39140 
                 1.45345 
               
               
                   
                 Exp. Bond Length 
                 1.391 
                 1.455 
               
               
                   
                 (Å) 
                 (C 60 ) 
                 (C 60 ) 
               
               
                   
                 b, c (a 0 ) 
                 0.66540 
                 1.29266 
               
               
                   
                 e 
                 0.89223 
                 0.72817 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 17.10 
               
             
             
               
                   
               
               
                 The energy parameters (eV) of functional groups of C 60 . 
               
             
          
           
               
                   
                   
                 C═C 
                 C—C 
               
               
                   
                 Parameters 
                 Group 
                 Group 
               
               
                   
                   
               
             
          
           
               
                   
                 f 1   
                 1 
                 1 
               
               
                   
                 n 1   
                 2 
                 1 
               
               
                   
                 n 2   
                 0 
                 0 
               
               
                   
                 n 3   
                 0 
                 0 
               
               
                   
                 C 1   
                 0.5 
                 0.5 
               
               
                   
                 C 2   
                 0.85252 
                 1 
               
               
                   
                 c 1   
                 1 
                 1 
               
               
                   
                 c 2   
                 0.85252 
                 0.91771 
               
               
                   
                 c 3   
                 0 
                 1 
               
               
                   
                 c 4   
                 4 
                 2 
               
               
                   
                 c 5   
                 0 
                 0 
               
               
                   
                 C 1o   
                 0.5 
                 0.5 
               
               
                   
                 C 2o   
                 0.85252 
                 1 
               
               
                   
                 V e  (eV) 
                 −101.12679 
                 −33.63376 
               
               
                   
                 V p  (eV) 
                 20.69825 
                 9.90728 
               
               
                   
                 T (eV) 
                 34.31559 
                 8.91674 
               
               
                   
                 V m  (eV) 
                 −17.15779 
                 −4.45837 
               
               
                   
                 E (AO/HO) (eV) 
                 0 
                 −14.63489 
               
               
                   
                 ΔE H     2     MO  (AO/HO) (eV) 
                 0 
                 −2.26759 
               
               
                   
                 E T  (AO/HO) (eV) 
                 0 
                 −12.36730 
               
               
                   
                 E T  (H 2 MO) (eV) 
                 −63.27075 
                 −31.63541 
               
               
                   
                 E T  (atom-atom, msp 3  · AO) (eV) 
                 −2.26759 
                 −0.85034 
               
               
                   
                 E T  (MO) (eV) 
                 −65.53833 
                 −32.48571 
               
               
                   
                 ω (10 15  rad/s) 
                 49.7272 
                 19.8904 
               
               
                   
                 E K  (eV) 
                 32.73133 
                 13.09221 
               
               
                   
                 Ē D  (eV) 
                 −0.35806 
                 −0.23254 
               
               
                   
                 Ē Kvib  (eV) 
                 0.17727 [6] 
                 0.14667 [6] 
               
               
                   
                 Ē osc  (eV) 
                 −0.26942 
                 −0.15921 
               
               
                   
                 E mag  (eV) 
                 0.14803 
                 0.14803 
               
               
                   
                 E T  (Group) (eV) 
                 −66.07718 
                 −32.49689 
               
               
                   
                 E initial  (c 4  AO/HO) (eV) 
                 −14.63489 
                 −14.63489 
               
               
                   
                 E initial  (c 5  AO/HO) (eV) 
                 0 
                 0 
               
               
                   
                 E D  (Group) (eV) 
                 7.53763 
                 3.22711 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 17.11 
               
             
             
               
                   
               
               
                 The total bond energies of C 60  calculated using 
               
               
                 the functional group composition and the energies of 
               
               
                 Table 17.10 compared to the experimental values [7]. 
               
             
          
           
               
                   
                   
                   
                   
                 Calculated 
                   
                   
               
               
                   
                   
                   
                   
                 Total 
                 Experimental 
               
               
                   
                   
                   
                   
                 Bond 
                 Total Bond 
               
               
                   
                   
                   
                   
                 Energy 
                 Energy 
                 Relative 
               
               
                 Formula 
                 Name 
                 C═C 
                 C—C 
                 (eV) 
                 (eV) 
                 Error 
               
               
                   
               
               
                 C 60   
                 Fullerene 
                 30 
                 60 
                 419.75539 
                 419.73367 
                 −0.00005 
               
               
                   
               
             
          
         
       
     
       Fullerene Dihedral Angles 
       [0233]    For C 60  the bonding at each vertex atom C b  comprises two single bonds, C a —C b —C a , and a double bond, C b ═C c . The dihedral angle θ ∠C═C/C—C—C  between the plane defined by the C a —C b —C a  moiety and the line defined by the corresponding C b ═C c  moiety is calculated using the results given in Table 17.12 (as shown in the priority document) and Eqs. (15.114-15.117). The distance d 1  along the bisector of θ ∠C     a     —C     b     —C     a    from C b  to the internuclear-distance line between one C a  and the other C a , 2c′ C     a     —C     a   , is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           d 
                           1 
                         
                         = 
                         
                           2 
                            
                           
                             c 
                             
                               
                                 C 
                                 b 
                               
                               - 
                               
                                 C 
                                 a 
                               
                             
                             ′ 
                           
                            
                           cos 
                            
                           
                             
                               θ 
                               
                                 
                                   ∠C 
                                   a 
                                 
                                 - 
                                 
                                   C 
                                   b 
                                 
                                 - 
                                 Ca 
                               
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           2.74663 
                            
                           
                             a 
                             0 
                           
                            
                           cos 
                            
                           
                             
                               180.00 
                               ° 
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1.61443 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17.8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where 2c′ C     b     —C     a    is the internuclear distance between C b  and C a . The atoms C a , C a , and C c  define the base of a pyramid. Then, the pyramidal angle θ ∠C     a     C     c     C     a    can be solved from the internuclear distances between C c  and C a , 2c′ C     a     —C     a   , and between C a  and C a , 2c′ C     a     —C     a   , using the law of cosines (Eq. (15.115)): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           θ 
                           
                             
                               ∠C 
                               a 
                             
                              
                             
                               C 
                               b 
                             
                              
                             
                               C 
                               a 
                             
                           
                         
                         = 
                         
                           
                             cos 
                             
                               - 
                               1 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       2 
                                        
                                       
                                         c 
                                         
                                           
                                             C 
                                             c 
                                           
                                           - 
                                           
                                             C 
                                             a 
                                           
                                         
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       2 
                                        
                                       
                                         c 
                                         
                                           
                                             C 
                                             c 
                                           
                                           - 
                                           
                                             C 
                                             a 
                                           
                                         
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       2 
                                        
                                       
                                         c 
                                         
                                           
                                             C 
                                             a 
                                           
                                           - 
                                           
                                             C 
                                             a 
                                           
                                         
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                               
                                 2 
                                  
                                 
                                   ( 
                                   
                                     2 
                                      
                                     
                                       c 
                                       
                                         
                                           C 
                                           c 
                                         
                                         - 
                                         
                                           C 
                                           a 
                                         
                                       
                                       ′ 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     2 
                                      
                                     
                                       c 
                                       
                                         
                                           C 
                                           c 
                                         
                                         - 
                                         
                                           C 
                                           a 
                                         
                                       
                                       ′ 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             cos 
                             
                               - 
                               1 
                             
                           
                           ( 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           4.65618 
                                            
                                           
                                             a 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     + 
                                     
                                       
                                         ( 
                                         
                                           4.65618 
                                            
                                           
                                             a 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         4.4441 
                                          
                                         
                                           a 
                                           0 
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                             
                             
                               2 
                                
                               
                                 ( 
                                 
                                   4.65618 
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   4.65618 
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           57.01 
                           ° 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17.9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Then, the distance d 2  along the bisector of θ ∠C     a     C     c     C     a    from C c  to the internuclear-distance line 2c′ C     a     —C     a   , is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           d 
                           2 
                         
                         = 
                         
                           2 
                            
                           
                             c 
                             
                               
                                 C 
                                 c 
                               
                               - 
                               
                                 C 
                                 a 
                               
                             
                             ′ 
                           
                            
                           cos 
                            
                           
                             
                               θ 
                               
                                 
                                   ∠C 
                                   a 
                                 
                                  
                                 
                                   C 
                                   c 
                                 
                                  
                                 
                                   C 
                                   a 
                                 
                               
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           4.65618 
                            
                           
                             a 
                             0 
                           
                            
                           cos 
                            
                           
                             
                               57.01 
                               ° 
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           4.09176 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17.10 
                   ) 
                 
               
             
           
         
       
     
         [0234]    The lengths d 1 , d 2 , and 2c′ C     b     =C     c    define a triangle wherein the angle between d 1  and the internuclear distance between C b  and C c , 2c′ C     b     =C     c   , is the dihedral angle θ ∠C═C/C—C—C  that can be solved using the law of cosines (Eq. (15.117)): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           θ 
                           
                             ∠C 
                             = 
                             
                               
                                 C 
                                 / 
                                 C 
                               
                               - 
                               C 
                               - 
                               C 
                             
                           
                         
                         = 
                         
                           
                             cos 
                             
                               - 
                               1 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   d 
                                   1 
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       2 
                                        
                                       
                                         c 
                                         
                                           
                                             C 
                                             b 
                                           
                                           = 
                                           
                                             C 
                                             c 
                                           
                                         
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 - 
                                 
                                   d 
                                   2 
                                   2 
                                 
                               
                               
                                 2 
                                  
                                 
                                   
                                     d 
                                     1 
                                   
                                    
                                   
                                     ( 
                                     
                                       2 
                                        
                                       
                                         c 
                                         
                                           
                                             C 
                                             b 
                                           
                                           = 
                                           
                                             C 
                                             c 
                                           
                                         
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             cos 
                             
                               - 
                               1 
                             
                           
                           ( 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           1.61443 
                                            
                                           
                                             a 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     + 
                                     
                                       
                                         ( 
                                         
                                           2.62936 
                                            
                                           
                                             a 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         4.09176 
                                          
                                         
                                           a 
                                           0 
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                             
                             
                               2 
                                
                               
                                 ( 
                                 
                                   1.61443 
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   2.62936 
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           148.29 
                           ° 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17.11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The dihedral angle for a truncated icosahedron corresponding to θ ∠C═C/C—C—C  is 
         [0000]      θ ∠C═C/C—C—C   (17.12)
 
         [0235]    The dihedral angle θ ∠C—C/C—C═C  between the plane defined by the C a —C b ═C c  moiety and the line defined by the corresponding C b —C a  moiety is calculated using the results given in Table 17.12 (as shown in the priority document) and Eqs. (15.118-15.127). The parameter d 1  is the distance from C b  to the internuclear-distance line between C a  and C c , 2c′ C     a     —C     c   . The angle between d 1  and the C b —C a  bond, θ ∠C     a     C     b     d     1   , can be solved reiteratively using Eq. (15.121): 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     ( 
                     17.13 
                     ) 
                   
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       ( 
                       
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     2 
                                      
                                     
                                       c 
                                       
                                         
                                           C 
                                           b 
                                         
                                         - 
                                         
                                           C 
                                           a 
                                         
                                       
                                       ′ 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             2 
                                              
                                             
                                               c 
                                               
                                                 
                                                   C 
                                                   b 
                                                 
                                                 - 
                                                 
                                                   C 
                                                   a 
                                                 
                                               
                                               ′ 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                       - 
                                       
                                         
                                           ( 
                                           
                                             2 
                                              
                                             
                                               c 
                                               
                                                 
                                                   C 
                                                   b 
                                                 
                                                 - 
                                                 
                                                   C 
                                                   c 
                                                 
                                               
                                               ′ 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                     
                                       2 
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     ( 
                                                     
                                                       2 
                                                        
                                                       
                                                         c 
                                                         
                                                           
                                                             C 
                                                             b 
                                                           
                                                           - 
                                                           
                                                             C 
                                                             a 
                                                           
                                                         
                                                         ′ 
                                                       
                                                     
                                                     ) 
                                                   
                                                    
                                                   cosine 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     θ 
                                                     
                                                       
                                                         ∠C 
                                                         a 
                                                       
                                                        
                                                       
                                                         C 
                                                         b 
                                                       
                                                        
                                                       
                                                         d 
                                                         1 
                                                       
                                                     
                                                   
                                                 
                                                 - 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   ( 
                                                   
                                                     2 
                                                      
                                                     
                                                       c 
                                                       
                                                         
                                                           C 
                                                           b 
                                                         
                                                         - 
                                                         
                                                           C 
                                                           c 
                                                         
                                                       
                                                       ′ 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 cosine 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     θ 
                                                     
                                                       
                                                         ∠C 
                                                         a 
                                                       
                                                        
                                                       
                                                         C 
                                                         b 
                                                       
                                                        
                                                       
                                                         C 
                                                         c 
                                                       
                                                     
                                                   
                                                   - 
                                                   
                                                     θ 
                                                     
                                                       
                                                         ∠C 
                                                         a 
                                                       
                                                        
                                                       
                                                         C 
                                                         b 
                                                       
                                                        
                                                       
                                                         d 
                                                         1 
                                                       
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                     c 
                                     
                                       
                                         C 
                                         b 
                                       
                                       - 
                                       
                                         C 
                                         a 
                                       
                                     
                                     ′ 
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           2 
                                            
                                           
                                             c 
                                             
                                               
                                                 C 
                                                 b 
                                               
                                               - 
                                               
                                                 C 
                                                 a 
                                               
                                             
                                             ′ 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     - 
                                     
                                       
                                         ( 
                                         
                                           2 
                                            
                                           
                                             c 
                                             
                                               
                                                 C 
                                                 b 
                                               
                                               - 
                                               
                                                 C 
                                                 c 
                                               
                                             
                                             ′ 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                   
                                     2 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   ( 
                                                   
                                                     2 
                                                      
                                                     
                                                       c 
                                                       
                                                         
                                                           C 
                                                           b 
                                                         
                                                         - 
                                                         
                                                           C 
                                                           a 
                                                         
                                                       
                                                       ′ 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 cosine 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   θ 
                                                   
                                                     
                                                       ∠C 
                                                       a 
                                                     
                                                      
                                                     
                                                       C 
                                                       b 
                                                     
                                                      
                                                     
                                                       d 
                                                       1 
                                                     
                                                   
                                                 
                                               
                                               - 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   2 
                                                    
                                                   
                                                     c 
                                                     
                                                       
                                                         C 
                                                         b 
                                                       
                                                       - 
                                                       
                                                         C 
                                                         c 
                                                       
                                                     
                                                     ′ 
                                                   
                                                 
                                                 ) 
                                               
                                                
                                               cosine 
                                             
                                           
                                         
                                         
                                           
                                             
                                               ( 
                                               
                                                 
                                                   θ 
                                                   
                                                     
                                                       ∠C 
                                                       a 
                                                     
                                                      
                                                     
                                                       C 
                                                       b 
                                                     
                                                      
                                                     
                                                       C 
                                                       c 
                                                     
                                                   
                                                 
                                                 - 
                                                 
                                                   θ 
                                                   
                                                     
                                                       ∠C 
                                                       a 
                                                     
                                                      
                                                     
                                                       C 
                                                       b 
                                                     
                                                      
                                                     
                                                       d 
                                                       1 
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                                
                               cosine 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 
                                   
                                     ∠C 
                                     a 
                                   
                                    
                                   
                                     C 
                                     b 
                                   
                                    
                                   
                                     d 
                                     1 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                         c 
                                         
                                           
                                             C 
                                             a 
                                           
                                           - 
                                           
                                             C 
                                             c 
                                           
                                         
                                         ′ 
                                       
                                     
                                     2 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     0 
                   
                 
               
             
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   2.74663 
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           2.74663 
                                            
                                           
                                             a 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     - 
                                     
                                       
                                         ( 
                                         
                                           2.62936 
                                            
                                           
                                             a 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                   
                                     2 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   ( 
                                                   
                                                     2.74663 
                                                      
                                                     
                                                       a 
                                                       0 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 cosine 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   θ 
                                                   
                                                     
                                                       ∠C 
                                                       a 
                                                     
                                                      
                                                     
                                                       C 
                                                       b 
                                                     
                                                      
                                                     
                                                       d 
                                                       1 
                                                     
                                                   
                                                 
                                               
                                               - 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   2.62936 
                                                    
                                                   
                                                     a 
                                                     0 
                                                   
                                                 
                                                 ) 
                                               
                                                
                                               cosine 
                                             
                                           
                                         
                                         
                                           
                                             
                                               ( 
                                               
                                                 
                                                   120.00 
                                                   ° 
                                                 
                                                 - 
                                                 
                                                   θ 
                                                   
                                                     
                                                       ∠C 
                                                       a 
                                                     
                                                      
                                                     
                                                       C 
                                                       b 
                                                     
                                                      
                                                     
                                                       d 
                                                       1 
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       
                         
                           
                             - 
                             
                               ( 
                               
                                 2 
                                  
                                 
                                   ( 
                                   
                                     2.74663 
                                      
                                     
                                       a 
                                       0 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             2.74663 
                                              
                                             
                                               a 
                                               0 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                       - 
                                       
                                         
                                           ( 
                                           
                                             2.62936 
                                              
                                             
                                               a 
                                               0 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                     
                                       2 
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     ( 
                                                     
                                                       2.74663 
                                                        
                                                       
                                                         a 
                                                         0 
                                                       
                                                     
                                                     ) 
                                                   
                                                    
                                                   cosine 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     θ 
                                                     
                                                       
                                                         ∠C 
                                                         a 
                                                       
                                                        
                                                       
                                                         C 
                                                         b 
                                                       
                                                        
                                                       
                                                         d 
                                                         1 
                                                       
                                                     
                                                   
                                                 
                                                 - 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   ( 
                                                   
                                                     2.62936 
                                                      
                                                     
                                                       a 
                                                       0 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 cosine 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     120.00 
                                                     ° 
                                                   
                                                   - 
                                                   
                                                     θ 
                                                     
                                                       
                                                         ∠C 
                                                         a 
                                                       
                                                        
                                                       
                                                         C 
                                                         b 
                                                       
                                                        
                                                       
                                                         d 
                                                         1 
                                                       
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 cosine 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   
                                     
                                       ∠C 
                                       a 
                                     
                                      
                                     
                                       C 
                                       b 
                                     
                                      
                                     
                                       d 
                                       1 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     4.6562 
                                      
                                     
                                       a 
                                       0 
                                     
                                   
                                   2 
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                     
                     ) 
                   
                   = 
                   0 
                 
               
             
           
         
       
     
         [0000]    The solution of Eq. (17.13) is 
         [0000]      θ C     a     C     a     d     1   =57.810°  (17.14)
 
         [0000]    Eq. (17.14) can be substituted into Eq. (15.120) to give d 1 : 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           d 
                           1 
                         
                         = 
                         
                           
                             
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                     c 
                                     
                                       
                                         C 
                                         b 
                                       
                                       - 
                                       
                                         C 
                                         a 
                                       
                                     
                                     ′ 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             - 
                             
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                     c 
                                     
                                       
                                         C 
                                         b 
                                       
                                       - 
                                       
                                         C 
                                         c 
                                       
                                     
                                     ′ 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           
                             2 
                              
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             2 
                                              
                                             
                                               c 
                                               
                                                 
                                                   C 
                                                   b 
                                                 
                                                 - 
                                                 
                                                   C 
                                                   a 
                                                 
                                               
                                               ′ 
                                             
                                           
                                           ) 
                                         
                                          
                                         cosine 
                                          
                                         
                                             
                                         
                                          
                                         
                                           θ 
                                           
                                             
                                               ∠C 
                                               a 
                                             
                                              
                                             
                                               C 
                                               b 
                                             
                                              
                                             
                                               d 
                                               1 
                                             
                                           
                                         
                                       
                                       - 
                                       
                                         ( 
                                         
                                           2 
                                            
                                           
                                             c 
                                             
                                               
                                                 C 
                                                 b 
                                               
                                               - 
                                               
                                                 C 
                                                 c 
                                               
                                             
                                             ′ 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       cosine 
                                        
                                       
                                         ( 
                                         
                                           
                                             θ 
                                             
                                               
                                                 ∠C 
                                                 a 
                                               
                                                
                                               
                                                 C 
                                                 b 
                                               
                                                
                                               
                                                 C 
                                                 c 
                                               
                                             
                                           
                                           - 
                                           
                                             θ 
                                             
                                               
                                                 ∠C 
                                                 a 
                                               
                                                
                                               
                                                 C 
                                                 b 
                                               
                                                
                                               
                                                 d 
                                                 1 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 ( 
                                 
                                   2.74663 
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             - 
                             
                               
                                 ( 
                                 
                                   2.62936 
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           
                             2 
                              
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             2.74663 
                                              
                                             
                                               a 
                                               0 
                                             
                                           
                                           ) 
                                         
                                          
                                         
                                           cosine 
                                            
                                           
                                             ( 
                                             
                                               57.810 
                                               ° 
                                             
                                             ) 
                                           
                                         
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           2.62936 
                                            
                                           
                                             a 
                                             0 
                                           
                                         
                                         ) 
                                       
                                        
                                       
                                         cosine 
                                          
                                         
                                           ( 
                                           
                                             
                                               120.00 
                                               ° 
                                             
                                             - 
                                             
                                               57.810 
                                               ° 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1.33278 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17.15 
                   ) 
                 
               
             
           
         
       
     
         [0236]    The atoms C a , C a , and C c  define the base of a pyramid. Then, the pyramidal angle θ∠ C     a     C     a     C     c    can be solved from the internuclear distances between C a  and C a , 2c′ C     a     —C     a   , and between C a  and C c , 2c′ C     a     —C     c   , using the law of cosines (Eq. (15.115)): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           θ 
                           
                             
                               ∠C 
                               a 
                             
                              
                             
                               C 
                               a 
                             
                              
                             
                               C 
                               c 
                             
                           
                         
                         = 
                         
                           
                             cos 
                             
                               - 
                               1 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       2 
                                        
                                       
                                         c 
                                         
                                           
                                             C 
                                             a 
                                           
                                           - 
                                           
                                             C 
                                             a 
                                           
                                         
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       2 
                                        
                                       
                                         c 
                                         
                                           
                                             C 
                                             a 
                                           
                                           - 
                                           
                                             C 
                                             c 
                                           
                                         
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       2 
                                        
                                       
                                         c 
                                         
                                           
                                             C 
                                             a 
                                           
                                           - 
                                           
                                             C 
                                             c 
                                           
                                         
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                               
                                 2 
                                  
                                 
                                   ( 
                                   
                                     2 
                                      
                                     
                                       c 
                                       
                                         
                                           C 
                                           a 
                                         
                                         - 
                                         
                                           C 
                                           a 
                                         
                                       
                                       ′ 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     2 
                                      
                                     
                                       c 
                                       
                                         
                                           C 
                                           a 
                                         
                                         - 
                                         
                                           C 
                                           c 
                                         
                                       
                                       ′ 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             cos 
                             
                               - 
                               1 
                             
                           
                           ( 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           4.44410 
                                            
                                           
                                             a 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     + 
                                     
                                       
                                         ( 
                                         
                                           4.65618 
                                            
                                           
                                             a 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         4.65618 
                                          
                                         
                                           a 
                                           0 
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                             
                             
                               2 
                                
                               
                                 ( 
                                 
                                   4.44410 
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   4.65618 
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           61.50 
                           ° 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17.16 
                   ) 
                 
               
             
           
         
       
     
         [0237]    The parameter d 2  is the distance from C a  to the bisector of the internuclear-distance line between C a  and C c , 2c′C   a     —C     c   . The angle between d 2  and the C a —C a  axis, θ ∠C     a     C     a     d     2   , can be solved reiteratively using Eq. (15.126): 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     ( 
                     17.17 
                     ) 
                   
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       ( 
                       
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     2 
                                      
                                     
                                       c 
                                       
                                         
                                           C 
                                           a 
                                         
                                         - 
                                         
                                           C 
                                           a 
                                         
                                       
                                       ′ 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             2 
                                              
                                             
                                               c 
                                               
                                                 
                                                   C 
                                                   a 
                                                 
                                                 - 
                                                 
                                                   C 
                                                   a 
                                                 
                                               
                                               ′ 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                       - 
                                       
                                         
                                           ( 
                                           
                                             2 
                                              
                                             
                                               c 
                                               
                                                 
                                                   C 
                                                   a 
                                                 
                                                 - 
                                                 
                                                   C 
                                                   c 
                                                 
                                               
                                               ′ 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                     
                                       2 
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     ( 
                                                     
                                                       2 
                                                        
                                                       
                                                         c 
                                                         
                                                           
                                                             C 
                                                             a 
                                                           
                                                           - 
                                                           
                                                             C 
                                                             a 
                                                           
                                                         
                                                         ′ 
                                                       
                                                     
                                                     ) 
                                                   
                                                    
                                                   cosine 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     θ 
                                                     
                                                       
                                                         ∠C 
                                                         a 
                                                       
                                                        
                                                       
                                                         C 
                                                         a 
                                                       
                                                        
                                                       
                                                         d 
                                                         2 
                                                       
                                                     
                                                   
                                                 
                                                 - 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   ( 
                                                   
                                                     2 
                                                      
                                                     
                                                       c 
                                                       
                                                         
                                                           C 
                                                           a 
                                                         
                                                         - 
                                                         
                                                           C 
                                                           c 
                                                         
                                                       
                                                       ′ 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 cosine 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     θ 
                                                     
                                                       
                                                         ∠C 
                                                         a 
                                                       
                                                        
                                                       
                                                         C 
                                                         a 
                                                       
                                                        
                                                       
                                                         C 
                                                         c 
                                                       
                                                     
                                                   
                                                   - 
                                                   
                                                     θ 
                                                     
                                                       
                                                         ∠C 
                                                         a 
                                                       
                                                        
                                                       
                                                         C 
                                                         a 
                                                       
                                                        
                                                       
                                                         d 
                                                         2 
                                                       
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                     c 
                                     
                                       
                                         C 
                                         b 
                                       
                                       - 
                                       
                                         C 
                                         a 
                                       
                                     
                                     ′ 
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           2 
                                            
                                           
                                             c 
                                             
                                               
                                                 C 
                                                 a 
                                               
                                               - 
                                               
                                                 C 
                                                 a 
                                               
                                             
                                             ′ 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     - 
                                     
                                       
                                         ( 
                                         
                                           2 
                                            
                                           
                                             c 
                                             
                                               
                                                 C 
                                                 a 
                                               
                                               - 
                                               
                                                 C 
                                                 c 
                                               
                                             
                                             ′ 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                   
                                     2 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   ( 
                                                   
                                                     2 
                                                      
                                                     
                                                       c 
                                                       
                                                         
                                                           C 
                                                           a 
                                                         
                                                         - 
                                                         
                                                           C 
                                                           a 
                                                         
                                                       
                                                       ′ 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 cosine 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   θ 
                                                   
                                                     
                                                       ∠C 
                                                       a 
                                                     
                                                      
                                                     
                                                       C 
                                                       a 
                                                     
                                                      
                                                     
                                                       d 
                                                       2 
                                                     
                                                   
                                                 
                                               
                                               - 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   2 
                                                    
                                                   
                                                     c 
                                                     
                                                       
                                                         C 
                                                         a 
                                                       
                                                       - 
                                                       
                                                         C 
                                                         c 
                                                       
                                                     
                                                     ′ 
                                                   
                                                 
                                                 ) 
                                               
                                                
                                               cosine 
                                             
                                           
                                         
                                         
                                           
                                             
                                               ( 
                                               
                                                 
                                                   θ 
                                                   
                                                     
                                                       ∠C 
                                                       a 
                                                     
                                                      
                                                     
                                                       C 
                                                       a 
                                                     
                                                      
                                                     
                                                       C 
                                                       c 
                                                     
                                                   
                                                 
                                                 - 
                                                 
                                                   θ 
                                                   
                                                     
                                                       ∠C 
                                                       a 
                                                     
                                                      
                                                     
                                                       C 
                                                       a 
                                                     
                                                      
                                                     
                                                       d 
                                                       2 
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                                
                               cosine 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 
                                   
                                     ∠C 
                                     a 
                                   
                                    
                                   
                                     C 
                                     b 
                                   
                                    
                                   
                                     d 
                                     2 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                         c 
                                         
                                           
                                             C 
                                             a 
                                           
                                           - 
                                           
                                             C 
                                             c 
                                           
                                         
                                         ′ 
                                       
                                     
                                     2 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     0 
                   
                 
               
             
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   4.44410 
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           4.44410 
                                            
                                           
                                             a 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     - 
                                     
                                       
                                         ( 
                                         
                                           4.65618 
                                            
                                           
                                             a 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                   
                                     2 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   ( 
                                                   
                                                     4.44410 
                                                      
                                                     
                                                       a 
                                                       0 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 cosine 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   θ 
                                                   
                                                     
                                                       ∠C 
                                                       a 
                                                     
                                                      
                                                     
                                                       C 
                                                       a 
                                                     
                                                      
                                                     
                                                       d 
                                                       2 
                                                     
                                                   
                                                 
                                               
                                               - 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   4.65618 
                                                    
                                                   
                                                     a 
                                                     0 
                                                   
                                                 
                                                 ) 
                                               
                                                
                                               cosine 
                                             
                                           
                                         
                                         
                                           
                                             
                                               ( 
                                               
                                                 
                                                   61.50 
                                                   ° 
                                                 
                                                 - 
                                                 
                                                   θ 
                                                   
                                                     
                                                       ∠C 
                                                       a 
                                                     
                                                      
                                                     
                                                       C 
                                                       a 
                                                     
                                                      
                                                     
                                                       d 
                                                       2 
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       
                         
                           
                             - 
                             
                               ( 
                               
                                 2 
                                  
                                 
                                   ( 
                                   
                                     4.44410 
                                      
                                     
                                       a 
                                       0 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             4.44410 
                                              
                                             
                                               a 
                                               0 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                       - 
                                       
                                         
                                           ( 
                                           
                                             4.65618 
                                              
                                             
                                               a 
                                               0 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                     
                                       2 
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     ( 
                                                     
                                                       4.44410 
                                                        
                                                       
                                                         a 
                                                         0 
                                                       
                                                     
                                                     ) 
                                                   
                                                    
                                                   cosine 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     θ 
                                                     
                                                       
                                                         ∠C 
                                                         a 
                                                       
                                                        
                                                       
                                                         C 
                                                         a 
                                                       
                                                        
                                                       
                                                         d 
                                                         2 
                                                       
                                                     
                                                   
                                                 
                                                 - 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   ( 
                                                   
                                                     4.65618 
                                                      
                                                     
                                                       a 
                                                       0 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 cosine 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     61.50 
                                                     ° 
                                                   
                                                   - 
                                                   
                                                     θ 
                                                     
                                                       
                                                         ∠C 
                                                         a 
                                                       
                                                        
                                                       
                                                         C 
                                                         a 
                                                       
                                                        
                                                       
                                                         d 
                                                         2 
                                                       
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 cosine 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   
                                     
                                       ∠C 
                                       a 
                                     
                                      
                                     
                                       C 
                                       a 
                                     
                                      
                                     
                                       d 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     4.6562 
                                      
                                     
                                       a 
                                       0 
                                     
                                   
                                   2 
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                     
                     ) 
                   
                   = 
                   0 
                 
               
             
           
         
       
     
         [0000]    The solution of Eq. (17.17) is 
         [0000]      θ ∠C     a     C     a     d     2   =31.542°  (17.18) Eq. (17.18) can be substituted into Eq. (15.125) to give d 2 :
 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           d 
                           2 
                         
                         = 
                         
                           
                             
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                     c 
                                     
                                       
                                         C 
                                         a 
                                       
                                       - 
                                       
                                         C 
                                         a 
                                       
                                     
                                     ′ 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             - 
                             
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                     c 
                                     
                                       
                                         C 
                                         a 
                                       
                                       - 
                                       
                                         C 
                                         c 
                                       
                                     
                                     ′ 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           
                             2 
                              
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             2 
                                              
                                             
                                               c 
                                               
                                                 
                                                   C 
                                                   a 
                                                 
                                                 - 
                                                 
                                                   C 
                                                   a 
                                                 
                                               
                                               ′ 
                                             
                                           
                                           ) 
                                         
                                          
                                         cosine 
                                          
                                         
                                             
                                         
                                          
                                         
                                           θ 
                                           
                                             
                                               ∠C 
                                               a 
                                             
                                              
                                             
                                               C 
                                               a 
                                             
                                              
                                             
                                               d 
                                               2 
                                             
                                           
                                         
                                       
                                       - 
                                       
                                         ( 
                                         
                                           2 
                                            
                                           
                                             c 
                                             
                                               
                                                 C 
                                                 a 
                                               
                                               - 
                                               
                                                 C 
                                                 c 
                                               
                                             
                                             ′ 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       cosine 
                                        
                                       
                                         ( 
                                         
                                           
                                             θ 
                                             
                                               
                                                 ∠C 
                                                 a 
                                               
                                                
                                               
                                                 C 
                                                 a 
                                               
                                                
                                               
                                                 C 
                                                 c 
                                               
                                             
                                           
                                           - 
                                           
                                             θ 
                                             
                                               
                                                 ∠C 
                                                 a 
                                               
                                                
                                               
                                                 C 
                                                 a 
                                               
                                                
                                               
                                                 d 
                                                 2 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 ( 
                                 
                                   4.44410 
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             - 
                             
                               
                                 ( 
                                 
                                   4.65618 
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           
                             2 
                              
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             4.44410 
                                              
                                             
                                               a 
                                               0 
                                             
                                           
                                           ) 
                                         
                                          
                                         
                                           cosine 
                                            
                                           
                                             ( 
                                             
                                               31.542 
                                               ° 
                                             
                                             ) 
                                           
                                         
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           4.65618 
                                            
                                           
                                             a 
                                             0 
                                           
                                         
                                         ) 
                                       
                                        
                                       
                                         cosine 
                                          
                                         
                                           ( 
                                           
                                             
                                               61.50 
                                               ° 
                                             
                                             - 
                                             
                                               31.542 
                                               ° 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           3.91101 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17.19 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The lengths d 1 , d 2 , and 2c′ C     b     —C     a    define a triangle wherein the angle between d 1  and the internuclear distance between C b  and C a , 2c′ C     b     —C     a   , is the dihedral angle θ ∠C—C/C—C═C  that can be solved using the law of cosines (Eq. (15.117)): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           θ 
                           
                             
                               ∠C 
                               - 
                               
                                 C 
                                 / 
                                 C 
                               
                               - 
                               C 
                             
                             = 
                             C 
                           
                         
                         = 
                         
                           
                             cos 
                             
                               - 
                               1 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   d 
                                   1 
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       2 
                                        
                                       
                                         c 
                                         
                                           
                                             C 
                                             b 
                                           
                                           - 
                                           
                                             C 
                                             a 
                                           
                                         
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 - 
                                 
                                   d 
                                   2 
                                   2 
                                 
                               
                               
                                 2 
                                  
                                 
                                   
                                     d 
                                     1 
                                   
                                    
                                   
                                     ( 
                                     
                                       2 
                                        
                                       
                                         c 
                                         
                                           
                                             C 
                                             b 
                                           
                                           - 
                                           
                                             C 
                                             a 
                                           
                                         
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             cos 
                             
                               - 
                               1 
                             
                           
                           ( 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           1.33278 
                                            
                                           
                                             a 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     + 
                                     
                                       
                                         ( 
                                         
                                           2.74663 
                                            
                                           
                                             a 
                                             0 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         3.91101 
                                          
                                         
                                           a 
                                           0 
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                             
                             
                               2 
                                
                               
                                 ( 
                                 
                                   1.33278 
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   2.74663 
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           144.71 
                           ° 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17.16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The dihedral angle for a truncated icosahedron corresponding to θ ∠C—C/C—C═C  is 
         [0000]      θ ∠C—C/C—C═C =144.24°  (17.20)
 
       Graphite 
       [0238]    In addition to fullerene and diamond described in the corresponding sections, graphite is the third allotrope of carbon. It comprises planar sheets of covalently bound carbon atoms arranged in hexagonal aromatic rings of a macromolecule of indefinite size. The sheets, in turn, are bound together by weaker intermolecular forces. It was demonstrated in the Fullerene (C 60 ) section, that a very complex macromolecule, fullerene, could be simply solved from the groups at each vertex carbon atom of the structure. Specifically, a C═C group is bound to two C—C bonds at each vertex carbon atom of C 60 . The solution of the macromolecule is given by superposition of the geometrical and energy parameters of the corresponding two groups. Similarly, diamond comprising, in principle, an infinite network of carbons was also solved in the Diamond section using the functional group solutions, the diamond C—C functional group which is the only functional group of diamond. 
         [0239]    The structure of the indefinite network of aromatic hexagons of a sheet of graphite can also be solved by considering the vertex atom. As in the case of fullerene, each sheet of joined hexagons can be constructed with a C═C group bound to two C—C bonds at each vertex carbon atom of graphite. However, an alternative bonding to that C 60  is possible for graphite due to the structure comprising repeating hexagonal units. In this case, the lowest energy structure is achieved with a single functional group, one which has aromatic character. The aromatic bond is uniquely stable and requires the sharing of the electrons of multiple H 2 -type MOs. The results of the derivation of the parameters of the benzene molecule given in the Benzene Molecule (C 6 H 6 ) section was generalized to any aromatic functional group of aromatic and heterocyclic compounds in the Aromatic and Heterocyclic Compounds section. Ethylene serves as a basis element for the C 3e ═C bonding of the aromatic bond wherein each of the C 3e ═C aromatic bonds comprises (0.75)(4)=3 electrons according to Eq. (15.161) wherein C 2  of Eq. (15.51) for the aromatic 
         [0000]    
       
         
           
             C 
              
             
               = 
               
                 3 
                  
                 e 
               
             
              
             C 
           
         
       
     
         [0000]    -bond MO given by Eq. (15.162) is C 2 (aromaticC2sp 3 HO)=c 2 (aromaticC2sp 3  HO)=0.85252 and E T (atom−atom,msp 3 .AO)=−2.26759 eV. 
         [0240]    In graphite, the minimum energy structure with equivalent carbon atoms wherein each carbon forms bonds with three other such carbons requires a redistribution of charge within an aromatic system of bonds. Considering that each carbon contributes four bonding electrons, the sum of electrons of a vertex-atom group is four from the vertex atom plus two from each of the two atoms bonded to the vertex atom where the latter also contribute two each to the juxtaposed group. These eight electrons are distributed equivalently over the three bonds of the group such that the electron number assignable to each bond is 8/3. Thus, the 
         [0000]    
       
         
           
             C 
              
             
               = 
               
                 
                   8 
                   / 
                   3 
                 
                  
                 e 
               
             
              
             C 
           
         
       
     
         [0000]    functional group of graphite comprises the aromatic bond with the exception that the electron-number per bond is 8/3. E T (Group) and E D  (Group) are given by Eqs. (15.165) and (15.166), respectively, with 
         [0000]    
       
         
           
             
               f 
               1 
             
             = 
             
               
                 
                   2 
                   3 
                 
                  
                 
                     
                 
                  
                 and 
                  
                 
                     
                 
                  
                 
                   c 
                   4 
                 
               
               = 
               
                 
                   8 
                   3 
                 
                 . 
               
             
           
         
       
     
         [0000]    As in the case of diamond comprising equivalent carbon atoms, the C2sp 3  HO magnetic energy E mag  given by Eq. (15.67) was subtracted due to a set of unpaired electrons being created by bond breakage such that c 3  of Eqs. (15.165) and (15.166) is one. 
         [0241]    The symbol of the functional group of graphite is given in Table 17.13. The geometrical (Eqs. (15.1-15.5) and (15.51)), intercept (Eqs. (15.80-15.87)), and energy (Eqs. (15.6-15.11), (15.17-15.65), and (15.165-15.166)) parameters of graphite are given in Tables 17.14, 17.15 (as shown in the priority document), and 17.16, respectively. The total energy of graphite given in Table 17.17 was calculated as the sum over the integer multiple of each E D  (Group) of Table 17.16 corresponding to functional-group composition of the molecular solid. The experimental 
         [0000]    
       
         
           
             C 
              
             
               = 
               
                 
                   8 
                   / 
                   3 
                 
                  
                 e 
               
             
              
             C 
           
         
       
     
         [0000]    bond energy of graphite at 0 K, 
         [0000]    
       
         
           
             
               
                 E 
                 
                   D 
                   exp 
                 
               
               ( 
               
                 C 
                  
                 
                   = 
                   
                     
                       8 
                       / 
                       3 
                     
                      
                     e 
                   
                 
                  
                 C 
               
               ) 
             
             , 
           
         
       
     
         [0000]    is given by the difference between the enthalpy of formation of gaseous carbon atoms from graphite, ΔH f  (C graph (gas)), and the interplanar binding energy, E x , wherein graphite solid has a defined heat of formation of zero (ΔH f  (C (graphite)=0): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       
                         D 
                         exp 
                       
                     
                     ( 
                     
                       C 
                        
                       
                         = 
                         
                           
                             8 
                             / 
                             3 
                           
                            
                           e 
                         
                       
                        
                       C 
                     
                     ) 
                   
                   = 
                   
                     
                       2 
                       3 
                     
                      
                     
                       [ 
                       
                         
                           Δ 
                            
                           
                               
                           
                            
                           
                             
                               H 
                               f 
                             
                              
                             
                               ( 
                               
                                 
                                   C 
                                   graphite 
                                 
                                  
                                 
                                   ( 
                                   gas 
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                         - 
                         
                           E 
                           x 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   17.21 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The factor of 2/3 corresponds to the ratio of 8/3 electrons per bond and 4 electrons per carbon atom. The heats of formation of atomic carbon from graphite [9] and E x [ 10] are: 
         [0000]      Δ H   f ( C   graphite (gas))=711.185 kJ/mole(7.37079 eV/atom)  (17.22)
 
         [0000]        E   x =0.0228 eV/atom  (17.23)
 
       Using Eqs. (17.21-17.23), 
       [0242]    
       
         
           
             
               E 
               
                 D 
                 exp 
               
             
             ( 
             
               C 
                
               
                 = 
                 
                   
                     8 
                     / 
                     3 
                   
                    
                   e 
                 
               
                
               C 
             
             ) 
           
         
       
     
         [0000]    is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               D 
                               exp 
                             
                           
                           ( 
                           
                             C 
                              
                             
                               = 
                               
                                 
                                   8 
                                   / 
                                   3 
                                 
                                  
                                 e 
                               
                             
                              
                             C 
                           
                           ) 
                         
                         = 
                         
                           
                             2 
                             3 
                           
                            
                           
                             [ 
                             
                               
                                 7.37079 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               - 
                               
                                 0.0228 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           4.89866 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17.24 
                   ) 
                 
               
             
           
         
       
     
         [0243]    The bond angle parameters of graphite determined using Eqs. (15.87-15.117) are given in Table 17.18 (as shown in the priority document). The inter-plane distance for graphite of 3.5 Å is calculated using the same equation as used to determine the bond angles (Eq. (15.99)). The structure of graphite is shown in  FIG. 8 . The graphite 
         [0000]    
       
         
           
             C 
              
             
               = 
               
                 
                   8 
                   / 
                   3 
                 
                  
                 e 
               
             
              
             C 
           
         
       
     
         [0000]    functional group can serve as a basis element to form additional complex polycyclic aromatic carbon structures such as nanotubes [11-15]. 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 17.13 
               
             
             
               
                   
               
               
                 The symbols of the functional froup of graphite. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
                   
               
               
                   
                 CC bond (graphite-C) 
                 
                   
                     
                       
                         C 
                          
                         
                           
                             — 
                             — 
                           
                           
                               
                           
                           
                             
                               8 
                               / 
                               3 
                             
                              
                             e 
                           
                         
                          
                         C 
                       
                     
                   
                 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
             
           
               
                 TABLE 17.14 
               
             
             
               
                   
               
               
                 The geometrical bond parameters of graphite and experimental values. 
               
             
          
           
               
                 Parameter 
                         C          —   —                 8   /   3        e          C       
 Group 
               
               
                   
               
               
                 a (a 0 ) 
                 1.47348 
               
               
                 c′ (a 0 ) 
                 1.31468 
               
               
                 Bond Length 2c′ (Å) 
                 1.39140 
               
               
                 Exp. Bond Length (Å) 
                 1.42 
               
               
                   
                 (graphite) [11] 
               
               
                   
                 1.399 
               
               
                   
                 (benzene) [16] 
               
               
                 b, c (a 0 ) 
                 0.66540 
               
               
                 e 
                 0.89223 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
             
           
               
                 TABLE 17.16 
               
             
             
               
                   
               
               
                 The energy parameters (eV) of the functional group of graphite. 
               
             
          
           
               
                 Parameters 
                         C          —   —                 8   /   3        e          C       
 Group 
               
               
                   
               
               
                 f 1   
                 2/3 
               
               
                 n 1   
                 2 
               
               
                 n 2   
                 0 
               
               
                 n 3   
                 0 
               
               
                 C 1   
                 0.5 
               
               
                 C 2   
                 0.85252 
               
               
                 c 1   
                 1 
               
               
                 c 2   
                 0.85252 
               
               
                 c 3   
                 1 
               
               
                 c 4   
                 8/3 
               
               
                 c 5   
                 0 
               
               
                 C 1o   
                 0.5 
               
               
                 C 2o   
                 0.85252 
               
               
                 V e  (eV) 
                 −101.12679 
               
               
                 V p  (eV) 
                 20.69825 
               
               
                 T (eV) 
                 34.31559 
               
               
                 V m  (eV) 
                 −17.15779 
               
               
                 E( AO/HO ) (eV) 
                 0 
               
               
                 ΔE H     2     MO ( AO/HO ) (eV) 
                 0 
               
               
                 E T  ( AO/HO ) (eV) 
                 0 
               
               
                 E T  ( H     2     MO ) (eV) 
                 −63.27075 
               
               
                 E T  (atom − atom, msp 3 .AO) (eV) 
                 −2.26759 
               
               
                 E T  ( MO ) (eV) 
                 −65.53833 
               
               
                 ω (10 15  rad/s) 
                 49.7272 
               
               
                 E K  (eV) 
                 32.73133 
               
               
                 Ē D  (eV) 
                 −0.35806 
               
               
                 Ē Kvib  (eV) 
                 0.19649 
               
               
                   
                 [17] 
               
               
                 Ē osc  (eV) 
                 −0.25982 
               
               
                 E mag  (eV) 
                 0.14803 
               
               
                 E T  ( Group ) (eV) 
                 −43.93995 
               
               
                 E initial  ( c     4     AO/HO ) (eV) 
                 −14.63489 
               
               
                 E initial  ( c     5     AO/HO ) (eV) 
                 0 
               
               
                 E D  ( Group ) (eV) 
                 4.91359 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 17.17 
               
               
                   
               
             
             
               
                 The total bond energy of graphite calculated using 
               
               
                 the functional group composition and the energy of 
               
               
                 Table 17.16 compared to the experimental value [9-10]. 
               
             
          
           
               
                     Formula 
                     Name 
                   
       C          —   —                 8   /   3        e          C       
 
                 Calculated Total Bond Energy (eV) 
                 Experimental Total Bond Energy (eV) 
                     Relative Error 
               
               
                   
               
               
                 C n   
                 Graphite 
                 1 
                 4.91359 
                 4.89866 
                 −0.00305 
               
               
                   
               
             
          
         
       
     
       REFERENCES 
       [0000]    
       
         1. http://newton.ex.ac.uk/research/qsystems/people/sque/diamond/. 
         2. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), pp. 5-18; 5-45. 
         3. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), p. 4-150. 
         4. J. Wagner, Ch. Wild, P. Koidl, “Resonance effects in scattering from polycrystalline diamond films”, Appl. Phys. Lett. Vol. 59, (1991), pp. 779-781. 
         5. W. I. F. David, R. M. Ibberson, J. C. Matthewman, K. Prassides, T. J. S. Dennis, J. P. Hare, H. W. Kroto, R. Taylor, D. R. M. Walton, “Crystal structure and bonding of C 60 ”, Nature, Vol. 353, (1991), pp. 147-149. 
         6. B. Chase, N. Herron, E. Holler, “Vibrational spectroscopy of C 60  and C 70  temperature-dependent studies”, J. Phys. Chem., Vol. 96, (1992), pp. 4262-4266. 
         7. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), pp. 9-63; 5-18 to 5-42. 
         8. J. M. Hawkins, “Osmylation of C 60 : proof and characterization of the soccer-ball framework”, Acc. Chem. Res., (1992), Vol. 25, pp. 150-156. 
         9. M. W. Chase, Jr., C. A. Davies, J. R. Downey, Jr., D. J. Frurip, R. A. McDonald, A. N. Syverud, JANAF Thermochemical Tables, Third Edition, Part II, Cr—Zr, J. Phys. Chem. Ref. Data, Vol. 14, Suppl. 1, (1985), p. 536. 
         10. M. C. Schabel, J. L. Martins, “Energetics of interplanar binding in graphite”, Phys. Rev. B, Vol. 46, No. 11, (1992), pp. 7185-7188. 
         11. J. -C. Charlier, J. -P. Michenaud, “Energetics of multilayered carbon tubules”, Phys. Rev. Ltts., Vol. 70, No. 12, (19930, pp. 1858-1861. 
         12. J. P. Lu, “Elastic properties of carbon nanotubes and nanoropes,” Phys. Rev. Letts., (1997), Vol. 79, No. 7, pp. 1297-1300. 
         13. G. Zhang, X. Jiang, E. Wang, “Tubular graphite cones,” Science, (2003), vol. 300, pp. 472-474. 
         14. A. N. Kolmogorov, V. H. Crespi, M. H. Schleier-Smith, J. C. Ellenbogen, “Nanotube-substrate interactions: Distinguishing carbon nanotubes by the helical angle,” Phys. Rev. Letts., (2004), Vol. 92, No. 8, pp. 085503-1-085503-4. 
         15. J.-W. Jiang, H. Tang, B.-S. Wang, Z.-B. Su, “A lattice dynamical rreatment for the total potential of single-walled carbon nanontubes and its applications: Relaxed equilibrium structure, elastic properties, and vibrational modes of ultra-narrow tubes,” available at http://arxiv.org/PS_cache/cond-mat/pdf/0610/0610792.pdf, Oct. 28, 2006. 
         16. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), p. 9-29. 
         17. G. Herzberg,  Molecular Spectra and Molecular Structure II Infrared and Raman Spectra of Polyatomic Molecules , Van Nostrand Reinhold Company, New York, N.Y., (1945), pp. 362-369. 
         18. D. R. McKenzie, D. Muller, B. A. Pailthorpe, “Compressive-stress-induced formation of thin-film tetrahedral amorphous carbon”, Phys. Rev. Lett., (1991), Vol. 67, No. 6, pp. 773-776. 
         19. W. I. F. David, R. M. Ibberson, G. A. Jeffrey, J. R. Ruble, “The structure analysis of deuterated benzene and deuterated nitromethane by pulsed-neutron powder diffraction: a comparison with single crystal neutron analysis”, Physica B (1992), 180 &amp; 181, pp. 597-600. 
         20. G. A. Jeffrey, J. R. Ruble, R. K. McMullan, J. A. Pople, “The crystal structure of deuterated benzene,” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 414, No. 1846, (Nov. 9, 1987), pp. 47-57. 
         21. H. B. Burgi, S. C. Capelli, “Getting more out of crystal-structure analyses,” Helvetica Chimica Acta, Vol. 86, (2003), pp. 1625-1640. 
       
     
       The Nature of the Metallic Bond of Alkali Metals 
     Generalization of the Nature of the Metallic Bond 
       [0265]    Common metals comprise alkali, alkaline earth, and transition elements and have the properties of high electrical and thermal conductivity, opacity, surface luster, ductility, and malleability. From Maxwell&#39;s equations, the electric field inside of a metal conductor is zero. As shown in Appendix IV, the bound electron exhibits this feature. The charge is confined to a two dimensional layer and the field is normal and discontinuous at the surface. The relationship between the electric field equation and the electron source charge-density function is given by Maxwell&#39;s equation in two dimensions [1-3]. 
         [0000]    
       
         
           
             
               
                 
                   
                     n 
                     · 
                     
                       ( 
                       
                         
                           E 
                           1 
                         
                         - 
                         
                           E 
                           2 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     σ 
                     
                       ɛ 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   19.1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where n is the normal unit vector, E 1 =0 (E 1  is the electric field inside of the MO), E 2  is the electric field outside of the MO and σ is the surface charge density. The properties of metals can be accounted for the existence of free electrons bound to the corresponding lattice of positive ions. Based on symmetry, the natural coordinates are Cartesian. Then, the problem of the solution of the nature of the metal bonds reduces to a familiar electrostatics problem—the fields and the two-dimensional surface charge density induced on a planar conductor by a point charge such that a zero potential inside of the conductor is maintained according to Maxwell&#39;s equations. 
         [0266]    There are many examples of charges located near a conductor such as an electron emitted from a cathode or a power line suspended above the conducting earth. Consider a point charge +e at a position (0,0,d) near an infinite planar conductor as shown in  FIG. 9 . 
         [0267]    With the potential of the conductor set equal to zero, the potential Φ in the upper half space (z&gt;0) is given by the Poisson equation (Eq. (I.30)), subject to the boundary condition that Φ=0 at z=0 and at z=∞. The potential for the point charge in free space is 
         [0000]    
       
         
           
             
               
                 
                   
                     Φ 
                      
                     
                       ( 
                       
                         x 
                         , 
                         y 
                         , 
                         z 
                       
                       ) 
                     
                   
                   = 
                   
                     
                        
                       
                         4 
                          
                         
                           πɛ 
                           0 
                         
                       
                     
                      
                     
                       ( 
                       
                         1 
                         
                           
                             
                               x 
                               2 
                             
                             + 
                             
                               y 
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   z 
                                   - 
                                   d 
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   19.2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The Poisson solution that meets the boundary condition that the potential is zero at the surface of the infinite planar conductor is that due to the point charge and an image charge of −e at the position (0,0,−d) as shown in  FIG. 10 .
 
The potential for the corresponding electrostatic dipole in the positive half space is
 
         [0000]    
       
         
           
             
               
                 
                   
                     Φ 
                      
                     
                       ( 
                       
                         x 
                         , 
                         y 
                         , 
                         z 
                       
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                                
                               
                                 4 
                                  
                                 
                                   πɛ 
                                   0 
                                 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         1 
                                         
                                           
                                             
                                               x 
                                               2 
                                             
                                             + 
                                             
                                               y 
                                               2 
                                             
                                             + 
                                             
                                               
                                                 ( 
                                                 
                                                   z 
                                                   - 
                                                   d 
                                                 
                                                 ) 
                                               
                                               2 
                                             
                                           
                                         
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       1 
                                       
                                         
                                           
                                             x 
                                             2 
                                           
                                           + 
                                           
                                             y 
                                             2 
                                           
                                           + 
                                           
                                             
                                               ( 
                                               
                                                 z 
                                                 + 
                                                 d 
                                               
                                               ) 
                                             
                                             2 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               z 
                             
                             ≥ 
                             0 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               z 
                             
                             ≤ 
                             0 
                           
                         
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   19.3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The electric field shown in  FIG. 11  is nonzero only in the positive half space and is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         E 
                         = 
                         
                           - 
                           
                             ∇ 
                             Φ 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                              
                             
                               4 
                                
                               
                                 πɛ 
                                 0 
                               
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     xi 
                                     x 
                                   
                                   + 
                                   
                                     yi 
                                     y 
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         z 
                                         - 
                                         d 
                                       
                                       ) 
                                     
                                      
                                     
                                       i 
                                       z 
                                     
                                   
                                 
                                 
                                   
                                     ( 
                                     
                                       
                                         x 
                                         2 
                                       
                                       + 
                                       
                                         y 
                                         2 
                                       
                                       + 
                                       
                                         
                                           ( 
                                           
                                             z 
                                             - 
                                             d 
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                     ) 
                                   
                                   
                                     3 
                                     / 
                                     2 
                                   
                                 
                               
                               - 
                               
                                 
                                   
                                     xi 
                                     x 
                                   
                                   + 
                                   
                                     yi 
                                     y 
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         z 
                                         + 
                                         d 
                                       
                                       ) 
                                     
                                      
                                     
                                       i 
                                       z 
                                     
                                   
                                 
                                 
                                   
                                     ( 
                                     
                                       
                                         x 
                                         2 
                                       
                                       + 
                                       
                                         y 
                                         2 
                                       
                                       + 
                                       
                                         
                                           ( 
                                           
                                             z 
                                             + 
                                             d 
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                     ) 
                                   
                                   
                                     3 
                                     / 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    At the surface (z=0), the electric field is normal to the conductor as required by Gauss&#39; and Faraday&#39;s laws: 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       ( 
                       
                         x 
                         , 
                         y 
                         , 
                         0 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         - 
                          
                       
                        
                       
                           
                       
                        
                       
                         di 
                         z 
                       
                     
                     
                       2 
                        
                       
                         
                           
                             πɛ 
                             0 
                           
                            
                           
                             ( 
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                               + 
                               
                                 d 
                                 2 
                               
                             
                             ) 
                           
                         
                         
                           3 
                           / 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The surface charge density shown in  FIG. 12  is given by Eq. (19.1) with n=i z  and E 2 =0: 
         [0000]    
       
         
           
             
               
                 
                   σ 
                   = 
                   
                     
                       
                         
                           - 
                            
                         
                          
                         
                             
                         
                          
                         d 
                       
                       
                         2 
                          
                         
                           
                             π 
                              
                             
                               ( 
                               
                                 
                                   x 
                                   2 
                                 
                                 + 
                                 
                                   y 
                                   2 
                                 
                                 + 
                                 
                                   d 
                                   2 
                                 
                               
                               ) 
                             
                           
                           
                             3 
                             / 
                             2 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           - 
                            
                         
                          
                         
                             
                         
                          
                         d 
                       
                       
                         2 
                          
                         
                           
                             π 
                              
                             
                               ( 
                               
                                 
                                   ρ 
                                   2 
                                 
                                 + 
                                 
                                   d 
                                   2 
                                 
                               
                               ) 
                             
                           
                           
                             3 
                             / 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The total induced charge is given by the integral of the density over the surface: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           q 
                           induced 
                         
                         = 
                         
                           ∫ 
                           
                             σ 
                              
                             
                                
                               s 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ∫ 
                             
                               - 
                               ∞ 
                             
                             ∞ 
                           
                            
                           
                             
                               ∫ 
                               
                                 - 
                                 ∞ 
                               
                               ∞ 
                             
                              
                             
                               
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                       
                                   
                                    
                                   d 
                                 
                                 
                                   2 
                                    
                                   
                                     
                                       π 
                                        
                                       
                                         ( 
                                         
                                           
                                             x 
                                             2 
                                           
                                           + 
                                           
                                             y 
                                             2 
                                           
                                           + 
                                           
                                             d 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                     
                                       3 
                                       / 
                                       2 
                                     
                                   
                                 
                               
                                
                               
                                   
                               
                                
                               
                                  
                                 y 
                               
                                
                               
                                   
                               
                                
                               
                                  
                                 x 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               d 
                             
                             
                               2 
                                
                               π 
                             
                           
                            
                           
                             
                               ∫ 
                               
                                 - 
                                 ∞ 
                               
                               ∞ 
                             
                              
                             
                               
                                 ∫ 
                                 
                                   - 
                                   
                                     π 
                                     2 
                                   
                                 
                                 
                                   π 
                                   2 
                                 
                               
                                
                               
                                 
                                   
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     θ 
                                   
                                   
                                     
                                       x 
                                       2 
                                     
                                     + 
                                     
                                       d 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                    
                                   θ 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                    
                                   x 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               d 
                             
                             π 
                           
                            
                           
                             
                               ∫ 
                               
                                 - 
                                 ∞ 
                               
                               ∞ 
                             
                              
                             
                               
                                 1 
                                 
                                   
                                     x 
                                     2 
                                   
                                   + 
                                   
                                     d 
                                     2 
                                   
                                 
                               
                                
                               
                                   
                               
                                
                               
                                  
                                 x 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               d 
                             
                             π 
                           
                            
                           
                             
                               ∫ 
                               
                                 - 
                                 
                                   π 
                                   2 
                                 
                               
                               
                                 π 
                                 2 
                               
                             
                              
                             
                               
                                 1 
                                 d 
                               
                                
                               
                                   
                               
                                
                               
                                  
                                 
                                   θ 
                                   ′ 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           - 
                            
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    wherein the change of variables 
         [0000]    
       
         
           
             y 
             = 
             
               
                 ( 
                 
                   
                     x 
                     2 
                   
                   + 
                   
                     d 
                     2 
                   
                 
                 ) 
               
               
                 1 
                 2 
               
             
           
         
       
     
         [0000]    tan θ and x=d tan θ′ were used. The total surface charge induced on the surface of the conductor is exactly equal to the negative of the point charge located above the conductor. 
         [0268]    Now consider the case where the infinite planar conductor is charged with a surface charge density σ corresponding to a total charge of a single electron, −e, and the point charge of +e is due to a metal ion M + . Then, according to Maxwell&#39;s equations, the potential function of M+ is given by Eq. (19.3), the electric field between M+ and σ is given by Eqs. (19.4-19.5), and σ is given by Eq. (19.6). The field lines of M+end on σ, and the electric field is zero in the metal and in the negative half space. The potential energy between M+ and σ at the surface (z=0) given by the product of Eq. (19.2) and Eq. (19.6) is 
         [0000]    
       
         
           
             
               
                 
                   V 
                   = 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       ∞ 
                     
                      
                     
                       
                         ∫ 
                         
                           - 
                           ∞ 
                         
                         ∞ 
                       
                        
                       
                         
                            
                           
                             4 
                              
                             
                               πɛ 
                               o 
                             
                           
                         
                          
                         
                           ( 
                           
                             1 
                             
                               
                                 
                                   x 
                                   2 
                                 
                                 + 
                                 
                                   y 
                                   2 
                                 
                                 + 
                                 
                                   d 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               d 
                             
                             
                               2 
                                
                               
                                 
                                   π 
                                    
                                   
                                     ( 
                                     
                                       
                                         x 
                                         2 
                                       
                                       + 
                                       
                                         y 
                                         2 
                                       
                                       + 
                                       
                                         d 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   3 
                                   / 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                             
                         
                          
                         
                            
                           x 
                         
                          
                         
                             
                         
                          
                         
                            
                           y 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.8 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     V 
                     = 
                     
                       
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                            
                           d 
                         
                         
                           8 
                            
                           
                             π 
                             2 
                           
                            
                           
                             ɛ 
                             0 
                           
                         
                       
                        
                       
                         
                           ∫ 
                           
                             - 
                             ∞ 
                           
                           ∞ 
                         
                          
                         
                           
                             ∫ 
                             
                               - 
                               ∞ 
                             
                             ∞ 
                           
                            
                           
                             
                               1 
                               
                                 
                                   ( 
                                   
                                     
                                       x 
                                       2 
                                     
                                     + 
                                     
                                       y 
                                       2 
                                     
                                     + 
                                     
                                       d 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                              
                             
                                
                               x 
                             
                              
                             
                                
                               y 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using a change of coordinates to cylindrical and integral # 47 of Lide [4] gives: 
         [0000]    
       
         
           
             
               
                 
                   V 
                   = 
                   
                     
                       ∫ 
                       0 
                       ∞ 
                     
                      
                     
                       
                         ∫ 
                         0 
                         
                           2 
                            
                           π 
                         
                       
                        
                       
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                              
                             d 
                           
                           
                             8 
                              
                             
                               π 
                               2 
                             
                              
                             
                               
                                 
                                   ɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       ρ 
                                       2 
                                     
                                     + 
                                     
                                       d 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                               2 
                             
                           
                         
                          
                         ρ 
                          
                         
                             
                         
                          
                         
                            
                           φ 
                         
                          
                         
                             
                         
                          
                         
                            
                           ρ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.10 
                   ) 
                 
               
             
             
               
                 
                   V 
                   = 
                   
                     
                       
                         
                           - 
                           
                              
                             2 
                           
                         
                          
                         d 
                       
                       
                         4 
                          
                         
                           πɛ 
                           0 
                         
                       
                     
                      
                     
                       
                         ∫ 
                         0 
                         ∞ 
                       
                        
                       
                         
                           ρ 
                           
                             
                               ( 
                               
                                 
                                   ρ 
                                   2 
                                 
                                 + 
                                 
                                   d 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           ρ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.11 
                   ) 
                 
               
             
             
               
                 
                   V 
                   = 
                   
                     
                       
                         
                           - 
                           
                              
                             2 
                           
                         
                          
                         d 
                       
                       
                         4 
                          
                         
                           πɛ 
                           0 
                         
                       
                     
                      
                     
                       
                         ( 
                         
                           
                             - 
                             1 
                           
                           
                             2 
                              
                             
                               ( 
                               
                                 
                                   ρ 
                                   2 
                                 
                                 + 
                                 
                                   d 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                       0 
                       ∞ 
                     
                   
                 
               
               
                 
                   ( 
                   19.12 
                   ) 
                 
               
             
             
               
                 
                   V 
                   = 
                   
                     
                       - 
                       
                          
                         2 
                       
                     
                     
                       4 
                        
                       π 
                        
                       
                           
                       
                        
                       
                         
                           ɛ 
                           0 
                         
                          
                         
                           ( 
                           
                             2 
                              
                             
                                 
                             
                              
                             d 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The corresponding force from the negative gradient as well as the integral of the product of the electric field (Eq. (19.5)) and the charge density (Eq. (19.6)) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         F 
                         = 
                         
                           - 
                           
                             ∇ 
                             V 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ∫ 
                             A 
                           
                            
                           
                             
                               E 
                                
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                   , 
                                   0 
                                 
                                 ) 
                               
                             
                              
                             σ 
                              
                             
                                
                               A 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     e 
                                     2 
                                   
                                    
                                   
                                     d 
                                     2 
                                   
                                 
                                 
                                   
                                     
                                       ( 
                                       
                                         2 
                                          
                                         π 
                                       
                                       ) 
                                     
                                     2 
                                   
                                    
                                   
                                     ɛ 
                                     0 
                                   
                                 
                               
                                
                               
                                 i 
                                 z 
                               
                             
                             ) 
                           
                            
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                              
                             
                               
                                 ∫ 
                                 0 
                                 
                                   2 
                                    
                                   π 
                                 
                               
                                
                               
                                 
                                   ρ 
                                    
                                   
                                      
                                     φ 
                                   
                                    
                                   
                                      
                                     ρ 
                                   
                                 
                                 
                                   
                                     ( 
                                     
                                       
                                         ρ 
                                         2 
                                       
                                       + 
                                       
                                         d 
                                         2 
                                       
                                     
                                     ) 
                                   
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           2 
                            
                           
                             π 
                             ( 
                             
                               
                                 
                                   
                                     e 
                                     2 
                                   
                                    
                                   
                                     d 
                                     2 
                                   
                                 
                                 
                                   
                                     
                                       ( 
                                       
                                         2 
                                          
                                         π 
                                       
                                       ) 
                                     
                                     2 
                                   
                                    
                                   
                                     ɛ 
                                     0 
                                   
                                 
                               
                                
                               
                                 i 
                                 z 
                               
                             
                             ) 
                           
                            
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                              
                             
                               
                                 ρ 
                                  
                                 
                                    
                                   ρ 
                                 
                               
                               
                                 
                                   ( 
                                   
                                     
                                       ρ 
                                       2 
                                     
                                     + 
                                     
                                       d 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 3 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           2 
                            
                           
                             π 
                             ( 
                             
                               
                                 
                                   
                                     e 
                                     2 
                                   
                                    
                                   
                                     d 
                                     2 
                                   
                                 
                                 
                                   
                                     
                                       ( 
                                       
                                         2 
                                          
                                         π 
                                       
                                       ) 
                                     
                                     2 
                                   
                                    
                                   
                                     ɛ 
                                     0 
                                   
                                 
                               
                                
                               
                                 i 
                                 z 
                               
                             
                             ) 
                           
                            
                           
                             1 
                             
                               4 
                                
                               
                                 d 
                                 4 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               e 
                               2 
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 d 
                                 2 
                               
                             
                           
                            
                           
                             i 
                             z 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where d is treated as a variable to be solved as discussed below. The potential is equivalent to that of the charge and its image charge located a distance 2d apart. In addition, the potential and force are equivalent to those of the charge +e and an image charge 
         [0000]    
       
         
           
             
               - 
               e 
             
             2 
           
         
       
     
         [0000]    located a distance d apart. 
         [0269]    In addition to the infinite planar conductor at z=0 and the point charge +e at a position (0,0,d) near the infinite planar conductor as shown in  FIG. 9 , next consider the introduction of a second infinite planar conductor located at position z=2d as shown in  FIG. 13 . 
         [0270]    As shown, by Kong [5], an image charge at (0,0,−d) meets the boundary condition of zero potential at the bottom plate, but it gives rise to a potential at the top. Similarly, an image charge at (0,0,3d), meets the boundary condition of zero potential at the top plate, but it gives rise to a potential at the bottom. Satisfaction of the boundary condition of zero potential at both plates due to the presence of the initial real charge requires an infinite series of alternating positive and negative image charges spaced a distance d apart with the potential given by the summation over the real point source and its point-source image charges of +e and −e. Since fields superimpose, by adding real charges in a periodic lattice, the image charges cancel except for one per each real charge at a distance 2d apart as in the original case considered in  FIG. 9 . 
         [0271]    In the real world, the idealized infinite planar conductor is a planar metal sheet experimentally comprised of an essentially infinite lattice of metal ions M +  and free electrons that provide surface densities σ in response to an applied external field such as that due to an external charge of +e due to a metal ion M + . Then, it is required that the solutions of the external point charge at an infinite planar conductor are also those of the metal ions and free electrons of metals based on the uniqueness of solutions of Maxwell&#39;s equations and the constraint that the individual electrons in a metal conserve the classical physical laws of the macro-scale conductor. In metals, a superposition of planar free electrons given in the Electron in Free Space section replaces the infinite planar conductor. Then, the nature of the metal bond is a lattice of metal ions with field lines that end on the corresponding lattice of electrons wherein each has the two-dimensional charge density σ given by Eq. (19.6) to match the boundary conditions of equipotential, minimum energy, and conservation of charge and angular momentum for an ionized electron. Consider an infinite lattice of positive charges in the hollow Cartesian cavities whose walls are the intersecting planes of conductors and that each planar conductor comprises an electron. By Gauss&#39; law, the field lines of each real charge end on each of the n planar-electron walls of the cavity wherein the surface charge density of contribution of each electron is that of image charge of 
         [0000]    
       
         
           
             
               - 
               e 
             
             n 
           
         
       
     
         [0000]    equidistance across each wall from a given charge +e. Then, each electron contributes the charge 
         [0000]    
       
         
           
             
               - 
               e 
             
             n 
           
         
       
     
         [0000]    to the corresponding ion where each is equivalent electrostatically to an image point charge at twice the distance from the point charge of +e due to M + . 
         [0272]    Thus, the metallic bond is equivalent to the ionic bond given in the Alkali-Hydride Crystal Structures section with a Madelung constant of one with each negative ion at a position of one half the distance between the corresponding positive ions, but electrostatically equivalent to being positioned at twice this distance, the M + -M + -separation distance. The surface charge density of a planar electron having an electric field equivalent to that of image point charge for the corresponding positive ion of the lattice is shown in  FIG. 14 . 
       Alkali-Metal Crystal Structures 
       [0273]    The alkali metals are lithium (Li), sodium (Na), potassium (K), rubidium (Rb), and cesium (Cs). These alkali metals each comprise an equal number of alkali cations and electrons in unit cells of a crystalline lattice. The crystal structure of these metals is the body-centered cubic CsCl structure [6-8]. This close-packed structure is expected since it gives the optimal approach of the positive ions and negative electrons. For a body-centered cell, there is an identical atom at 
         [0000]    
       
         
           
             
               x 
               + 
               
                 a 
                 2 
               
             
             , 
             
               y 
               + 
               
                 a 
                 2 
               
             
             , 
             
               z 
               + 
               
                 a 
                 2 
               
             
           
         
       
     
         [0000]    for each atom at x, y, z. The structure of the ions with lattice parameters a=b=c and electrons at the diagonal positions centered at 
         [0000]    
       
         
           
             ( 
             
               
                 x 
                 + 
                 
                   a 
                   4 
                 
               
               , 
               
                 y 
                 + 
                 
                   a 
                   4 
                 
               
               , 
               
                 z 
                 + 
                 
                   a 
                   4 
                 
               
             
             ) 
           
         
       
     
         [0000]    are shown in  FIG. 15 . In this case n=8 electron planes per body-centered ion are perpendicular to the four diagonal axes running from each corner of the cube through the center to the opposite corner. The planes intersect these diagonals at one half the distance from each corner to the center of the body-centered atom. The mutual intersection of the planes forms a hexagonal cavity about each ion of the lattice. The length l 1  to a perpendicular electron plane along the axis from a corner atom to a body-centered atom that is the midpoint of this axis is 
         [0000]    
       
         
           
             
               
                 
                   
                     l 
                     1 
                   
                   = 
                   
                     
                       
                         
                           
                             ( 
                             
                               a 
                               4 
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               a 
                               4 
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               a 
                               4 
                             
                             ) 
                           
                           2 
                         
                       
                     
                     = 
                     
                       
                         a 
                          
                         
                           3 
                         
                       
                       4 
                     
                   
                 
               
               
                 
                   ( 
                   19.15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The angle θ d  of each diagonal axis from the xy-plane of the unit cell is 
         [0000]    
       
         
           
             
               
                 
                   
                     θ 
                     d 
                   
                   = 
                   
                     
                       
                         tan 
                         
                           - 
                           1 
                         
                       
                       ( 
                       
                         
                           1 
                           4 
                         
                         
                           
                             2 
                           
                           4 
                         
                       
                       ) 
                     
                     = 
                     
                       35.26 
                        
                       ° 
                     
                   
                 
               
               
                 
                   ( 
                   19.16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The angle θ p  from the horizontal to the electron plane that is perpendicular to the diagonal axis is 
         [0000]      θ p =180°−90°−35.26°=54.73°  (19.17)
 
         [0000]    The length l 3  along a diagonal axis in the xy-plane from a corner atom to another at which point an electron plane intersects the xy-plane is 
         [0000]    
       
         
           
             
               
                 
                   
                     l 
                     3 
                   
                   = 
                   
                     
                       
                         l 
                         1 
                       
                       
                         cos 
                          
                         
                             
                         
                          
                         
                           θ 
                           d 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             a 
                              
                             
                               3 
                             
                           
                           4 
                         
                         
                           cos 
                            
                           
                             ( 
                             
                               35.26 
                                
                               ° 
                             
                             ) 
                           
                         
                       
                       = 
                       
                         
                           
                             
                               a 
                                
                               
                                 3 
                               
                             
                             4 
                           
                           
                             
                               2 
                               3 
                             
                           
                         
                         = 
                         
                           a 
                            
                           
                             3 
                             
                               4 
                                
                               
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.18 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The length l 2  of the octagonal edge of the electron plane from a body-centered atom to the xy-plane defined by four corner atoms is 
         [0000]    
       
         
           
             
               
                 
                   
                     l 
                     2 
                   
                   = 
                   
                     
                       
                         l 
                         3 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         d 
                       
                     
                     = 
                     
                       
                         a 
                          
                         
                           3 
                           
                             4 
                              
                             
                               2 
                             
                           
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               35.26 
                                
                               ° 
                             
                             ) 
                           
                         
                       
                       = 
                       
                         
                           a 
                            
                           
                             3 
                             
                               4 
                                
                               
                                 2 
                               
                             
                           
                            
                           
                             1 
                             
                               3 
                             
                           
                         
                         = 
                         
                           
                             a 
                             4 
                           
                            
                           
                             
                               3 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.19 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The length l 4  along the edge of the unit cell in the xy-plane from a corner atom to another at which point an electron plane intersects the xy-plane at this axis is 
         [0000]    
       
         
           
             
               
                 
                   
                     l 
                     4 
                   
                   = 
                   
                     
                       
                         l 
                         3 
                       
                       
                         cos 
                          
                         
                           ( 
                           
                             45 
                              
                             ° 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         
                           a 
                            
                           
                             3 
                             
                               4 
                                
                               
                                 2 
                               
                             
                           
                         
                         
                           cos 
                            
                           
                             ( 
                             
                               45 
                                
                               ° 
                             
                             ) 
                           
                         
                       
                       = 
                       
                         
                           3 
                           4 
                         
                          
                         a 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.20 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The dimensions and angles given by Eqs. (19.15-19.20) are shown in  FIG. 15 . 
         [0274]    Each M +  is surrounded by six planar two-dimensional membranes that are comprised of electron density σ on which the electric field lines of the positive charges end. The resulting unit cell consists cations at the end of each edge and at the center of the cell with an electron membrane as the perpendicular bisector of the axis from an identical atom at 
         [0000]    
       
         
           
             
               x 
               + 
               
                 a 
                 2 
               
             
             , 
             
               y 
               + 
               
                 a 
                 2 
               
             
             , 
             
               z 
               + 
               
                 a 
                 2 
               
             
           
         
       
     
         [0000]    for each atom at x, y, z such that the unit cell contains two cations and two electrons. The ions and electrons of the unit cell are also shown in  FIG. 15 . The electron membranes exist throughout the metal, but they terminate on metal atomic orbitals or MOs of bonds between metal atoms and other reacted atoms such as the MOs of metal oxide bonds at the edges of the metal. 
         [0275]    The interionic radius of each cation and electron membrane can be derived by considering the electron energies at these radii and by calculating the corresponding forces of the electrons with the ions. Then, the lattice energy is given by the sum over the crystal of the energy of the interacting ion and electron pairs at the radius of force balance between the electrons and ions. 
         [0276]    For each point charge of +e due to a metal ion M + , the planar two-dimensional membrane comprised of electrons contributes a surface charge density a given by Eq. (19.6) corresponding to that of a point image charge having a total charge of a single electron, −e. The potential of each electron is double that of Eq. (19.13) since there are two mirror-image M +  ions per planar electron membrane: 
         [0000]    
       
         
           
             
               
                 
                   V 
                   = 
                   
                     
                       - 
                       
                         e 
                         2 
                       
                     
                     
                       4 
                        
                       
                         πɛ 
                         0 
                       
                        
                       d 
                     
                   
                 
               
               
                 
                   ( 
                   19.21 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where d is treated as a variable to be solved. The same result is obtained from considering the integral of the product of two times the electric field (Eq. (19.5)) and the charge density (Eq. (19.6)) according to Eq. (19.14). In order to conserve angular momentum and maintain current continuity, the kinetic energy has two components. Since the free electron of a metal behaves as a point mass, one component using Eq. (1.47) with r=d is 
         [0000]    
       
         
           
             
               
                 
                   T 
                   = 
                   
                     
                       
                         1 
                         2 
                       
                        
                       
                         m 
                         e 
                       
                        
                       
                         v 
                         2 
                       
                     
                     = 
                     
                       
                         1 
                         2 
                       
                        
                       
                         
                           ℏ 
                           2 
                         
                         
                           
                             m 
                             e 
                           
                            
                           
                             d 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.22 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The other component of kinetic energy is given by integrating the mass density σ m  (r) (Eq. (19.6) with e replaced by m e  and velocity v(r) (Eq. (1.47)) over their radial dependence (r=√{square root over (x 2 +y 2 +z 2 )}=√{square root over (ρ 2 +d 2 )}): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         T 
                         = 
                         
                           
                             1 
                             2 
                           
                            
                           
                             ∫ 
                             
                               σ 
                                
                               
                                   
                               
                                
                               
                                 v 
                                 2 
                               
                                
                               
                                  
                                 A 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             1 
                             2 
                           
                            
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                              
                             
                               
                                 ∫ 
                                 0 
                                 
                                   2 
                                    
                                   π 
                                 
                               
                                
                               
                                 
                                   
                                     
                                       m 
                                       e 
                                     
                                      
                                     d 
                                   
                                   
                                     2 
                                      
                                     
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             
                                               ρ 
                                               2 
                                             
                                             + 
                                             
                                               d 
                                               2 
                                             
                                           
                                           ) 
                                         
                                       
                                       
                                         3 
                                         / 
                                         2 
                                       
                                     
                                   
                                 
                                  
                                 
                                   
                                     ℏ 
                                     2 
                                   
                                   
                                     
                                       m 
                                       e 
                                       2 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           ρ 
                                           2 
                                         
                                         + 
                                         
                                           d 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                  
                                 ρ 
                                  
                                 
                                    
                                   φ 
                                 
                                  
                                 
                                    
                                   ρ 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 ℏ 
                                 2 
                               
                                
                               d 
                             
                             
                               4 
                                
                               π 
                                
                               
                                   
                               
                                
                               
                                 m 
                                 e 
                               
                             
                           
                            
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                              
                             
                               
                                 ∫ 
                                 0 
                                 
                                   2 
                                    
                                   π 
                                 
                               
                                
                               
                                 
                                   ρ 
                                   
                                     
                                       ( 
                                       
                                         
                                           ρ 
                                           2 
                                         
                                         + 
                                         
                                           d 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     
                                       5 
                                       / 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                    
                                   φ 
                                 
                                  
                                 
                                    
                                   ρ 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               2 
                                
                               
                                 πℏ 
                                 2 
                               
                                
                               d 
                             
                             
                               4 
                                
                               π 
                                
                               
                                   
                               
                                
                               
                                 m 
                                 e 
                               
                             
                           
                            
                           
                             
                               ( 
                               
                                 
                                   - 
                                   1 
                                 
                                 
                                   2 
                                    
                                   
                                     ( 
                                     
                                       3 
                                       2 
                                     
                                     ) 
                                   
                                    
                                   
                                     
                                       ( 
                                       
                                         
                                           ρ 
                                           2 
                                         
                                         + 
                                         
                                           d 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     
                                       3 
                                       / 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                             0 
                             ∞ 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ( 
                             
                               1 
                               3 
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                 1 
                                 2 
                               
                                
                               
                                 
                                   ℏ 
                                   2 
                                 
                                 
                                   
                                     m 
                                     e 
                                   
                                    
                                   
                                     d 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.23 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where integral #47 of Lide [4] was used. Thus, the total kinetic energy given by the sum of Eqs. (19.22) and (19.23) is 
         [0000]    
       
         
           
             
               
                 
                   T 
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             1 
                             3 
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             1 
                             2 
                           
                            
                           
                             
                               ℏ 
                               2 
                             
                             
                               
                                 m 
                                 e 
                               
                                
                               
                                 d 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         4 
                         3 
                       
                        
                       
                         ( 
                         
                           
                             1 
                             2 
                           
                            
                           
                             
                               ℏ 
                               2 
                             
                             
                               
                                 m 
                                 e 
                               
                                
                               
                                 d 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.24 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Each metal M (M=Li, Na, K, Rb, Cs) is comprised of M +  and e −  ions. The structure of the ions comprises lattice parameters a=b=c and electrons at the diagonal positions centered at 
         [0000]    
       
         
           
             
               ( 
               
                 
                   x 
                   + 
                   
                     a 
                     4 
                   
                 
                 , 
                 
                   y 
                   + 
                   
                     a 
                     4 
                   
                 
                 , 
                 
                   z 
                   + 
                   
                     a 
                     4 
                   
                 
               
               ) 
             
             . 
           
         
       
     
         [0000]    Thus, the separation distance d between each M +  and the corresponding electron membrane is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           d 
                           = 
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       x 
                                     
                                     2 
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       y 
                                     
                                     2 
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       z 
                                     
                                     2 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       1 
                                       3 
                                     
                                      
                                     a 
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       1 
                                       4 
                                     
                                      
                                     a 
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       1 
                                       4 
                                     
                                      
                                     a 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               
                                 3 
                               
                               4 
                             
                              
                             a 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       where 
                        
                       
                           
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       x 
                     
                     = 
                     
                       
                         Δ 
                          
                         
                             
                         
                          
                         y 
                       
                       = 
                       
                         
                           Δ 
                            
                           
                               
                           
                            
                           z 
                         
                         = 
                         
                           
                             a 
                             2 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.25 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, the lattice parameter a is given by 
         [0000]    
       
         
           
             
               
                 
                   a 
                   = 
                   
                     
                       4 
                        
                       d 
                     
                     
                       3 
                     
                   
                 
               
               
                 
                   ( 
                   19.26 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The molar metal bond energy E D  is given by Avogadro&#39;s number N times the negative sum of the potential energy, kinetic energy, and ionization or binding energy (BE(M)) of M: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                           D 
                         
                         = 
                         
                           - 
                           
                             N 
                              
                             
                               ( 
                               
                                 V 
                                 + 
                                 T 
                                 + 
                                 
                                   BE 
                                    
                                   
                                     ( 
                                     M 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           N 
                           ( 
                           
                             
                               
                                  
                                 2 
                               
                               
                                 4 
                                  
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 d 
                               
                             
                             - 
                             
                               
                                 4 
                                 3 
                               
                                
                               
                                 ( 
                                 
                                   
                                     1 
                                     2 
                                   
                                    
                                   
                                     
                                       ℏ 
                                       2 
                                     
                                     
                                       
                                         m 
                                         e 
                                       
                                        
                                       
                                         d 
                                         2 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                             - 
                             
                               BE 
                                
                               
                                 ( 
                                 M 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.27 
                   ) 
                 
               
             
           
         
       
     
         [0277]    The separation distance d between each M +  and the corresponding electron membrane is given by the force balance between the outward centrifugal force and the sum of the electric, paramagnetic and diamagnetic forces as given in the Three-Through Twenty-Electron Atoms section. The electric force F ele  corresponding to Eq. (19.21) given by its negative gradient is 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     ele 
                   
                   = 
                   
                     
                       
                         e 
                         2 
                       
                       
                         4 
                          
                         
                           πɛ 
                           0 
                         
                          
                         
                           d 
                           2 
                         
                       
                     
                      
                     
                       i 
                       z 
                     
                   
                 
               
               
                 
                   ( 
                   19.28 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where inward is taken as the positive direction. The centrifugal force F centrifugal  is given by negative gradient of Eq. (19.24) times two since the charge and mass density are doubled due to the presence of mirror image M +  ion pairs across the electron membrane at the origin for any given ion. 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     centrifugal 
                   
                   = 
                   
                     
                       - 
                       
                         8 
                         3 
                       
                     
                      
                     
                       
                         ℏ 
                         2 
                       
                       
                         
                           m 
                           e 
                         
                          
                         
                           d 
                           3 
                         
                       
                     
                      
                     
                       i 
                       z 
                     
                   
                 
               
               
                 
                   ( 
                   19.29 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where d is treated as a variable to be solved. In addition, there is an outward spin-pairing force F mag  between the electron density elements of two opposing ions that is given by Eqs. (7.24) and (10.52): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       F 
                       mag 
                     
                     = 
                     
                       
                         - 
                         
                           1 
                           Z 
                         
                       
                        
                       
                         
                           ℏ 
                           2 
                         
                         
                           
                             m 
                             e 
                           
                            
                           
                             d 
                             3 
                           
                         
                       
                        
                       
                         
                           s 
                            
                           
                             ( 
                             
                               s 
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                        
                       
                         i 
                         z 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       where 
                        
                       
                           
                       
                        
                       s 
                     
                     = 
                     
                       
                         1 
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   19.30 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The remaining magnetic forces are determined by the electron configuration of the particular atom as given for the examples of lithium, sodium, and potassium metals in the corresponding sections. 
       Lithium Metal 
       [0278]    For Li + , there are two spin-paired electrons in an orbitsphere with 
         [0000]    
       
         
           
             
               
                 
                   
                     r 
                     1 
                   
                   = 
                   
                     
                       r 
                       2 
                     
                     = 
                     
                       
                         a 
                         0 
                       
                       [ 
                       
                         
                           1 
                           2 
                         
                         - 
                         
                           
                             
                               3 
                               4 
                             
                           
                           6 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   19.31 
                   ) 
                 
               
             
           
         
       
     
         [0000]    as given by Eq. (7.35) where r n  is the radius of electron n which has velocity v n . For the next electron that contributes to the metal-electron membrane, the outward centrifugal force on electron 3 is balanced by the electric force and the magnetic forces (on electron 3). The radius of the metal-band electron is calculated by equating the outward centrifugal force (Eq. (19.29)) to the sum of the electric (Eq. (19.28)) and diamagnetic (Eq. (19.30)) forces as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       8 
                       3 
                     
                      
                     
                       
                         ℏ 
                         2 
                       
                       
                         
                           m 
                           e 
                         
                          
                         
                           d 
                           3 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         e 
                         2 
                       
                       
                         4 
                          
                         
                           πɛ 
                           0 
                         
                          
                         
                           d 
                           2 
                         
                       
                     
                     - 
                     
                       
                         
                           ℏ 
                           2 
                         
                         
                           
                             Zm 
                             e 
                           
                            
                           
                             d 
                             3 
                           
                         
                       
                        
                       
                         
                           3 
                           4 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.32 
                   ) 
                 
               
             
             
               
                 
                   d 
                   = 
                   
                     
                       
                         ( 
                         
                           
                             8 
                             3 
                           
                           + 
                           
                             
                               
                                 3 
                                 4 
                               
                             
                             3 
                           
                         
                         ) 
                       
                        
                       
                         a 
                         0 
                       
                     
                     = 
                     
                       
                         2.95534 
                          
                         
                           a 
                           0 
                         
                       
                       = 
                       
                         1.56390 
                          
                         X 
                          
                         
                             
                         
                          
                         
                           10 
                           
                             - 
                             10 
                           
                         
                          
                         m 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.33 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Z=3. Using Eq. (19.26), the lattice parameter a is 
         [0000]        a= 6.82507 a   0 =3.61167×10 −10  m  (19.34)
 
         [0000]    The experimental lattice parameter a [7] is 
         [0000]        a= 6.63162 a   0 =3.5093×10 −10  m  (19.35)
 
         [0000]    The calculated Li—Li distance is in reasonable agreement with the experimental distance given the experimental difficulty of performing X-ray diffraction on lithium due to the low electron densities. 
         [0279]    Using Eq. (19.27) and the experimental binding energy of lithium, BE(Li)=5.39172 eV=8.63849×10 −19  J [9], the molar metal bond energy E D  is 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                     D 
                   
                   = 
                   
                     
                       N 
                        
                       
                         ( 
                         
                           
                             
                               
                                 
                                   
                                     e 
                                     2 
                                   
                                   
                                     4 
                                      
                                     
                                       πɛ 
                                       0 
                                     
                                      
                                     1.56390 
                                     × 
                                     
                                       10 
                                       
                                         - 
                                         10 
                                       
                                     
                                      
                                     m 
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     4 
                                     3 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         1 
                                         2 
                                       
                                        
                                       
                                         
                                           ℏ 
                                           2 
                                         
                                         
                                           
                                             
                                               m 
                                               e 
                                             
                                              
                                             
                                               ( 
                                               
                                                 1.56390 
                                                 × 
                                                 
                                                   10 
                                                   
                                                     - 
                                                     10 
                                                   
                                                 
                                                  
                                                 m 
                                               
                                               ) 
                                             
                                           
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 8.63849 
                                 × 
                                 
                                   10 
                                   
                                     - 
                                     19 
                                   
                                 
                                  
                                 J 
                               
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       167.76 
                        
                       
                           
                       
                        
                       kJ 
                        
                       
                         / 
                       
                        
                       mole 
                     
                   
                 
               
               
                 
                   ( 
                   19.36 
                   ) 
                 
               
             
           
         
       
     
         [0000]    This agrees well with the experimental lattice [10] energy of 
         [0000]        E   D =159.3 kJ/mole  (19.37)
 
         [0000]    and confirms that Li metal comprises a precise packing of discrete ions, Li +  and e − . Using the Li—Li and Li + −e −  distances and the calculated (Eq. (7.35)) Li +  ionic radius of 0.35566a 0 =0.18821 Å, the crystalline lattice structure of the unit cell of Li metal is shown in  FIG. 16 , a portion of the crystalline lattice of Li metal is shown in  FIG. 17 , and the Li unit cell is shown relative to the other alkali metals in  FIG. 18 . 
       Sodium Metal 
       [0280]    For Na + , there are two indistinguishable spin-paired electrons in an orbitsphere with radii r i  and r 2  both given by Eq. (7.35) (Eq. (10.51)), two indistinguishable spin-paired electrons in an orbitsphere with radii r 3  and r 4  both given by Eq. (10.62), and three sets of paired electrons in an orbitsphere at r 10  given by Eq. (10.212). For Z=11, the next electron which binds to contribute to the metal electron membrane to form the metal bond is attracted by the central Coulomb field and is repelled by diamagnetic forces due to the 3 sets of spin-paired inner electrons. 
         [0281]    In addition to the spin-spin interaction between electron pairs, the three sets of 2p electrons are orbitally paired. The metal electron of the sodium atom produces a magnetic field at the position of the three sets of spin-paired 2p electrons. In order for the electrons to remain spin and orbitally paired, a corresponding diamagnetic force, F diamagnetic 3 , on electron eleven from the three sets of spin-paired electrons follows from Eqs. (10.83-10.84) and (10.220): 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     
                       diamagnetic 
                        
                       
                           
                       
                        
                       3 
                     
                   
                   = 
                   
                     
                       - 
                       
                         1 
                         Z 
                       
                     
                      
                     
                       
                         10 
                          
                         
                           ℏ 
                           2 
                         
                       
                       
                         
                           m 
                           e 
                         
                          
                         
                           d 
                           3 
                         
                       
                     
                      
                     
                       
                         s 
                          
                         
                           ( 
                           
                             s 
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                      
                     
                       i 
                       z 
                     
                   
                 
               
               
                 
                   ( 
                   19.38 
                   ) 
                 
               
             
           
         
       
     
         [0000]    corresponding to the p x  and p y  electrons with no spin-orbit coupling of the orthogonal p z  electrons (Eq. (10.84)). The outward centrifugal force on electron 11 is balanced by the electric force and the magnetic forces (on electron 11). The radius of the outer electron is calculated by equating the outward centrifugal force (Eq. (19.29)) to the sum of the electric (Eq. (19.28)) and diamagnetic (Eqs. (19.30) and (19.38)) forces as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       8 
                       3 
                     
                      
                     
                       
                         ℏ 
                         2 
                       
                       
                         
                           m 
                           e 
                         
                          
                         
                           d 
                           3 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         e 
                         2 
                       
                       
                         4 
                          
                         
                           πɛ 
                           0 
                         
                          
                         
                           d 
                           2 
                         
                       
                     
                     - 
                     
                       
                         
                           ℏ 
                           2 
                         
                         
                           
                             Zm 
                             e 
                           
                            
                           
                             d 
                             3 
                           
                         
                       
                        
                       
                         
                           3 
                           4 
                         
                       
                     
                     - 
                     
                       
                         1 
                         Z 
                       
                        
                       
                         
                           10 
                            
                           
                             ℏ 
                             2 
                           
                         
                         
                           
                             m 
                             e 
                           
                            
                           
                             d 
                             3 
                           
                         
                       
                        
                       
                         
                           3 
                           4 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.39 
                   ) 
                 
               
             
             
               
                 
                   
                     d 
                     = 
                     
                       
                         
                           ( 
                           
                             
                               8 
                               3 
                             
                             + 
                             
                               
                                 11 
                                  
                                 
                                   
                                     3 
                                     4 
                                   
                                 
                               
                               11 
                             
                           
                           ) 
                         
                          
                         
                           a 
                           0 
                         
                       
                       = 
                       
                         
                           3.53269 
                            
                           
                             a 
                             0 
                           
                         
                         = 
                         
                           1.86942 
                           × 
                           
                             10 
                             
                               - 
                               10 
                             
                           
                            
                           m 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       where 
                        
                       
                           
                       
                        
                       Z 
                     
                     = 
                     
                       
                         11 
                          
                         
                             
                         
                          
                         and 
                          
                         
                             
                         
                          
                         s 
                       
                       = 
                       
                         
                           1 
                           2 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.40 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eq. (19.26), the lattice parameter a is 
         [0000]        a= 8.15840 a   0 =4.31724×10 −10  m  (19.41)
 
         [0000]    The experimental lattice parameter a [7] is 
         [0000]        a= 8.10806 a   0 =4.2906×10 −10  m  (19.42)
 
         [0282]    The calculated Na—Na distance is in good agreement with the experimental distance. 
         [0283]    Using Eq. (19.27) and the experimental binding energy of sodium, BE(Na)=5.13908 eV=8.23371×10 −19  J [9], the molar metal bond energy E D  is 
         [0000]    
       
         
           
             
               E 
               D 
             
             = 
             
               
                 N 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             
                               e 
                               2 
                             
                             
                               4 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               1.86942 
                               × 
                               
                                 10 
                                 
                                   - 
                                   10 
                                 
                               
                                
                               m 
                             
                           
                           - 
                         
                       
                     
                     
                       
                         
                           
                             
                               4 
                               3 
                             
                              
                             
                               ( 
                               
                                 
                                   1 
                                   2 
                                 
                                  
                                 
                                   
                                     ℏ 
                                     2 
                                   
                                   
                                     
                                       
                                         m 
                                         e 
                                       
                                        
                                       
                                         ( 
                                         
                                           1.86942 
                                           × 
                                           
                                             10 
                                             
                                               - 
                                               10 
                                             
                                           
                                            
                                           m 
                                         
                                         ) 
                                       
                                     
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             8.23371 
                             × 
                             
                               10 
                               
                                 - 
                                 19 
                               
                             
                              
                             J 
                           
                         
                       
                     
                   
                   ) 
                 
               
               = 
               
                 107.10 
                  
                 
                     
                 
                  
                 kJ 
                  
                 
                   / 
                 
                  
                 mole 
               
             
           
         
       
     
         [0000]    This agrees well with the experimental lattice [10] energy of 
         [0000]        E   D =107.5 kJ/mole  (19.44)
 
         [0000]    and confirms that Na metal comprises a precise packing of discrete ions, Na +  and e − . Using the Na—Na and Na +  −e −  distances and the calculated (Eq. (10.212)) Na +  ionic radius of 0.56094a 0 =0.29684 Å, the crystalline lattice structure of Na metal is shown in  FIG. 18B . 
       Potassium Metal 
       [0284]    For K + , there are two indistinguishable spin-paired electrons in an orbitsphere with radii r 1  and r 2  both given by Eq. (7.35) (Eq. (10.51)), two indistinguishable spin-paired electrons in an orbitsphere with radii r 3  and r 4  both given by Eq. (10.62), three sets of paired electrons in an orbitsphere at r 10 , given by Eq. (10.212), two indistinguishable spin-paired electrons in an orbitsphere with radii r 11  and r 12  both given by Eq. (10.255), and three sets of paired electrons in an orbitsphere with radius r 18  given by Eq. (10.399). With Z=19, the next electron which binds to contribute to the metal electron membrane to form the metal bond is attracted by the central Coulomb field and is repelled by diamagnetic forces due to the 3 sets of spin-paired inner 3p electrons. 
         [0285]    The spherically symmetrical closed 3p shell of nineteen-electron atoms produces a diamagnetic force, F diamagnetic , that is equivalent to that of a closed s shell given by Eq. (10.11) with the appropriate radii. The inner electrons remain at their initial radii, but cause a diamagnetic force according to Lenz&#39;s law that is 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     diamagnetic 
                   
                   = 
                   
                     
                       - 
                       
                         
                           ℏ 
                           2 
                         
                         
                           4 
                            
                           
                             m 
                             e 
                           
                            
                           
                             d 
                             2 
                           
                            
                           
                             r 
                             18 
                           
                         
                       
                     
                      
                     
                       
                         s 
                          
                         
                           ( 
                           
                             s 
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                      
                     
                       i 
                       z 
                     
                   
                 
               
               
                 
                   ( 
                   19.45 
                   ) 
                 
               
             
           
         
       
     
         [0286]    The diamagnetic force, F diamagnetic 3 , on electron nineteen from the three sets of spin-paired electrons given by Eq. (10.409) is 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     
                       diamagnetic 
                        
                       
                           
                       
                        
                       3 
                     
                   
                   = 
                   
                     
                       - 
                       
                         1 
                         Z 
                       
                     
                      
                     
                       
                         12 
                          
                         ℏ 
                       
                       
                         
                           m 
                           e 
                         
                          
                         
                           d 
                           3 
                         
                       
                     
                      
                     
                       
                         s 
                          
                         
                           ( 
                           
                             s 
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                      
                     
                       i 
                       z 
                     
                   
                 
               
               
                 
                   ( 
                   19.46 
                   ) 
                 
               
             
           
         
       
     
         [0000]    corresponding to the 3 p x , p y , and p z , electrons. 
         [0287]    The outward centrifugal force on electron 19 is balanced by the electric force and the magnetic forces (on electron 19). The radius of the outer electron is calculated by equating the outward centrifugal force (Eq. (19.29)) to the sum of the electric (Eq. (19.28)) and diamagnetic (Eqs. (19.30), (19.45), and (19.46)) forces as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       8 
                       3 
                     
                      
                     
                       
                         ℏ 
                         2 
                       
                       
                         
                           m 
                           e 
                         
                          
                         
                           d 
                           3 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           e 
                           2 
                         
                         
                           4 
                            
                           π 
                            
                           
                               
                           
                            
                           
                             ɛ 
                             0 
                           
                            
                           
                             d 
                             2 
                           
                         
                       
                       - 
                       
                         
                           
                             ℏ 
                             2 
                           
                           
                             
                               Zm 
                               e 
                             
                              
                             
                               d 
                               3 
                             
                           
                         
                          
                         
                           
                             3 
                             4 
                           
                         
                       
                       - 
                       
                         
                           1 
                           Z 
                         
                          
                         
                           
                             12 
                              
                             
                               ℏ 
                               2 
                             
                           
                           
                             
                               m 
                               e 
                             
                              
                             
                               d 
                               3 
                             
                           
                         
                          
                         
                           
                             3 
                             4 
                           
                         
                       
                       - 
                       
                         
                           
                             ℏ 
                             2 
                           
                           
                             4 
                              
                             
                               m 
                               e 
                             
                              
                             
                               d 
                               2 
                             
                              
                             
                               r 
                               18 
                             
                           
                         
                          
                         
                           
                             3 
                             4 
                           
                         
                          
                         
                             
                         
                          
                         where 
                          
                         
                             
                         
                          
                         s 
                       
                     
                     = 
                     
                       
                         1 
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   19.47 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     d 
                     = 
                     
                       
                         
                           
                             a 
                             0 
                           
                           ( 
                           
                             
                               8 
                               3 
                             
                             + 
                             
                               
                                 13 
                                 Z 
                               
                                
                               
                                 
                                   3 
                                   4 
                                 
                               
                             
                           
                           ) 
                         
                         
                           
                             ( 
                             
                               Z 
                               - 
                               18 
                             
                             ) 
                           
                           - 
                           
                             
                               
                                 3 
                                 4 
                               
                             
                             
                               4 
                                
                               
                                 
                                   r 
                                   18 
                                 
                                 
                                   a 
                                   0 
                                 
                               
                             
                           
                         
                       
                       = 
                       
                         
                           
                             a 
                             0 
                           
                           ( 
                           
                             
                               8 
                               3 
                             
                             + 
                             
                               
                                 13 
                                 19 
                               
                                
                               
                                 
                                   3 
                                   4 
                                 
                               
                             
                           
                           ) 
                         
                         
                           1 
                           - 
                           
                             
                               
                                 3 
                                 4 
                               
                             
                             
                               4 
                                
                               
                                 
                                   r 
                                   18 
                                 
                                 
                                   a 
                                   0 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.48 
                   ) 
                 
               
             
           
         
       
     
       Substitution of 
       [0288]    
       
         
           
             
               
                 r 
                 18 
               
               
                 a 
                 0 
               
             
             = 
             0.85215 
           
         
       
     
         [0000]    (Eq. (10.399) with Z=19) into Eq. (19.48) gives 
         [0000]        d= 4.36934 a   0 =2.31215×10 −10  m  (19.49)
 
         [0000]    Using Eq. (19.26), the lattice parameter a is 
         [0000]        a= 10.09055 a   0 =5.33969×10 −1    m   (19.50)
 
         [0000]    The experimental lattice parameter a [7] is 
         [0000]        a= 10.05524 a   0 =5.321×10 −10  m  (19.51)
 
         [0000]    The calculated K—K distance is in good agreement with the experimental distance. 
         [0289]    Using Eq. (19.27) and the experimental binding energy of potassium, BE(K)=4.34066 eV=6.9545×10 −19  J [9], the molar metal bond energy E D  is 
         [0000]    
       
         
           
             
               E 
               D 
             
             = 
             
               
                 N 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             
                               e 
                               2 
                             
                             
                               4 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               2.31215 
                               × 
                               
                                 10 
                                 
                                   - 
                                   10 
                                 
                               
                                
                               m 
                             
                           
                           - 
                         
                       
                     
                     
                       
                         
                           
                             
                               4 
                               3 
                             
                              
                             
                               ( 
                               
                                 
                                   1 
                                   2 
                                 
                                  
                                 
                                   
                                     ℏ 
                                     2 
                                   
                                   
                                     
                                       
                                         m 
                                         e 
                                       
                                        
                                       
                                         ( 
                                         
                                           2.31215 
                                           × 
                                           
                                             10 
                                             
                                               - 
                                               10 
                                             
                                           
                                            
                                           m 
                                         
                                         ) 
                                       
                                     
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                         
                       
                     
                     
                       
                         
                           6.9545 
                           × 
                           
                             10 
                             
                               - 
                               19 
                             
                           
                            
                           J 
                         
                       
                     
                   
                   ) 
                 
               
               = 
               
                 90.40 
                  
                 
                     
                 
                  
                 kJ 
                  
                 
                   / 
                 
                  
                 mole 
               
             
           
         
       
     
         [0000]    This agrees well with the experimental lattice [10] energy of 
         [0000]        E   D =89 kJ/mole  (19.53)
 
         [0000]    and confirms that K metal comprises a precise packing of discrete ions, K +  and e − . Using the K—K and K +  −e −  distances and the calculated (Eq. (10.399)) K +  ionic radius of 0.85215a 0 =0.45094 Å, the crystalline lattice structure of K metal is shown in  FIG. 18C . 
       Rubidium and Cesium Metals 
       [0290]    Rubidium and cesium provide further examples of the nature of the bonding in alkali metals. The distance d between each metal ion M +  and the corresponding electron membrane is calculated from the experimental parameter a, and then the molar metal bond energy E D  is calculated using Eq. (19.27). 
         [0291]    The experimental lattice parameter a [7] for rubidium is 
         [0000]        a= 10.78089 a   0 =5.705×10 −10  m  (19.54)
 
         [0000]    Using Eq. (19.25), the lattice parameter d is 
         [0000]        d= 4.66826 a   0 =2.47034×10 −10  m  (19.55)
 
         [0000]    Using Eqs. (19.27) and (19.55) and the experimental binding energy of rubidium, BE(Rb)=4.17713 eV=6.6925×10 −19  J [9], the molar metal bond energy E D  is 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                     D 
                   
                   = 
                   
                     
                       N 
                        
                       
                         ( 
                         
                           
                             
                               
                                 
                                   
                                     e 
                                     2 
                                   
                                   
                                     4 
                                      
                                     
                                       πɛ 
                                       0 
                                     
                                      
                                     2.47034 
                                     × 
                                     
                                       10 
                                       
                                         - 
                                         10 
                                       
                                     
                                      
                                     m 
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     4 
                                     3 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         1 
                                         2 
                                       
                                        
                                       
                                         
                                           ℏ 
                                           2 
                                         
                                         
                                           
                                             
                                               m 
                                               e 
                                             
                                              
                                             
                                               ( 
                                               
                                                 2.47034 
                                                 × 
                                                 
                                                   10 
                                                   
                                                     - 
                                                     10 
                                                   
                                                 
                                                  
                                                 m 
                                               
                                               ) 
                                             
                                           
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 6.6925 
                                 × 
                                 
                                   10 
                                   
                                     - 
                                     19 
                                   
                                 
                                  
                                 J 
                               
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       79.06 
                        
                       
                           
                       
                        
                       kJ 
                        
                       
                         / 
                       
                        
                       mole 
                     
                   
                 
               
               
                 
                   ( 
                   19.56 
                   ) 
                 
               
             
           
         
       
     
         [0000]    This agrees well with the experimental lattice [10] energy of 
         [0000]        E   D =80.9 kJ/mole  (19.57)
 
         [0000]    and confirms that Rb metal comprises a precise packing of discrete ions, Rb +  and e − . Using the Rb—Rb and Rb + -e −  distances and the Rb +  ionic radius of 0.52766 Å calculated using Eq. (10.102) and the experimental ionization energy of Rb + , 27.2895 eV [9], the crystalline lattice structure of Rb metal is shown in  FIG. 18D . 
         [0292]    The experimental lattice parameter a [7] for cesium is 
         [0000]        a= 11.60481 a   0 =6.141×10 −10  m  (19.58)
 
         [0000]    Using Eq. (19.25), the lattice parameter d is 
         [0000]        d= 5.02503 a   0 =2.65913×10 −10  m  (19.59)
 
         [0000]    Using Eqs. (19.27) and (19.59) and the experimental binding energy of cesium, BE(Cs)=3.8939 eV=6.23872×10 −19  J [9], the molar metal bond energy E D  is 
         [0000]    
       
         
           
             
               E 
               D 
             
             = 
             
               
                 N 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             
                               e 
                               2 
                             
                             
                               4 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               2.65913 
                               × 
                               
                                 10 
                                 
                                   - 
                                   10 
                                 
                               
                                
                               m 
                             
                           
                           - 
                         
                       
                     
                     
                       
                         
                           
                             
                               4 
                               3 
                             
                              
                             
                               ( 
                               
                                 
                                   1 
                                   2 
                                 
                                  
                                 
                                   
                                     ℏ 
                                     2 
                                   
                                   
                                     
                                       
                                         m 
                                         e 
                                       
                                        
                                       
                                         ( 
                                         
                                           2.65913 
                                           × 
                                           
                                             10 
                                             
                                               - 
                                               10 
                                             
                                           
                                            
                                           m 
                                         
                                         ) 
                                       
                                     
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                         
                       
                     
                     
                       
                         
                           6.23872 
                           × 
                           
                             10 
                             
                               - 
                               19 
                             
                           
                            
                           J 
                         
                       
                     
                   
                   ) 
                 
               
               = 
               
                 77.46 
                  
                 
                     
                 
                  
                 kJ 
                  
                 
                   / 
                 
                  
                 mole 
               
             
           
         
       
     
         [0000]    This agrees well with the experimental lattice [10] energy of 
         [0000]        E   D =76.5 kJ/mole  (19.61)
 
         [0000]    and confirms that Cs metal comprises a precise packing of discrete ions, Cs +  and e − . Using the Cs—Cs and Cs + −e −  distances and the Cs +  ionic radius of 0.62182 Å calculated using Eq. (10.102) and the experimental ionization energy of Cs + , 23.15744 eV [9], the crystalline lattice structure of Cs metal is shown in  FIG. 18E . 
         [0293]    Other metals can be solved in a similar manner. Iron, for example, is also a body-centered cubic lattice, and the solution of the lattice spacing and energies are given by Eqs. (19.21-19.30). The parameter d is given by the iron force balance which has a corresponding form to those of alkali metals such as that of lithium given by Eqs. (19.32-19.35). In addition, the changes in radius and energy of the second 4s electron due to the ionization of the first of the two 4s electrons to the metal band is calculated in the similar manner as those of the atoms of diatomic molecules such as N 2  given by Eqs. (19.621-19.632). This energy term is added to those of Eq. (19.27) to give the molar metal bond energy E D . 
       Physical Implications of the Nature of Free Electrons in Metals 
       [0294]    The extension of the free-electron membrane throughout the crystalline lattice is the reason for the high thermal and electrical conductivity of metals. Electricity can be conduced on the extended electron membranes by the application of an electron field and a connection with a source of electrons to maintain current continuity. Heat can be transferred by radiation or by collisions, or by infrared-radiation-induced currents propagated through the crystal. The surface luster and opacity is due to the reflection of electromagnetic radiation by mirror currents on the surfaces of the free-planar electron membranes. Ductility and malleability result from the feature that the field lines of a given ion end on the induced electron surface charge of the planar, perfectly conducting electron membrane. Thus, layers of the metal lattice can slide over each other without juxtaposing charges of the same sign which causes ionic crystals to fracture. 
         [0295]    The electrons in metals have surface-charge distributions that are merely equivalent to the image charges of the ions. When there is vibration of the ions, the thermal electron kinetic energy can be directed through channels of least resistance from collisions. The resulting kinetic energy distribution over the population of electrons can be modeled using Fermi Dirac statistics wherein the specific heat of a metal is dominated by the motion of the ions since the electrons behave as image charges. Based on the physical solution of the nature of the metallic bond, the small electron contribution to the specific heat of a metal is predicted to be proportional to the ratio of the temperature to the electron kinetic energy [11]. Based on Fermi-Dirac statistics, the electron contribution to the specific heat of a metal given by Eq. (23.68) is 
         [0000]    
       
         
           
             
               
                 
                   
                     C 
                     Ve 
                   
                   = 
                   
                     
                       
                         π 
                         2 
                       
                       2 
                     
                      
                     
                       ( 
                       
                         kT 
                         
                           ɛ 
                           F 
                         
                       
                       ) 
                     
                      
                     R 
                   
                 
               
               
                 
                   ( 
                   19.62 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Now that the true structure of metals has been solved, it is interesting to relate the Fermi energy to the electron kinetic energy. The relationships between the electron velocity, the de Broglie wavelength, and the lattice spacing used calculate the Fermi energy in the Electron-Energy Distribution section are also used in the kinetic energy derivation. The Fermi energy given by Eq. (23.61) is 
         [0000]    
       
         
           
             
               
                 
                   
                     ɛ 
                     F 
                   
                   = 
                   
                     
                       
                         
                           h 
                           2 
                         
                         
                           2 
                            
                           m 
                         
                       
                        
                       
                         
                           ( 
                           
                             
                               3 
                                
                               N 
                             
                             
                               8 
                                
                               π 
                                
                               
                                   
                               
                                
                               V 
                             
                           
                           ) 
                         
                         
                           2 
                           / 
                           3 
                         
                       
                     
                     = 
                     
                       
                         
                           h 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                         
                       
                        
                       
                         
                           ( 
                           
                             3 
                             
                               8 
                                
                               π 
                             
                           
                           ) 
                         
                         
                           2 
                           / 
                           3 
                         
                       
                        
                       
                         n 
                         
                           2 
                           / 
                           3 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.63 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the electron density parameter for alkali metals is two electrons per body-centered cubic cell of lattice spacing a. Since in the physical model, the field lines of two mirror-image ions M +  end on opposite sides per section of the two-dimensional electron membrane, the kinetic energy equivalent to the Fermi energy is twice that given by Eq. (19.24). Then, the ratio R ε     F     /T  of the Fermi energy to the kinetic energy provides a comparison of the statistical model to the solution of the nature of the metallic bond in the determination of electron contribution to the specific heat: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           R 
                           
                             
                               ɛ 
                               F 
                             
                             / 
                             T 
                           
                         
                         = 
                         
                           
                             ɛ 
                             F 
                           
                           T 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 
                                   h 
                                   2 
                                 
                                 
                                   2 
                                    
                                   
                                     m 
                                     e 
                                   
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     3 
                                     
                                       8 
                                        
                                       π 
                                     
                                   
                                   ) 
                                 
                                 
                                   2 
                                   / 
                                   3 
                                 
                               
                                
                               
                                 n 
                                 
                                   2 
                                   / 
                                   3 
                                 
                               
                             
                             
                               
                                 8 
                                 3 
                               
                                
                               
                                 ( 
                                 
                                   
                                     1 
                                     2 
                                   
                                    
                                   
                                     
                                       ℏ 
                                       2 
                                     
                                     
                                       
                                         m 
                                         e 
                                       
                                        
                                       
                                         d 
                                         2 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           = 
                           
                             
                               
                                 
                                   h 
                                   2 
                                 
                                 
                                   2 
                                    
                                   
                                     m 
                                     e 
                                   
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     3 
                                     
                                       8 
                                        
                                       π 
                                     
                                   
                                   ) 
                                 
                                 
                                   2 
                                   / 
                                   3 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     2 
                                     
                                       a 
                                       3 
                                     
                                   
                                   ) 
                                 
                                 
                                   2 
                                   / 
                                   3 
                                 
                               
                             
                             
                               
                                 8 
                                 3 
                               
                                
                               
                                 ( 
                                 
                                   
                                     1 
                                     2 
                                   
                                    
                                   
                                     
                                       ℏ 
                                       2 
                                     
                                     
                                       
                                         m 
                                         e 
                                       
                                        
                                       
                                         d 
                                         2 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 
                                   h 
                                   2 
                                 
                                 
                                   2 
                                    
                                   
                                     m 
                                     e 
                                   
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     3 
                                     
                                       8 
                                        
                                       π 
                                     
                                   
                                   ) 
                                 
                                 
                                   2 
                                   / 
                                   3 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     2 
                                     
                                       
                                         ( 
                                         
                                           
                                             4 
                                              
                                             d 
                                           
                                           
                                             3 
                                           
                                         
                                         ) 
                                       
                                       3 
                                     
                                   
                                   ) 
                                 
                                 
                                   2 
                                   / 
                                   3 
                                 
                               
                             
                             
                               
                                 ( 
                                 
                                   8 
                                   3 
                                 
                                 ) 
                               
                                
                               
                                 
                                   ( 
                                   
                                     1 
                                     
                                       2 
                                        
                                       π 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     h 
                                     2 
                                   
                                   
                                     2 
                                      
                                     
                                       m 
                                       e 
                                     
                                      
                                     
                                       d 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           = 
                           1.068 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.64 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Eq. (19.26) was used to convert the parameter a to d. 
         [0296]    From the physical nature of the current, the electrical and thermal conductivities corresponding to the currents can be determined. The electrical current is classically given by 
         [0000]    
       
         
           
             
               
                 
                   i 
                   = 
                   
                     
                       e 
                        
                       
                           
                       
                        
                       ν 
                     
                     = 
                     
                       σ 
                        
                       
                         
                           ɛ 
                           F 
                         
                         he 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.65 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the energy and angular momentum of the conduction electrons are quantized according to  and Planck&#39;s equation (Eq. (4.8)), respectively. From Eq. (19.65), the electrical conductivity is given by 
         [0000]    
       
         
           
             
               
                 
                   σ 
                   = 
                   
                     
                       
                         e 
                         2 
                       
                        
                       h 
                        
                       
                           
                       
                        
                       ν 
                     
                     
                       ɛ 
                       F 
                     
                   
                 
               
               
                 
                   ( 
                   19.66 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where v is the frequency of the unit current carried by each electron. The thermal current is also carried by the kinetic energy of the electron plane waves. Since there are two degrees of freedom in the plane of each electron rather than three, the thermal conductivity κ is given by 
         [0000]    
       
         
           
             
               
                 
                   κ 
                   = 
                   
                     
                       
                         2 
                         3 
                       
                        
                       
                         
                           C 
                           Ve 
                         
                         
                           
                             N 
                             0 
                           
                            
                           h 
                         
                       
                     
                     = 
                     
                       
                         
                           π 
                           2 
                         
                         3 
                       
                        
                       
                         ( 
                         
                           
                             
                               k 
                               B 
                               2 
                             
                              
                             T 
                           
                           
                             
                               ɛ 
                               F 
                             
                             / 
                             h 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.67 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The Wiedemann-Franz law gives the relationship of the thermal conductivity κ to the electrical conductivity σ and absolute temperature T. Thus, using Eqs. (19.66-19.67), the constant L 0  is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     L 
                     0 
                   
                   = 
                   
                     
                       κ 
                       
                         σ 
                          
                         
                             
                         
                          
                         T 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               π 
                               2 
                             
                             3 
                           
                            
                           
                             ( 
                             
                               
                                 hk 
                                 B 
                                 2 
                               
                               
                                 ɛ 
                                 F 
                               
                             
                             ) 
                           
                         
                         
                           
                             he 
                             2 
                           
                           
                             ɛ 
                             F 
                           
                         
                       
                       = 
                       
                         
                           
                             π 
                             2 
                           
                           3 
                         
                          
                         
                           
                             ( 
                             
                               
                                 k 
                                 B 
                               
                               e 
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19.68 
                   ) 
                 
               
             
           
         
       
     
         [0000]    From Eqs. (19.64) and (19.68), the statistical model is reasonably close to the physical model to be useful in modeling the specific-heat contribution of electrons in metals based on their inventory of thermal energy and the thermal-energy distribution in the crystal. However, the correct physical nature of the current carriers comprising two-dimensional electron planes is required in cases where the simplistic statistical model fails as in the case of the anisotropic violation of the Wiedemann-Franz law [12-13]. 
         [0297]    Semiconductors comprise covalent bonds wherein the electrons are of sufficiently high energy that excitation creates an ion and a free electron. The free electron forms a membrane as in the case of metals. This membrane has the same planar structure throughout the crystal. This feature accounts for the high conductivity of semiconductors when the electrons are excited by the application of external fields or electromagnetic energy that causes ion-pair (M + -e − ) formation. 
         [0298]    Superconductors comprise free-electron membranes wherein current flows in a reduced dimensionality of two or one dimensions with the bonding being covalent along the remaining directions such that electron scattering from other planes does not interfere with the current flow. In addition, the spacing of the electrons along the membrane is such that the energy is band-passed with respect to magnetic interactions of conducting electrons as given in the superconductivity section. 
       REFERENCES 
       [0000]    
       
         1. J. D. Jackson, Classical Electrodynamics, Second Edition, John Wiley &amp; Sons, New York, (1975), pp. 17-22. 
         2. H. A. Haus, J. R. Melcher, “Electromagnetic Fields and Energy”, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, (1985), Sec. 5.3. 
         3. J. A. Stratton,  Electromagnetic Theory , McGraw-Hill Book Company, (1941), p. 195. 
         4. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), p. A-23. 
         5. J. A. Kong,  Electromagnetic Wave Theory , Second Edition, John Wiley &amp; Sons, Inc., New York, (1990), pp. 330-331. 
         6. A. Beiser,  Concepts of Modern Physics , Fourth Edition, McGraw-Hill, New York, (1987), p. 372. 
         7. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), pp. 12-15 to 12-18. 
         8. A. K. Cheetham, P. Day, Editors, Solid State Chemistry Techniques, Clarendon Press, Oxford, (1987), pp. 52-57. 
         9. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), pp. 10-202 to 10-204. 
         10. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), pp. 5-4 to 5-18. 
         11. E. C. Stoner, “Collective electron specific heat and spin paramagnetism in metals”, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 154, No. 883 (May 1, 1936), pp. 656-678. 
         12. M. A. Tanatar, J. Paglione, C. Petrovic, L. Taillefer, “Anisotropic violation of the Wiedemann-Franz law at a quantum critical point,” Science, Vol. 316, (2007), pp. 1320-1322. 
         13. P. Coleman, “Watching electrons break up,” Science, Vol. 316, (2007), pp. 1290-1291. 
       
     
       Silicon Molecular Functional Groups and Molecules 
     General Considerations of the Silicon Molecular Bond 
       [0312]    Silane molecules comprising an arbitrary number of atoms can be solved using similar principles and procedures as those used to solve organic molecules of arbitrary length and complexity. Silanes can be considered to be comprised of functional groups such as SiH 3 , SiH 2 , SiH, Si—Si, and C—Si. The solutions of these functional groups or any others corresponding to the particular silane can be conveniently obtained by using generalized forms of the force balance equation given in the Force Balance of the σ MO of the Carbon Nitride Radical section for molecules comprised of silicon and hydrogen only and the geometrical and energy equations given in the Derivation of the General Geometrical and Energy Equations of Organic Chemistry section for silanes further comprised of heteroatoms such as carbon. The appropriate functional groups with the their geometrical parameters and energies can be added as a linear sum to give the solution of any silane. 
       Silanes (Si n H 2n+2 ) 
       [0313]    As in the case of carbon, the bonding in the silicon atom involves four sp 3  hybridized orbitals formed from the 3p and 3s electrons of the outer shells. Si—Si and Si—H bonds form between Si3sp 3  HOs and between a Si3sp 3  HO and a H1s AO to yield silanes. The geometrical parameters of each Si—Si and SiH n=123  functional group is solved from the force balance equation of the electrons of the corresponding σ-MO and the relationships between the prolate spheroidal axes. Then, the sum of the energies of the H 2 -type ellipsoidal MOs is matched to that of the Si3sp 3  shell as in the case of the corresponding carbon molecules. As in the case of ethane given in the Ethane Molecule section, the energy of the Si—Si functional group is determined for the effect of the donation of 25% electron density from the each participating Si3sp 3  HO to the Si—Si-bond MO. 
         [0314]    The energy of silicon is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). A minimum energy is achieved while matching the potential, kinetic, and orbital energy relationships given in the Hydroxyl Radical (OH) section with the donation of 75% electron density from the participating Si3sp 3  HO to each Si—H-bond MO. As in the case of acetylene given in the Acetylene Molecule section, the energy of each Si—H n  functional group is determined for the effect of the charge donation. 
         [0315]    The 3sp 3  hybridized orbital arrangement after Eq. (13.422) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         3 
                          
                         
                           sp 
                           3 
                         
                          
                         
                             
                         
                          
                         state 
                       
                     
                   
                   
                     
                       
                         
                           ↑ 
                           
                             0 
                             , 
                             0 
                           
                         
                          
                         
                             
                         
                          
                         
                           ↑ 
                           
                             1 
                             , 
                             
                               - 
                               1 
                             
                           
                         
                          
                         
                             
                         
                          
                         
                           ↑ 
                           
                             1 
                             , 
                             0 
                           
                         
                          
                         
                             
                         
                          
                         
                           ↑ 
                           
                             1 
                             , 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the quantum numbers (l, m t ) are below each electron. The total energy of the state is given by the sum over the four electrons. The sum E T (Si 3sp 3 ) of experimental energies [1] of Si, Si + , Si 2+ , and Si 3+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               Si 
                               , 
                               
                                 3 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             45.14181 
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             33.49302 
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             8.15168 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           103.13235 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    By considering that the central field decreases by an integer for each successive electron of the shell, the radius r 3sp     3    of the Si3sp 3  shell may be calculated from the Coulombic energy using Eq. (15.13): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             3 
                              
                             
                               sp 
                               3 
                             
                           
                         
                         = 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               10 
                             
                             13 
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   Z 
                                   - 
                                   n 
                                 
                                 ) 
                               
                                
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     103.13235 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             10 
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   103.13235 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1.31926 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Z=14 for silicon. Using Eq. (15.14), the Coulombic energy E Coulomb (Si,3sp 3 ) of the outer electron of the Si3sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               Si 
                               , 
                               
                                 3 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 3 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.31926 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             10.31324 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, one of the spin-paired 3s electrons is promoted to Si3sp 3  shell as an unpaired electron. The energy for the promotion is the magnetic energy given by Eq. (15.15) at the initial radius of the 3s electrons. From Eq. (10.255) with Z=14, the radius r u  of Si3s shell is 
         [0000]      r 12 =1.25155a 0   (20.5)
 
         [0000]    Using Eqs. (15.15) and (20.5), the unpairing energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       ( 
                       magnetic 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         2 
                          
                         
                           πμ 
                           0 
                         
                          
                         
                            
                           2 
                         
                          
                         
                           ℏ 
                           2 
                         
                       
                       
                         
                           
                             m 
                             e 
                             2 
                           
                            
                           
                             ( 
                             
                               r 
                               12 
                             
                             ) 
                           
                         
                         3 
                       
                     
                     = 
                     
                       
                         
                           8 
                            
                           
                             πμ 
                             o 
                           
                            
                           
                             μ 
                             B 
                             2 
                           
                         
                         
                           
                             ( 
                             
                               1.25155 
                                
                               
                                 a 
                                 0 
                               
                             
                             ) 
                           
                           3 
                         
                       
                       = 
                       
                         0.05836 
                          
                         
                             
                         
                          
                         eV 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (20.4) and (20.6), the energy E(Si,3sp 3 ) of the outer electron of the Si3sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               Si 
                               , 
                               
                                 3 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   3 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     12 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             10.31324 
                              
                             e 
                              
                             
                                 
                             
                              
                             V 
                           
                           + 
                           
                             0.05836 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             10.25487 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.7 
                   ) 
                 
               
             
           
         
       
     
         [0316]    Next, consider the formation of the Si—Si-bond MO of silanes wherein each silicon atom has a Si3sp 3  electron with an energy given by Eq. (20.7). The total energy of the state of each silicon atom is given by the sum over the four electrons. The sum E T (Si silane ,3sp 3 ) of energies of Si3sp 3  (Eq. (20.7)), Si + , Si 2+ , and Si 3+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Si 
                                 silane 
                               
                               , 
                               
                                 3 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       45.14181 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       33.49302 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       16.34584 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       E 
                                        
                                       
                                         ( 
                                         
                                           Si 
                                           , 
                                           
                                             3 
                                              
                                             
                                               sp 
                                               3 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       45.14181 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       33.39302 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       16.34584 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       10.25487 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             105.23554 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E(Si,3sp 3 ) is the sum of the energy of Si, −8.15168 eV, and the hybridization energy. 
         [0317]    The sharing of electrons between two Si3sp 3  HOs to form a Si—Si-bond MO permits each participating orbital to decrease in size and energy. In order to further satisfy the potential, kinetic, and orbital energy relationships, each Si3sp 3  HO donates an excess of 25% of its electron density to the Si—Si-bond MO to form an energy minimum. By considering this electron redistribution in the silane molecule as well as the fact that the central field decreases by an integer for each successive electron of the shell, the radius r silane3sp     3   , of the Si3sp 3  shell may be calculated from the Coulombic energy using Eq. (15.18): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             silane 
                              
                             
                                 
                             
                              
                             3 
                              
                             
                               sp 
                               3 
                             
                           
                         
                         = 
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     10 
                                   
                                   13 
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               0.25 
                             
                             ) 
                           
                            
                           
                             
                                
                               2 
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     105.23554 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             9.75 
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   105.23554 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1.26057 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (20.9), the Coulombic energy E Coulomb (Si silane ,3sp 3 ) of the outer electron of the Si3sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 Si 
                                 silane 
                               
                               , 
                               
                                 3 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 silane 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.26057 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             10.79339 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, one of the spin-paired 3s electrons is promoted to Si3sp 3  shell as an unpaired electron. The energy for the promotion is the magnetic energy given by Eq. (20.6). Using Eqs. (20.6) and (20.10), the energy E(S silane ,3sp 3 ) of the outer electron of the Si3sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 Si 
                                 silane 
                               
                               , 
                               
                                 3 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   silane 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     12 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               10.79339 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.05836 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             10.73503 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (Si—Si,3sp 3 ), the energy change of each Si3sp 3  shell with the formation of the Si—Si-bond MO is given by the difference between Eq. (20.11) and Eq. (20.7): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Si 
                                 - 
                                 Si 
                               
                               , 
                               
                                 3 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   Si 
                                   silane 
                                 
                                 , 
                                 
                                   3 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 Si 
                                 , 
                                 
                                   3 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               10.73503 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 10.25487 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             0.48015 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.12 
                   ) 
                 
               
             
           
         
       
     
         [0318]    Next, consider the formation of the Si—H-bond MO of silanes wherein each silicon atom contributes a Si3sp 3  electron having the sum E T (Si silane 3sp 3 ) of energies of Si3sp 3  (Eq. (20.7)), Si + , Si 2+ , and Si 3+  given by Eq. (20.8). Each Si—H-bond MO of each functional group SiH n=123  forms with the sharing of electrons between each Si3sp 3  HO and each H1s AO. As in the case of C—H, the H 2 -type ellipsoidal MO comprises 75% of the Si—H-bond MO according to Eq. (13.429). Furthermore, the donation of electron density from each Si3sp 3  HO to each Si—H-bond MO permits the participating orbital to decrease in size and energy. In order to further satisfy the potential, kinetic, and orbital energy relationships, each Si3sp 3  HO donates an excess of 75% of its electron density to the Si—H-bond MO to form an energy minimum. By considering this electron redistribution in the silane molecule as well as the fact that the central field decreases by an integer for each successive electron of the shell, the radius r silane3sp     3    of the Si3sp 3  shell may be calculated from the Coulombic energy using Eq. (15.18): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             silane 
                              
                             
                                 
                             
                              
                             3 
                              
                             
                               sp 
                               3 
                             
                           
                         
                         = 
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     10 
                                   
                                   13 
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               0.75 
                             
                             ) 
                           
                            
                           
                             
                                
                               2 
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     105.23554 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             9.25 
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   105.23554 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1.19592 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (20.13), the Coulombic energy E Coulomb (Si silane ,3sp 3 ) of the outer electron of the Si3sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 Si 
                                 silane 
                               
                               , 
                               
                                 3 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 silane 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.19592 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             11.37682 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, one of the spin-paired 3s electrons is promoted to Si3sp 3  shell as an unpaired electron. The energy for the promotion is the magnetic energy given by Eq. (20.6). Using Eqs. (20.6) and (20.14), the energy E(Si silane ,3sp 3 ) of the outer electron of the Si3sp3 shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   Si 
                                   silane 
                                 
                                 , 
                                 
                                   3 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 - 
                                 
                                    
                                   2 
                                 
                               
                               
                                 8 
                                  
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   r 
                                   
                                     silane 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                      
                                     
                                       sp 
                                       3 
                                     
                                   
                                 
                               
                             
                             + 
                             
                               
                                 2 
                                  
                                 
                                   πμ 
                                   0 
                                 
                                  
                                 
                                    
                                   2 
                                 
                                  
                                 
                                   ℏ 
                                   2 
                                 
                               
                               
                                 
                                   
                                     m 
                                     e 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       r 
                                       12 
                                     
                                     ) 
                                   
                                 
                                 3 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               11.37682 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.05836 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             11.31845 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (Si—H,3sp 3 ), the energy change of each Si3sp 3  shell with the formation of the Si—H-bond MO is given by the difference between Eq. (20.15) and Eq. (20.7): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Si 
                                 - 
                                 H 
                               
                               , 
                               
                                 3 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   Si 
                                   silane 
                                 
                                 , 
                                 
                                   3 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 Si 
                                 , 
                                 
                                   3 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               11.31845 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 10.25487 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             1.06358 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.16 
                   ) 
                 
               
             
           
         
       
     
         [0319]    Silane (SiH 4 ) involves only Si—H-bond MOs of equivalent tetrahedral structure to form a minimum energy surface involving a linear combination of all four hydrogen MOs. Here, the donation of electron density from the Si3sp 3  HO to each Si—H-bond MO permits the participating orbital to decrease in size and energy as well. However, given the resulting continuous electron-density surface and the equivalent MOs, the Si3sp 3  HO donates an excess of 100% of its electron density to the Si—H-bond MO to form an energy minimum. By considering this electron redistribution in the silane molecule as well as the fact that the central field decreases by an integer for each successive electron of the shell, the radius r silane3sp     3   , of the Si3sp 3  shell may be calculated from the Coulombic energy using Eq. (15.18): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             silane 
                              
                             
                                 
                             
                              
                             3 
                              
                             
                               sp 
                               3 
                             
                           
                         
                         = 
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     10 
                                   
                                   13 
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               1 
                             
                             ) 
                           
                            
                           
                             
                                
                               2 
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     105.23554 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             9 
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   105.23554 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1.16360 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.17 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (20.17), the Coulombic energy E Coulomb (Si silane ,3sp 3 ) of the outer electron of the Si3sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 Si 
                                 silane 
                               
                               , 
                               
                                 3 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 silane 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.16360 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             11.69284 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.18 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, one of the spin-paired 3s electrons is promoted to Si3sp 3  shell as an unpaired electron. The energy for the promotion is the magnetic energy given by Eq. (20.6). Using Eqs. (20.6) and (20.18), the energy E(Si silane ,3sp 3 ) of the outer electron of the Si3sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 Si 
                                 silane 
                               
                               , 
                               
                                 3 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   silane 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     12 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               11.69284 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.05836 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             11.63448 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.19 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (Si—H,3sp 3 ), the energy change of each Si3sp 3  shell with the formation of the Si—H-bond MO is given by the difference between Eq. (20.19) and Eq. (20.7): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Si 
                                 - 
                                 H 
                               
                               , 
                               
                                 3 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   Si 
                                   silane 
                                 
                                 , 
                                 
                                   3 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 Si 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               11.63448 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 10.25487 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             1.37960 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.20 
                   ) 
                 
               
             
           
         
       
     
         [0320]    Consider next the radius of the HO due to the contribution of charge to more than one bond. The energy contribution due to the charge donation at each silicon atom superimposes linearly. In general, the radius r mol3sp     3    of the Si3sp 3  HO of a silicon atom of a given silane molecule is calculated after Eq. (15.32) by considering ΣE T     mol   (MO,3sp 3 ), the total energy donation to all bonds with which it participates in bonding. The general equation for the radius is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             mol 
                              
                             
                                 
                             
                              
                             3 
                              
                             
                               sp 
                               3 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         
                                           
                                             E 
                                             Coulomb 
                                           
                                            
                                           
                                             ( 
                                             
                                               Si 
                                               , 
                                               
                                                 3 
                                                  
                                                 
                                                   sp 
                                                   3 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         ∑ 
                                         
                                             
                                         
                                          
                                         
                                           
                                             E 
                                             
                                               T 
                                               mol 
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               MO 
                                               , 
                                               
                                                 3 
                                                  
                                                 
                                                   sp 
                                                   3 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                              
                             2 
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         
                                           e 
                                            
                                           
                                               
                                           
                                            
                                           10.31324 
                                            
                                           
                                               
                                           
                                            
                                           eV 
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         ∑ 
                                         
                                             
                                         
                                          
                                         
                                            
                                           
                                             
                                               E 
                                               
                                                 T 
                                                 mol 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 MO 
                                                 , 
                                                 
                                                   3 
                                                    
                                                   
                                                     sp 
                                                     3 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                            
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.21 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E Coulomb (Si,3sp 3 ) is given by Eq. (20.4). The Coulombic energy E Coulomb (Si,3sp 3 ) of the outer electron of the Si 3sp 3  shell considering the charge donation to all participating bonds is given by Eq. (15.14) with Eq. (20.4). The energy E(Si,3sp 3 ) of the outer electron of the Si 3sp 3  shell is given by the sum of E Coulomb (Si,3sp 3 ) and E(magnetic) (Eq. (20.6)). The final values of the radius of the Si3sp 3  HO, r 3sp     3   , E Coulomb (Si,3sp 3 ), and E(Si silane 3sp 3 ) calculated using ΣE T     mol    (MO,3sp 3 ), the total energy donation to each bond with which an atom participates in bonding are given in Table 20.1. These hybridization parameters are used in Eqs. (15.88-15.117) for the determination of bond angles given in Table 20.7 (as shown in the priority document). 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 20.1 
               
             
             
               
                   
               
               
                 Hybridization parameters of atoms for determination of bond angles with final 
               
               
                 values of r 3sp     3   , E Coulomb  (Si, 3sp 3 ), and E(Si silane  3SP 3 ) calculated 
               
               
                 using the appropriate values of ΣE T     mol    (MO, 3sp 3 ) (E T     mol    (MO, 3sp 3 ) 
               
               
                 designated as E T ) for each corresponding terminal bond spanning each angle. 
               
             
          
           
               
                 Atom 
                   
                   
                   
                   
                   
                   
                 E Coulomb  (Si, 3sp 3 ) 
                 E(Si, 3sp 3 ) 
               
               
                 Hybridization 
                   
                   
                   
                   
                   
                 r 3sp     3     
                 (eV) 
                 (eV) 
               
               
                 Designation 
                 E T   
                 E T   
                 E T   
                 E T   
                 E T   
                 Final 
                 Final 
                 Final 
               
               
                   
               
             
          
           
               
                 1 
                 0 
                 0 
                 0 
                 0 
                 0 
                 1.31926 
                 −10.31324 
                 −10.25487 
               
               
                 2 
                 −0.48015 
                 0 
                 0 
                 0 
                 0 
                 1.26057 
                 −10.79339 
                 −10.73503 
               
               
                   
               
             
          
         
       
     
         [0321]    The MO semimajor axis of each functional group of silanes is determined from the force balance equation of the centrifugal, Coulombic, and magnetic forces as given in the Polyatomic Molecular Ions and Molecules section and the More Polyatomic Molecules and Hydrocarbons section. The distance from the origin of the H 2 -type-ellipsoidal-MO to each focus c′, the internuclear distance 2c′, and the length of the semiminor axis of the prolate spheroidal H 2 -type MO b=c are solved from the semimajor axis a. Then, the geometric and energy parameters of the MO are calculated using Eqs. (15.1-15.117). 
         [0322]    The force balance of the centrifugal force equated to the Coulombic and magnetic forces is solved for the length of the semimajor axis. The Coulombic force on the pairing electron of the MO is 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     Coulomb 
                   
                   = 
                   
                     
                       
                          
                         2 
                       
                       
                         8 
                          
                         
                           πɛ 
                           0 
                         
                          
                         
                           ab 
                           2 
                         
                       
                     
                      
                     
                       Di 
                       ξ 
                     
                   
                 
               
               
                 
                   ( 
                   20.22 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The spin pairing force is 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     
                       spin 
                       - 
                       pairing 
                     
                   
                   = 
                   
                     
                       
                         ℏ 
                         2 
                       
                       
                         2 
                          
                         
                           m 
                           e 
                         
                          
                         
                           a 
                           2 
                         
                          
                         
                           b 
                           2 
                         
                       
                     
                      
                     
                       Di 
                       ξ 
                     
                   
                 
               
               
                 
                   ( 
                   20.23 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The diamagnetic force is: 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     
                       diamagneticMO 
                        
                       
                           
                       
                        
                       1 
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           
                             n 
                             e 
                           
                            
                           
                             ℏ 
                             2 
                           
                         
                         
                           4 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                     
                      
                     
                       Di 
                       ξ 
                     
                   
                 
               
               
                 
                   ( 
                   20.24 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where n e  is the total number of electrons that interact with the binding σ-MO electron. The diamagnetic force F diamagneticMO2  on the pairing electron of the σ MO is given by the sum of the contributions over the components of angular momentum: 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     
                       diamagneticMO 
                        
                       
                           
                       
                        
                       2 
                     
                   
                   = 
                   
                     - 
                     
                       
                         ∑ 
                         
                           i 
                           , 
                           j 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             
                                
                               
                                 L 
                                 i 
                               
                                
                             
                              
                             ℏ 
                           
                           
                             
                               Z 
                               j 
                             
                              
                             2 
                              
                             
                               m 
                               e 
                             
                              
                             
                               a 
                               2 
                             
                              
                             
                               b 
                               2 
                             
                           
                         
                          
                         
                           Di 
                           ξ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.25 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where |L| is the magnitude of the angular momentum of each atom at a focus that is the source of the diamagnetism at the σ-MO. The centrifugal force is 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     centrifugalMO 
                   
                   = 
                   
                     
                       - 
                       
                         
                           ℏ 
                           2 
                         
                         
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                     
                      
                     
                       Di 
                       ξ 
                     
                   
                 
               
               
                 
                   ( 
                   20.26 
                   ) 
                 
               
             
           
         
       
     
         [0323]    The force balance equation for the σ-MO of the Si—Si-bond MO with n e =3 and 
         [0000]    
       
         
           
             
                
               L 
                
             
             = 
             
               4 
                
               
                 
                   3 
                   4 
                 
               
                
               ℏ 
             
           
         
       
     
         [0000]    corresponding to four electrons of the Si3sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ℏ 
                         2 
                       
                       
                         
                           m 
                           e 
                         
                          
                         
                           a 
                           2 
                         
                          
                         
                           b 
                           2 
                         
                       
                     
                      
                     D 
                   
                   = 
                   
                     
                       
                         
                            
                           2 
                         
                         
                           8 
                            
                           
                             πɛ 
                             0 
                           
                            
                           
                             ab 
                             2 
                           
                         
                       
                        
                       D 
                     
                     + 
                     
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                     - 
                     
                       
                         ( 
                         
                           
                             3 
                             2 
                           
                           + 
                           
                             
                               4 
                                
                               
                                 
                                   3 
                                   4 
                                 
                               
                             
                             Z 
                           
                         
                         ) 
                       
                        
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                   
                 
               
               
                 
                   ( 
                   20.27 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     a 
                     = 
                     
                       
                         ( 
                         
                           
                             5 
                             2 
                           
                           + 
                           
                             
                               4 
                                
                               
                                 
                                   3 
                                   4 
                                 
                               
                             
                             Z 
                           
                         
                         ) 
                       
                        
                       
                         a 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.28 
                   ) 
                 
               
             
           
         
       
     
         [0000]    With Z=14, the semimajor axis of the Si—Si-bond MO is 
         [0000]      a=2.74744a 0   (20.29)
 
         [0324]    The force balance equation for each σ-MO of the Si—H-bond MO with n e =2 and 
         [0000]    
       
         
           
             
                
               L 
                
             
             = 
             
               4 
                
               
                 
                   3 
                   4 
                 
               
                
               ℏ 
             
           
         
       
     
         [0000]    corresponding to four electrons of the Si3sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ℏ 
                         2 
                       
                       
                         
                           m 
                           e 
                         
                          
                         
                           a 
                           2 
                         
                          
                         
                           b 
                           2 
                         
                       
                     
                      
                     D 
                   
                   = 
                   
                     
                       
                         
                            
                           2 
                         
                         
                           8 
                            
                           
                             πɛ 
                             0 
                           
                            
                           
                             ab 
                             2 
                           
                         
                       
                        
                       D 
                     
                     + 
                     
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                     - 
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               4 
                                
                               
                                 
                                   3 
                                   4 
                                 
                               
                             
                             Z 
                           
                         
                         ) 
                       
                        
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                   
                 
               
               
                 
                   ( 
                   20.30 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     a 
                     = 
                     
                       
                         ( 
                         
                           2 
                           + 
                           
                             
                               4 
                                
                               
                                 
                                   3 
                                   4 
                                 
                               
                             
                             Z 
                           
                         
                         ) 
                       
                        
                       
                         a 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.31 
                   ) 
                 
               
             
           
         
       
     
         [0000]    With Z=14, the semimajor axis of the Si—H-bond MO is 
         [0000]      a=2.24744a 0   (20.32)
 
         [0325]    Using the semimajor axis, the geometric and energy parameters of the MO are calculated using Eqs. (15.1-15.117) in the same manner as the organic functional groups given in the Organic Molecular Functional Groups and Molecules section. For the Si—Si functional group, the Si3sp 3  HOs are equivalent; thus, c 1 =1 in both the geometry relationships (Eqs. (15.2-15.5)) and the energy equation (Eq. (15.61)). In order for the bridging MO to intersect the Si3sp 3  HOs while matching the potential, kinetic, and orbital energy relationships given in the Hydroxyl Radical (OH) section, for the Si—Si functional group, 
         [0000]    
       
         
           
             
               C 
               1 
             
             = 
             
               0.75 
               2 
             
           
         
       
     
         [0000]    in both the geometry relationships (Eqs. (15.2-15.5)) and the energy equation (Eq. (15.61)). This is the same value as C 1  of the chlorine molecule given in the corresponding section. The hybridization factor gives the parameters c 2  and C 2  for both as well. To meet the equipotential condition of the union of the two Si3sp 3  HOs, c 2  and C 2  of Eqs. (15.2-15.5) and Eq. (15.61) for the Si—Si-bond MO is given by Eq. (15.72) as the ratio of 10.31324 eV, the magnitude of E Coulomb (Si silane ,3sp 3 ) (Eq. (20.4)), and 13.605804 eV, the magnitude of the Coulombic energy between the electron and proton of H (Eq. (1.243)): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 silane 
                                  
                                 Si 
                               
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 silane 
                                  
                                 Si 
                               
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             10.31324 
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             13.605804 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.75800 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.33 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The energy of the MO is matched to that of the Si3sp 3  HO such that E(AO/HO) is E(Si,3sp 3 ) given by Eq. (20.7) and E T (atom−atom,msp 3 .AO) is two times E T (Si—Si,3sp 3 ) given by Eq. (20.12). 
         [0326]    For the Si—H-bond MO of the SiH n=123  functional groups, c 1  is one and C 1 =0.75 based on the orbital composition as in the case of the C—H-bond MO. In silanes, the energy of silicon is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). Thus, c 2  in Eq. (15.61) is also one, and the energy matching condition is determined by the C 2  parameter, the hybridization factor for the Si—H-bond MO given by Eq. (20.33). Since the energy of the MO is matched to that of the Si3sp 3  HO, E(AO/HO) is E(Si,3sp 3 ) given by Eq. (20.7) and E T (atom−atom,msp 3 .AO) is E T (Si—H,3sp 3 ) given by Eq. (20.16). The energy E D  (SiH n=123 ) of the functional groups SiH n=123  is given by the integer n times that of Si—H: 
         [0000]      E D (SiH n=1,2,3 )= n E D (SiH)  (20.34)
 
         [0327]    Similarly, for silane, E T (atom−atom,msp 3 .AO) is E T (Si—H,3sp 3 ) given by Eq. (20.20). The energy E D  (SiH 4 ) of SiH 4  is given by the integer 4 times that of the SiH n=4  functional group: 
         [0000]      E D (SiH 4 )=4E D (SiH n=4 )  (20.35)
 
         [0328]    The symbols of the functional groups of silanes are given in Table 20.2. The geometrical (Eqs. (15.1-15.5), (20.1-20.16), (20.29), and (20.32-20.33)), intercept (Eqs. (15.80-15.87) and (20.21)), and energy (Eqs. (15.61), (20.1-20.16), and (20.33-20.35)) parameters of silanes are given in Tables 20.3, 20.4 (as shown in the priority document), and 20.5, respectively. The total energy of each silane given in Table 20.6 (as shown in the priority document) was calculated as the sum over the integer multiple of each E D  (Group) of Table 20.5 corresponding to functional-group composition of the molecule. E mag  of Table 20.5 is given by Eqs. (15.15) and (20.3). The bond angle parameters of silanes determined using Eqs. (15.88-15.117) are given in Table 20.7 (as shown in the priority document). In particular for silanes, the bond angle ∠HSiH is given by Eq. (15.99) wherein E T (atom−atom,msp 3 .AO) is given by Eq. (20.16) in order to match the energy donated from the Si3sp 3  HO to the Si—H-bond MO due to the energy of silicon being less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). The parameter c′ 2  is given by Eq. (15.100) as in the case of a H—H terminal bond of an alkyl or alkenyl group, except that c 2 (Si3sp 3 ) is given by Eq. (15.63) such that c′ 2  is the ratio of c 2  of Eq. (15.72) for the H—H bond which is one and c 2  of the silicon of the corresponding Si—H bond considering the effect of the formation of the H—H terminal bond: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           c 
                           2 
                           ′ 
                         
                         = 
                         
                           1 
                           
                             
                               c 
                               2 
                             
                              
                             
                               ( 
                               
                                 Si 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             13.605804 
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               E 
                               Coulomb 
                             
                              
                             
                               ( 
                               
                                 Si 
                                 - 
                                 
                                   H 
                                    
                                   
                                       
                                   
                                    
                                   Si 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.36 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The color scale, translucent view of the charge-densities of the series Si comprising the concentric shells of the central Si atom of each member with the outer shell joined with one or more hydrogen MOs are shown in  FIGS. 19A-D . The charge-density of disilane is shown in  FIG. 20 . 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 20.2 
               
             
             
               
                   
               
               
                 The symbols of the functional groups of silanes. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
                   
               
               
                   
                 SiH group of SiH n=1,2,3   
                 Si—H (i) 
               
               
                   
                 SiH group of SiH n=4   
                 Si—H (ii) 
               
               
                   
                 SiSi bond (n-Si) 
                 Si—Si 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 20.3 
               
             
             
               
                   
               
               
                 The geometrical bond parameters of silanes 
               
               
                 and experimental values [2]. 
               
             
          
           
               
                 Parameter 
                 Si—H (i) and (ii)Group 
                 Si—Si Group 
               
               
                   
               
               
                 a (a 0 ) 
                 2.24744 
                 2.74744 
               
               
                 c′ (a 0 ) 
                 1.40593 
                 2.19835 
               
               
                 Bond Length 2c′ (Å) 
                 1.48797 
                 2.32664 
               
               
                 Exp. Bond Length (Å) 
                 1.492 (Si 2 H 6 ) 
                 2.331 (Si 2 H 6 ) 
               
               
                   
                   
                 2.32 (Si 2 Cl 6 ) 
               
               
                 b, c (a 0 ) 
                 1.75338 
                 1.64792 
               
               
                 e 
                 0.62557 
                 0.80015 
               
               
                   
               
             
          
         
       
     
         [0000]                                                        TABLE 20.4                   The energy parameters (eV) of the functional groups of silanes.                Si—H (i)   Si—H (ii)   Si—Si       Parameters   Group   Group   Group                    n 1     1   1   1       n 2     0   0   0       n 3     0   0   0       C 1     0.75   0.75   0.37500       C 2     0.75800   0.75800   0.75800       c 1     1   1   1       c 2     1   1   0.75800       c 3     0   0   0       c 4     1   1   2       c 5     1   1   0       C 1o     0.75   0.75   0.37500       C 2o     0.75800   0.75800   0.75800       V e  (eV)   −28.41703   −28.41703   −20.62357       V p  (eV)   9.67746   9.67746   6.18908       T (eV)   6.32210   6.32210   3.75324       V m  (eV)   −3.16105   −3.16105   −1.87662       E (AO/HO) (eV)   −10.25487   −10.25487   −10.25487       ΔE H     2     MO  (AO/HO) (eV)   0   0   0       E T  (AO/HO) (eV)   −10.25487   −10.25487   −10.25487       E T  (H 2 MO) (eV)   −25.83339   −25.83339   −22.81274       E T  (atom-atom, msp 3  · AO) (eV)   −1.06358   −1.37960   −0.96031       E T  (MO) (eV)   −26.89697   −27.21299   −23.77305       ω (10 15  rad/s)   13.4257   13.4257   4.83999       E K  (eV)   8.83703   8.83703   3.18577       Ē D  (eV)   −0.15818   −0.16004   −0.08395       Ē Kvib  (eV)   0.25315 [3]   0.25315 [3]   0.06335 [3]       Ē osc  (eV)   −0.03161   −0.03346   −0.05227       E mag  (eV)   0.04983   0.04983   0.04983       E T  (Group) (eV)   −26.92857   −27.24646   −23.82532       E initial  (c 4  AO/HO) (eV)   −10.25487   −10.25487   −10.25487       E initial  (c 5  AO/HO) (eV)   −13.59844   −13.59844   0       E D  (Group) (eV)   3.07526   3.39314   3.31557       Exp. E D  (Group) (eV)   3.0398 (Si—H [4])       3.3269 (H 3 Si—SiH 3  [5])                    
Alkyl silanes and disilanes (Si m ,C n H 2(m+n+2 , m,n=1,2,3,4,5 . . . ∞)
 
         [0329]    The branched-chain alkyl silanes and disilanes, Si m ,C n H 2(m+n)+2 , comprise at least a terminal methyl group (CH 3 ) and at least one Si bound by a carbon-silicon single bond comprising a C—Si group, and may comprise methylene (CH 2 ), methylyne (CH), C—C, SiH n=1,2,3 , and Si—Si functional groups. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C—C bonds can be identified. The n-alkane C—C bond is the same as that of straight-chain alkanes. In addition, the C—C bonds within isopropyl ((CH 3 ) 2  CH) and t-butyl ((CH 3 ) 3 C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C—C bonds comprise functional groups. These groups in branched-chain alkyl silanes and disilanes are equivalent to those in branched-chain alkanes, and the SiH n=1,2,3  functional groups of alkyl silanes are equivalent to those in silanes (Si n H 2n+2 ). The Si—Si functional group of alkyl silanes is equivalent to that in silanes; however, in dialkyl silanes, the Si—Si functional group is different due to an energy matching condition with the C—Si bond having a mutual silicon atom. 
         [0330]    For the C—Si functional group, hybridization of the 2s and 2p AOs of each C and the 3s and 3p AOs of each Si to form single 2sp 3  and 3sp 3  shells, respectively, forms an energy minimum, and the sharing of electrons between the C2sp 3  and Si3sp 3  HOs to form a MO permits each participating orbital to decrease in radius and energy. In branched-chain alkyl silanes, the energy of silane is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). Thus, c 2  in Eq. (15.61) is one, and the energy matching condition is determined by the C 2  parameter. Then, the C2sp 3  HO has an energy of E(C,2sp 3 )=−14.63489 eV (Eq. (15.25)), and the Si3sp 3  HO has an energy of E(Si,3sp 3 )=−10.25487 eV (Eq. (20.7)). To meet the equipotential condition of the union of the C—Si H 2 -type-ellipsoidal-MO with these orbitals, the hybridization factor C 2  of Eq. (15.61) for the C—Si-bond MO given by Eq. (15.77) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               C 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Si 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Si 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 C 
                                 , 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               10.25487 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               14.63489 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.70071 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.37 
                   ) 
                 
               
             
           
         
       
     
         [0000]    For monosilanes, E T (atom−atom,msp 3 .AO) of the C—Si-bond MO is −1.20473 eV corresponding to the single-bond contributions of carbon and silicon of −0.72457 eV given by Eq. (14.151) and −0.48015 eV given by Eq. (14.151) with s=1 in Eq. (15.18). The energy of the C—Si-bond MO is the sum of the component energies of the H 2 -type ellipsoidal MO given in Eq. (15.51) with E(AO/HO)=E(Si,3sp 3 ) given by Eq. (20.7) and ΔE H     2     MO (AO/HO)=E T (atom−atom,msp 3 .AO) in order to match the energies of the carbon and silicon HOs. 
         [0331]    For the co-bonded Si—Si group of the C—Si group of disilanes, 
         [0000]    E T (atom−atom,msp 3 .AO) is −0.96031 eV, two times E T (Si—Si,3sp 3 ) given by Eq. (20.12). Thus, in order to match the energy between these groups,
 
E T (atom−atom,msp 3 .AO) of the C—Si-bond MO is −0.92918 eV corresponding to the single-bond methylene-type contribution of carbon given by Eq. (14.513). As in the case of monosilanes, E(AO/HO)=E(Si,3sp 3 ) given by Eq. (20.7) and
 
ΔE H     2     MO (AO/HO)=E T (atom−atom,msp 3 .AO) in order to match the energies of the carbon and silicon HOs.
 
         [0332]    The symbols of the functional groups of alkyl silanes and disilanes are given in Table 20.8. The geometrical (Eqs. (15.1-15.5), (20.1-20.16), (20.29), (20.32-20.33) and (20.37)) and intercept (Eqs. (15.80-15.87) and (20.21)) parameters of alkyl silanes and disilanes are given in Tables 20.9 and 20.10 (as shown in the priority document), respectively. Since the energy of the Si3sp 3  HO is matched to that of the C2sp 3  HO, the radius r mol2sp     3    of the Si3sp 3  HO of the silicon atom and the C2sp 3  HO of the carbon atom of a given C—Si-bond MO is calculated after Eq. (15.32) by considering ΣE T     mol   (MO,2sp 3 ), the total energy donation to all bonds with which each atom participates in bonding. In the case that the MO does not intercept the Si HO due to the reduction of the radius from the donation of Si 3sp 3  HO charge to additional MO&#39;s, the energy of each MO is energy matched as a linear sum to the Si HO by contacting it through the bisector current of the intersecting MOs as described in the Methane Molecule (CH 4 ) section. The energy (Eqs. (15.61), (20.1-20.16), and (20.33-20.37)) parameters of alkyl silanes and disilanes are given in Table 20.11 (as shown in the priority document). The total energy of each alkyl silane and disilane given in Table 20.12 (as shown in the priority document) was calculated as the sum over the integer multiple of each E D  (Group) of Table 20.11 (as shown in the priority document) corresponding to functional-group composition of the molecule. The bond angle parameters of alkyl silanes and disilanes determined using Eqs. (15.88-15.117) and Eq. (20.36) are given in Table 20.13 (as shown in the priority document). The charge-densities of exemplary alkyl silane, dimethylsilane and alkyl disilane, hexamethyldisilane comprising the concentric shells of atoms with the outer shell bridged by one or more H 2 -type ellipsoidal MOs or joined with one or more hydrogen MOs are shown in  FIGS. 21 and 22 , respectively. 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 20.8 
               
             
             
               
                   
               
               
                 The symbols of functional groups of alkyl silanes and disilanes. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
                   
               
               
                   
                 CSi bond (monosilanes) 
                 C—Si (i) 
               
               
                   
                 CSi bond (disilanes) 
                 C—Si (ii) 
               
               
                   
                 SiSi bond (n-Si) 
                 Si—Si 
               
               
                   
                 SiH group of SiH n=1,2,3   
                 Si—H 
               
               
                   
                 CH 3  group 
                 C—H (CH 3 ) 
               
               
                   
                 CH 2  group 
                 C—H (CH 2 ) 
               
               
                   
                 CH 
                 C—H 
               
               
                   
                 CC bond (n-C) 
                 C—C (a) 
               
               
                   
                 CC bond (iso-C) 
                 C—C (b) 
               
               
                   
                 CC bond (tert-C) 
                 C—C (c) 
               
               
                   
                 CC (iso to iso-C) 
                 C—C (d) 
               
               
                   
                 CC (t to t-C) 
                 C—C (e) 
               
               
                   
                 CC (t to iso-C) 
                 C—C (f) 
               
               
                   
                   
               
             
          
         
       
     
       Silicon Oxides, Silicic Acids, Silanols, Siloxanes and Disiloxanes 
       [0333]    The silicon oxides, silicic acids, silanols, siloxanes, and disiloxanes each comprise at least one Si—O group, and this group in disiloxanes is part of the Si—O—Si moiety. Silicic acids may have up to three Si—H bonds corresponding to the SiH n=1,2,3  functional groups of alkyl silanes, and silicic acids and silanols further comprise at least one OH group equivalent to that of alcohols. In addition to the SiH n=1,2,3  group of alkyl silanes, silanols, siloxanes, and disiloxanes may comprise the functional groups of organic molecules as well as the C—Si group of alkyl silanes. The alkyl portion of the alkyl silanol, siloxane, or disiloxane may comprise at least one terminal methyl group (CH 3 ) the end of each alkyl chain, and may comprise methylene (CH 2 ), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C—C bonds can be identified. The n-alkane C—C bond is the same as that of straight-chain alkanes. In addition, the C—C bonds within isopropyl ((CH 3 ) 2  CH) and t-butyl ((CH 3 ) 3 C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C—C bonds comprise functional groups. The branched-chain-alkane groups in silanols, siloxanes, and disiloxanes are equivalent to those in branched-chain alkanes. The alkene groups when present such as the C═C group are equivalent to those of the corresponding alkene. Siloxanes further comprise two types of C—O functional groups, one for methyl or t-butyl groups corresponding to the C and the other for general alkyl groups as given for ethers. 
         [0334]    The distinguishing aspect of silicon oxides, silicic acids, silanols, siloxanes, and disiloxane is the nature of the corresponding Si—O functional group. In general, the sharing of electrons between a Si3sp 3  HO and an O2p AO to form a Si—O-bond MO permits each participating orbital to decrease in size and energy. Consider the case wherein the Si3sp 3  HO donates an excess of 50% of its electron density to the Si—O-bond MO to form an energy minimum while further satisfying the potential, kinetic, and orbital energy relationships. By considering this electron redistribution in the molecule comprising a Si—O bond as well as the fact that the central field decreases by an integer for each successive electron of the shell, the radius r Si—O3sp     3    of the Si3sp 3  shell may be calculated from the Coulombic energy using Eq. (15.18): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             Si 
                             - 
                             
                               O 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                             
                           
                         
                         = 
                           
                          
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     10 
                                   
                                   13 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               0.5 
                             
                             ) 
                           
                            
                           
                             
                               e 
                               2 
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     105.23554 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             9.5 
                              
                             
                                 
                             
                              
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   105.23554 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           1.22825 
                            
                           
                               
                           
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.38 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (20.38), the Coulombic energy E Coulomb (Si Si—O ,3sp 3 ) of the outer electron of the Si3sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 Si 
                                 
                                   Si 
                                   - 
                                   O 
                                 
                               
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 Si 
                                 - 
                                 
                                   O 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.22825 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             11.07743 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.39 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, the spin-paired 3s electrons are promoted to Si3sp 3  shell as unpaired electrons. The energy for the promotion is the magnetic energy given by Eq. (20.6). Using Eqs. (20.6) and (20.39), the energy E(Si Si—O ,3sp 3 ) of the outer electron of the Si3sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 Si 
                                 
                                   Si 
                                   - 
                                   O 
                                 
                               
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   silane 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 e 
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     12 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               11.07743 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.05836 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             11.01906 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.40 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (Si—O,3sp 3 ), the energy change of each Si3sp 3  shell with the formation of the Si—O-bond MO is given by the difference between Eq. (20.40) and Eq. (20.7): 
         [0000]      E T (Si—O,3sp 3 )=E(Si Si—O ,3sp 3 )−E(Si,3sp 3 )=−11.01906 eV−(−10.25487 eV)=−0.76419 eV  (20.41)
 
         [0335]    Using Eq. (15.28), to meet the energy matching condition in silanols and siloxanes for all σ MOs at the Si3sp 3  HO and O2p AO of each Si—O-bond MO as well as with the C2sp 3  HOs of the molecule, the energy E(Si RSi—OR′ ,3sp 3 ) (R,R′ are alkyl or H) of the outer electron of the Si3sp 3  shell of the silicon atom must be the average of E(Si silane ,3sp 3 ) (Eq. (20.11)) and E T (Si—O,3sp 3 ) (Eq. (20.40)): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 Si 
                                 
                                   RSi 
                                   - 
                                   
                                     OR 
                                     ′ 
                                   
                                 
                               
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               E 
                                
                               
                                 ( 
                                 
                                   
                                     Si 
                                     silane 
                                   
                                   , 
                                   
                                     3 
                                      
                                     
                                         
                                     
                                      
                                     
                                       sp 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               E 
                                
                               
                                 ( 
                                 
                                   
                                     Si 
                                     
                                       Si 
                                       - 
                                       O 
                                     
                                   
                                   , 
                                   
                                     3 
                                      
                                     
                                         
                                     
                                      
                                     
                                       sp 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           2 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ( 
                               
                                 
                                   - 
                                   10.73503 
                                 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                             + 
                             
                               ( 
                               
                                 
                                   - 
                                   11.01906 
                                 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                           2 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             10.87705 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.42 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eq. (15.29), E T     silanol, silazane   (Si—O,3sp 3 ), the energy change of each Si3sp 3  shell with the formation of each RSi—OR′-bond MO, must be the average of E T (Si—Si,3sp 3 ) (Eq. (20.12)) and E T (Si—O,3sp 3 ) (Eq. (20.41)): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               T 
                               
                                 
                                   silanol 
                                   , 
                                 
                                 siloxane 
                               
                             
                           
                            
                           
                             ( 
                             
                               
                                 Si 
                                 - 
                                 O 
                               
                               , 
                               
                                 3 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               
                                 E 
                                 T 
                               
                                
                               
                                 ( 
                                 
                                   
                                     Si 
                                     - 
                                     Si 
                                   
                                   , 
                                   
                                     3 
                                      
                                     
                                       sp 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 E 
                                 T 
                               
                                
                               
                                 ( 
                                 
                                   
                                     Si 
                                     - 
                                     O 
                                   
                                   , 
                                   
                                     3 
                                      
                                     
                                       sp 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           2 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               ( 
                               
                                 
                                   - 
                                   0.48015 
                                 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                             + 
                             
                               ( 
                               
                                 
                                   - 
                                   0.76419 
                                 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                           2 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             0.62217 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.43 
                   ) 
                 
               
             
           
         
       
     
         [0336]    To meet the energy matching condition in silicic acids for all σ MOs at the Si3sp 3  HO and O2p AO of each Si—O-bond MO as well as all H AOs, the energy E(Si H     n     Si—(OH)     4-n   ,3sp 3 ) of the outer electron of the Si3sp 3  shell of the silicon atom must be the average of E(Si silane ,3sp 3 )(Eq. (20.15)) and E T (Si—O,3sp 3 ) (Eq. (20.40)): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 Si 
                                 
                                   
                                     
                                       H 
                                       n 
                                     
                                      
                                     Si 
                                   
                                   - 
                                   
                                     
                                       ( 
                                       OH 
                                       ) 
                                     
                                     
                                       4 
                                       - 
                                       n 
                                     
                                   
                                 
                               
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               E 
                                
                               
                                 ( 
                                 
                                   
                                     Si 
                                     silane 
                                   
                                   , 
                                   
                                     3 
                                      
                                     
                                         
                                     
                                      
                                     
                                       sp 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               E 
                                
                               
                                 ( 
                                 
                                   
                                     Si 
                                     
                                       Si 
                                       - 
                                       O 
                                     
                                   
                                   , 
                                   
                                     3 
                                      
                                     
                                         
                                     
                                      
                                     
                                       sp 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           2 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ( 
                               
                                 
                                   - 
                                   11.37682 
                                 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                             + 
                             
                               ( 
                               
                                 
                                   - 
                                   11.01906 
                                 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                           2 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             11.16876 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.44 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eq. (15.29), E T     silicic acid   (Si—O,3sp 3 ), the energy change of each Si3sp 3  shell with the formation of each RSi—OR′-bond MO, must be the average of E T (Si—H, 3sp 3 ) (Eq. (20.16)) and E T (Si—O, 3sp 3 ) (Eq. (20.41)): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               T 
                               
                                 silicic 
                                  
                                 
                                     
                                 
                                  
                                 acid 
                               
                             
                           
                            
                           
                             ( 
                             
                               
                                 Si 
                                 - 
                                 O 
                               
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               
                                 E 
                                 T 
                               
                                
                               
                                 ( 
                                 
                                   
                                     Si 
                                     - 
                                     H 
                                   
                                   , 
                                   
                                     3 
                                      
                                     
                                         
                                     
                                      
                                     
                                       sp 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 E 
                                 T 
                               
                                
                               
                                 ( 
                                 
                                   
                                     Si 
                                     - 
                                     O 
                                   
                                   , 
                                   
                                     3 
                                      
                                     
                                         
                                     
                                      
                                     
                                       sp 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           2 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ( 
                               
                                 
                                   - 
                                   1.06358 
                                 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                             + 
                             
                               ( 
                               
                                 
                                   - 
                                   0.76419 
                                 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                           2 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             0.91389 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.45 
                   ) 
                 
               
             
           
         
       
     
         [0337]    Using Eqs. (20.22-22.26), the general force balance equation for the σ-MO of the silicon to oxygen Si—O-bond MO in terms of n e  and |L i | corresponding to the angular momentum terms of the 3sp 3  HO shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ℏ 
                         2 
                       
                       
                         
                           m 
                           e 
                         
                          
                         
                           a 
                           2 
                         
                          
                         
                           b 
                           2 
                         
                       
                     
                      
                     D 
                   
                   = 
                   
                     
                       
                         
                           e 
                           2 
                         
                         
                           8 
                            
                           
                             πɛ 
                             0 
                           
                            
                           
                             ab 
                             2 
                           
                         
                       
                        
                       D 
                     
                     + 
                     
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                     - 
                     
                       
                         ( 
                         
                           
                             
                               n 
                               e 
                             
                             2 
                           
                           + 
                           
                             
                               ∑ 
                               i 
                             
                              
                             
                                 
                             
                              
                             
                               
                                  
                                 
                                   L 
                                   i 
                                 
                                  
                               
                               Z 
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                   
                 
               
               
                 
                   ( 
                   20.46 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Having a solution for the semimajor axis a of 
         [0000]    
       
         
           
             
               
                 
                   a 
                   = 
                   
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             n 
                             e 
                           
                           2 
                         
                         + 
                         
                           
                             ∑ 
                             i 
                           
                            
                           
                               
                           
                            
                           
                             
                                
                               
                                 L 
                                 i 
                               
                                
                             
                             Z 
                           
                         
                       
                       ) 
                     
                      
                     
                       a 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   20.47 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In terms of the angular momentum L, the semimajor axis a is 
         [0000]    
       
         
           
             
               
                 
                   a 
                   = 
                   
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             n 
                             e 
                           
                           2 
                         
                         + 
                         
                           L 
                           Z 
                         
                       
                       ) 
                     
                      
                     
                       a 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   20.48 
                   ) 
                 
               
             
           
         
       
     
         [0338]    Using the semimajor axis, the geometric and energy parameters of the MO are calculated using Eqs. (15.1-15.117) in the same manner as the organic functional groups given in the Organic Molecular Functional Groups and Molecules section. The semimajor axis a solutions given by Eq. (20.48) of the force balance equation, Eq. (20.46), for the σ-MO of the Si—O-bond MO of each functional group of silicon oxide, silicon dioxide, silicic acids, silanols, siloxanes, and disiloxanes are given in Table 20.15 (as shown in the priority document) with the force-equation parameters Z=14, n e , and L corresponding to the angular momentum of the Si3sp 3  HO shell. 
         [0339]    For the Si—O functional groups, hybridization of the 3s and 3p AOs of Si to form a single 3sp 3  shell forms an energy minimum, and the sharing of electrons between the Si3sp 3  HO and the O AO to form σ MO permits each participating orbital to decrease in radius and energy. The O AO has an energy of E(O)=−13.61805 eV, and the Si3sp 3  HO has an energy of E(Si,3sp 3 )=−10.25487 eV (Eq. (20.7)). To meet the equipotential condition of the union of the Si—O H 2 -type-ellipsoidal-MO with these orbitals, the corresponding hybridization factors c 2  and C 2  of Eq. (15.61) for silicic acids, silanols, siloxanes, and disiloxanes and the hybridization factor C 2  of silicon oxide and silicon dioxide given by Eq. (15.77) are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               O 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Si 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               O 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Si 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Si 
                                 , 
                                 
                                   3 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               O 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               10.25487 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               13.61805 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.75304 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20.49 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Each bond of silicon oxide and silicon dioxide is a double bond such that c 1 =2 and C 1 =0.75 in the geometry relationships (Eqs. (15.2-15.5)) and the energy equation (Eq. (15.61)). Each Si—O bond in silicic acids, silanols, siloxanes, and disiloxanes is a single bond corresponding to c 1 =1 and C 1 =0.5 as in the case of alkanes (Eq. (14.152))). 
         [0340]    Since the energy of the MO is matched to that of the Si3sp 3  HO, E(AO/HO) in Eq. (15.61) is E(Si,3sp 3 ) given by Eq. (20.7) and twice this value for double bonds. E T (atom−atom,msp 3 .AO) of the Si—O-bond MO of each functional group is determined by energy matching in the molecule while achieving an energy minimum. For silicon oxide and silicon dioxide, E T (atom−atom,msp 3 .AO) is three and two times −1.37960 eV given by Eq. (20.20), respectively. E T (atom−atom,msp 3 .AO) of silicic acids is two times −0.91389 eV given by Eq. (20.45). E T (atom−atom,msp 3 .AO) of silanols, siloxanes, and disiloxanes is two times −0.62217 eV given by Eq. (20.43). 
         [0341]    The symbols of the functional groups of silicon oxides, silicic acids, silanols, siloxanes, and disiloxanes are given in Table 20.14. The geometrical (Eqs. (15.1-15.5), (20.1-20.21), (20.29), (20.32-20.33), (20.37), and (20.46-20.49)) and intercept (Eqs. (15.80-15.87) and (20.21)) parameters are given in Tables 20.15 and 20.16, respectively (as shown in the priority document). The energy (Eqs. (15.61), (20.1-20.20), (20.33-20.35), (20.37-45), and (20.49)) parameters are given in Table 20.17 (as shown in the priority document). The total energy of each silicon oxide, silicic acid, silanol, siloxane, or disiloxane given in Table 20.18 (as shown in the priority document) was calculated as the sum over the integer multiple of each E D  (Group) of Table 20.17 (as shown in the priority document) corresponding to functional-group composition of the molecule. The bond angle parameters determined using Eqs. (15.88-15.117) are given in Table 20.19 (as shown in the priority document). The charge-densities of exemplary siloxane, ((CH 3 ) 2 SiO) 3  and disiloxane, hexamethyldisiloxane comprising the concentric shells of atoms with the outer shell bridged by one or more H 2 -type ellipsoidal MOs or joined with one or more hydrogen MOs are shown in  FIGS. 23 and 24 , respectively. 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 20.14 
               
             
             
               
                   
               
               
                 The symbols of functional groups of silicon oxides, silicic 
               
               
                 acids, silanols, siloxanes and disiloxanes. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
                   
               
               
                   
                 SiO bond (silicon oxide) 
                 Si—O (i) 
               
               
                   
                 SiO bond (silicon dioxide) 
                 Si—O (ii) 
               
               
                   
                 SiO bond (silicic acid) 
                 Si—O (iii) 
               
               
                   
                 SiO bond (silanol and siloxane) 
                 Si—O (iv) 
               
               
                   
                 Si—OSi bond (disiloxane) 
                 Si—O (v) 
               
               
                   
                 SiH group of SiH n=1,2,3   
                 Si—H 
               
               
                   
                 CSi bond 
                 C—Si (i) 
               
               
                   
                 OH group 
                 OH 
               
               
                   
                 CO (CH 3 —O—and (CH 3 ) 3 C—O—) 
                 C—O (i) 
               
               
                   
                 CO (alkyl) 
                 C—O (ii) 
               
               
                   
                 CH 3  group 
                 C—H (CH 3 ) 
               
               
                   
                 CH 2  group 
                 C—H (CH 2 ) 
               
               
                   
                 CH 
                 C—H 
               
               
                   
                 CC bond (n-C) 
                 C—C (a) 
               
               
                   
                 CC bond (iso-C) 
                 C—C (b) 
               
               
                   
                 CC bond (tert-C) 
                 C—C (c) 
               
               
                   
                 CC (iso to iso-C) 
                 C—C (d) 
               
               
                   
                 CC (t to t-C) 
                 C—C (e) 
               
               
                   
                 CC (t to iso-C) 
                 C—C (f) 
               
               
                   
                   
               
             
          
         
       
     
       REFERENCES 
       [0000]    
       
         1. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), pp. 10-202 to 10-204. 
         2. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), p. 9-26. 
         3. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), p. 9-86. 
         4. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), p. 9-57. 
         5. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), p. 9-71. 
         6. B. H. Boo, P. B. Armentrout, “Reaction of silicon ion ( 2 P) with silane (SiH 4 , SiD 4 ). Heats of formation of SiH n , SiH n   +  (n=1, 2, 3), and Si 2 H n   +  (n=0, 1, 2, 3). Remarkable isotope exchange reaction involving four hydrogen shifts,” J. Am. Chem. Soc., (1987), Vol. 109, pp. 3549-3559. 
         7. G. Katzer, M. C. Ernst, A. F. Sax, J. Kalcher, “Computational thermochemistry of medium-sized silicon hydrides,” J. Phys. Chem. A, (1997), Vol. 101, pp. 3942-3958. 
         8. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), pp. 9-19 to 9-45. 
         9. M. R. Frierson, M. R. Imam, V. B. Zalkow, N. L. Allinger, “The MM2 force field for silanes and polysilanes,” J. Org. Chem., Vol. 53, (1988), pp. 5248-5258. 
         10. D. Lin-Vien. N. B. Colthup, W. G. Fateley, J. G. Grasselli,  The Handbook of Infrared and Raman Frequencies of Organic Molecules , Academic Press, Inc., Harcourt Brace Jovanovich, Boston, (1991), p. 256. 
         11. G. Herzberg,  Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules , Van Nostrand Reinhold Company, New York, N.Y., (1945), p. 344. 
         12. R. J. Fessenden, J. S. Fessenden,  Organic Chemistry , Willard Grant Press. Boston, Mass., (1979), p. 320. 
         13. cyclohexane at http://webbook.nist.gov/. 
         14. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), p. 5-28. 
         15. M. J. S. Dewar, C. Jie, “AM 1 calculations for compounds containing silicon”, Organometallics, Vol. 6, (1987), pp. 1486-1490. 
         16. R. Walsh, “Certainties and uncertainties in the heats of formation of the methylsilylenes”, Organometallics, Vol. 8, (1989), pp. 1973-1978. 
         17. R. W. Kilb, L. Pierce, “Microwave spectrum, structure, and internal barrier of methyl silane,” J. Chem. Phys., Vol. 27, No. 1, (1957), pp. 108-112. 
         18. M. W. Chase, Jr., C. A. Davies, J. R. Downey, Jr., D. J. Frurip, R. A. McDonald, A. N. Syverud, JANAF Thermochemical Tables, Third Edition, Part II, Cr—Zr, J. Phys. Chem. Ref Data, Vol. 14, Suppl. 1, (1985), p. 1728. 
         19. M. W. Chase, Jr., C. A. Davies, J. R. Downey, Jr., D. J. Frurip, R. A. McDonald, A. N. Syverud, JANAF Thermochemical Tables, Third Edition, Part II, Cr—Zr, J. Phys. Chem. Ref Data, Vol. 14, Suppl. 1, (1985), p. 1756. 
         20. D. Nyfeler, T. Armbruster, “Silanol groups in minerals and inorganic compounds”, American Mineralogist, Vol. 83, (1998), pp. 119-125. 
         21. K. P. Huber, G. Herzberg,  Molecular Spectra and Molecular Structure, IV. Constants of Diatomic Molecules , Van Nostrand Reinhold Company, New York, (1979). 
         22. J. Crovisier, Molecular Database—Constants for molecules of astrophysical interest in the gas phase: photodissociation, microwave and infrared spectra, Ver. 4.2, Observatoire de Paris, Section de Meudon, Meudon, France, May 2002, pp. 34-37, available at http://wwwusr.obspm.fr/˜crovisie/. 
         23. dimethyl ether at http://webbook.nist.gov/. 
         24. R. J. Fessenden, J. S. Fessenden, Organic Chemistry, Willard Grant Press. Boston, Mass., (1979), p. 320. 
         25. fluoroethane at http://webbook.nist.gov/. 
         26. G. Herzberg,  Molecular Spectra and Molecular Structure II Infrared and Raman Spectra of Polyatomic Molecules , Van Nostrand Reinhold Company, New York, N.Y., (1945), p. 326. 
         27. M. D. Allendorf, C. F. Melius, P. Ho, M. R. Zachariah, “Theoretical study of the thermochemistry of molecules in the Si—O—H system,” J. Phys. Chem., Vol. 99, (1995), pp. 15285-15293. 
         28. N. S. Jacobson, E. J. Opila, D. L. Myers, E. H. Copeland, “Thermodynamics of gas phase species in the Si—O—H system,” J. Chem. Thermodynamics, Vol. 37, (2005), pp. 1130-1137. 
         29. J. D. Cox, G. Pilcher,  Thermochemistry of Organometallic Compounds , Academic Press, New York, (1970), pp. 468-469. 
         30. J. C. S. Chu, R. Soller, M. C. Lin, C. F. Melius, “Thermal decomposition of tetramethyl orthoscilicate in the gas phase: An experimental and theoretical study of the initiation process, J. Phys. Chem., Vol. 99, (1995), pp. 663-672. 
         31. R. Becerra, R. Walsh, In  The Chemistry of Organic Silicon Compounds ; Z. Rappaport, Y. Apeloig, Eds.; Thermochemistry, Vol. 2; Wiley, New York, (1998), Chp. 4. 
         32. C. L. Darling, H. B. Schlegel, “Heats of formation of SiH n O and SiH n O 2  calculated by ab initio molecular orbital methods at the G-2 level of theory,” J. Phys. Chem. Vol. 97, (1993), 8207-8211. 
         33. A. C. M. Kuo,  In Polymer Data Handbook ; Poly(dimethylsiloxane); Oxford University Press, (1999), p 419. 
       
     
       The Nature of the Semiconductor Bond of Silicon 
     Generalization of the Nature of the Semiconductor Bond 
       [0375]    Semiconductors are solids that have properties intermediate between insulators and metals. 
         [0376]    For an insulator to conduct, high energy and power are required to excite electrons into a conducing state in sufficient numbers. Application of high energy to cause electron ionization to the continuum level or to cause electrons to transition to conducing molecular orbitals (MOs) will give rise to conduction when the power is adequate to maintain a high population density of such states. Only high temperatures or extremely high-strength electric fields will provide enough energy and power to achieve an excited state population permissive of conduction. In contrast, metals are highly conductive at essentially any field strength and power. Diamond and alkali metals given in the corresponding sections are representative of insulator and metal classes of solids at opposite extremes of conductivity. It is apparent from the bonding of diamond comprising a network of highly stable MOs that it is an insulator, and the planar free-electron membranes in metals give rise to their high conductivity. 
         [0377]    Column IV elements silicon, germanium, and α-gray tin all have the diamond structure and are insulators under standard conditions. However, the electrons of these materials can be exited into a conducting excited state with modest amounts of energy compared to a pure insulator. As opposed to the 5.2 eV excitation energy for carbon, silicon, germanium, and α-gray tin have excitation energies for conduction of only 1.1 eV, 0.61 eV, and 0.078 eV, respectively. Thus, a semiconductor can carry a current by providing the relatively small amount of energy required to excite electrons to conducting excited states. As in the case of insulators, excitation can occur thermally by a temperature increase. Since the number of excited electrons increases with temperature, a concomitant increase in conductance is observed. This behavior is the opposite of that of metals. Alternatively, the absorption of photons of light causes the electrons in the ground state to be excited to a conducting state which is the basis of conversion of solar power into electricity in solar cells and detection and reception in photodetectors and fiber optic communications, respectively. In certain semiconductors, rather than decay by internal conversion to phonons, the energy of excited-state electrons is emitted as light as the electrons transition from the excited conducting state to the ground state. This photon emission process is the basis of light emitting diodes (LEDs) and semiconductor lasers which have broad application in industry. 
         [0378]    In addition to elemental materials such as silicon and germanium, semiconductors may be compound materials such as gallium arsenide and indium phosphide, or alloys such as silicon germanium or aluminum arsenide. Conduction in materials such as silicon and germanium crystals can be enhanced by adding small amounts (e.g. 1-10 parts per million) of dopants such as boron or phosphorus as the crystals are grown. Phosphorous with five valance electrons has a free electron even after contributing four electrons to four singe bond-MOs of the diamond structure of silicon. Since this fifth electron can be ionized from a phosphorous atom with only 0.011 eV provided by an applied electric field, phosphorous as an electron donor makes silicon a conductor. 
         [0379]    In an opposite manner to that of the free electrons of the dopant carrying electricity, an electron acceptor may also transform silicon to a conductor. Atomic boron has only three valance electrons rather than the four needed to replace a silicon atom in the diamond structure of silicon. Consequently, a neighboring silicon atom has an unpaired electron per boron atom. These electrons can be ionized to carry electricity as well. Alternatively, a valance electron of a silicon atom neighboring a boron atom can be excited to ionize and bind to the boron. The resulting negative boron ion can remain stationary as the corresponding positive center on silicon migrates from atom to atom in response to an applied electric field. This occurs as an electron transfers from a silicon atom with four electrons to one with three to fill the vacant silicon orbital. Concomitantly, the positive center is transferred in the opposite direction. Thus, inter-atomic electron transfer can carry current in a cascade effect as the propagation of a “hole” in the opposite direction as the sequentially transferring electrons. 
         [0380]    The ability of the conductivity of semiconductors to transition from that of insulators to that of metals with the application of sufficient excitation energy implies a transition of the excited electrons from covalent to a metallic-bond electrons. The bonding in diamond shown in the Nature of the Molecular Bond of Diamond section is a network of covalent bonds. Semiconductors comprise covalent bonds wherein the electrons are of sufficiently high energy that excitation creates an ion and a free electron. The free electron forms a membrane as in the case of metals given in the Nature of the Metallic Bond of Alkali Metals section. This membrane has the same planar structure throughout the crystal. This feature accounts for the high conductivity of semiconductors when the electrons are excited by the application of external fields or electromagnetic energy that causes ion-pair (M+−e − ) formation. 
         [0381]    It was demonstrated in the Nature of the Metallic Bond of Alkali Metals section that the solutions of the external point charge at an infinite planar conductor are also those of the metal ions and free electrons of metals based on the uniqueness of solutions of Maxwell&#39;s equations and the constraint that the individual electrons in a metal conserve the classical physical laws of the macro-scale conductor. The nature of the metal bond is a lattice of metal ions with field lines that end on the corresponding lattice of electrons comprising two-dimensional charge density σ given by Eq. (19.6) where each is equivalent electrostatically to a image point charge at twice the distance from the point charge of +e due to M + . Thus, the metallic bond is equivalent to the ionic bond given in the Alkali-Hydride Crystal Structures section with a Madelung constant of one with each negative ion at a position of one half the distance between the corresponding positive ions, but electrostatically equivalent to being positioned at twice this distance, the M + -M + -separation distance. Then, the properties of semiconductors can be understood as due to the excitation of a bound electron from a covalent state such as that of the diamond structure to a metallic state such as that of an alkali metal. The equations are the same as those of the corresponding insulators and metals. 
       Nature of the Insulator-Type Semiconductor Bond 
       [0382]    As given in the Nature of the Solid Molecular Bond of Diamond section, diamond C—C bonds are all equivalent, and each C—C bond can be considered bound to a t-butyl group at the corresponding vertex carbon. Thus, the parameters of the diamond C—C functional group are equivalent to those of the t-butyl C—C group of branched alkanes given in the Branched Alkanes section. Silicon also has the diamond structure. The diamond Si—Si bonds are all equivalent, and each Si—Si bond can be considered bound to three other Si—Si bonds at the corresponding vertex silicon. Thus, the parameters of the crystalline silicon Si—Si functional group are equivalent to those of the Si—Si group of silanes given in the Silanes (Si n H 2n+2 ) section except for the E T (atom−atom,msp 3 .AO) term of Eq. (15.61). Since bonds in pure crystalline silicon are only between Si3sp 3  HOs having energy less than the Coulombic energy between the electron and proton of H given by Eq. (1.243) E T (atom−atom,msp 3 .AO)=0. Also, as in the case of the C—C functional group of diamond, the Si3sp 3  HO magnetic energy E mag  is subtracted due to a set of unpaired electrons being created by bond breakage such that c 3  of Eq. (15.65) is one, and E mag  is given by Eqs. (15.15) and (20.3): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             mag 
                           
                            
                           
                             ( 
                             
                               Si 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                 sp 
                                 3 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             c 
                             3 
                           
                            
                           
                             
                               8 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 μ 
                                 B 
                                 2 
                               
                             
                             
                               r 
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             c 
                             3 
                           
                            
                           
                             
                               8 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 μ 
                                 B 
                                 2 
                               
                             
                             
                               
                                 ( 
                                 
                                   1.31926 
                                    
                                   
                                       
                                   
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             c 
                             3 
                           
                            
                           0.04983 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21.1 
                   ) 
                 
               
             
           
         
       
     
         [0383]    The symbols of the functional group of crystalline silicon is given in Table 21.1. The geometrical (Eqs. (15.1-15.5), (20.3-20.7), (20.29), and (20.33)) parameters of crystalline silicon are given in Table 21.2. Using the internuclear distance 2c′, the lattice parameter a of crystalline silicon is given by Eq. (17.3). The intercept (Eqs. (15.80-15.87), (20.3), and (20.21)) and energy (Eqs. (15.61), (20.3-20.7), and (20.33)) parameters of crystalline silicon are given in Tables 21.2, 21.3 (as shown in the priority document), and 21.4, respectively. 
         [0384]    The total energy of crystalline silicon given in Table 21.5 was calculated as the sum over the integer multiple of each E D  (Group) of Table 21.4 corresponding to functional-group composition of the solid. The bond angle parameters of crystalline silicon determined using Eqs. (15.88-15.117), (20.4), (20.33), and (21.1) are given in Table 21.6 (as shown in the priority document). The diamond structure of silicon in the insulator state is shown in  FIG. 25 . The predicted structure matches the experimental images of silicon determined using STM [1] as shown in  FIG. 26 . 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 21.1 
               
             
             
               
                   
               
               
                 The symbols of the functional group of crystalline silicon. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
                   
               
               
                   
                 SiSi bond (diamond-type-Si) 
                 Si—Si 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 21.2 
               
             
             
               
                   
               
               
                 The geometrical bond parameters of crystalline 
               
               
                 silicon and experimental values. 
               
             
          
           
               
                   
                   
                 Si—Si 
               
               
                   
                 Parameter 
                 Group 
               
               
                   
                   
               
               
                   
                 a (a 0 ) 
                 2.74744 
               
               
                   
                 c′ (a 0 ) 
                 2.19835 
               
               
                   
                 Bond Length 2c′ (Å) 
                 2.32664 
               
               
                   
                 Exp. Bond Length (Å) 
                 2.35 [2] 
               
               
                   
                 b, c (a 0 ) 
                 1.64792 
               
               
                   
                 e 
                 0.80015 
               
               
                   
                 Lattice Parameter a 1  (Å) 
                 5.37409 
               
               
                   
                 Exp. Lattice Parameter a 1  (Å) 
                 5.4306 [3] 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 21.4 
               
             
             
               
                   
               
               
                 The energy parameters (eV) of the functional 
               
               
                 group of crystalline silicon. 
               
             
          
           
               
                   
                   
                 Si—Si 
               
               
                   
                 Parameters 
                 Group 
               
               
                   
                   
               
             
          
           
               
                   
                 n 1   
                 1 
               
               
                   
                 n 2   
                 0 
               
               
                   
                 n 3   
                 0 
               
               
                   
                 C 1   
                 0.37500 
               
               
                   
                 C 2   
                 0.75800 
               
               
                   
                 c 1   
                 1 
               
               
                   
                 c 2   
                 0.75800 
               
               
                   
                 c 3   
                 0 
               
               
                   
                 c 4   
                 2 
               
               
                   
                 c 5   
                 0 
               
               
                   
                 C 1o   
                 0.37500 
               
               
                   
                 C 2o   
                 0.75800 
               
               
                   
                 V e  (eV) 
                 −20.62357 
               
               
                   
                 V p  (eV) 
                 6.18908 
               
               
                   
                 T (eV) 
                 3.75324 
               
               
                   
                 V m  (eV) 
                 −1.87662 
               
               
                   
                 E (AO/HO) (eV) 
                 −10.25487 
               
               
                   
                 ΔE H     2     MO  (AO/HO) (eV) 
                 0 
               
               
                   
                 E T  (AO/HO) (eV) 
                 −10.25487 
               
               
                   
                 E T  (H 2 MO) (eV) 
                 −22.81274 
               
               
                   
                 E T  (atom-atom, msp 3  · AO) (eV) 
                 0 
               
               
                   
                 E T  (MO) (eV) 
                 −22.81274 
               
               
                   
                 ω (10 15  rad/s) 
                 4.83999 
               
               
                   
                 E K  (eV) 
                 3.18577 
               
               
                   
                 Ē D  (eV) 
                 −0.08055 
               
               
                   
                 Ē Kvib  (eV) 
                 0.06335 [4] 
               
               
                   
                 Ē osc  (eV) 
                 −0.04888 
               
               
                   
                 E mag  (eV) 
                 0.04983 
               
               
                   
                 E T  (Group) (eV) 
                 −22.86162 
               
               
                   
                 E initial  (c 4  AO/HO) (eV) 
                 −10.25487 
               
               
                   
                 E initial  (c 5  AO/HO) (eV) 
                 0 
               
               
                   
                 E D  (Group) (eV) 
                 2.30204 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 21.5 
               
             
             
               
                   
               
               
                 The total bond energy of crystalline silicon calculated using 
               
               
                 the functional group composition and the energy of Table 21.4 
               
               
                 compared to the experimental value [5]. 
               
             
          
           
               
                   
                   
                   
                 Calculated 
                 Experimental 
                   
               
               
                   
                   
                   
                 Total Bond 
                 Total Bond 
                 Relative 
               
               
                 Formula 
                 Name 
                 Si—Si 
                 Energy (eV) 
                 Energy (eV) 
                 Error 
               
               
                   
               
               
                 Si n   
                 Crystalline 
                 1 
                 2.30204 
                 2.3095 
                 0.003 
               
               
                   
                 silicon 
               
               
                   
               
             
          
         
       
     
       Nature of the Conductor-Type Semiconductor Bond 
       [0385]    With the application of excitation energy equivalent to at least the band gap in the form of photons for example, electrons in silicon transition to conducting states. The nature of these states are equivalent to those of the electrons of metals with the appropriate lattice parameters and boundary conditions of silicon. Since the planar electron membranes are in contact throughout the crystalline matrix, the Maxwellian boundary condition that an equipotential must exist between contacted perfect conductors maintains that all of the planar electrons are at the energy of the highest energy state electron. This condition with the availability of a multitude of states with different ion separation distances and corresponding energies coupled with a near continuum of phonon states and corresponding energies gives rise to a continuum energy band or conduction band in the excitation spectrum. Thus, the conducting state of silicon comprises a background covalent diamond structure with free metal-type electrons and an equal number of silicon cations dispersed in the covalent lattice wherein excitation has occurred. The band gap can be calculated from the difference between the energy of the free electrons at the minimum electron-ion separation distance (the parameter d given in the Nature of the Metallic Bond of Alkali Metals section) and the energy of the covalent-type electrons of the diamond-type bonds given in the Nature of the Insulator-Type Semiconductor Bond section. 
         [0386]    The band gap is the lowest energy possible to form free electrons and corresponding Si +  ions. Since the gap is the energy difference between the total energy of the free electrons and the MO electrons, a minimum gap corresponds to the lowest energy state of the free electrons. With the ionization of silicon atoms, planar electron membranes form with the corresponding ions at initial positions of the corresponding bond in the silicon lattice. The potential energy between the electrons and ions is a maximum if the electron membrane comprises the superposition of the two electrons ionized from a corresponding Si—Si bond, and the orientation of the membrane is the transverse bisector of the former bond axis such that the magnitude of the potential is four times that of a single Si+−e −  pair. In this case, the potential is given by two times Eq. (19.21). Furthermore, all of the field lines of the silicon ions end on the intervening electrons. Thus, the repulsion energy between Si +  ions is zero such the energy of the ionized state is a minimum. Using the parameters from Tables 21.1 and 21.6 (as shown in the priority document), the Si + −e −  distance of c′=1.16332 Å, and the calculated Si +  ionic radius of r si+3sp     3   =1.16360a 0 =0.61575 Å (Eq. 20.17), the lattice structure of crystalline silicon in a conducting state is shown in  FIG. 27 . 
         [0387]    The optimal Si +  ion-electron separation distance parameter d is given by 
         [0000]        d=c′= 2.19835 a   0 =1.16332×10 −10  m  (21.2)
 
         [0000]    The band gap is given by the difference in the energy of the free electrons at the optimal Si + -electron separation distance parameter d given by Eq. (21.2) and the energy of the electrons in the initial state of the Si—Si-bond MO. The total energy of electrons of a covalent Si—Si-bond MO E T (Si Si—SiMO ) given by Eq. (15.65) and Table 20.4 is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               Si 
                               
                                 Si 
                                 - 
                                 SiMO 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               E 
                               T 
                             
                              
                             
                               ( 
                               MO 
                               ) 
                             
                           
                           + 
                           
                             
                               E 
                               _ 
                             
                             osc 
                           
                           - 
                           
                             E 
                             mag 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               22.81274 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           0.04888 
                           - 
                           
                             0.04983 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             22.81179 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21.3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The minimum energy of a free-conducting electron in silicon for the determination of the band gap E T(band gap) (free e −  in Si) is given by the sum twice the potential energy and the kinetic energy given by Eqs. (19.21) and (19.24), respectively. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               T 
                                
                               
                                 ( 
                                 
                                   band 
                                    
                                   
                                       
                                   
                                    
                                   gap 
                                 
                                 ) 
                               
                             
                           
                            
                           
                             ( 
                             
                               free 
                                
                               
                                   
                               
                                
                               
                                 e 
                                 - 
                               
                                
                               
                                   
                               
                                
                               in 
                                
                               
                                   
                               
                                
                               Si 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           V 
                           + 
                           T 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 - 
                                 2 
                               
                                
                               
                                   
                               
                                
                               
                                 e 
                                 2 
                               
                             
                             
                               4 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               d 
                             
                           
                           + 
                           
                             
                               4 
                               3 
                             
                              
                             
                               ( 
                               
                                 
                                   1 
                                   2 
                                 
                                  
                                 
                                   
                                     ℏ 
                                     2 
                                   
                                   
                                     
                                       m 
                                       e 
                                     
                                      
                                     
                                       d 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21.4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In addition, the ionization of the MO electrons increases the charge on the two corresponding Si3sp 3  HO with a corresponding energy decrease, E T (atom−atom,msp 3 .AO) given by one half that of Eq. (20.20). With d given by Eq. (21.2), E T(band gap) (free e −  in Si) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       
                         T 
                          
                         
                           ( 
                           
                             band 
                              
                             
                                 
                             
                              
                             gap 
                           
                           ) 
                         
                       
                     
                      
                     
                       ( 
                       
                         free 
                          
                         
                             
                         
                          
                         
                           e 
                           - 
                         
                          
                         
                             
                         
                          
                         in 
                          
                         
                             
                         
                          
                         Si 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               
                                 
                                   
                                     - 
                                     2 
                                   
                                    
                                   
                                     e 
                                     2 
                                   
                                 
                                 
                                   4 
                                    
                                   
                                     
                                       πɛ 
                                       0 
                                     
                                     ( 
                                     
                                       1.16332 
                                       × 
                                       
                                         10 
                                         
                                           - 
                                           10 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       m 
                                     
                                     ) 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   4 
                                   3 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       1 
                                       2 
                                     
                                      
                                     
                                       
                                         ℏ 
                                         2 
                                       
                                       
                                         
                                           
                                             m 
                                             e 
                                           
                                           ( 
                                           
                                             1.16332 
                                             × 
                                             
                                               10 
                                               
                                                 - 
                                                 10 
                                               
                                             
                                              
                                             
                                                 
                                             
                                              
                                             m 
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 E 
                                 T 
                               
                                
                               
                                 ( 
                                 
                                   
                                     atom 
                                     - 
                                     atom 
                                   
                                   , 
                                   
                                     
                                       msp 
                                       3 
                                     
                                     · 
                                     AO 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         
                           
                             - 
                             24.75614 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                         + 
                         
                           3.75374 
                            
                           
                               
                           
                            
                           eV 
                         
                         - 
                         
                           
                             1.37960 
                             2 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                       = 
                       
                         
                           - 
                           21.69220 
                         
                          
                         
                             
                         
                          
                         eV 
                       
                     
                   
                 
               
               
                 
                   ( 
                   21.5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The band gap in silicon E g  given by the difference between E T(band gap) (free e −  in Si) (Eq. (21.5)) and E T (Si Si—SiMO ) (Eq. (213)) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                           g 
                         
                         = 
                           
                          
                         
                           
                             
                               E 
                               
                                 T 
                                  
                                 
                                   ( 
                                   
                                     band 
                                      
                                     
                                         
                                     
                                      
                                     gap 
                                   
                                   ) 
                                 
                               
                             
                              
                             
                               ( 
                               
                                 free 
                                  
                                 
                                     
                                 
                                  
                                 
                                   e 
                                   - 
                                 
                                  
                                 
                                     
                                 
                                  
                                 in 
                                  
                                 
                                     
                                 
                                  
                                 Si 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               E 
                               T 
                             
                              
                             
                               ( 
                               
                                 Si 
                                 
                                   Si 
                                   - 
                                   SiMO 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               21.69220 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 22.81179 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           1.120 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21.6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The experimental band gap for silicon [6] is 
         [0000]      E=1.12 eV  (21.7)
 
         [0000]    The calculated band gap is in excellent agreement with the experimentally measured value. This result along with the prediction of the correct lattice parameters, cohesive energy, and bond angles given in Tables 21.2, 21.5, and 21.6 (as shown in the priority document), respectively, confirms that conductivity in silicon is due the creation of discrete ions, Si+ and e − , with the excitation of electrons from covalent bonds. The current carriers are free metal-type electrons that exist as planar membranes with current propagation along these structures shown in  FIG. 27 . Since the conducting electrons are equivalent to those of metals, the resulting kinetic energy distribution over the population of electrons can be modeled using the statistics of electrons in metals, Fermi Dirac statistics given in the Fermi-Dirac section and the Physical Implications of Free Electrons in Metals section. 
       REFERENCES 
       [0000]    
       
         1. H. N. Waltenburg, J. T. Yates, “Surface chemistry of silicon”, Chem. Rev., Vol. 95, (1995), pp. 1589-1673. 
         2. D. W. Palmer, www.semiconductors.co.uk, (2006), September. 
         3. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), p. 12-18. 
         4. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), p. 9-86. 
         5. B. Farid, R. W. Godby, “Cohesive energies of crystals”, Physical Review B, Vol. 43 (17), (1991), pp. 14248-14250. 
         6. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), p. 12-82. 
       
     
       Boron Molecular Functional Groups and Molecules 
     General Considerations of the Boron Molecular Bond 
       [0394]    Boron molecules comprising an arbitrary number of atoms can be solved using similar principles and procedures as those used to solve organic molecules of arbitrary length and complexity. Boron molecules can be considered to be comprised of functional groups such as B—B, B—C, B—H, B—O, B—N, B—X (X is a halogen atom), and the alkyl functional groups of organic molecules. The solutions of these functional groups or any others corresponding to the particular boron molecule can be conveniently obtained by using generalized forms of the force balance equation given in the Force Balance of the σ MO of the Carbon Nitride Radical section for molecules comprised of boron and hydrogen only and the geometrical and energy equations given in the Derivation of the General Geometrical and Energy Equations of Organic Chemistry section for boron molecules further comprised of heteroatoms such as carbon. The appropriate functional groups with the their geometrical parameters and energies can be added as a linear sum to give the solution of any molecule containing boron. 
       Boranes (B x H y ) 
       [0395]    As in the case of carbon, silicon, and aluminum, the bonding in the boron atom involves four sp 3  hybridized orbitals formed from the 2p and 2s electrons of the outer shells except that only three HOs are filled. Bonds form between the B2sp 3  HOs of two boron atoms and between a B2sp 3  HO and a H1s AO to yield boranes. The geometrical parameters of each B—H and B—B functional group is solved from the force balance equation of the electrons of the corresponding σ-MO and the relationships between the prolate spheroidal axes. Then, the sum of the energies of the H 2 -type ellipsoidal MOs is matched to that of the B2sp 3  shell as in the case of the corresponding carbon molecules. As in the case of ethane (C—C functional group given in the Ethane Molecule section) and silane (Si—Si functional group given in the Silanes section), the energy of the B—B functional group is determined for the effect of the donation of 25% electron density from the each participating B2sp 3  HO to the B—B-bond MO. 
         [0396]    The energy of boron is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). A minimum energy is achieved while matching the potential, kinetic, and orbital energy relationships given in the Hydroxyl Radical (OH) section with the donation of 25% electron density from each participating B2sp 3  HO to each B—H and B—B-bond MO. As in the case of acetylene given in the Acetylene Molecule section, the energies of the B—H and B—B functional groups are determined for the effect of the charge donation. 
         [0397]    The 2sp 3  hybridized orbital arrangement is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         2 
                          
                         
                           sp 
                           3 
                         
                          
                         
                             
                         
                          
                         state 
                       
                     
                   
                   
                     
                       
                         
                           ↑ 
                           
                             0 
                             , 
                             0 
                           
                         
                          
                         
                             
                         
                          
                         
                           ↑ 
                           
                             1 
                             , 
                             
                               - 
                               1 
                             
                           
                         
                          
                         
                             
                         
                          
                         
                           ↑ 
                           
                             1 
                             , 
                             0 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                               
                           
                           
                             1 
                             , 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the quantum numbers (l, m t ) are below each electron. The total energy of the state is given by the sum over the four electrons. The sum E T (B, 2sp 3 ) of experimental energies [1] of B, B + , and B 2+  is 
         [0000]      E T (B,2 sp   3 )=37.93064 eV+25.1548 eV+8.29802 eV=71.38346 eV  (22.2)
 
         [0000]    By considering that the central field decreases by an integer for each successive electron of the shell, the radius r 2sp     3   , of the B2sp 3  shell may be calculated from the Coulombic energy using Eq. (15.13): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             2 
                              
                             
                               sp 
                               3 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               2 
                             
                             4 
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   Z 
                                   - 
                                   n 
                                 
                                 ) 
                               
                                
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     71.38346 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             6 
                              
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   71.38346 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           1.14361 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Z=5 for boron. Using Eq. (15.14), the Coulombic energy E Coulomb (B,2sp 3 ) of the outer electron of the B2sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               B 
                               , 
                               
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.14361 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             11.89724 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, one of the spin-paired 2s electrons is promoted to B2sp 3  shell as an unpaired electron. The energy for the promotion is the magnetic energy given by Eq. (15.15) at the initial radius of the 2s electrons. From Eq. (10.62) with Z=5, the radius r 3  of B2s shell is 
         [0000]      r 3 =1.07930a 0   (22.5)
 
         [0000]    Using Eqs. (15.15) and (22.5), the unpairing energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             2 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                                
                               2 
                             
                              
                             
                               ℏ 
                               2 
                             
                           
                           
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   3 
                                 
                                 ) 
                               
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             8 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                               μ 
                               B 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 1.07930 
                                  
                                 
                                   a 
                                   0 
                                 
                               
                               ) 
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.09100 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (24.4) and (22.6), the energy E(B,2sp 3 ) of the outer electron of the B2sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               B 
                               , 
                               
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 e 
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     3 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               11.89724 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.09100 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             11.80624 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.7 
                   ) 
                 
               
             
           
         
       
     
         [0398]    Next, consider the formation of the B—H and B—B-bond MOs of boranes wherein each boron atom has a B2sp 3  electron with an energy given by Eq. (22.7). The total energy of the state of each boron atom is given by the sum over the three electrons. The sum E T (B borane ,2sp 3 ) of energies of B2sp 3  (Eq. (22.7)), and B 2 + is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 B 
                                 borane 
                               
                               , 
                               
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           - 
                           
                             ( 
                             
                               
                                 37.93064 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               + 
                               
                                 25.1548 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               + 
                               
                                 E 
                                  
                                 
                                   ( 
                                   
                                     B 
                                     , 
                                     
                                       2 
                                        
                                       
                                         sp 
                                         3 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           - 
                           
                             ( 
                             
                               
                                 37.93064 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               + 
                               
                                 25.1548 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               + 
                               
                                 11.80624 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             74.89168 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E(B,2sp 3 ) is the sum of the energy of B, −8.29802 eV, and the hybridization energy. 
         [0399]    Each B—H-bond MO forms with the sharing of electrons between each B2sp 3  HO and each H1s AO. As in the case of C—H, the H 2 -type ellipsoidal MO comprises 75% of the B—H-bond MO according to Eq. (13.429) and Eq. (13.59). Similarly to the case of C—C, the B—B H 2 -type ellipsoidal MO comprises 50% contribution from the participating B2sp 3  HOs according to Eq. (14.152). The sharing of electrons between a B2sp 3  HO and one or more H1s AOs to form B—H-bond MOs or between two B2sp 3  HOs to form a B—B-bond MO permits each participating orbital to decrease in size and energy. As shown below, the boron HOs have spin and orbital angular momentum terms in the force balance which determines the geometrical parameters of each σ MO. The angular momentum term requires that each σ MO be treated independently in terms of the charge donation. In order to further satisfy the potential, kinetic, and orbital energy relationships, each B2sp 3  HO donates an excess of 25% of its electron density to the B—H or B—B-bond MO to form an energy minimum. By considering this electron redistribution in the borane molecule as well as the fact that the central field decreases by an integer for each successive electron of the shell, the radius r borane2sp     3   , of the B2sp 3  shell may be calculated from the Coulombic energy using Eq. (15.18): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             borane 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                               sp 
                               3 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     2 
                                   
                                   4 
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               0.25 
                             
                             ) 
                           
                            
                           
                             
                               e 
                               2 
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     74.89168 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             5.75 
                              
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   74.89168 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           1.04462 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (22.9), the Coulombic energy E Coulomb (B borane ,2sp 3 ) of the outer electron of the B2sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 B 
                                 borane 
                               
                               , 
                               
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 borane 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.04462 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             13.02464 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, one of the spin-paired 2s electrons are promoted to B2sp 3  shell as an unpaired electron. The energy for the promotion is the magnetic energy given by Eq. (22.6). Using Eqs. (22.6) and (22.10), the energy E(B borane ,2sp 3 ) of the outer electron of the B2sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 B 
                                 borane 
                               
                               , 
                               
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   borane 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 e 
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     3 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               13.02464 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.09100 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             12.93364 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (B—H,2sp 3 ) and E T (B —B,2sp 3 ), the energy change of each B2sp 3  shell with the formation of the B—H and B—B-bond MO, respectively, is given by the difference between Eq. (22.11) and Eq. (22.7): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 B 
                                 - 
                                 H 
                               
                               , 
                               
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 B 
                                 - 
                                 B 
                               
                               , 
                               
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   B 
                                   borane 
                                 
                                 , 
                                 
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 B 
                                 , 
                                 
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               12.93364 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 11.80624 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             1.12740 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.12 
                   ) 
                 
               
             
           
         
       
     
         [0400]    Next, consider the case that each B2sp 3  HO donates an excess of 50% of its electron density to the σ MO to form an energy minimum. By considering this electron redistribution in the borane molecule as well as the fact that the central field decreases by an integer for each successive electron of the shell, the radius r borane2sp     3    of the B2sp 3  shell may be calculated from the Coulombic energy using Eq. (15.18): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             borane 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                               sp 
                               3 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     2 
                                   
                                   4 
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               0.5 
                             
                             ) 
                           
                            
                           
                             
                               e 
                               2 
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     74.89168 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             5.5 
                              
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   74.89168 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.99920 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (22.13), the Coulombic energy E Coulomb (B borane ,2sp 3 ) of the outer electron of the B2sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 B 
                                 borane 
                               
                               , 
                               
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 borane 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             0.99920 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             13.61667 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, one of the spin-paired 2s electrons is promoted to B2sp 3  shell as an unpaired electron. The energy for the promotion is the magnetic energy given by Eq. (22.6). Using Eqs. (22.6) and (22.14), the energy E(B borane ,2sp 3 ) of the outer electron of the B2sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 B 
                                 borane 
                               
                               , 
                               
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   borane 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 e 
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     3 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               13.61667 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.09100 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             13.52567 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (B−atom,2sp 3 ), the energy change of each B2sp 3  shell with the formation of the B−atom-bond MO is given by the difference between Eq. (22.15) and Eq. (22.7): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 B 
                                 - 
                                 atom 
                               
                               , 
                               
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   B 
                                   borane 
                                 
                                 , 
                                 
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 B 
                                 , 
                                 
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               13.52567 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 11.80624 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             1.711943 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.16 
                   ) 
                 
               
             
           
         
       
     
         [0401]    Consider next the radius of the HO due to the contribution of charge to more than one bond. The energy contribution due to the charge donation at each boron atom superimposes linearly. In general, the radius r mol2sp     3    of the B2sp 3  HO of a boron atom of a given borane molecule is calculated after Eq. (15.32) by considering ΣE T     mol   (MO,2sp 3 ), the total energy donation to all bonds with which it participates in bonding. The general equation for the radius is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             mol 
                              
                             
                                 
                             
                              
                             3 
                              
                             
                               sp 
                               3 
                             
                           
                         
                         = 
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       E 
                                       Coulomb 
                                     
                                      
                                     
                                       ( 
                                       
                                         B 
                                         , 
                                         
                                           2 
                                            
                                           
                                             sp 
                                             3 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     ∑ 
                                     
                                       
                                         E 
                                         
                                           T 
                                           mol 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           MO 
                                           , 
                                           
                                             2 
                                              
                                             
                                               sp 
                                               3 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             e 
                             2 
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     11.89724 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   + 
                                   
                                     ∑ 
                                     
                                        
                                       
                                         
                                           E 
                                           
                                             T 
                                             mol 
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             MO 
                                             , 
                                             
                                               2 
                                                
                                               
                                                 sp 
                                                 3 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.17 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E Coulomb (B, 2sp 3 ) is given by Eq. (22.4). The Coulombic energy E Coulomb (B, 2sp 3 ) of the outer electron of the B 2sp 3  shell considering the charge donation to all participating bonds is given by Eq. (15.14) with Eq. (22.4). The energy E(B,2sp 3 ) of the outer electron of the B 2sp 3  shell is given by the sum of E Coulomb (B, 2sp 3 ) and E(magnetic) (Eq. (22.6)). The final values of the radius of the B2sp 3  HO, r 2sp     3   , E Coulomb (B,2sp3), and E(B borane 2sp 3 ) calculated using ΣE T     mol   (MO,2sp 3 ), the total energy donation to each bond with which an atom participates in bonding are given in Table 22.1. These hybridization parameters are used in Eqs. (15.88-15.117) for the determination of bond angles given in Table 22.7 (as shown in the priority document). 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 22.1 
               
             
             
               
                   
               
               
                 Atom hybridization designation (# first column) and hybridization 
               
               
                 parameters of atoms for determination of bond angles with final values 
               
               
                 of r 2sp     3   , E Coulomb  (B, 2sp 3 ) (designated as E Coulomb ), and E(B borane  2sp 3 ) 
               
               
                 (designated as E) calculated using the appropriate values of ΣE T     mol    (MO, 2sp 3 ) 
               
               
                 (designated as E T ) for each corresponding terminal bond spanning each angle. 
               
             
          
           
               
                   
                   
                   
                   
                   
                   
                 r 3sp     3     
                 E Coulomb  (eV) 
                 E (eV) 
               
               
                 # 
                 E T   
                 E T   
                 E T   
                 E T   
                 E T   
                 Final 
                 Final 
                 Final 
               
               
                   
               
             
          
           
               
                 1 
                 0 
                 0 
                 0 
                 0 
                 0 
                 1.14361 
                 11.89724 
                 11.80624 
               
               
                 2 
                 −1.71943 
                 0 
                 0 
                 0 
                 0 
                 0.99920 
                 −13.61667 
                 −13.52567 
               
               
                 3 
                 −1.18392 
                 −1.18392 
                 0 
                 0 
                 0 
                 0.95378 
                 −14.26508 
                 −14.17408 
               
               
                 4 
                 −1.12740 
                 −1.12740 
                 −0.56370 
                 0 
                 0 
                 0.92458 
                 −14.71574 
                 −14.62474 
               
               
                   
               
             
          
         
       
     
         [0402]    The MO semimajor axes of the B—H and B—B functional groups of boranes are determined from the force balance equation of the centrifugal, Coulombic, and magnetic forces as given in the Polyatomic Molecular Ions and Molecules section and the More Polyatomic Molecules and Hydrocarbons section. In each case, the distance from the origin of the H 2 -type-ellipsoidal-MO to each focus c′, the internuclear distance 2c′, and the length of the semiminor axis of the prolate spheroidal H 2 -type MO b=c are solved from the semimajor axis a. Then, the geometric and energy parameters of each MO are calculated using Eqs. (15.1-15.117). 
         [0403]    The force balance of the centrifugal force equated to the Coulombic and magnetic forces is solved for the length of the semimajor axis. The Coulombic force on the pairing electron of the MO is 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     Coulomb 
                   
                   = 
                   
                     
                       
                         e 
                         2 
                       
                       
                         8 
                          
                         
                           πɛ 
                           0 
                         
                          
                         a 
                          
                         
                             
                         
                          
                         
                           b 
                           2 
                         
                       
                     
                      
                     D 
                      
                     
                         
                     
                      
                     
                       i 
                       ξ 
                     
                   
                 
               
               
                 
                   ( 
                   22.18 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The spin-pairing force is 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     
                       spin 
                        
                       
                         - 
                       
                        
                       pairing 
                     
                   
                   = 
                   
                     
                       
                         ℏ 
                         2 
                       
                       
                         2 
                          
                         
                           m 
                           e 
                         
                          
                         
                           a 
                           2 
                         
                          
                         
                           b 
                           2 
                         
                       
                     
                      
                     D 
                      
                     
                         
                     
                      
                     
                       i 
                       ξ 
                     
                   
                 
               
               
                 
                   ( 
                   22.19 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The diamagnetic force is: 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     
                       diamagneticMO 
                        
                       
                           
                       
                        
                       1 
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           
                             n 
                             e 
                           
                            
                           
                             ℏ 
                             2 
                           
                         
                         
                           4 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                     
                      
                     D 
                      
                     
                         
                     
                      
                     
                       i 
                       ξ 
                     
                   
                 
               
               
                 
                   ( 
                   22.20 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where n e  is the total number of electrons that interact with the binding σ-MO electron. The diamagnetic force F diamagneticMO2  on the pairing electron of the σ MO is given by the sum of the contributions over the components of angular momentum: 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     
                       diamagneticMO 
                        
                       
                           
                       
                        
                       2 
                     
                   
                   = 
                   
                     - 
                     
                       
                         ∑ 
                         
                           i 
                           , 
                           j 
                         
                       
                        
                       
                         
                           
                             
                                
                               
                                 L 
                                 i 
                               
                                
                             
                              
                             ℏ 
                           
                           
                             
                               Z 
                               j 
                             
                              
                             2 
                              
                             
                               m 
                               e 
                             
                              
                             
                               a 
                               2 
                             
                              
                             
                               b 
                               2 
                             
                           
                         
                          
                         D 
                          
                         
                             
                         
                          
                         
                           i 
                           ξ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.21 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where |L| is the magnitude of the angular momentum of each atom at a focus that is the source of the diamagnetism at the σ-MO. The centrifugal force is 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     centrifugalMO 
                   
                   = 
                   
                     
                       - 
                       
                         
                           ℏ 
                           2 
                         
                         
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                     
                      
                     D 
                      
                     
                         
                     
                      
                     
                       i 
                       ξ 
                     
                   
                 
               
               
                 
                   ( 
                   22.22 
                   ) 
                 
               
             
           
         
       
     
         [0404]    The force balance equation for the σ-MO of the two-center B—H-bond MO is the given by centrifugal force given by Eq. (22.22) equated to the sum of the Coulombic (Eq. (22.18)), spin-pairing (Eq. (22.19)), and F diamagneticMO2  (Eq. (22.21)) with 
         [0000]    
       
         
           
             
                
               L 
                
             
             = 
             
               4 
                
               
                 
                   3 
                   4 
                 
               
                
               ℏ 
             
           
         
       
     
         [0000]    corresponding to the four B2sp 3  HOs: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ℏ 
                         2 
                       
                       
                         
                           m 
                           e 
                         
                          
                         
                           a 
                           2 
                         
                          
                         
                           b 
                           2 
                         
                       
                     
                      
                     D 
                   
                   = 
                   
                     
                       
                         
                           e 
                           2 
                         
                         
                           8 
                            
                           
                             πɛ 
                             0 
                           
                            
                           a 
                            
                           
                               
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                     + 
                     
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                     - 
                     
                       
                         
                           4 
                            
                           
                             
                               3 
                               4 
                             
                           
                         
                         Z 
                       
                        
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                   
                 
               
               
                 
                   ( 
                   22.23 
                   ) 
                 
               
             
             
               
                 
                   a 
                   = 
                   
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             4 
                              
                             
                               
                                 3 
                                 4 
                               
                             
                           
                           Z 
                         
                       
                       ) 
                     
                      
                     
                       a 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   22.24 
                   ) 
                 
               
             
           
         
       
     
         [0000]    With Z=5, the semimajor axis of the B—H-bond MO is 
         [0000]      a=1.69282a 0   (22.25)
 
         [0405]    The force balance equation for each σ-MO of the B—B-bond MO with n e =2 and 
         [0000]    
       
         
           
             
                
               L 
                
             
             = 
             
               3 
                
               
                 
                   3 
                   4 
                 
               
                
               ℏ 
             
           
         
       
     
         [0000]    corresponding to three electrons of the B2sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ℏ 
                         2 
                       
                       
                         
                           m 
                           e 
                         
                          
                         
                           a 
                           2 
                         
                          
                         
                           b 
                           2 
                         
                       
                     
                      
                     D 
                   
                   = 
                   
                     
                       
                         
                           e 
                           2 
                         
                         
                           8 
                            
                           
                             πɛ 
                             0 
                           
                            
                           a 
                            
                           
                               
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                     + 
                     
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                     - 
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               3 
                                
                               
                                 
                                   3 
                                   4 
                                 
                               
                             
                             Z 
                           
                         
                         ) 
                       
                        
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                   
                 
               
               
                 
                   ( 
                   22.26 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     a 
                     = 
                     
                       
                         ( 
                         
                           2 
                           + 
                           
                             
                               3 
                                
                               
                                 
                                   3 
                                   4 
                                 
                               
                             
                             Z 
                           
                         
                         ) 
                       
                        
                       
                         a 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.27 
                   ) 
                 
               
             
           
         
       
     
         [0000]    With Z=5, the semimajor axis of the B—B-bond MO is 
         [0000]      a=2.51962a 0   (22.28)
 
         [0406]    Using the semimajor axis, the geometric and energy parameters of the MO are calculated using Eqs. (15.1-15.127) in the same manner as the organic functional groups given in the Organic Molecular Functional Groups and Molecules section. For the B—H functional group, c 1  is one and C 1 =0.75 based on the MO orbital composition as in the case of the C—H-bond MO. In boranes, the energy of boron is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). Thus, the energy matching condition is determined by the c 2  and C 2  parameters in Eqs. (15.51) and (15.61). Then, the hybridization factor for the B—H-bond MO given by the ratio of 11.89724 eV, the magnitude of E Coulomb (B borane ,2sp 3 ) (Eq. (22.4)), and 13.605804 eV, the magnitude of the Coulombic energy between the electron and proton of H (Eq. (1.243)): 
         [0000]    
       
         
           
             
               
                 
                   
                     c 
                     2 
                   
                   = 
                   
                     
                       
                         C 
                         2 
                       
                        
                       
                         ( 
                         
                           borane 
                            
                           
                               
                           
                            
                           2 
                            
                           
                             sp 
                             3 
                           
                            
                           HO 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           11.89724 
                            
                           
                               
                           
                            
                           eV 
                         
                         
                           13.605804 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                       = 
                       0.87442 
                     
                   
                 
               
               
                 
                   ( 
                   22.29 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Since the energy of the MO is matched to that of the B2sp 3  HO, E(AO/HO) in Eqs. (15.51) and (15.61) is E(B,2sp 3 ) given by Eq. (22.7), and E T (atom−atom,msp 3 .AO) is one half of −1.12740 eV corresponding the independent single-bond charge contribution (Eq. (22.12)) of one center. 
         [0407]    For the B—B functional group, c 1  is one and C 1 =0.5 based on the MO orbital composition as in the case of the C—C-bond MO. The energy matching condition is determined by the c 2  and C 2  parameters in Eqs. (15.51) and (15.61), and the hybridization factor for the B—B-bond MO given is by Eq. (22.29). Since the energy of the MO is matched to that of the B2sp 3  HO, E(AO/HO) in Eqs. (15.51) and (15.61) is E(B,2sp 3 ) given by Eq. (22.7), and E T (atom−atom,msp 3 .AO) is two times −1.12740 eV corresponding the independent single-bond charge contributions (Eq. (22.12)) from each of the two B2sp 3  HOs. 
       Bridging Bonds of Boranes (B—H—B and B—B—B) 
       [0408]    As in the case of the Al 3sp 3  HOs given in the Organoaluminum Hydrides (Al—H—Al and Al—C—Al) section, the B2sp 3  HOs comprise four orbitals containing three electrons as given by Eq. (23.1) that can form three-center as well as two-center bonds. The designation for a three-center bond involving two B2sp 3  HOs and a H1s AO is B—H—B, and the designation for a three-center bond involving three B2sp 3  HOs is B—B—B. 
         [0409]    The parameters of the force balance equation for the σ-MO of the B—H—B-bond MO are n e =2 and |L|=0 due to the cancellation of the angular momentum between borons: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ℏ 
                         2 
                       
                       
                         
                           m 
                           e 
                         
                          
                         
                           a 
                           2 
                         
                          
                         
                           b 
                           2 
                         
                       
                     
                      
                     D 
                   
                   = 
                   
                     
                       
                         
                           e 
                           2 
                         
                         
                           8 
                            
                           
                             πɛ 
                             0 
                           
                            
                           a 
                            
                           
                               
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                     + 
                     
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                     - 
                     
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                   
                 
               
               
                 
                   ( 
                   22.30 
                   ) 
                 
               
             
           
         
       
     
         [0000]    From Eq. (22.30), the semimajor axis of the B—H—B-bond MO is 
         [0000]      a=2a 0   (22.31)
 
         [0000]    The parameters in Eqs. (15.51) and (15.61) are the same as those of the B—H—B functional group except that E T (atom−atom,msp 3 .AO) is two times −1.12740 eV corresponding the independent single-bond charge contributions (Eq. (22.12)) from each of the two B2sp 3  HOs. The force balance equation and the semimajor axis for the σ-MO of the B—B—B-bond MO are the same as those of the B—B-bond MO given by Eqs. (22.30) and (22.31), respectively. The parameters in Eqs. (15.51) and (15.61) are the same as those of the B—B functional group except that E T (atom−atom,msp 3 .AO) is three times −1.12740 eV corresponding the independent single-bond charge contributions (Eq. (22.12)) from each of the three B2sp 3  HOs. 
         [0410]    The H 2 -type ellipsoidal MOs of the B—H—B three-center intersect and form a continuous single surface. However, in the case of the B—B—B-bond MO the current of each B—B MO forms a bisector current described in the Methane Molecule (CH 4 ) section that is continuous with the center B2sp 3 -HO shell (Eqs. (15.36-15.44)). Based on symmetry, the polar angle φ at which the B—H—B H 2 -type ellipsoidal MOs intersect is given by the bisector of the external angle between the B—H bonds: 
         [0000]    
       
         
           
             
               
                 
                   φ 
                   = 
                   
                     
                       
                         
                           360 
                            
                           ° 
                         
                         - 
                         
                           θ 
                           
                             ∠ 
                              
                             
                                 
                             
                              
                             BHB 
                           
                         
                       
                       2 
                     
                     = 
                     
                       
                         
                           
                             360 
                              
                             ° 
                           
                           - 
                           
                             85.4 
                              
                             ° 
                           
                         
                         2 
                       
                       = 
                       
                         137.3 
                          
                         ° 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.32 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where [2] 
         [0000]      θ ∠BHB =85.4°  (22.33)
 
         [0000]    The polar radius r i  at this angle is given by Eqs. (13.84-13.85): 
         [0000]    
       
         
           
             
               
                 
                   
                     r 
                     i 
                   
                   = 
                   
                     
                       ( 
                       
                         a 
                         - 
                         
                           c 
                           ′ 
                         
                       
                       ) 
                     
                      
                     
                       
                         1 
                         + 
                         
                           
                             c 
                             ′ 
                           
                           a 
                         
                       
                       
                         1 
                         + 
                         
                           
                             
                               c 
                               ′ 
                             
                             a 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.34 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Substitution of the parameters of Table 22.2 into Eq. (22.34) gives 
         [0000]        r   i =2.26561 a   0 =1.19891×10 −10  m  (22.35)
 
         [0411]    The polar angle φ at which the B—B—B H 2 -type ellipsoidal MOs intersect is given by the bisector of the external angle between the B—B bonds: 
         [0000]    
       
         
           
             
               
                 
                   φ 
                   = 
                   
                     
                       
                         
                           360 
                            
                           ° 
                         
                         - 
                         
                           θ 
                           
                             ∠ 
                              
                             
                                 
                             
                              
                             BBB 
                           
                         
                       
                       2 
                     
                     = 
                     
                       
                         
                           
                             360 
                              
                             ° 
                           
                           - 
                           
                             58.9 
                              
                             ° 
                           
                         
                         2 
                       
                       = 
                       
                         150.6 
                          
                         ° 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.36 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where [3] 
         [0000]      θ ∠BHB =58.9°  (22.37)
 
         [0000]    The polar radius r i  at this angle is given by Eqs. (13.84-13.85): 
         [0000]    
       
         
           
             
               
                 
                   
                     r 
                     i 
                   
                   = 
                   
                     
                       ( 
                       
                         a 
                         - 
                         
                           c 
                           ′ 
                         
                       
                       ) 
                     
                      
                     
                       
                         1 
                         + 
                         
                           
                             c 
                             ′ 
                           
                           a 
                         
                       
                       
                         1 
                         + 
                         
                           
                             
                               c 
                               ′ 
                             
                             a 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             φ 
                             ′ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.38 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Substitution of the parameters of Table 22.2 into Eq. (22.38) gives 
         [0000]        r   i =3.32895 a   0 =1.76160×10 −10  m  (22.39)
 
         [0412]    The symbols of the functional groups of boranes are given in Table 22.2. The geometrical (Eqs. (15.1-15.5) and (22.23-22.39)), intercept (Eqs. (15.80-15.87) and (22.17)), and energy (Eq. (15.61), (22.4), (22.7), (22.12), and (22.29)) parameters of boranes are given in Tables 22.3, 22.4 (as shown in the priority document), and 22.5, respectively. In the case that the MO does not intercept the B HO due to the reduction of the radius from the donation of Bsp 3  HO charge to additional MO&#39;s, the energy of each MO is energy matched as a linear sum to the B HO by contacting it through the bisector current of the intersecting MOs as described in the Methane Molecule (CH 4 ) section. The total energy of each borane given in Table 22.6 (as shown in the priority document) was calculated as the sum over the integer multiple of each E D  (Group) of Table 22.5 corresponding to functional-group composition of the molecule. E mag  of Table 22.5 is given by Eqs. (15.15) and (22.3). The bond angle parameters of boranes determined using Eqs. (15.88-15.117) and (20.36) with B2sp 3  replacing Si3sp 3  are given in Table 22.7 (as shown in the priority document). The charge-density in diborane is shown in  FIG. 28 . 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 22.2 
               
             
             
               
                   
               
               
                 The symbols of the functional groups of boranes. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
                   
               
               
                   
                 BH group 
                 B—H 
               
               
                   
                 BHB (bridged H) 
                 B—H—B 
               
               
                   
                 BB bond 
                 B—B 
               
               
                   
                 BBB (bridged B) 
                 B—B—B 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
             
           
               
                 TABLE 22.3 
               
             
             
               
                   
               
               
                 The geometrical bond parameters of boranes 
               
               
                 and experimental values. 
               
             
          
           
               
                   
                   
                   
                   
                 B—B 
               
               
                   
                   
                   
                   
                 and 
               
               
                   
                   
                 B—H 
                 B—H—B 
                 B—B—B 
               
               
                   
                 Parameter 
                 Group 
                 Group 
                 Groups 
               
               
                   
                   
               
               
                   
                 a (a 0 ) 
                 1.69282 
                 2.00000 
                 2.51962 
               
               
                   
                 c′ (a 0 ) 
                 1.13605 
                 1.23483 
                 1.69749 
               
               
                   
                 Bond Length 
                 1.20235 
                 1.30689 
                 1.79654 
               
               
                   
                 2c′ (Å) 
               
               
                   
                 Exp. Bond 
                 1.19 [4] 
                 1.32 [4] 
                 1.798 [3] 
               
               
                   
                 Length 
                 (diborane) 
                 (diborane) 
                 (B 13 H 19 ) 
               
               
                   
                 (Å) 
               
               
                   
                 b, c (a 0 ) 
                 1.25500 
                 1.57327 
                 1.86199 
               
               
                   
                 e 
                 0.67110 
                 0.61742 
                 0.67371 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 22.5 
               
             
             
               
                   
               
               
                 The energy parameters (eV) of functional groups of boranes. 
               
             
          
           
               
                   
                 B—H 
                 B—H—B 
                 B—B 
                 B—B—B 
               
               
                 Parameters 
                 Group 
                 Group 
                 Group 
                 Group 
               
               
                   
               
             
          
           
               
                 n 1   
                 1 
                 1 
                 1 
                 1 
               
               
                 n 2   
                 0 
                 0 
                 0 
                 0 
               
               
                 n 3   
                 0 
                 0 
                 0 
                 0 
               
               
                 C 1   
                 0.75 
                 0.75 
                 0.5 
                 0.5 
               
               
                 C 2   
                 0.87442 
                 0.87442 
                 0.87442 
                 0.87442 
               
               
                 c 1   
                 1 
                 1 
                 1 
                 1 
               
               
                 c 2   
                 0.87442 
                 0.87442 
                 0.87442 
                 0.87442 
               
               
                 c 3   
                 0 
                 0 
                 0 
                 0 
               
               
                 c 4   
                 1 
                 1 
                 2 
                 2 
               
               
                 c 5   
                 1 
                 1 
                 0 
                 0 
               
               
                 C 1o   
                 0.75 
                 0.75 
                 0.5 
                 0.5 
               
               
                 C 2o   
                 0.87442 
                 0.87442 
                 0.87442 
                 0.87442 
               
               
                 V e  (eV) 
                 −34.04561 
                 −27.77951 
                 −22.91867 
                 −22.91867 
               
               
                 V p  (eV) 
                 11.97638 
                 11.01833 
                 8.01527 
                 8.01527 
               
               
                 T (eV) 
                 10.05589 
                 6.94488 
                 4.54805 
                 4.54805 
               
               
                 V m  (eV) 
                 −5.02794 
                 −3.47244 
                 −2.27402 
                 −2.27402 
               
               
                 E (AO/HO) (eV) 
                 −11.80624 
                 −11.80624 
                 −11.80624 
                 −11.80624 
               
               
                 ΔE H     2     MO  (AO/HO) (eV) 
                 0 
                 0 
                 0 
                 0 
               
               
                 E T  (AO/HO) (eV) 
                 −11.80624 
                 −11.80624 
                 −11.80624 
                 −11.80624 
               
               
                 E T  (H 2 MO) (eV) 
                 −28.84754 
                 −25.09498 
                 −24.43561 
                 −24.43561 
               
               
                 E T  (atom-atom, msp 3  · AO) (eV) 
                 −0.56370 
                 −2.25479 
                 −2.25479 
                 −3.38219 
               
               
                 E T  (MO) (eV) 
                 −29.41123 
                 −29.60457 
                 −26.69041 
                 −27.81781 
               
               
                 ω (10 15  rad/s) 
                 15.2006 
                 23.9931 
                 6.83486 
                 6.83486 
               
               
                 E K  (eV) 
                 10.00529 
                 15.79265 
                 4.49882 
                 4.49882 
               
               
                 Ē D  (eV) 
                 −0.18405 
                 −0.23275 
                 −0.11200 
                 −0.11673 
               
               
                 Ē Kvib  (eV) 
                 0.29346 [5] 
                 0.09844 [6] 
                 0.13035 [5] 
                 0.13035 [5] 
               
               
                 Ē osc  (eV) 
                 −0.03732 
                 −0.18353 
                 −0.04682 
                 −0.05156 
               
               
                 E mag  (eV) 
                 0.07650 
                 0.07650 
                 0.07650 
                 0.07650 
               
               
                 E T  (Group) (eV) 
                 −29.44855 
                 −29.78809 
                 −26.73723 
                 −27.86936 
               
               
                 E initial  (c 4  AO/HO) (eV) 
                 −11.80624 
                 −11.80624 
                 −11.80624 
                 −11.80624 
               
               
                 E initial  (c 5  AO/HO) (eV) 
                 −13.59844 
                 −13.59844 
                 0 
                 0 
               
               
                 E D  (Group) (eV) 
                 4.04387 
                 4.38341 
                 3.12475 
                 4.25687 
               
               
                   
               
             
          
         
       
     
       Alkyl Boranes (R x B y H z ; R=Alkyl) 
       [0413]    The alkyl boranes may comprise at least a terminal methyl group (CH 3 ) and at least one B bound by a carbon-boron single bond comprising a C—B group, and may comprise methylene (CH 2 ), methylyne (CH), C—C, B—H, B—B, B—H—B, and B—B—B functional groups. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C—C bonds can be identified. The n-alkane C—C bond is the same as that of straight-chain alkanes. In addition, the C—C bonds within isopropyl ((CH 3 ) 2  CH) and t-butyl ((CH 3 ) 3 C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C—C bonds comprise functional groups. Additional groups include aromatics such as phenyl. These groups in alkyl boranes are equivalent to those in branched-chain alkanes and aromatics, and the B—H, B—B, B—H—B, and B—B—B functional groups of alkyl boranes are equivalent to those in boranes. 
         [0414]    For the C—B functional group, hybridization of the 2s and 2p AOs of each C and B to form single 2sp 3  shells forms an energy minimum, and the sharing of electrons between the C2sp 3  and B2sp 3  HOs to form σ MO permits each participating orbital to decrease in radius and energy. In alkyl boranes, the energy of boron is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). Thus, c 1  in Eq. (15.61) is one, and the energy matching condition is determined by the c 2  and C 2  parameters. Then, the C2sp 3  HO has an energy of E(C,2sp 3 )=−14.63489 eV (Eq. (15.25)), and the B2sp 3  HOs has an energy of E(B,2sp 3 =−11.80624 eV (Eq. (22.7)). To meet the equipotential condition of the union of the C—B H 2 -type-ellipsoidal-MO with these orbitals, the hybridization factors c 2  and C 2  of Eq. (15.61) for the C—B-bond MO given by Eq. (15.77) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               C 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               B 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               C 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               B 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 B 
                                 , 
                                 
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 C 
                                 , 
                                 
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               11.80624 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               14.63489 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.80672 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.40 
                   ) 
                 
               
             
           
         
       
     
         [0000]    E T (atom−atom,msp 3 .AO) of the C—B-bond MO is −1.44915 eV corresponding to the single-bond contributions of carbon and boron of −0.72457 eV given by Eq. (14.151). The energy of the C—B-bond MO is the sum of the component energies of the H 2 -type ellipsoidal MO given in Eq. (15.51) with E(AO/HO)=E(B,2sp 3 ) given by Eq. (22.7) and ΔE H     2     MO (AO/HO)=E T (atom−atom,msp 3 .AO) in order to match the energies of the carbon and boron HOs. 
         [0415]    Consider next the radius of the HO due to the contribution of charge to more than one bond. The energy contribution due to the charge donation at each boron atom and carbon atom superimposes linearly. In general, since the energy of the B2sp 3  HO is matched to that of the C2sp 3  HO, the radius r mol2sp     3    of the B2sp 3  HO of a boron atom and the C2sp 3  HO of a carbon atom of a given alkyl borane molecule is calculated after Eq. (15.32) by considering ΣE T     mol   (MO,2sp 3 ), the total energy donation to all bonds with which it participates in bonding. The Coulombic energy E Coulomb (atom, 2sp 3 ) of the outer electron of the atom 2sp 3  shell considering the charge donation to all participating bonds is given by Eq. (15.14). The hybridization parameters used in Eqs. (15.88-15.117) for the determination of bond angles of alkyl boranes are given in Table 22.8. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 22.8 
               
             
             
               
                   
               
               
                 Atom hybridization designation (# first column) and hybridization parameters 
               
               
                 of atoms for determination of bond angles with final values of r 2sp     3   , E Coulomb   
               
               
                 (atom, 2sp 3 ) (designated as E Coulomb ), and E Coulomb (atom alkylborane 2sp 3 ) 
               
               
                 (designated as E) calculated using the appropriate values of ΣE T     mol    (MO, 2sp 3 ) 
               
               
                 (designated as E T ) for each corresponding terminal bond spanning each angle. 
               
             
          
           
               
                   
                   
                   
                   
                   
                   
                   
                 E Coulomb  (eV) 
                 E (eV) 
               
               
                 # 
                 E T   
                 E T   
                 E T   
                 E T   
                 E T   
                 r 3sp     3    Final 
                 Final 
                 Final 
               
               
                   
               
               
                 1 
                 −0.36229 
                 −0.92918 
                 0 
                 0 
                 0 
                 0.84418 
                 −16.11722 
                 −15.92636 
               
               
                   
               
             
          
         
       
     
         [0416]    The symbols of the functional groups of alkyl boranes are given in Table 22.9. The geometrical (Eqs. (15.1-15.5) and (22.23-22.40)), intercept (Eqs. (15.32) and (15.80-15.87)), and energy (Eq. (15.61), (22.4), (22.7), (22.12), (22.29), and (22.40)) parameters of alkyl boranes are given in Tables 22.10, 22.11, and 22.12, respectively (all as shown in the priority document). In the case that the MO does not intercept the B HO due to the reduction of the radius from the donation of B 2sp 3  HO charge to additional MO&#39;s, the energy of each MO is energy matched as a linear sum to the B HO by contacting it through the bisector current of the intersecting MOs as described in the Methane Molecule (CH 4 ) section. The total energy of each alkyl borane given in Table 22.13 (as shown in the priority document) was calculated as the sum over the integer multiple of each E D  (Group) of Table 22.12 (as shown in the priority document) corresponding to functional-group composition of the molecule. E mag  of Table 22.13 (as shown in the priority document) is given by Eqs. (15.15) and (22.3) for B—H. The bond angle parameters of alkyl boranes determined using Eqs. (15.88-15.117) are given in Table 22.14 (as shown in the priority document). The charge-densities of exemplary alkyl borane, trimethylborane and alkyl diborane, tetramethyldiborane comprising the concentric shells of atoms with the outer shell bridged by one or more H 2 -type ellipsoidal MOs or joined with one or more hydrogen MOs are shown in  FIGS. 29 and 30 , respectively. 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 22.9 
               
             
             
               
                   
               
               
                 The symbols of the functional groups of alkyl boranes. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
               
               
                   
                 C—B bond 
                 C—B 
               
               
                   
                 BH bond 
                 B—H 
               
               
                   
                 BHB (bridged H) 
                 B—H—B 
               
               
                   
                 BB bond 
                 B—B 
               
               
                   
                 BBB (bridged B) 
                 B—B—B 
               
               
                   
                 CC (aromatic bond) 
                 C 3e ═C 
               
               
                   
                 CH (aromatic) 
                 CH (i) 
               
               
                   
                 CH 3  group 
                 C—H (CH 3 ) 
               
               
                   
                 CH 2  group 
                 C—H (CH 2 ) 
               
               
                   
                 CH 
                 C—H (ii) 
               
               
                   
                 CC bond (n-C) 
                 C—C (a) 
               
               
                   
                 CC bond (iso-C) 
                 C—C (b) 
               
               
                   
                 CC bond (tert-C) 
                 C—C (c) 
               
               
                   
                 CC (iso to iso-C) 
                 C—C (d) 
               
               
                   
                 CC (t to t-C) 
                 C—C (e) 
               
               
                   
                 CC (t to iso-C) 
                 C—C (f) 
               
               
                   
               
             
          
         
       
     
       Alkoxy Boranes ((RO) x  B y H z ; R=Alkyl) and Aklyl Borinic Acids ((RO) q  B r H s (HO) t ) 
       [0417]    The alkoxy boranes and borinic acids each comprise a B—O functional group, at least one boron-alkyl-ether moiety or a one or more hydroxyl groups, respectively, and in some cases one or more alkyl groups and borane moieties. Each alkoxy moiety, C n H 2n+1 O, of alkoxy boranes comprises one of two types of C—O functional groups that are equivalent to those give in the Ethers (C n H 2n+2 O m , n=2,3,4,5 . . . ∞) section. One is for methyl or t-butyl groups, and the other is for general alkyl groups. Each hydroxyl functional group of borinic acids and alkyl borinic acids is equivalent to that given in the Alcohols (C n H 2n+2 O m , n=1,2,3,4,5 . . . ∞) section. The alkyl portion may be part of the alkoxy moiety, or an alkyl group may be bound to the central boron atom by a carbon-boron single bond comprising the C—B group of the Alkyl Boranes (R x B y H z ; R=alkyl) section. Each alkyl portion may comprise at least a terminal methyl group (CH 3 ) and methylene (CH 2 ), methylyne (CH), and C—C functional groups. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C—C bonds can be identified. The n-alkane C—C bond is the same as that of straight-chain alkanes. In addition, the C—C bonds within isopropyl ((CH 3 ) 2  CH) and t-butyl ((CH 3 ) 3 C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C—C bonds comprise functional groups. Additional R groups include aromatics such as phenyl. These groups in alkoxy boranes and alkyl borinic acids are equivalent to those in branched-chain alkanes and aromatics given in the corresponding sections. Furthermore, B—H, B—B, B—H—B, and B—B—B groups may be present that are equivalent to those in boranes as given in the Boranes (B z H y ) section. 
         [0418]    The MO semimajor axes of the B—O functional groups of alkoxy alkanes and borinic acids are determined from the force balance equation of the centrifugal, Coulombic, and magnetic forces as given in the Boranes (B x H y ) section. In each case, the distance from the origin of the H 2 -type-ellipsoidal-MO to each focus c′, the internuclear distance 2c′, and the length of the semiminor axis of the prolate spheroidal H 2 -type MO b=c are solved from the semimajor axis a. Then, the geometric and energy parameters of each MO are calculated using Eqs. (15.1-15.117). 
         [0419]    The parameters of the force balance equation for the σ-MO of the B—O-bond MO in Eqs. (22.18-22.22) are n e =2 and |L|=0: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ℏ 
                         2 
                       
                       
                         
                           m 
                           e 
                         
                          
                         
                           a 
                           2 
                         
                          
                         
                           b 
                           2 
                         
                       
                     
                      
                     D 
                   
                   = 
                   
                     
                       
                         
                           e 
                           2 
                         
                         
                           8 
                            
                           
                             πɛ 
                             0 
                           
                            
                           a 
                            
                           
                               
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                     + 
                     
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                     - 
                     
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                   
                 
               
               
                 
                   ( 
                   22.41 
                   ) 
                 
               
             
           
         
       
     
         [0000]    From Eq. (22.41), the semimajor axis of the B—O-bond MO is 
         [0000]      a=2a 0   (22.42)
 
         [0420]    For the B—O functional groups, hybridization of the 2s and 2p AOs of each C and B to form single 2sp 3  shells forms an energy minimum, and the sharing of electrons between the C2sp 3  and B2sp 3  HOs to form σ MO permits each participating orbital to decrease in radius and energy. The energy of boron is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). Thus, in c 1  and c 2  in Eq. (15.61) is one, and the energy matching condition is determined by the C 2  parameter. The approach to the hybridization factor of O to B in boric acids is similar to that of the O to S bonding in the SO group of sulfoxides. The O AO has an energy of E(O)=−13.61805 eV, and the B2sp 3  HOs has an energy of E(B,2sp 3 )=−11.80624 eV (Eq. (22.7)). To meet the equipotential condition of the union of the B—O H 2 -type-ellipsoidal-MO with these orbitals in borinic acids and to energy match the OH group, the hybridization factor C 2  of Eq. (15.61) for the B—O-bond MO given by Eq. (15.77) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               OAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               B 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               OAO 
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 B 
                                 , 
                                 
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               13.61805 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               11.80624 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         1.15346 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.43 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Since the energy of the MO is matched to that of the B2sp 3  HO, E(AO/HO) in Eqs. (15.51) and (15.61) is E(B,2sp 3 ) given by Eq. (22.7), and E T (atom−atom,msp 3 .AO) is −1.12740 eV corresponding to the independent single-bond charge contribution (Eq. (22.12)) of one center. 
         [0421]    The parameters of the B—O functional group of alkoxy boranes are the same as those of borinic acids except for C 1  and C 2 . Rather than being bound to an H, the oxygen is bound to a C2sp 3  HO, and consequently, the hybridization of the C—O given by Eq. (15.133) includes the C2sp 3  HO hybridization factor of 0.91771 (Eq. (13.430)). To meet the equipotential condition of the union of the B—O H 2 -type-ellipsoidal-MO with the B2sp 3  HOs having an energy of E(B,2sp 3 )=−11.80624 eV (Eq. (22.7)) and the O AO having an energy of E(O)=−13.61805 eV such that the hybridization matches that of the C—O-bond MO, the hybridization factor C 2  of Eq. (15.61) for the B—O-bond MO given by Eqs. (15.77) and (15.79) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               B 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               O 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               E 
                                
                               
                                 ( 
                                 
                                   B 
                                   , 
                                   
                                     2 
                                      
                                     
                                       sp 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                             
                               E 
                                
                               
                                 ( 
                                 O 
                                 ) 
                               
                             
                           
                            
                           
                             
                               c 
                               2 
                             
                              
                             
                               ( 
                               
                                 C 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                                  
                                 HO 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 
                                   - 
                                   11.80624 
                                 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               
                                 
                                   - 
                                   13.61805 
                                 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                             
                              
                             
                               ( 
                               0.91771 
                               ) 
                             
                           
                           = 
                           0.79562 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.44 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Furthermore, in order to form an energy minimum in the B—O-bond MO, oxygen acts as an H in bonding with B since the 2p shell of 0 is at the Coulomb energy between an electron and a proton (Eq. (10.163)). In this case, k′ is 0.75 as given by Eq. (13.59) such that C 1 =0.75 in Eq. (15.61). 
         [0422]    Consider next the radius of the HO due to the contribution of charge to more than one bond. The energy contribution due to the charge donation at each boron atom and oxygen atom superimposes linearly. In general, since the energy of the B2sp 3  HO and O AO is matched to that of the C2sp 3  HO when a the molecule contains a C—B-bond MO and a C—O-bond MO, respectively, the corresponding radius r molsp     3    of the B2sp 3  HO of a boron atom, the C2sp 3  HO of a carbon atom, and the O AO of a given alkoxy borane or borinic acid molecule is calculated after Eq. (15.32) by considering ΣE T     mol   (MO,2sp 3 ), the total energy donation to all bonds with which it participates in bonding. The Coulombic energy E Coulomb (atom,2sp 3 ) of the outer electron of the atom 2sp 3  shell considering the charge donation to all participating bonds is given by Eq. (15.14). In the case that the boron or oxygen atom is not bound to a C2sp 3  HO, r mol2sp     3    is calculated using Eq. (15.31) where E Coulomb (atom,msp 3 ) is E Coulomb (B2sp 3 )=11.89724 eV and E(O)=−13.61805 eV, respectively. 
         [0423]    The symbols of the functional groups of alkoxy boranes and borinic acids are given in Table 22.15. The geometrical (Eqs. (15.1-15.5) and (22.42-22.44)), intercept (Eqs. (15.31-15.32) and (15.80-15.87)), and energy (Eq. (15.61), (22.4), (22.7), (22.12), (22.29), and (22.43-22.44)) parameters of alkoxy boranes and borinic acids are given in Tables 22.16, 22.17, and 22.18, respectively (all as shown in the priority document). In the case that the MO does not intercept the B HO due to the reduction of the radius from the donation of B 2sp 3  HO charge to additional MO&#39;s, the energy of each MO is energy matched as a linear sum to the B HO by contacting it through the bisector current of the intersecting MOs as described in the Methane Molecule (CH 4 ) section. The total energy of each alkyl borane given in Table 22.19 (as shown in the priority document) was calculated as the sum over the integer multiple of each E D  (Group) of Table 22.18 (as shown in the priority document) corresponding to functional-group composition of the molecule. E mag  of Table 22.18 (as shown in the priority document) is given by Eqs. (15.15) and (22.3) for the B—O groups and the B—H, B—B, B—H—B, and B—B—B groups. E mag  of Table 22.18 (as shown in the priority document) is given by Eqs. (15.15) and (10.162) for the OH group. The bond angle parameters of alkoxy boranes and borinic acids determined using Eqs. (15.88-15.117) are given in Table 22.20 (as shown in the priority document). The charge-densities of exemplary alkoxy borane, trimethoxyborane, boric acid, and phenylborinic anhydride comprising the concentric shells of atoms with the outer shell bridged by one or more I I 2 -type ellipsoidal MOs or joined with one or more hydrogen MOs are shown in  FIGS. 31 ,  32 , and  33 , respectively. 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 22.15 
               
             
             
               
                   
               
               
                 The symbols of the functional groups of alkoxy boranes and borinic 
               
               
                 acids. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
               
               
                   
                 B—O bond (borinic acid) 
                 B—O (i) 
               
               
                   
                 B—O bond (alkoxy borane) 
                 B—O (ii) 
               
               
                   
                 OH group 
                 OH 
               
               
                   
                 C—O (CH 3 —O—and (CH 3 ) 3 C—O—) 
                 C—O (i) 
               
               
                   
                 C—O (alkyl) 
                 C—O (ii) 
               
               
                   
                 C—B bond 
                 C—B 
               
               
                   
                 BH bond 
                 B—H 
               
               
                   
                 BHB (bridged H) 
                 B—H—B 
               
               
                   
                 BB bond 
                 B—B 
               
               
                   
                 BBB (bridged B) 
                 B—B—B 
               
               
                   
                 CC (aromatic bond) 
                 C 3e ═C 
               
               
                   
                 CH (aromatic) 
                 CH (i) 
               
               
                   
                 CH 3  group 
                 C—H (CH 3 ) 
               
               
                   
                 CH 2  group 
                 C—H (CH 2 ) 
               
               
                   
                 CH 
                 C—H (ii) 
               
               
                   
                 CC bond (n-C) 
                 C—C (a) 
               
               
                   
                 CC bond (iso-C) 
                 C—C (b) 
               
               
                   
               
             
          
         
       
     
       Tertiary and Quaternary Animoboranes and Borane Amines (R q B r N s R t ; R═H; Alkyl) 
       [0424]    The tertiary and quaternary amino boranes and borane amines each comprise at least one B bound by a boron-nitrogen single bond comprising a B—N group, and may comprise at least a terminal methyl group (CH 3 ), as well other alkyl and borane groups such as methylene (CH 2 ), methylyne (CH), C—C, B—H, B—C, B—H, B—B, B—H—B, and B—B—B functional groups. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C—C bonds can be identified. The n-alkane C—C bond is the same as that of straight-chain alkanes. In addition, the C—C bonds within isopropyl ((CH 3 ) 2  CH) and t-butyl ((CH 3 ) 3 C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C—C bonds comprise functional groups. These groups in tertiary and quaternary amino boranes and borane amines are equivalent to those in branched-chain alkanes, the B—C group is equivalent to that of alkyl boranes, and the B—H, B—B, B—H—B, and B—B—B functional groups are equivalent to those in boranes. 
         [0425]    In tertiary amino boranes and borane amines, the nitrogen atom of each B—N bond is bound to two other atoms such that there are a total of three bounds per atom. The amino or amine moiety may comprise NH 2 , N(H)R, and NR 2 . The corresponding functional group for the NH 2  moiety is the NH 2  functional group given in the Primary Amines (C n H 2n+2+m N m , n=1,2,3,4,5 . . . ∞) section. The N(H)R moiety comprises the NH functional group of the Secondary Amines (C n H 2n+2+m N m , n=1,2,3,4,5 . . . ∞) section and the C—N functional group of the Primary Amines (C n H 2n+2+m N m , n=1,2,3,4,5 . . . ∞) section. The NR 2  moiety comprises two types of C—N functional groups, one for the methyl group corresponding to the C of C—N and the other for general alkyl secondary amines given in the Secondary Amines (C n H 2n+2+ N m , n=2,3,4,5 . . . ∞) section. 
         [0426]    In quaternary amino boranes and borane amines, the nitrogen atom of each B—N bond is bound to three other atoms such that there are a total of four bounds per atom. The amino or amine moiety may comprise NH 3 , N(H 2 )R, N(H)R 2 , and NR 3 . The corresponding functional group for the NH 3  moiety is ammonia given in the Ammonia (NH 3 ) section. The N(H 2 )R moiety comprises the NH 2  and the C—N functional groups given in the Primary Amines (C n H 2n+2+m N m , n=1,2,3,4,5 . . . ∞) section. The N(H)R 2  moiety comprises the NH functional group and two types of C—N functional groups, one for the methyl group corresponding to the C of C—N and the other for general alkyl secondary amines given in the Secondary Amines (C n H 2n+2+m N m , n=2,3,4,5 . . . ∞) section. The NR 3  moiety comprises the C—N functional group of tertiary amines given in the Tertiary Amines (C n H 2n+3 N, n=3,4,5 . . . ∞) section. 
         [0427]    The bonding in the B—N functional groups of tertiary and quaternary amino boranes and borane amines is similar to that of the B—O groups of alkoxy boranes and borinic acids given in the corresponding section. The MO semimajor axes of the B—N functional groups are determined from the force balance equation of the centrifugal, Coulombic, and magnetic forces as given in the Boranes (B x H y ) section. In each case, the distance from the origin of the H 2 -type-ellipsoidal-MO to each focus c′, the internuclear distance 2c′, and the length of the semiminor axis of the prolate spheroidal H 2 -type MO b=c are solved from the semimajor axis a. Then, the geometric and energy parameters of each MO are calculated using Eqs. (15.1-15.117). 
         [0428]    As in the case of the B—O-bond MOs, the σ-MOs of the tertiary and quaternary B—N-bond MOs is energy matched to the B2sp 3  HO which determines that the parameters of the force balance equation based on electron angular momentum are determined by those of the boron atom. Thus, the parameters of the force balance equation for the σ-MO of the B—N-bond MOs in Eqs. (22.18-22.22) are n e =1 and 
         [0000]    
       
         
           
             
                
               L 
                
             
             = 
             
               
                 3 
                  
                 
                   
                     3 
                     4 
                   
                 
               
               Z 
             
           
         
       
     
         [0000]    corresponding to the three electrons of the boron atom: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ℏ 
                         2 
                       
                       
                         
                           m 
                           e 
                         
                          
                         
                           a 
                           2 
                         
                          
                         
                           b 
                           2 
                         
                       
                     
                      
                     D 
                   
                   = 
                   
                     
                       
                         
                           e 
                           2 
                         
                         
                           8 
                            
                           
                             πɛ 
                             0 
                           
                            
                           a 
                            
                           
                               
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                     + 
                     
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                     - 
                     
                       
                         ( 
                         
                           
                             1 
                             2 
                           
                           + 
                           
                             
                               3 
                                
                               
                                 
                                   3 
                                   4 
                                 
                               
                             
                             Z 
                           
                         
                         ) 
                       
                        
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                   
                 
               
               
                 
                   ( 
                   22.45 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     a 
                     = 
                     
                       
                         ( 
                         
                           
                             3 
                             2 
                           
                           + 
                           
                             
                               3 
                                
                               
                                 
                                   3 
                                   4 
                                 
                               
                             
                             Z 
                           
                         
                         ) 
                       
                        
                       
                         a 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.46 
                   ) 
                 
               
             
           
         
       
     
         [0000]    With Z=5, the semimajor axis of the tertiary B—N-bond MO is 
         [0000]      a=2.01962a 0   (22.47)
 
         [0429]    For the B—N functional groups, hybridization of the 2s and 2p AOs of B to form single 2sp 3  shells forms an energy minimum, and the sharing of electrons between the B2sp 3  HO and N AO to form σ MO permits each participating orbital to decrease in radius and energy. The energy of boron is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). Thus, in c 1  and c 2  in Eq. (15.61) is one, and the energy matching condition is determined by the C 1  and C 2  parameters. The N AO has an energy of E(N)=−14.53414 eV, and the B2sp 3  HOs has an energy of E(B,2sp 3 )=−11.80624 eV (Eq. (22.7)). To meet the equipotential condition of the union of the B—N H 2 -type-ellipsoidal-MO with these orbitals, the hybridization factor C 2  of Eq. (15.61) for the B—N-bond MO given by Eq. (15.77) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               NAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               B 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 B 
                                 , 
                                 
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               NAO 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               11.80624 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               14.53414 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.81231 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.48 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Since the energy of the MO is matched to that of the B2sp 3  HO, E(AO/HO) in Eqs. (15.51) and (15.61) is E(B,2sp 3 ) given by Eq. (22.7), and E T (atom−atom,msp 3 .AO) for ternary B—N is −1.12740 eV corresponding to the independent single-bond charge contribution (Eq. (22.12)) of one center as in the case of the alkoxy borane B—O functional group. Furthermore, k′ is 0.75 as given by Eq. (13.59) such that C 1 =0.75 in Eq. (15.61) which is also equivalent to C 1  of the B—O alkoxy borane group. E T (atom−atom,msp 3 .AO) of the quaternary B—N-bond MO is determined by considering that the bond involves an electron transfer from the nitrogen atom to the boron atom to form zwitterions such as R 3 N + —B − R′ 3 . By considering the electron redistribution in the quaternary amino borane and borane amine molecule as well as the fact that the central field decreases by an integer for each successive electron of the shell, the radius r B-Nborane2sp     3    of the B2sp 3  shell may be calculated from the Coulombic energy using Eq. (15.18), except that the sign of the charge donation is positive: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             B 
                             - 
                             
                               Nborane 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                             
                           
                         
                         = 
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     2 
                                   
                                   4 
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               + 
                               1 
                             
                             ) 
                           
                            
                           
                             
                               e 
                               2 
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     74.89168 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               7 
                                
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     74.89168 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                           = 
                           
                             1.27171 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.49 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (22.49), the Coulombic energy E Coulomb (B B-Nborane ,2sp 3 ) of the outer electron of the B2sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 B 
                                 
                                   B 
                                   - 
                                   Nborane 
                                 
                               
                               , 
                               
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 B 
                                 - 
                                 
                                   Nborane 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.27171 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             10.69881 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.50 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, one of the spin-paired 2s electrons is promoted to B2sp 3  shell as an unpaired electron. The energy for the promotion is the magnetic energy given by Eq. (22.6). Using Eqs. (22.6) and (22.50), the energy E(B B-Nborane ,2sp 3 ) of the outer electron of the B2sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 B 
                                 
                                   B 
                                   - 
                                   Nborane 
                                 
                               
                               , 
                               
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               - 
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   B 
                                   - 
                                   
                                     Nborane 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                      
                                     
                                       sp 
                                       3 
                                     
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 e 
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     3 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               10.69881 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.09100 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             10.60781 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.51 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (B—N,2sp 3 ), the energy change of each B2sp 3  shell with the formation of the B—N-bond MO is given by the difference between Eq. (22.51) and Eq. (22.7): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 B 
                                 - 
                                 N 
                               
                               , 
                               
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   B 
                                   
                                     B 
                                     - 
                                     Nborane 
                                   
                                 
                                 , 
                                 
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 B 
                                 , 
                                 
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               10.60781 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 11.80624 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1.19843 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.52 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (atom−atom,msp 3 .AO) of the quaternary B—N-bond MO is 1.19843 eV. 
         [0430]    Consider next the radius of the HO due to the contribution of charge to more than one bond. The energy contribution due to the charge donation at each boron atom and nitrogen atom superimposes linearly. In general, since the energy of the B2sp 3  HO and N AO is matched to that of the C2sp 3  HO when a the molecule contains a C—B-bond MO and a C—N-bond MO, respectively, the corresponding radius r mol2sp     3    of the B2sp 3  HO of a boron atom, the C2sp 3  HO of a carbon atom, and the N AO of a given B—N-containing borane molecule is calculated after Eq. (15.32) by considering Z E mol (MO,2sp 3 ), the total energy donation to all bonds with which it participates in bonding. The Coulombic energy E Coulomb (atom, 2sp 3 ) of the outer electron of the atom 2sp 3  shell considering the charge donation to all participating bonds is given by Eq. (15.14). In the case that the boron or nitrogen atom is not bound to a C2sp 3  HO, r mol2sp     3    is calculated using Eq. (15.31) where E Coulomb (atom,msP 3 ) is E Coulomb (B2sp 3 )=−11.89724 eV and E(N)=−14.53414 eV, respectively. The hybridization parameters used in Eqs. (15.88-15.117) for the determination of bond angles of tertiary and quaternary amino boranes and borane amines are given in Table 22.21. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 22.21 
               
             
             
               
                   
               
               
                 Atom hybridization designation (# first column) and hybridization parameters 
               
               
                 of atoms for determination of bond angles with final values of r 2sp     3   , E Coulomb   
               
               
                 (atom,2sp 3 ) (designated as E Coulomb ), and E(atom B-Nborane  2sp 3 ) (designated as E) 
               
               
                 calculated using the appropriate values of  Σ E T     mol    (MO,2sp 3 ) (designated as E T ) 
               
               
                 for each corresponding terminal bond spanning each angle. 
               
             
          
           
               
                   
                   
                   
                   
                   
                   
                   
                 E Coulomb   
                 E 
               
               
                   
                   
                   
                   
                   
                   
                 r 3sp     3     
                 (eV) 
                 (eV) 
               
               
                 # 
                 E T   
                 E T   
                 E T   
                 E T   
                 E T   
                 Final 
                 Final 
                 Final 
               
               
                   
               
             
          
           
               
                 1 
                 −0.46459 
                 0 
                 0 
                 0 
                 0 
                 0.88983 
                 −15.29034 
                 −15.09948 
               
               
                   
                   
                   
                   
                   
                   
                 (Eq. (15.32)) 
               
               
                 2 
                 −0.56370 
                 −0.56370 
                 −0.56370 
                 0 
                 0 
                 0.82343 
                 −16.52324 
               
               
                   
                   
                   
                   
                   
                   
                 (Eq. (15.32)) 
               
               
                   
               
             
          
         
       
     
         [0431]    The symbols of the functional groups of tertiary and quaternary amino boranes and borane amines are given in Table 22.22. The geometrical (Eqs. (15.1-15.5) and (22.47)), intercept (Eqs. (15.31-15.32) and (15.80-15.87)), and energy (Eq. (15.61), (22.4), (22.7), (22.12), (22.48), and (22.52)) parameters of tertiary and quaternary amino boranes and borane amines are given in Tables 22.23, 22.24, and 22.25, respectively (all as shown in the priority document). In the case that the MO does not intercept the B HO due to the reduction of the radius from the donation of B 2sp 3  HO charge to additional MO&#39;s, the energy of each MO is energy matched as a linear sum to the B HO by contacting it through the bisector current of the intersecting MOs as described in the Methane Molecule (CH 4 ) section. The total energy of each tertiary and quaternary amino borane or borane amine given in Table 22.26 ((as shown in the priority document) was calculated as the sum over the integer multiple of each E D  (Group) of Table 22.25 (as shown in the priority document) corresponding to functional-group composition of the molecule. E mag  of Table 22.26 (as shown in the priority document) is given by Eqs. (15.15) and (22.3) for the B—N groups and the B—H, B—B, B—H—B, and B—B—B groups. E mag  of Table 22.26 (as shown in the priority document) is given by Eqs. (15.15) and (10.142) for NH 3 . The bond angle parameters of tertiary and quaternary amino boranes and borane amines determined using Eqs. (15.88-15.117) are given in Table 22.27 (as shown in the priority document). The charge-densities of exemplary tertiary amino borane, tris(dimethylamino)borane and quaternary amino borane, trimethylaminotrimethylborane comprising the concentric shells of atoms with the outer shell bridged by one or more H 2 -type ellipsoidal MOs or joined with one or more hydrogen MOs are shown in  FIGS. 34 and 35 , respectively. 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 22.22 
               
             
             
               
                   
               
               
                 The symbols of the functional groups of tertiary and quaternary amino 
               
               
                 boranes and borane amines. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
               
               
                   
                 B—N bond 3° 
                 B—N (i) 
               
               
                   
                 B—N bond 4° 
                 B—N (ii) 
               
               
                   
                 C—N bond 1° amine 
                 C—N (i) 
               
               
                   
                 C—N bond 2° amine (methyl) 
                 C—N (ii) 
               
               
                   
                 C—N bond 2° amine (alkyl) 
                 C—N (iii) 
               
               
                   
                 C—N bond 3° amine 
                 C—N (iv) 
               
               
                   
                 NH 3  group 
                 NH 3   
               
               
                   
                 NH 2  group 
                 NH 2   
               
               
                   
                 NH group 
                 NH 
               
               
                   
                 C—B bond 
                 C—B 
               
               
                   
                 BH bond 
                 B—H 
               
               
                   
                 BHB (bridged H) 
                 B—H—B 
               
               
                   
                 BB bond 
                 B—B 
               
               
                   
                 CH 3  group 
                 C—H (CH 3 ) 
               
               
                   
                 CH 2  group 
                 C—H (CH 2 ) 
               
               
                   
                 CH 
                 C—H (i) 
               
               
                   
                 CC bond (n-C) 
                 C—C (a) 
               
               
                   
               
             
          
         
       
     
       Halidoboranes 
       [0432]    The halidoboranes each comprise at least one B bound by a boron-halogen single bond comprising a B—X group where X═F, Cl, Br, I, and may further comprise one or more alkyl groups and borane moieties. The latter comprise alkyl and aryl moieties and B—C, B—H, B—B, B—H—B, and B—B—B functional groups wherein the B—C group is equivalent to that of alkyl boranes, and the B—H, B—B, B—H—B, and B—B—B functional groups are equivalent to those in boranes given in the corresponding sections. Alkoxy boranes and borinic acids moieties given in the Alkoxy Boranes and Alkyl Borinic Acids ((RO) q B r H s (HO) t ) section may be bound to the B—X group by a B—O functional groups. The former further comprise at least one boron-alkyl-ether moiety, and the latter comprise one or more hydroxyl groups, respectively. Each alkoxy moiety, C n H 2n+1 O, comprises one of two types of C—O functional groups that are equivalent to those give in the Ethers (C n H 2n+2 O m , n=2,3,4,5 . . . ∞) section. One is for methyl or t-butyl groups, and the other is for general alkyl groups. Each borinic acid hydroxyl functional group is equivalent to that given in the Alcohols (C n H 2n+2 O m n=1,2,3,4,5 . . . ∞) section. 
         [0433]    Tertiary amino-borane and borane-amine moieties given in the Tertiary and Quaternary Aminoboranes and Borane Amines (R q B r N s R t ; R═H; alkyl) section can be bound to the B—X group by a B—N functional group. The nitrogen atom of each B—N functional group is bound to two other atoms such that there are a total of three bounds per atom. The amino or amine moiety may comprise NH 2 , N(H)R, and NR 2 . The corresponding functional group for the NH 2  moiety is the NH 2  functional group given in the Primary Amines (C n H 2n+2+m N m , n=1,2,3,4,5 . . . ∞) section. The N(H)R moiety comprises the NH functional group of the Secondary Amines (C n H 2n+2+m N m , n=2,3,4,5 . . . ∞) section and the C—N functional group of the Primary Amines (C n H 2n+2+m N m , n=1,2,3,4,5 . . . ∞) section. The NR 2  moiety comprises two types of C—N functional groups, one for the methyl group corresponding to the C of C—N and the other for general alkyl secondary amines given in the Secondary Amines (C n H 2n+2+m N m , n=2,3,4,5 . . . ∞) section. 
         [0434]    Quaternary amino-borane and boraneamine moieties given in the Tertiary and Quaternary Aminoboranes and Borane Amines (R q B r N s R t ; R═H; alkyl) section can be bound to the B—X group by a B—N functional group. The nitrogen atom of each B—N bond is bound to three other atoms such that there are a total of four bounds per atom. The amino or amine moiety may comprise NH 3 , N(H 2 ) R, N(H)R 2 , and NR 3 . The corresponding functional group for the NH 3  moiety is ammonia given in the Ammonia (NH 3 ) section. The N(H 2 ) R moiety comprises the NH 2  and the C—N functional groups given in the Primary Amines (C n H 2n+2+m N m , n=1,2,3,4,5 . . . ∞) section. The N(H)R 2  moiety comprises the NH functional group and two types of C—N functional groups, one for the methyl group corresponding to the C of C—N and the other for general alkyl secondary amines given in the Secondary Amines (C n H 2n+2+m N m , n=2,3,4,5 . . . ∞) section. The NR 3  moiety comprises the C—N functional group of tertiary amines given in the Tertiary Amines (C n H 2N+3 N, n=3,4,5 . . . ∞) section. 
         [0435]    The alkyl portion may be part of the alkoxy moiety, amino or amine moiety, or an alkyl group, or it may be bound to the central boron atom by a carbon-boron single bond comprising the C—B group of the Alkyl Boranes (R X B y H z ; R=alkyl) section. Each alkyl portion may comprise at least a terminal methyl group (CH 3 ) and methylene (CH 2 ), methylyne (CH), and C—C functional groups. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C—C bonds can be identified. The n-alkane C—C bond is the same as that of straight-chain alkanes. In addition, the C—C bonds within isopropyl ((CH 3 ) 2  CH) and t-butyl ((CH 3 ) 3 C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C—C bonds comprise functional groups. Additional R groups include aromatics such as phenyl and —HC═CH 2 . These groups in halidobroanes are equivalent to those in branched-chain alkanes, aromatics, and alkenes given in the corresponding sections. 
         [0436]    The bonding in the B—X functional groups of halidoboranes is similar to that of the B—O and B—N groups of alkoxy boranes and borinic acids and tertiary and quaternary amino boranes and borane amines given in the corresponding sections. The MO semimajor axes of the B—X functional groups are determined from the force balance equation of the centrifugal, Coulombic, and magnetic forces as given in the Boranes (B X H Y ) section. In each case, the distance from the origin of the H 2 -type-ellipsoidal-MO to each focus c′, the internuclear distance 2c′, and the length of the semiminor axis of the prolate spheroidal H 2 -type MO b=c are solved from the semimajor axis a. Then, the geometric and energy parameters of each MO are calculated using Eqs. (15.1-15.117). 
         [0437]    As in the case of the B—O— and B—N-bond MOs, the σ-MOs of the B—X-bond MOs are energy matched to the B2sp 3  HO which determines that the parameters of the force balance equation based on electron angular momentum are determined by those of the boron atom. The parameters of the force balance equation for the σ-MO of the B—F-bond MO in Eqs. (22.18-22.22) are n e =1 and |L|=0: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ℏ 
                         2 
                       
                       
                         
                           m 
                           e 
                         
                          
                         
                           a 
                           2 
                         
                          
                         
                           b 
                           2 
                         
                       
                     
                      
                     D 
                   
                   = 
                   
                     
                       
                         
                           e 
                           2 
                         
                         
                           8 
                            
                           
                             πɛ 
                             0 
                           
                            
                           a 
                            
                           
                               
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                     + 
                     
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                     - 
                     
                       
                         ( 
                         
                           1 
                           2 
                         
                         ) 
                       
                        
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                   
                 
               
               
                 
                   ( 
                   22.53 
                   ) 
                 
               
             
           
         
       
     
         [0000]    From Eq. (22.53), the semimajor axis of the tertiary B—F-bond MO is 
         [0000]      a=1.5a 0   (22.54)
 
         [0438]    The force balance equation for each σ-MO of the B—Cl is equivalent to that of the B—B-bond MO with n e =2 and 
         [0000]    
       
         
           
             
                
               L 
                
             
             = 
             
               3 
                
               
                 
                   3 
                   4 
                 
               
                
               ℏ 
             
           
         
       
     
         [0000]    corresponding to three electrons of the B2sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ℏ 
                         2 
                       
                       
                         
                           m 
                           e 
                         
                          
                         
                           a 
                           2 
                         
                          
                         
                           b 
                           2 
                         
                       
                     
                      
                     D 
                   
                   = 
                   
                     
                       
                         
                           e 
                           2 
                         
                         
                           8 
                            
                           
                               
                           
                            
                           
                             πɛ 
                             0 
                           
                            
                           
                             ab 
                             2 
                           
                         
                       
                        
                       D 
                     
                     + 
                     
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                               
                           
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                     - 
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               3 
                                
                               
                                 
                                   3 
                                   4 
                                 
                               
                             
                             Z 
                           
                         
                         ) 
                       
                        
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                               
                           
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                   
                 
               
               
                 
                   ( 
                   22.55 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     a 
                     = 
                     
                       
                         ( 
                         
                           2 
                           + 
                           
                             
                               3 
                                
                               
                                 
                                   3 
                                   4 
                                 
                               
                             
                             Z 
                           
                         
                         ) 
                       
                        
                       
                         a 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.56 
                   ) 
                 
               
             
           
         
       
     
         [0000]    With Z=5, the semimajor axis of the B—Cl-bond MO is 
         [0000]      a=2.51962a 0   (22.57)
 
         [0439]    The hybridization of the bonding in the B—X functional groups of halidoboranes is similar to that of the C—X groups of alkyl halides given in the corresponding sections. For the B—X functional groups, hybridization of the 2s and 2p AOs of B to form single 2sp 3  shells forms an energy minimum, and the sharing of electrons between the B2sp 3  HO and X AO to form σ MO permits each participating orbital to decrease in radius and energy. The F AO has an energy of E(F)=−17.42282 eV, and the B2sp 3  HOs has an energy of E(B, 2sp 3 )=−11.80624 eV (Eq. (22.7)). To meet the equipotential condition of the union of the B—F H 2 -type-ellipsoidal-MO with these orbitals, the hybridization factor c 2  of Eq. (15.61) for the B—F-bond MO given by Eq. (15.77) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               F 
                                
                               
                                   
                               
                                
                               AO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               B 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 B 
                                 , 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 F 
                                  
                                 
                                     
                                 
                                  
                                 AO 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               11.80624 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               17.42282 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.68285 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.58 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Since the energy of the MO is matched to that of the B2sp 3  HO, E(AO/HO) in Eqs. (15.51) and (15.61) is E(B,2sp 3 ) given by Eq. (22.7). 
         [0440]    E T (atom−atom,msp 3 .AO) of the B—F-bond MO is determined by considering that the bond involves an electron transfer from the boron atom to the fluorine atom to form zwitterions such as H 2 B + —F − . By considering the electron redistribution in the fluoroborane as well as the fact that the central field decreases by an integer for each successive electron of the shell, the radius r B-Fborane2sp     3    of the B2sp 3  shell may be calculated from the Coulombic energy using Eq. (15.18): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             B—F 
                              
                             
                                 
                             
                              
                             borane 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                                 
                             
                              
                             
                               sp 
                               3 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     2 
                                   
                                   4 
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               1 
                             
                             ) 
                           
                            
                           
                             
                               e 
                               2 
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     74.89168 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               5 
                                
                               
                                   
                               
                                
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     74.89168 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                           = 
                           
                             0.90837 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.59 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (22.13), the Coulombic energy E Coulomb (B B-Fboran ,2sp 3 ) of the outer electron of the B2sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 B 
                                 
                                   
                                     B 
                                      
                                     — 
                                      
                                     F 
                                   
                                    
                                   
                                       
                                   
                                    
                                   borane 
                                 
                               
                               , 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 B—F 
                                  
                                 
                                     
                                 
                                  
                                 borane 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             
                               πɛ 
                               0 
                             
                              
                             0.90837 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             14.97834 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.60 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, one of the spin-paired 2s electrons is promoted to B2sp 3  shell as an unpaired electron. The energy for the promotion is the magnetic energy given by Eq. (22.6). Using Eqs. (22.6) and (22.60), the energy E(B B-Xborane ,2sp 3 ) of the outer electron of the B2sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 B 
                                 
                                   B—F 
                                    
                                   
                                       
                                   
                                    
                                   borante 
                                 
                               
                               , 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   B—F 
                                    
                                   
                                       
                                   
                                    
                                   borane 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 e 
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     3 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               14.97834 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.09100 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             14.88734 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.61 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (B—F,2sp 3 ), the energy change of each B2sp 3  shell with the formation of the B—F-bond MO is given by the difference between Eq. (22.15) and Eq. (22.7): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 B 
                                  
                                 — 
                                  
                                 F 
                               
                               , 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   B 
                                   
                                     B—F 
                                      
                                     
                                         
                                     
                                      
                                     borane 
                                   
                                 
                                 , 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 B 
                                 , 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               14.88734 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 11.80624 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             3.08109 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.62 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (atom−atom,msp 3 .AO) for ternary B—F is −6.16219 eV corresponding to the maximum charge contribution of an electron given by two times Eq. (22.62). 
         [0441]    In chloroboranes, the energies of chorine and boron are less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). Thus, in c 1  and c 2  in Eq. (15.61) is one, and the energy matching condition is determined by the C 2  parameter. The Cl AO has an energy of E(Cl)=−12.96764 eV, and the B2sp 3  HOs has an energy of E(B,2sp 3 )=−11.80624 eV (Eq. (22.7)). To meet the equipotential condition of the union of the B—Cl H 2 -type-ellipsoidal-MO with these orbitals, the hybridization factor c 2  of Eq. (15.61) for the B—Cl-bond MO given by Eq. (15.77) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               Cl 
                                
                               
                                   
                               
                                
                               AO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               B 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 B 
                                 , 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 Cl 
                                  
                                 
                                     
                                 
                                  
                                 AO 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               11.80624 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               12.96764 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.91044 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.63 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Since the energy of the MO is matched to that of the B2sp 3  HO, E(AO/HO) in Eqs. (15.51) and (15.61) is E(B,2sp 3 ) given by Eq. (22.7), and E T (atom−atom,msp 3 .AO) is given by two times Eq. (22.12) corresponding to the two centers. 
         [0442]    Consider next the radius of the HO due to the contribution of charge to more than one bond. The energy contribution due to the charge donation at each boron atom and halogen atom superimposes linearly. In general, since the energy of the B2sp 3  HO and X AO is matched to that of the C2sp 3  HO when a the molecule contains a C—B-bond MO and a C—X-bond MO, respectively, the corresponding radius r mol2sp     3   , of the B2sp 3  HO of a boron atom, the C2sp 3  HO of a carbon atom, and the X AO of a given halidoborane molecule is calculated after Eq. (15.32) by considering ΣE T     mol   (MO,2sp 3 ), the total energy donation to all bonds with which it participates in bonding. The Coulombic energy E Coulomb (atom,2sp 3 ) of the outer electron of the atom 2sp 3  shell considering the charge donation to all participating bonds is given by Eq. (15.14). In the case that the boron or halogen atom is not bound to a C2sp 3  HO, r mol2sp     3    is calculated using Eq. (15.31) where E Coulomb (atom,msp 3 ) is E Coulomb (B2sp 3 )=−11.89724 eV, E(F)=−17.42282 eV, or E(Cl)=−12.96764 eV. The hybridization parameters used in Eqs. (15.88-15.117) for the determination of bond angles of halidoboranes are given in Table 22.28. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 22.28 
               
             
             
               
                   
               
               
                 Atom hybridization designation (# first column) and hybridization parameters 
               
               
                 of atoms for determination of bond angles with final values of r 2sp     3   , E Coulomb   
               
               
                 (atom, 2sp 3 ) (designated as E Coulomb ), and E(atom B-Xborane  2sp 3 ) 
               
               
                 (designated as E) calculated using the appropriate values of ΣE T     mol    (MO, 2sp 3 ) 
               
               
                 (designated as E T ) for each corresponding terminal bond spanning each angle. 
               
             
          
           
               
                   
                   
                   
                   
                   
                   
                 r 3sp     3     
                 E Coulomb  (eV) 
                 E (eV) 
               
               
                 # 
                 E T   
                 E T   
                 E T   
                 E T   
                 E T   
                 Final 
                 Final 
                 Final 
               
               
                   
               
             
          
           
               
                 1 
                 −0.56370 
                 0 
                 0 
                 0 
                 0 
                 0.95939 
                 −14.18175 
                   
               
               
                   
                   
                   
                   
                   
                   
                 (Eq. (15.31)) 
               
               
                 2 
                 −3.08109 
                 −3.08109 
                 0 
                 0 
                 0 
                 0.75339 
                 −18.05943 
                 −17.96843 
               
               
                   
                   
                   
                   
                   
                   
                 (Eq. (15.31)) 
               
               
                 3 
                 −3.08109 
                 0 
                 0 
                   
                 0 
                 0.66357 
                 −20.50391 
                 −20.26346 
               
               
                   
                   
                   
                   
                   
                   
                 (Eq. (15.31)) 
               
               
                   
               
             
          
         
       
     
         [0443]    The symbols of the functional groups of halidoboranes are given in Table 22.29. The geometrical (Eqs. (15.1-15.5) and (22.47)), intercept (Eqs. (15.31-15.32) and (15.80-15.87)), and energy (Eq. (15.61), (22.4), (22.7), (22.12), (22.48), and (22.52)) parameters of halidoboranes are given in Tables 22.30, 22.31, and 22.32, respectively (all as shown in the priority document). In the case that the MO does not intercept the B HO due to the reduction of the radius from the donation of B 2sp 3  HO charge to additional MO&#39;s, the energy of each MO is energy matched as a linear sum to the B HO by contacting it through the bisector current of the intersecting MOs as described in the Methane Molecule (CH 4 ) section. The total energy of each halidoborane given in Table 22.33 (as shown in the priority document) was calculated as the sum over the integer multiple of each E D  (Group) of Table 22.32 (as shown in the priority document) corresponding to functional-group composition of the molecule. E mag  of Table 22.33 (as shown in the priority document) is given by Eqs. (15.15) and (22.3) for the B—X groups and the B—O, B—N, B—H, B—B, B—H—B, and B—B—B groups. E mag  of Table 22.33 (as shown in the priority document) is given by Eqs. (15.15) and (10.162) for the OH group. The bond angle parameters of halidoboranes determined using Eqs. (15.88-15.117) are given in Table 22.34 (as shown in the priority document). The charge-densities of exemplary fluoroborane, boron trifluoride and choloroborane, boron trichloride comprising the concentric shells of atoms with the outer shell bridged by one or more H 2 -type ellipsoidal MOs or joined with one or more hydrogen MOs are shown in  FIGS. 36 and 28 , respectively. 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 22.29 
               
             
             
               
                   
               
               
                 The symbols of the functional groups of halidoboranes. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
               
               
                   
                 B—F bond 
                 B—F 
               
               
                   
                 B—Cl bond 
                 B—Cl 
               
               
                   
                 B—N bond 3° 
                 B—N (i) 
               
               
                   
                 B—N bond 4° 
                 B—N (ii) 
               
               
                   
                 C—N bond 1° amine 
                 C—N (i) 
               
               
                   
                 C—N bond 2° amine (methyl) 
                 C—N (ii) 
               
               
                   
                 C—N bond 2° amine (alkyl) 
                 C—N (iii) 
               
               
                   
                 C—N bond 3° amine 
                 C—N (iv) 
               
               
                   
                 NH 3  group 
                 NH 3   
               
               
                   
                 NH 2  group 
                 NH 2   
               
               
                   
                 NH group 
                 NH 
               
               
                   
                 B—O bond (borinic acid) 
                 B—O (i) 
               
               
                   
                 B—O bond (alkoxy borane) 
                 B—O (ii) 
               
               
                   
                 OH group 
                 OH 
               
               
                   
                 C—O (CH 3 —O— and (CH 3 ) 3 C—O—) 
                 C—O (i) 
               
               
                   
                 C—O (alkyl) 
                 C—O (ii) 
               
               
                   
                 C—B bond 
                 C—B 
               
               
                   
                 BH bond 
                 B—H 
               
               
                   
                 BHB (bridged H) 
                 B—H—B 
               
               
                   
                 BB bond 
                 B—B 
               
               
                   
                 BBB (bridged B) 
                 B—B—B 
               
               
                   
                 CC (aromatic bond) 
                 C 3e ═C 
               
               
                   
                 CH (aromatic) 
                 CH (i) 
               
               
                   
                 CH 3  group 
                 C—H (CH 3 ) 
               
               
                   
                 CH 2  alkyl group 
                 C—H (CH 2 ) (i) 
               
               
                   
                 CH 
                 C—H (ii) 
               
               
                   
                 CC bond (n-C) 
                 C—C (a) 
               
               
                   
                 CC bond (iso-C) 
                 C—C (b) 
               
               
                   
                 HC═CH 2  (ethylene bond) 
                 C═C 
               
               
                   
                 CH 2  alkenyl group 
                 CH 2  (ii) 
               
               
                   
               
             
          
         
       
     
       REFERENCES 
       [0000]    
       
         1. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), pp. 10-202 to 10-204. 
         2. V. Ramakrishna, B. J. Duke, “Can the bis(diboranyl) structure of be observed? The story”, Inorg. Chem., Vol. 43, No. 25, (2004), pp. 8176-8184. 
         3. J. C. Huffman, D. C. Moody, R. Schaeffer, “Studies of boranes. XLV. Crystal structure, improved synthesis, and reactions of tridecaborane(19)”, Inorg. Chem., Vol. 15, No. 1. (1976), pp. 227-232. 
         4. K. Kuchitsu, “Comparison of molecular structures determined by electron diffraction and spectroscopy. Ethane and diborane”, J. Chem. Phys., Vol. 49, No. 10, (1968), pp. 4456-4462. 
         5. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), p. 9-82. 
         6. diborane ( 11 B 2 H 6 ) at http://webbook.nist.gov/. 
         7. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), pp. 9-63; 5-4 to 5-42. 
         8. W. N. Lipscomb, “The boranes and their relatives”, Nobel Lecture, Dec. 11, 1976. 
         9. A. B. Burg, R. Kratzer, “The synthesis of nonaborane, B 9 H 11 ”, Inorg. Chem., Vol. 1, No. 4, (1962), pp. 725-730. 
         10. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), p. 9-19 to 9-45. 
         11. BCCB at http://webbook.nist.gov/. 
         12. G. Herzberg,  Molecular Spectra and Molecular Structure II Infrared and Raman Spectra of Polyatomic Molecules , Van Nostrand Reinhold Company, New York, N.Y., (1945), pp. 362-369. 
         13. G. Herzberg,  Molecular Spectra and Molecular Structure II Infrared and Raman Spectra of Polyatomic Molecules , Van Nostrand Reinhold Company, New York, N.Y., (1945), p. 344. 
         14. R. J. Fessenden, J. S. Fessenden,  Organic Chemistry , Willard Grant Press. Boston, Mass., (1979), p. 320. 
         15. cyclohexane at http://webbook.nist.gov/. 
         16. R. L. Hughes, I. C. Smith, E. W. Lawless,  Production of the Boranes and Related Research , Ed. R. T. Holzmann, Academic Press, New York, (1967), pp. 390-396. 
         17. M. J. S. Dewar, C. Jie, E. G. Zoebisch, “AM1 calculations for compounds containing boron”, Organometallics, Vol. 7, (1988), pp. 513-521. 
         18. J. D. Cox, G. Pilcher,  Thermochemistry of Organometallic Compounds , Academic Press, New York, (1970), pp. 454-465. 
         19. W. I. F. David, R. M. Ibberson, G. A. Jeffrey, J. R. Ruble, “The structure analysis of deuterated benzene and deuterated nitromethane by pulsed-neutron powder diffraction: a comparison with single crystal neutron analysis”, Physica B (1992), 180 &amp; 181, pp. 597-600. 
         20. G. A. Jeffrey, J. R. Ruble, R. K. McMullan, J. A. Pople, “The crystal structure of deuterated benzene,” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 414, No. 1846, (Nov. 9, 1987), pp. 47-57. 
         21. H. B. Burgi, S. C. Capelli, “Getting more out of crystal-structure analyses,” Helvetica Chimica Acta, Vol. 86, (2003), pp. 1625-1640. 
         22. K. P. Huber, G. Herzberg,  Molecular Spectra and Molecular Structure, IV. Constants of Diatomic Molecules , Van Nostrand Reinhold Company, New York, (1979). 
         23. J. Crovisier, Molecular Database—Constants for molecules of astrophysical interest in the gas phase: photodissociation, microwave and infrared spectra, Ver. 4.2, Observatoire de Paris, Section de Meudon, Meudon, France, May 2002, pp. 34-37, available at http://wwwusr.obspm.fr/˜crovisie/. 
         24. dimethyl ether at http://webbook.nist.gov/. 
         25. methylamine at http://webbook.nist.gov/. 
         26. D. Lin-Vien. N. B. Colthup, W. G. Fateley, J. G. Grasselli,  The Handbook of Infrared and Raman Frequencies of Organic Molecules , Academic Press, Inc., Harcourt Brace Jovanovich, Boston, (1991), p. 482. 
         27. W. S. Benedict, E. K. Plyler, “Vibration-rotation bands of ammonia”, Can. J. Phys., Vol. 35, (1957), pp. 1235-1241. 
         28. T. Amano, P. F. Bernath, R. W. McKellar, “Direct observation of the v 1  and v 3  fundamental bands of NH 2  by difference frequency laser spectroscopy”, J. Mol. Spectrosc., Vol. 94, (1982), pp. 100-113. 
         29. D. R. Lide,  CRC Handbook of Chemistry and Physics,  79th Edition, CRC Press, Boca Raton, Fla., (1998-9), pp. 9-80 to 9-85. 
         30. D. R. Lide,  CRC Handbook of Chemistry and Physics,  86th Edition, CRC Press, Taylor &amp; Francis, Boca Raton, (2005-6), p. 9-55. 
         31. G. Herzberg,  Molecular Spectra and Molecular Structure II Infrared and Raman Spectra of Polyatomic Molecules , Van Nostrand Reinhold Company, New York, N.Y., (1945), p. 326. 
       
     
       Organometallic and Coordinate Functional Groups and Molecules 
     General Considerations of the Organometallic and Coordinate Bond 
       [0475]    Organometallic and coordinate compounds comprising an arbitrary number of atoms can be solved using similar principles and procedures as those used to solve organic molecules of arbitrary length and complexity. Organometallic and coordinate compounds can be considered to be comprised of functional groups such as M—C, M—H, M—X (X═F, Cl, Br, I), M—OH, M—OR, and the alkyl functional groups of organic molecules. The solutions of these functional groups or any others corresponding to the particular organometallic or coordinate compound can be conveniently obtained by using generalized forms of the force balance equation given in the Force Balance of the σ MO of the Carbon Nitride Radical section for molecules comprised of metal and atoms other than carbon and the geometrical and energy equations given in the Derivation of the General Geometrical and Energy Equations of Organic Chemistry section for organometallic and coordinate compounds comprised of carbon. The appropriate functional groups with the their geometrical parameters and energies can be added as a linear sum to give the solution of any organometallic or coordinate compound. 
       Alkyl Aluminum Hydrides (R n AlH 3-n ) 
       [0476]    Similar to the case of carbon and silicon, the bonding in the aluminum atom involves four sp 3  hybridized orbitals formed from the outer 3p and 3s shells except that only three HOs are filled. In organoaluminum compounds, bonds form between a Al3sp 3  HO and at least one C2sp 3  HO and one or more H1s AOs. The geometrical parameters of each AlH functional group is solved from the force balance equation of the electrons of the corresponding σ-MO and the relationships between the prolate spheroidal axes. Then, the sum of the energies of the H 2 -type ellipsoidal MOs is matched to that of the Al3sp 3  shell as in the case of the corresponding carbon and silicon molecules. As in the case of alkyl silanes given in the corresponding section, the sum of the energies of the H 2 -type ellipsoidal MO of the Al—C functional group is matched to that of the Al3sp 3  shell, and Eq. (15.51) is solved for the semimajor axis with n 1 =1 in Eq. (15.50). 
         [0477]    The energy of aluminum is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). A minimum energy is achieved while matching the potential, kinetic, and orbital energy relationships given in the Hydroxyl Radical (OH) section with the donation of 25% electron density from the participating Al3sp 3  HO to each Al—H-bond MO. 
         [0478]    The 3sp 3  hybridized orbital arrangement after Eq. (13.422) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         3 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         
                             
                         
                          
                         state 
                       
                     
                   
                   
                     
                       
                         
                           ↑ 
                           
                             0 
                              
                             
                               , 
                             
                              
                             0 
                           
                         
                          
                         
                             
                         
                          
                         
                           ↑ 
                           
                             
                               1 
                                
                               
                                 , 
                               
                             
                             - 
                             1 
                           
                         
                          
                         
                           ↑ 
                           
                             1 
                              
                             
                               , 
                             
                              
                             0 
                           
                         
                          
                         
                           
                               
                           
                           
                             1 
                              
                             
                               , 
                             
                              
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the quantum numbers (l,m l ) are below each electron. The total energy of the state is given by the sum over the three electrons. The sum E T (Al,3sp 3 ) of experimental energies [1] of Al, Al + , and Al 2+  is 
         [0000]      E T (Al,3 sp   3 )=−(28.44765 eV+18.82856 eV+5.98577 eV)=−53.26198 eV(23.2)  (23.2)
 
         [0000]    By considering that the central field decreases by an integer for each successive electron of the shell, the radius r 3sp     3   , of the Al3sp 3  shell may be calculated from the Coulombic energy using Eq. (15.13): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             3 
                              
                             
                                 
                             
                              
                             
                               sp 
                               3 
                             
                           
                         
                         = 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               10 
                             
                             12 
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   Z 
                                   - 
                                   n 
                                 
                                 ) 
                               
                                
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     53.26198 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             6 
                              
                             
                                 
                             
                              
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   53.26198 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1.53270 
                            
                           
                               
                           
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Z=13 for aluminum. Using Eq. (15.14), the Coulombic energy E Coulomb (Al, 3sp 3 ) of the outer electron of the Al3sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               Al 
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.53270 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             8.87700 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, the spin-paired 3s electrons are promoted to Al3sp 3  shell as unpaired electrons. The energy for the promotion is the magnetic energy given by Eq. (15.15) at the initial radius of the 3s electrons. From Eq. (10.255) with Z=13, the radius r 12  of Al3s shell is 
         [0000]      r 12 =1.41133a 0   (23.5)
 
         [0000]    Using Eqs. (15.15) and (23.5), the unpairing energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       ( 
                       magnetic 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         2 
                          
                         
                             
                         
                          
                         
                           πμ 
                           0 
                         
                          
                         
                           e 
                           2 
                         
                          
                         
                           ℏ 
                           2 
                         
                       
                       
                         
                           
                             m 
                             e 
                             2 
                           
                            
                           
                             ( 
                             
                               r 
                               12 
                             
                             ) 
                           
                         
                         3 
                       
                     
                     = 
                     
                       
                         
                           8 
                            
                           
                               
                           
                            
                           
                             πμ 
                             0 
                           
                            
                           
                             μ 
                             B 
                             2 
                           
                         
                         
                           
                             ( 
                             
                               1.41133 
                                
                               
                                   
                               
                                
                               
                                 a 
                                 0 
                               
                             
                             ) 
                           
                           3 
                         
                       
                       = 
                       
                         0.04070 
                          
                         
                             
                         
                          
                         eV 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (23.4) and (23.6), the energy E(Al,3sp 3 ) of the outer electron of the Al3sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               Al 
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               - 
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 e 
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     12 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               8.87700 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.04070 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             8.83630 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.7 
                   ) 
                 
               
             
           
         
       
     
         [0479]    Next, consider the formation of the Al—H-bond MO of organoaluminum hydrides wherein each aluminum atom has an Al3sp 3  electron with an energy given by Eq. (23.7). The total energy of the state of each aluminum atom is given by the sum over the three electrons. The sum E T (Al organoAl 3sp 3 ) of energies of Al3sp 3  (Eq. (23.7)), Alt, and Al 2+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Al 
                                 organoAl 
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           - 
                           
                             ( 
                             
                               
                                 28.44765 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               + 
                               
                                 18.82856 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               + 
                               
                                 E 
                                  
                                 
                                   ( 
                                   
                                     Al 
                                     , 
                                     
                                       3 
                                        
                                       
                                           
                                       
                                        
                                       
                                         sp 
                                         3 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           - 
                           
                             ( 
                             
                               
                                 28.44765 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               + 
                               
                                 18.82856 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               + 
                               
                                 8.83630 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             56.11251 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E(Al,3sp 3 ) is the sum of the energy of Al, −5.98577 eV, and the hybridization energy. 
         [0480]    Each Al—H-bond MO of each functional group AlH n=1,2,3  forms with the sharing of electrons between each Al3sp 3  HO and each H1s AO. As in the case of C—H, the H 2 -type ellipsoidal MO comprises 75% of the Al—H-bond MO according to Eq. (13.429). Furthermore, the donation of electron density from each Al3sp 3  HO to each Al—H-bond MO permits the participating orbital to decrease in size and energy. As shown below, the aluminum HOs have spin and orbital angular momentum terms in the force balance which determines the geometrical parameters of the σ MO. The angular momentum term requires that each Al—H-bond MO be treated independently in terms of the charge donation. In order to further satisfy the potential, kinetic, and orbital energy relationships, each Al3sp 3  HO donates an excess of 25% of its electron density to each Al—H-bond MO to form an energy minimum. By considering this electron redistribution in the organoaluminum hydride molecule as well as the fact that the central field decreases by an integer for each successive electron of the shell, the radius r organoAlH3sp     3    of the Al3sp 3  shell may be calculated from the Coulombic energy using Eq. (15.18): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             organoAlH 
                              
                             
                                 
                             
                              
                             3 
                              
                             
                                 
                             
                              
                             
                               sp 
                               3 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     10 
                                   
                                   12 
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               0.25 
                             
                             ) 
                           
                            
                           
                             
                               e 
                               2 
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     56.11251 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               5.75 
                                
                               
                                   
                               
                                
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     56.11251 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                           = 
                           
                             1.39422 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (23.9), the Coulombic energy E Coulomb (Al organoAlH ,3sp 3 ) of the outer electron of the Al3sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 Al 
                                 organoAlH 
                               
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 organoAlH 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.39422 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             9.75870 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, the spin-paired 3s electrons are promoted to Al3sp 3  shell as unpaired electrons. The energy for the promotion is the magnetic energy given by Eq. (23.6). Using Eqs. (23.6) and (23.10), the energy E(Al organoAlH ,3sp 3 ) of the outer electron of the Al3sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 Al 
                                 organoAlH 
                               
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                           = 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     12 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               9.75870 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.04070 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             9.71800 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (Al—H,3sp 3 ), the energy change of each Al3sp 3  shell with the formation of the Al—H-bond MO is given by the difference between Eq. (23.11) and Eq. (23.7): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Al 
                                 - 
                                 H 
                               
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   Al 
                                   organoAlH 
                                 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 Al 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     
                                       3 
                                        
                                       
                                           
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               9.71800 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 8.83630 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             0.88170 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.12 
                   ) 
                 
               
             
           
         
       
     
         [0481]    The MO semimajor axis of the Al—H functional group of organoaluminum hydrides is determined from the force balance equation of the centrifugal, Coulombic, and magnetic forces as given in the Polyatomic Molecular Ions and Molecules section and the More Polyatomic Molecules and Hydrocarbons section. The distance from the origin of the H 2 -type-ellipsoidal-MO to each focus c′, the internuclear distance 2c′, and the length of the semiminor axis of the prolate spheroidal H 2 -type MO b=c are solved from the semimajor axis a. Then, the geometric and energy parameters of the MO are calculated using Eqs. (15.1-15.117). 
         [0482]    The force balance of the centrifugal force equated to the Coulombic and magnetic forces is solved for the length of the semimajor axis. The Coulombic force on the pairing electron of the MO is 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     Coulomb 
                   
                   = 
                   
                     
                       
                          
                         2 
                       
                       
                         8 
                          
                         
                             
                         
                          
                         
                           πɛ 
                           0 
                         
                          
                         
                           ab 
                           2 
                         
                       
                     
                      
                     
                       Di 
                       ξ 
                     
                   
                 
               
               
                 
                   ( 
                   23.13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The spin pairing force is 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     
                       spin 
                        
                       
                         - 
                       
                        
                       pairing 
                     
                   
                   = 
                   
                     
                       
                         ℏ 
                         2 
                       
                       
                         2 
                          
                         
                             
                         
                          
                         
                           m 
                           e 
                         
                          
                         
                           a 
                           2 
                         
                          
                         
                           b 
                           2 
                         
                       
                     
                      
                     
                       Di 
                       ξ 
                     
                   
                 
               
               
                 
                   ( 
                   23.14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The diamagnetic force is: 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     
                       diamagneticMO 
                        
                       
                           
                       
                        
                       1 
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           
                             n 
                             e 
                           
                            
                           
                             ℏ 
                             2 
                           
                         
                         
                           4 
                            
                           
                               
                           
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                     
                      
                     
                       Di 
                       ξ 
                     
                   
                 
               
               
                 
                   ( 
                   23.15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where n e  is the total number of electrons that interact with the binding σ-MO electron. The diamagnetic force F diamagneticMO2  on the pairing electron of the σ MO is given by the sum of the contributions over the components of angular momentum: 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     
                       diamagneticMO 
                        
                       
                           
                       
                        
                       2 
                     
                   
                   = 
                   
                     - 
                     
                       
                         ∑ 
                         
                           i 
                           , 
                           j 
                         
                       
                        
                       
                         
                           
                             
                                
                               
                                 L 
                                 i 
                               
                                
                             
                              
                             ℏ 
                           
                           
                             
                               Z 
                               j 
                             
                              
                             2 
                              
                             
                                 
                             
                              
                             
                               m 
                               e 
                             
                              
                             
                               a 
                               2 
                             
                              
                             
                               b 
                               2 
                             
                           
                         
                          
                         
                           Di 
                           ξ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where |L| is the magnitude of the angular momentum of each atom at a focus that is the source of the diamagnetism at the σ-MO. The centrifugal force is 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     centrifugalMO 
                   
                   = 
                   
                     
                       - 
                       
                         
                           ℏ 
                           2 
                         
                         
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                     
                      
                     
                       Di 
                       ξ 
                     
                   
                 
               
               
                 
                   ( 
                   23.17 
                   ) 
                 
               
             
           
         
       
     
         [0483]    The force balance equation for the σ-MO of the Al—H-bond MO is the same as that of the Si—H except that Z=13 and there are three spin-unpaired electron in occupied orbitals rather than four, and the orbital with l, m l  angular momentum quantum numbers of (1,1) is unoccupied. With 
         [0000]    
       
         
           
             
               n 
               e 
             
             = 
             
               
                 2 
                  
                 
                     
                 
                  
                 and 
                  
                 
                     
                 
                  
                 
                    
                   L 
                    
                 
               
               = 
               
                 
                   3 
                    
                   
                     
                       3 
                       4 
                     
                   
                    
                   ℏ 
                    
                   
                       
                   
                    
                   and 
                    
                   
                       
                   
                    
                   
                      
                     L 
                      
                   
                 
                 = 
                 
                   3 
                    
                   
                     
                       3 
                       4 
                     
                   
                    
                   ℏ 
                 
               
             
           
         
       
     
         [0000]    corresponding to the spin and orbital angular momentum of the three occupied HOs of the Al3sp 3  shell, the force balance equation is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ℏ 
                         2 
                       
                       
                         
                           m 
                           e 
                         
                          
                         
                           a 
                           2 
                         
                          
                         
                           b 
                           2 
                         
                       
                     
                      
                     D 
                   
                   = 
                   
                     
                       
                         
                            
                           2 
                         
                         
                           8 
                            
                           
                               
                           
                            
                           
                             πɛ 
                             0 
                           
                            
                           
                             ab 
                             2 
                           
                         
                       
                        
                       D 
                     
                     + 
                     
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                               
                           
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                     - 
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               6 
                                
                               
                                 
                                   3 
                                   4 
                                 
                               
                             
                             Z 
                           
                         
                         ) 
                       
                        
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                               
                           
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                   
                 
               
               
                 
                   ( 
                   23.18 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     a 
                     = 
                     
                       
                         ( 
                         
                           2 
                           + 
                           
                             
                               6 
                                
                               
                                 
                                   3 
                                   4 
                                 
                               
                             
                             Z 
                           
                         
                         ) 
                       
                        
                       
                         a 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.19 
                   ) 
                 
               
             
           
         
       
     
         [0000]    With Z=13, the semimajor axis of the Al—H-bond MO is 
         [0000]      a=2.39970a 0   (23.20)
 
         [0000]    Using the semimajor axis, the geometric and energy parameters of the MO are calculated using Eqs. (15.1-15.127) in the same manner as the organic functional groups given in the Organic Molecular Functional Groups and Molecules section. For the Al—H functional group, c 1  is one and C 1 =0.75 based on the orbital composition as in the case of the C—H-bond MO. In organoaluminum hydrides, the energy of aluminum is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). Thus, c 2  in Eqs. (15.51) and (15.61) is also one, and the energy matching condition is determined by the C 2  parameter. Then, the hybridization factor for the Al—H-bond MO is given by the ratio of 8.87700 eV, the magnitude of E Coulomb (Al organoAlH ,3sp 3 )(Eq. (23.4)), and 13.605804 eV, the magnitude of the Coulombic energy between the electron and proton of H (Eq. (1.243)): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       C 
                       2 
                     
                      
                     
                       ( 
                       
                         organo 
                          
                         
                             
                         
                          
                         
                           Al 
                            
                           H 
                         
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         8.87700 
                          
                         
                             
                         
                          
                         eV 
                       
                       
                         13.605804 
                          
                         
                             
                         
                          
                         eV 
                       
                     
                     = 
                     0.65244 
                   
                 
               
               
                 
                   ( 
                   23.21 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Since the energy of the MO is matched to that of the Al3sp 3  HO, E(AO/HO) in Eqs. (15.51) and (15.61) is E(Al,3sp 3 ) given by Eq. (23.7), and E T (atom−atom,msp 3 .AO) is −0.88170 eV corresponding the independent single-bond charge contribution (Eq. (23.12)). The energies E D  (AlH n=1,2 ) of the functional groups AlH n=1,2  of organoaluminum hydride molecules are each given by the corresponding integer n times that of Al—H: 
         [0000]      E D (AlH n=1,2 )= n E D (AlH)  (23.22)
 
         [0484]    The branched-chain organoaluminum hydrides, R n AlH 3-n , comprise at least a terminal methyl group (CH 3 ) and at least one Al bound by a carbon-aluminum single bond comprising a C—Al group, and may comprise methylene (CH 2 ), methylyne (CH), C—C, and AlH n=1,2  functional groups. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C—C bonds can be identified. The n-alkane C—C bond is the same as that of straight-chain alkanes. In addition, the C—C bonds within isopropyl ((CH 3 ) 2  CH) and t-butyl ((CH 3 ) 3 C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C—C bonds comprise functional groups. These groups in branched-chain organoaluminum hydrides are equivalent to those in branched-chain alkanes. 
         [0485]    For the C—Al functional group, hybridization of the 2s and 2p AOs of each C and the 3s and 3p AOs of Al to form single 2sp 3  and 3sp 3  shells, respectively, forms an energy minimum, and the sharing of electrons between the C2sp 3  and Al3sp 3  HOs to form σ MO permits each participating orbital to decrease in radius and energy. Furthermore, the energy of aluminum is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). Thus, in organoaluminum hydrides, the C2sp 3  HO has a hybridization factor of 0.91771 (Eq. (13.430)) with a corresponding energy of E(C,2sp 3 )=−14.63489 eV (Eq. (15.25)), and the Al HO has an energy of E(Al,3sp 3 )=−8.83630 eV. To meet the equipotential, minimum-energy condition of the union of the Al3sp 3  and C2sp 3  HOs, c 2  and C 2  of Eqs. (15.2-15.5), (15.51), and (15.61) for the Al—C-bond MO given by Eqs. (15.77) and (15.79) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               C 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Al 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               C 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Al 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               E 
                                
                               
                                 ( 
                                 
                                   Al 
                                   , 
                                   
                                     3 
                                      
                                     
                                         
                                     
                                      
                                     
                                       sp 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                             
                               E 
                                
                               
                                 ( 
                                 
                                   C 
                                   , 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       sp 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                            
                           
                             
                               c 
                               2 
                             
                              
                             
                               ( 
                               
                                 C 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                                  
                                 HO 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 - 
                                 8.83630 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             
                               
                                 - 
                                 14.63489 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                           
                            
                           
                             ( 
                             0.91771 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.55410 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.23 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The energy of the C—Al-bond MO is the sum of the component energies of the H 2 -type ellipsoidal MO given in Eq. (15.51). Since the energy of the MO is matched to that of the Al3sp 3  HO, E(AO/HO) in Eqs. (15.51) and (15.61) is E(Al,3sp 3 ) given by Eq. (23.7). Since the C2sp 3  HOs have four electrons with a corresponding total field of ten in Eq. (15.13); whereas, the Al3sp 3  HOs have three electrons with a corresponding total field of six, E T (atom−atom,msp 3 .AO) is −0.72457 eV corresponding to the single-bond contributions of carbon (Eq. (14.151)). ΔE H     2     MO (AO/HO=E T (atom−atom,msp 3 .AO) in order to match the energies of the carbon and aluminum HOs. 
       Bridging Bonds of Organoaluminum Hydrides (Al—H—Al and Al—C—Al) 
       [0486]    As given in the Nature of the Chemical Bond of Hydrogen-Type Molecules and Molecular Ions section, the Organic Molecular Functional Groups and Molecules section, and other sections on bonding in neutral molecules, the molecular chemical bond typically comprises an integer number of paired electrons. One exception given in the Benzene Molecule section and other sections on aromatic molecules such as naphthalene, toluene, chlorobenzene, phenol, aniline, nitrobenzene, benzoic acid, pyridine, pyrimidine, pyrazine, quinoline, isoquinoline, indole, adenine, fullerene, and graphite is that the paired electrons are distributed over a linear combination of bonds such that the bonding between two atoms involves less than an integer multiple of two electrons. In these aromatic cases, three electrons can be assigned to a given bond between two atoms wherein the electrons of the linear combination of bonded atoms are paired and comprise an integer multiple of two. 
         [0487]    The Al3sp 3  HOs comprise four orbitals containing three electrons as given by Eq. (23.1). These three occupied orbitals can form three single bonds with other atoms wherein each Al3sp 3  HO and each orbital from the bonding atom contribute one electron each to the pair of the corresponding bond. However, an alternative bonding is possible that further lowers the energy of the resulting molecule wherein the remaining unoccupied orbital participates in bonding. (Actually an unoccupied orbital has no physical basis. It is only a convenient concept for the bonding electrons in this case additionally having the electron angular momentum state with l, m l  quantum numbers of (1,1)). In this case the set of two paired electrons are distributed over three atoms and belong to two bonds. Such an electron deficient bonding involving two paired electrons centered on three atoms is called a three-center bond as opposed to the typical single bond called a two-center bond. The designation for a three-center bond involving two Al3sp 3  HOs and a H1s AO is Al—H—Al, and the designation for a three-center bond involving two Al3sp 3  HOs and a C2sp 3  HO is Al—C—Al. 
         [0488]    Each Al—H—Al-bond MO and Al—C—Al-bond MO comprises the corresponding single bond and forms with further sharing of electrons between each Al3sp 3  HO and each H1s AO and C2sp 3  HO, respectively. Thus, the geometrical and energy parameters of the three-center bond are equivalent to those of the corresponding two-center bonds except that the bond energy is increased in the former case since the donation of electron density from the unoccupied Al3sp 3  HO to each Al—H—Al-bond MO and Al—C—Al-bond MO permits the participating orbital to decrease in size and energy. In order to further satisfy the potential, kinetic, and orbital energy relationships, the Al3sp 3  HO donates an additional excess of 25% of its electron density to form the bridge (three-center-bond MO) to decrease the energy in the multimer. By considering this electron redistribution in the organoaluminum hydride molecule as well as the fact that the central field decreases by an integer for each successive electron of the shell, the radius r organoAlH3sp     3    of the Al3sp 3  shell calculated from the Coulombic energy, the Coulombic energy E Coulomb (Al organoAlH ,3sp 3 ) of the outer electron of the Al3sp 3  shell, and the energy E(Al organoAlH ,3sp 3 ) of the outer electron of the Al3sp 3  shell are given by Eqs. (23.9), (23.10), and (23.11), respectively. Thus, E(Al—H—Al, 3sp 3 ) and E T (Al—C—Al,3sp 3 ), the energy change with the formation of the three-center-bond MO from the corresponding two-center-bond MO and the unoccupied Al3sp 3  HO is given by the Eq. (23.12): 
         [0000]      E T (Al—H—Al,3sp 3 )=E T (Al—C—Al,3sp 3 )=−0.88170 eV  (23.24)
 
         [0000]    The upper range of the experimental association enthalpy per bridge for both of the reactions 
         [0000]      2AlH(CH 3 ) 2 [AlH(CH 3 ) 2 ] 2   (23.25)
 
         [0000]      and 
         [0000]      2Al(CH 3 ) 3 →[Al(CH 3 ) 3 ] 2   (23.26)
 
         [0000]    is [2] 
         [0000]      E T (Al—H—Al,3sp 3 )=E T (Al—C—Al,3sp 3 )=−0.867 eV  (23.27)
 
         [0000]    which agrees with Eq. (23.24) very well. 
         [0489]    The symbols of the functional groups of alkyl organoaluminum hydrides are given in Table 23.1. The geometrical (Eqs. (15.1-15.5), (23.20), and (23.23) and intercept (Eqs. (15.80-15.87)) parameters of alkyl organoaluminum hydrides are given in Tables 23.2 and 23.3, respectively (both as shown in the priority document). Since the energy of the Al3sp 3  HO is matched to that of the C2sp 3  HO, the radius r mol2sp     3    of the Al3sp 3  HO of the aluminum atom and the C2sp 3  HO of the carbon atom of a given C—Al-bond MO are calculated after Eq. (15.32) by considering ΣE T     mol   (MO,2sp 3 ), the total energy donation to all bonds with which each atom participates in bonding. In the case that the MO does not intercept the Al HO due to the reduction of the radius from the donation of Al 3sp 3  HO charge to additional MO&#39;s, the energy of each MO is energy matched as a linear sum to the Al HO by contacting it through the bisector current of the intersecting MOs as described in the Methane Molecule (CH 4 ) section. The energy (Eq. (15.61), (23.4), (23.7), and (23.21-23.23)) parameters of alkyl organoaluminum hydrides are given in Table 23.5 (as shown in the priority document). The total energy of each alkyl aluminum hydride given in Table 23.5 (as shown in the priority document) was calculated as the sum over the integer multiple of each E D  (Group) of Table 23.4 (as shown in the priority document) corresponding to functional-group composition of the molecule. E mag  of Table 23.4 (as shown in the priority document) is given by Eqs. (15.15) and (23.3). The bond angle parameters of organoaluminum hydrides determined using Eqs. (15.88-15.117) are given in Table 23.6 (as shown in the priority document). The charge-density in trimethyl aluminum is shown in  FIG. 38 . 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 23.1 
               
             
             
               
                   
               
               
                 The symbols of the functional groups of organoaluminum hydrides. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
               
               
                   
                 AlH group of AlH n=1,2   
                 Al—H 
               
               
                   
                 AlHAl (bridged H) 
                 Al—H—Al 
               
               
                   
                 CAl bond 
                 C—Al 
               
               
                   
                 ALCAl (bridged C) 
                 Al—C—Al 
               
               
                   
                 CH 3  group 
                 C—H (CH 3 ) 
               
               
                   
                 CH 2  group 
                 C—H (CH 2 ) 
               
               
                   
                 CH 
                 C—H 
               
               
                   
                 CC bond (n-C) 
                 C—C (a) 
               
               
                   
                 CC bond (iso-C) 
                 C—C (b) 
               
               
                   
                 CC bond (tert-C) 
                 C—C (c) 
               
               
                   
                 CC (iso to iso-C) 
                 C—C (d) 
               
               
                   
                 CC (t to t-C) 
                 C—C (e) 
               
               
                   
                 CC (t to iso-C) 
                 C—C (f) 
               
               
                   
               
             
          
         
       
     
       Transition Metal Organometallic and Coordinate Bond 
       [0490]    The transition-metal atoms fill the 3d orbitals in the series Sc to Zn. The 4s orbitals are filled except in the cases of Cr and Cu wherein one 4s electron occupies a 3d orbital to achieve a half-filled and filled 3d shell, respectively. Experimentally the transition-metal elements ionize successively from the 4s shell to the 3d shell [12]. Thus, bonding in the transition metals involves the hybridization of the 3d and 4s electrons to form the corresponding number of 3d 4s HOs except for Cu and Zn which each have a filled inner 3d shell and one and two outer 4s electrons, respectively. Cu may form a single bond involving the 4s electron or the 3d and shells may hybridize to form multiple bonds with one or more ligands. The 4s shell of Zn hybridizes to form two 4s HOs that provide for two possible bonds, typically two metal-alkyl bonds. 
         [0491]    For organometallic and coordinate compounds comprised of carbon, the geometrical and energy equations are given in the Derivation of the General Geometrical and Energy Equations of Organic Chemistry section. For metal-ligand bonds other than to carbon, the force balance equation is that developed in the Force Balance of the σ MO of the Carbon Nitride Radical section wherein the diamagnetic force terms include orbital and spin angular momentum contributions. The electrons of the 3d4s HOs may pair such that the binding energy of the HO is increased. The hybridization factor accordingly changes which effects the bond distances and energies. The diamagnetic terms of the force balance equations of the electrons of the MOs formed between the 3d4s HOs and the AOs of the ligands also changes depending on whether the nonbonding HOs are occupied by paired or unpaired electrons. The orbital and spin angular momentum of the HOs and MOs is then determined by the state that achieves a minimum energy including that corresponding to the donation of electron charge from the HOs and AOs to the MOs. Historically, according to “crystal field theory and molecular orbital theory [13] the possibility of a bonding metal atom achieving a so called “high-spin” or “low-spin” state having unpaired electrons occupying higher-energy orbitals versus paired electrons occupying lower-energy orbitals was due to the strength of the ligand crystal field or the interaction between metal orbitals and the ligands, respectively. Excited-state spectral data recorded on transition-metal organometallic and coordinate compounds has been misinterpreted. Excitation of an unpaired electron in a 3d4s HO to a 3d4s paired state is equivalent to an excitation of the molecule to a higher energy MO since the MOs change energy due to the corresponding change in the hybridization factor and diamagnetic force balance terms. But, levels misidentified as crystal field levels do not exist in the absence of excitation by a photon. 
         [0492]    The parameters of the 3d4s HOs are determined using Eqs. (15.12-15.21). For transition metal atoms with electron configuration 3d n 4s 2 , the spin-paired 4s electrons are promoted to 3d4s shell during hybridization as unpaired electrons. Also, for n&gt;5 the electrons of the 3d shell are spin-paired and these electrons are promoted to 3d4s shell during hybridization as unpaired electrons. The energy for each promotion is the magnetic energy given by Eq. (15.15) at the initial radius of the 4s electrons and the paired 3d electrons determined using Eq. (10.102) with the corresponding nuclear charge Z of the metal atom and the number electrons n of the corresponding ion with the filled outer shell from which the pairing energy is determined. Typically, the electrons from the 4s and 3d shells successively fill unoccupied HOs until the HO shell is filled with unpaired electrons, then the electrons pair per HO. The magnetic energy of paring given by Eqs. (15.13) and (15.15) is added to E Coulomb (atom,3d4s) the for each pair. Thus, after Eq. (15.16), the energy E(atom,3d4s) of the outer electron of the atom 3d4s shell is given by the sum of E Coulomb (atom,3d4s) and E(magnetic): 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       ( 
                       
                         atom 
                         , 
                         
                           3 
                            
                           
                               
                           
                            
                           d 
                            
                           
                               
                           
                            
                           4 
                            
                           
                               
                           
                            
                           s 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                            
                           2 
                         
                       
                       
                         8 
                          
                         
                             
                         
                          
                         
                           πɛ 
                           0 
                         
                          
                         
                           r 
                           
                             3 
                              
                             
                                 
                             
                              
                             d 
                              
                             
                                 
                             
                              
                             4 
                              
                             
                                 
                             
                              
                             s 
                           
                         
                       
                     
                     + 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         
                           πμ 
                           0 
                         
                          
                         
                            
                           2 
                         
                          
                         
                           ℏ 
                           2 
                         
                       
                       
                         
                           m 
                           e 
                           2 
                         
                          
                         
                           r 
                           
                             4 
                              
                             
                                 
                             
                              
                             s 
                           
                           3 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           3 
                            
                           
                               
                           
                            
                           d 
                            
                           
                               
                           
                            
                           pairs 
                         
                       
                        
                       
                         
                           2 
                            
                           
                               
                           
                            
                           
                             πμ 
                             0 
                           
                            
                           
                              
                             2 
                           
                            
                           
                             ℏ 
                             2 
                           
                         
                         
                           
                             m 
                             e 
                             2 
                           
                            
                           
                             r 
                             
                               3 
                                
                               
                                   
                               
                                
                               d 
                             
                             3 
                           
                         
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           HO 
                            
                           
                               
                           
                            
                           pairs 
                         
                       
                        
                       
                         
                           2 
                            
                           
                               
                           
                            
                           
                             πμ 
                             0 
                           
                            
                           
                              
                             2 
                           
                            
                           
                             ℏ 
                             2 
                           
                         
                         
                           
                             m 
                             e 
                             2 
                           
                            
                           
                             r 
                             
                               3 
                                
                               
                                   
                               
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                   
                               
                                
                               s 
                             
                             3 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.28 
                   ) 
                 
               
             
           
         
       
     
         [0493]    The sharing of electrons between the metal 3d4s HOs and the ligand AOs or HOs to form a M-L-bond MO (L not C) permits each participating hybridized or atomic orbital to decrease in radius and energy. Due to the low binding energy of the metal atom and the high electronegativity of the ligand, an energy minimum is achieved while further satisfying the potential, kinetic, and orbital energy relationships, each metal 3d4s HO donates an excess of an electron per bond of its electron density to the M-L-bond MO. In each case, the radius of the hybridized shell is calculated from the Coulombic energy equation by considering that the central field decreases by an integer for each successive electron of the shell and the total energy of the shell is equal to the total Coulombic energy of the initial AO electrons plus the hybridization energy. After Eq. (15.17), the total energy E T (mol.atom,3d4s) of the HO electrons is given by the sum of energies of successive ions of the atom over the n electrons comprising total electrons of the initial AO shell and the hybridization energy: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       T 
                     
                      
                     
                       ( 
                       
                         
                           mol 
                           . 
                           atom 
                         
                         , 
                         
                           3 
                            
                           
                               
                           
                            
                           d 
                            
                           
                               
                           
                            
                           4 
                            
                           
                               
                           
                            
                           s 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       E 
                        
                       
                         ( 
                         
                           atom 
                           , 
                           
                             3 
                              
                             
                                 
                             
                              
                             d 
                              
                             
                                 
                             
                              
                             4 
                              
                             
                                 
                             
                              
                             s 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           2 
                         
                         n 
                       
                        
                       
                         IP 
                         m 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.29 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where IP m  is the m th ionization energy (positive) of the atom and the sum −IP 1  of plus the hybridization energy is E(atom,3d4s). Thus, the radius r 3d4s  of the hybridized shell due to its donation of a total charge −Qe to the corresponding MO is given by is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             3 
                              
                             
                                 
                             
                              
                             d 
                              
                             
                                 
                             
                              
                             4 
                              
                             
                                 
                             
                              
                             s 
                           
                         
                         = 
                           
                          
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     q 
                                     = 
                                     
                                       Z 
                                       - 
                                       n 
                                     
                                   
                                   
                                     Z 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     q 
                                   
                                   ) 
                                 
                               
                               - 
                               Q 
                             
                             ) 
                           
                            
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 
                                   E 
                                   T 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       mol 
                                       . 
                                       atom 
                                     
                                     , 
                                     
                                       3 
                                        
                                       
                                           
                                       
                                        
                                       d 
                                        
                                       
                                           
                                       
                                        
                                       4 
                                        
                                       
                                           
                                       
                                        
                                       s 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     q 
                                     = 
                                     
                                       Z 
                                       - 
                                       n 
                                     
                                   
                                   
                                     Z 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     q 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 s 
                                  
                                 
                                   ( 
                                   0.25 
                                   ) 
                                 
                               
                             
                             ) 
                           
                            
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 
                                   E 
                                   T 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       mol 
                                       . 
                                       atom 
                                     
                                     , 
                                     
                                       3 
                                        
                                       
                                           
                                       
                                        
                                       d 
                                        
                                       
                                           
                                       
                                        
                                       4 
                                        
                                       
                                           
                                       
                                        
                                       s 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.30 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where −e is the fundamental electron charge, s=1,2,3 for a single, double, and triple bond, respectively, and s=4 for typical coordinate and organometallic compounds wherein L is not carbon. The Coulombic energy E Coulomb (mol.atom,3d4s) of the outer electron of the atom 3d4s shell is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       Coulomb 
                     
                      
                     
                       ( 
                       
                         
                           mol 
                           . 
                           atom 
                         
                         , 
                         
                           3 
                            
                           
                               
                           
                            
                           d 
                            
                           
                               
                           
                            
                           4 
                            
                           
                               
                           
                            
                           s 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       - 
                       
                          
                         2 
                       
                     
                     
                       8 
                        
                       
                           
                       
                        
                       
                         πɛ 
                         0 
                       
                        
                       
                         r 
                         
                           3 
                            
                           
                               
                           
                            
                           d 
                            
                           
                               
                           
                            
                           4 
                            
                           
                               
                           
                            
                           s 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.31 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In the case that during hybridization the metal spin-paired 4s AO electrons are unpaired to contribute electrons to the 3d4s HO, the energy change for the promotion to the unpaired state is the magnetic energy E(magnetic) at the initial radius r of the AO electron given by Eq. (15.15). In addition in the case that the 3d4s HO electrons are paired, the corresponding magnetic energy is added. Then, the energy E(mol.atom,3d4s) of the outer electron of the atom 3d4s shell is given by the sum of E Coulomb (mol.atom,3d4s) and E(magnetic): 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       ( 
                       
                         
                           mol 
                           · 
                           atom 
                         
                         , 
                         
                           3 
                            
                           d 
                            
                           
                               
                           
                            
                           4 
                            
                           s 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                            
                           2 
                         
                       
                       
                         8 
                          
                         
                           πɛ 
                           0 
                         
                          
                         
                           r 
                           
                             3 
                              
                             d 
                              
                             
                                 
                             
                              
                             4 
                              
                             s 
                           
                         
                       
                     
                     + 
                     
                       
                         2 
                          
                         
                           πμ 
                           0 
                         
                          
                         
                            
                           2 
                         
                          
                         
                           ℏ 
                           2 
                         
                       
                       
                         
                           m 
                           e 
                           2 
                         
                          
                         
                           r 
                           
                             4 
                              
                             s 
                           
                           3 
                         
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           HO 
                            
                           
                               
                           
                            
                           pairs 
                         
                       
                        
                       
                         
                           2 
                            
                           
                             πμ 
                             0 
                           
                            
                           
                              
                             2 
                           
                            
                           
                             ℏ 
                             2 
                           
                         
                         
                           
                             m 
                             e 
                             2 
                           
                            
                           
                             r 
                             
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               s 
                             
                             3 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.32 
                   ) 
                 
               
             
           
         
       
     
         [0000]    E T (atom−atom,3d4s), the energy change of each atom msp 3  shell with the formation of the atom-atom-bond MO is given by the difference between E(mol.atom,3d4s) and E(atom,3d4s): 
         [0000]        E   T (atom−atom,3 d 4 s )= E (mol.atom,3 d 4 s )− E (atom,3 d 4 s )  (23.33)
 
         [0494]    Any unpaired electrons of ligands typically pair with unpaired HO electrons of the metal. In the case that no such electrons of the metal are available, the ligand electrons pair and form a bond with an unpaired metal HO when available. An unoccupied HO may form by the pairing of the corresponding HO electrons to form an energy minimum due to the effect on the bond parameters such as the diamagnetic force term, hybridization factor, and the E T (atom−atom,msp 3 .AO) term. In the case of carbonyls, the two unpaired Csp 3  HO electrons on each carbonyl pair with any unpaired electrons of the metal HOs. Any excess carbonyl electrons pair in the formation of the corresponding MO and any remaining metal HO electrons pair where possible. In the latter case, the energy of the HO for the determination of the hybridization factor and other bonding parameters in Eqs. (15.51) and (15.65) is given by the Coulombic energy plus the pairing energy. 
         [0495]    The force balance of the centrifugal force equated to the Coulombic and magnetic forces is solved for the length of the semimajor axis. The Coulombic force on the pairing electron of the MO is 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     Coulomb 
                   
                   = 
                   
                     
                       
                          
                         2 
                       
                       
                         8 
                          
                         
                           πɛ 
                           0 
                         
                          
                         
                           ab 
                           2 
                         
                       
                     
                      
                     
                       Di 
                       ξ 
                     
                   
                 
               
               
                 
                   ( 
                   23.34 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The spin pairing force is 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     
                       spin 
                        
                       
                         - 
                       
                        
                       pairing 
                     
                   
                   = 
                   
                     
                       
                         ℏ 
                         2 
                       
                       
                         2 
                          
                         
                           m 
                           e 
                         
                          
                         
                           a 
                           2 
                         
                          
                         
                           b 
                           2 
                         
                       
                     
                      
                     
                       Di 
                       ξ 
                     
                   
                 
               
               
                 
                   ( 
                   23.35 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The diamagnetic force is: 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     
                       diamagneticMO 
                        
                       
                           
                       
                        
                       1 
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           
                             n 
                             e 
                           
                            
                           
                             ℏ 
                             2 
                           
                         
                         
                           4 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                     
                      
                     
                       Di 
                       ξ 
                     
                   
                 
               
               
                 
                   ( 
                   23.36 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where n e  is the total number of electrons that interact with the binding σ-MO electron. The diamagnetic force F diamagneticMO2  on the pairing electron of the σ-MO is given by the sum of the contributions over the components of angular momentum: 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     
                       diamagneticMO 
                        
                       
                           
                       
                        
                       2 
                     
                   
                   = 
                   
                     - 
                     
                       
                         ∑ 
                         i 
                       
                        
                       
                         
                           
                             
                                
                               
                                 L 
                                 i 
                               
                                
                             
                              
                             ℏ 
                           
                           
                             Z 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                               m 
                               e 
                             
                              
                             
                               a 
                               2 
                             
                              
                             
                               b 
                               2 
                             
                           
                         
                          
                         
                           Di 
                           ξ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.37 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where |L i | is the magnitude of the angular momentum component of the metal atom at a focus that is the source of the diamagnetism at the σ-MO. The centrifugal force is 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     centrifugalMO 
                   
                   = 
                   
                     
                       - 
                       
                         
                           ℏ 
                           2 
                         
                         
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                     
                      
                     
                       Di 
                       ξ 
                     
                   
                 
               
               
                 
                   ( 
                   23.38 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The general force balance equation for the σ-MO of the metal (M) to ligand (L) M-L-bond MO in terms of n e  and |L i | corresponding to the orbital and spin angular momentum terms of the 3d 4s HO shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ℏ 
                         2 
                       
                       
                         
                           m 
                           e 
                         
                          
                         
                           a 
                           2 
                         
                          
                         
                           b 
                           2 
                         
                       
                     
                      
                     D 
                   
                   = 
                   
                     
                       
                         
                            
                           2 
                         
                         
                           8 
                            
                           
                             πɛ 
                             0 
                           
                            
                           
                             ab 
                             2 
                           
                         
                       
                        
                       D 
                     
                     + 
                     
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                     - 
                     
                       
                         ( 
                         
                           
                             
                               n 
                               e 
                             
                             2 
                           
                           + 
                           
                             
                               ∑ 
                               i 
                             
                              
                             
                               
                                  
                                 
                                   L 
                                   i 
                                 
                                  
                               
                               Z 
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           ℏ 
                           2 
                         
                         
                           2 
                            
                           
                             m 
                             e 
                           
                            
                           
                             a 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                        
                       D 
                     
                   
                 
               
               
                 
                   ( 
                   23.39 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Having a solution for the semimajor axis a of 
         [0000]    
       
         
           
             
               
                 
                   a 
                   = 
                   
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             n 
                             e 
                           
                           2 
                         
                         + 
                         
                           
                             ∑ 
                             i 
                           
                            
                           
                             
                                
                               
                                 L 
                                 i 
                               
                                
                             
                             Z 
                           
                         
                       
                       ) 
                     
                      
                     
                       a 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   23.40 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In term of the total angular momentum L, the semimajor axis a is 
         [0000]    
       
         
           
             
               
                 
                   a 
                   = 
                   
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             n 
                             e 
                           
                           2 
                         
                         + 
                         
                           L 
                           Z 
                         
                       
                       ) 
                     
                      
                     
                       a 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   23.41 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using the semimajor axis, the geometric and energy parameters of the MO are calculated using Eqs. (15.1-15.117) in the same manner as the organic functional groups given in the Organic Molecular Functional Groups and Molecules section. 
         [0496]    Bond angles in organometallic and coordinate compounds are determined using the standard Eqs. (15.70-15.79) and (15.88-15.117) with the appropriate E T (atom−atom,msp 3 .AO) for energy matching with the B—C terminal bond of the corresponding angle ∠BAC. For bond angles in general, if the groups can be maximally displaced in terms of steric interactions and magnitude of the residual E T  term is less that the steric energy, then the geometry that minimizes the steric interactions is the lowest energy. Steric-energy minimizing geometries include tetrahedral (T d ) and octahedral symmetry (O h ). 
       Scandium Functional Groups and Molecules 
       [0497]    The electron configuration of scandium is [Ar]4s 2 3d having the corresponding term  2 D 3/2 . The total energy of the state is given by the sum over the three electrons. The sum E T (Sc,3d4s) of experimental energies [1] of Sc, Sc + , and Sc 2+  is 
         [0000]      E T (Sc,3 d 4 s )=−(24.75666 eV+12.79977 eV+6.56149 eV)=−44.11792 eV  (23.42)
 
         [0000]    By considering that the central field decreases by an integer for each successive electron of the shell, the radius r 3d4s  of the Sc3d4s shell may be calculated from the Coulombic energy using Eq. (15.13): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             3 
                              
                             d 
                              
                             
                                 
                             
                              
                             4 
                              
                             s 
                           
                         
                         = 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               18 
                             
                             20 
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   Z 
                                   - 
                                   n 
                                 
                                 ) 
                               
                                
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     44.11792 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             6 
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   44.11792 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1.85038 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.43 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Z=21 for scandium. Using Eq. (15.14), the Coulombic energy E Coulomb (SC,3d 45) of the outer electron of the Sc3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               Sc 
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.85038 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             7.35299 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.44 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, the spin-paired 4s electrons are promoted to Sc3d4s shell as unpaired electrons. The energy for the promotion is the magnetic energy given by Eq. (15.15) at the initial radius of the 4s electrons. From Eq. (10.102) with Z=21 and n=21, the radius r 21  of Sc4s shell is 
         [0000]      r 21 =2.07358a 0   (23.45)
 
         [0000]    Using Eqs. (15.15) and (23.45), the unpairing energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                         
                           
                             2 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                                
                               2 
                             
                              
                             
                               ℏ 
                               2 
                             
                           
                           
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   21 
                                 
                                 ) 
                               
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             8 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                               μ 
                               B 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 2.07358 
                                  
                                 
                                   a 
                                   0 
                                 
                               
                               ) 
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           0.01283 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.46 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (23.44) and (23.46), the energy E(Sc,3d4s) of the outer electron of the Sc3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               Sc 
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     21 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               7.352987 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.01283 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             7.34015 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.47 
                   ) 
                 
               
             
           
         
       
     
         [0498]    Next, consider the formation of the Sc-L-bond MO of wherein each scandium atom has an Sc3d4s electron with an energy given by Eq. (23.47). The total energy of the state of each scandium atom is given by the sum over the three electrons. The sum E T (SC Sc-L 3d4s) of energies of Sc3d4s (Eq. (23.47)), Sc + , and Sc 2+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Sc 
                                 
                                   Sc 
                                    
                                   
                                     - 
                                   
                                    
                                   L 
                                 
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               s 
                             
                             ) 
                           
                         
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       24.75666 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       12.79977 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     E 
                                      
                                     
                                       ( 
                                       
                                         Sc 
                                         , 
                                         
                                           3 
                                            
                                           d 
                                            
                                           
                                               
                                           
                                            
                                           4 
                                            
                                           s 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       24.75666 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       12.79977 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     7.34015 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             44.89658 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.48 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E(Sc,3d4s) is the sum of the energy of Sc, −6.56149 eV, and the hybridization energy. 
         [0499]    The scandium HO donates an electron to each MO. Using Eq. (23.30), the radius radius r 3d4s  of the Ti3d4s shell calculated from the Coulombic energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             Sc 
                              
                             
                               - 
                             
                              
                             L 
                              
                             
                                 
                             
                              
                             3 
                              
                             d 
                              
                             
                                 
                             
                              
                             4 
                              
                             s 
                           
                         
                         = 
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     18 
                                   
                                   20 
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               1 
                             
                             ) 
                           
                            
                           
                             
                                
                               2 
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     44.89658 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             5 
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   44.89658 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1.51524 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.49 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (23.49), the Coulombic energy E Coulomb (Sc Sc-L ,3d4s) of the outer electron of the Sc3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 Sc 
                                 
                                   Sc 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 Sc 
                                 - 
                                 
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.51524 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             8.97932 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.50 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The only magnetic energy term is that for unpairing of the 4s electrons given by Eq. (23.46). Using Eqs. (23.32), (23.46), and (23.50), the energy E(Sc Sc-L ,3d4s) of the outer electron of the Sc3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 Sc 
                                 
                                   Sc 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   Sc 
                                   - 
                                   
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     s 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     21 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               8.97932 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.01283 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             8.96648 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.51 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (Sc-L,3d4s), the energy change of each Sc3d4s shell with the formation of the Sc-L-bond MO is given by the difference between Eq. (23.51) and Eq. (23.47): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Sc 
                                 - 
                                 L 
                               
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   Sc 
                                   
                                     Sc 
                                     - 
                                     L 
                                   
                                 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 Sc 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               8.96648 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 7.34015 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             1.62633 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.52 
                   ) 
                 
               
             
           
         
       
     
         [0500]    The semimajor axis a solution given by Eq. (23.41) of the force balance equation, Eq. (23.39), for the σ-MO of the Sc-L-bond MO of ScL n  is given in Table 23.8 (as shown in the priority document) with the force-equation parameters Z=21, n e , and L corresponding to the orbital and spin angular momentum terms of the 3d4s HO shell. 
         [0501]    For the Sc-L functional groups, hybridization of the 4s and 3d AOs of Sc to form a single 3d4s shell forms an energy minimum, and the sharing of electrons between the Sc3d4s HO and L AO to form σ MO permits each participating orbital to decrease in radius and energy. The F AO has an energy of E(F)=−17.42282 eV, the CI AO has an energy of E(Cl)=−12.96764 eV, the O AO has an energy of E(O)=−13.61805 eV, and the Sc3d4s HOs has an energy of E(Sc,3d4s)=−7.34015 eV (Eq. (23.47)). To meet the equipotential condition of the union of the Sc-L H 2 -type-ellipsoidal-MO with these orbitals, the hybridization factor(s), at least one of c 2  and C 2  of Eq. (15.61) for the Sc-L-bond MO given by Eq. (15.77) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               FAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Sc 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               sHO 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               FAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Sc 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               sHO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Sc 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               FAO 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               7.34015 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               17.42282 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.42130 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.53 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               ClAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Sc 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               sHO 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               ClAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Sc 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               sHO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Sc 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               ClAO 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               7.34015 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               12.96764 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.56604 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.54 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               O 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Sc 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               sHO 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Sc 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               O 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               7.34015 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               13.61805 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.53900 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.55 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Since the energy of the MO is matched to that of the Sc3d4s HO, E(AO/HO) in Eq. (15.61) is E(Sc,3d4s) given by Eq. (23.47) and twice this value for double bonds. E T (atom−atom,msp 3 .AO) of the Sc-L-bond MO is determined by considering that the bond involves an electron transfer from the scandium atom to the ligand atom to form partial ionic character in the bond as in the case of the zwitterions such as H 2 B + —F −  given in the Halido Boranes section. E T (atom−atom,msp 3 .AO) is −3.25266 eV, two times the energy of Eq. (23.52) for single bonds, and −6.50532 eV, four times the energy of Eq. (23.52) for double bonds. 
         [0502]    The symbols of the functional groups of scandium coordinate compounds are given in Table 23.7. The geometrical (Eqs. (15.1-15.5) and (23.41)), intercept (Eqs. (15.31-15.32) and (15.80-15.87)), and energy (Eqs. (15.61) and (23.28-23.33)) parameters of scandium coordinate compounds are given in Tables 23.8, 23.9 (as shown in the priority document), and 23.10, respectively. The total energy of each scandium coordinate compounds given in Table 23.11 (as shown in the priority document) was calculated as the sum over the integer multiple of each E D  (Group) of Table 23.10 corresponding to functional-group composition of the compound. The charge-densities of exemplary scandium coordinate compound, scandium trifluoride comprising the concentric shells of atoms with the outer shell bridged by one or more H 2 -type ellipsoidal MOs or joined with one or more hydrogen MOs is shown in  FIG. 39 . 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 23.7 
               
             
             
               
                   
               
               
                 The symbols of the functional groups of scandium coordinate 
               
               
                 compounds. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
               
               
                   
                 ScF group of ScF 
                 Sc—F (a) 
               
               
                   
                 ScF group of ScF 2   
                 Sc—F (b) 
               
               
                   
                 ScF group of ScF 3   
                 Sc—F (c) 
               
               
                   
                 ScCl group of ScCl 
                 Sc—Cl 
               
               
                   
                 ScO group of ScO 
                 Sc—O 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 23.8 
               
             
             
               
                   
               
               
                 The geometrical bond parameters of scandium 
               
               
                 coordinate compounds and experimental values. 
               
             
          
           
               
                 Para- 
                 Sc—F (a) 
                 Sc—F (b) 
                 Sc—F (c) 
                 Sc—Cl 
                 Sc—O 
               
               
                 meter 
                 Group 
                 Groups 
                 Group 
                 Group 
                 Group 
               
               
                   
               
               
                 n e   
                 1 
                 2 
                 2 
                 2 
                 1 
               
               
                   
               
               
                 L 
                 
                   
                     
                       
                         2 
                         + 
                         
                           
                             3 
                             4 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         4 
                          
                         
                           
                             3 
                             4 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                         + 
                         
                           
                             3 
                             4 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         1 
                         + 
                         
                           3 
                            
                           
                             
                               3 
                               4 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         3 
                         + 
                         
                           2 
                            
                           
                             
                               3 
                               4 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 a (a 0 ) 
                 1.63648 
                 2.16496 
                 2.13648 
                 2.17134 
                 1.72534 
               
               
                 c′ (a 0 ) 
                 1.60922 
                 1.60294 
                 1.59236 
                 1.95858 
                 1.51672 
               
               
                 Bond 
                 1.70313 
                 1.69647 
                 1.68528 
                 2.07287 
                 1.60523 
               
               
                 Length 
                   
                   
                   
                   
                   
               
               
                 2c′ (Å) 
                   
                   
                   
                   
                   
               
               
                 Exp. 
                 1.788 [14] 
                 1.788 [14] 
                 1.788 [41] 
                 2.229 [15] 
                 1.668 [15] 
               
               
                 Bond 
                 (scandium 
                 (scandium 
                 (scandium 
                 (scandium 
                 (scandium 
               
               
                 Length 
                 fluoride) 
                 fluoride) 
                 fluoride) 
                 chloride) 
                 oxide) 
               
               
                 (Å) 
                   
                   
                   
                   
                   
               
               
                 b, c (a 0 ) 
                 0.29743 
                 1.45521 
                 1.45521 
                 0.93737 
                 0.82240 
               
               
                 e 
                 0.98335 
                 0.74040 
                 0.74040 
                 0.90202 
                 0.87909 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 23.10 
               
             
             
               
                   
               
               
                 The energy parameters (eV) of functional groups of 
               
               
                 scandium coordinate compounds. 
               
             
          
           
               
                   
                 Sc—F (a) 
                 Sc—F (b) 
                 Sc—F (c) 
                 Sc—Cl 
                 Sc—O 
               
               
                 Parameters 
                 Groups 
                 Groups 
                 Group 
                 Group 
                 Group 
               
               
                   
               
             
          
           
               
                 n 1   
                 1 
                 1 
                 1 
                 1 
                 2 
               
               
                 n 2   
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 n 3   
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 C 1   
                 0.75 
                 1 
                 1 
                 0.5 
                 0.375 
               
               
                 C 2   
                 0.42130 
                 0.42130 
                 0.42130 
                 0.56604 
                 1 
               
               
                 c 1   
                 1 
                 1 
                 1 
                 1 
                 1 
               
               
                 c 2   
                 0.42130 
                 1 
                 1 
                 0.56604 
                 0.53900 
               
               
                 c 3   
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 c 4   
                 1 
                 1 
                 1 
                 1 
                 2 
               
               
                 c 5   
                 1 
                 1 
                 1 
                 1 
                 2 
               
               
                 C 1o   
                 0.75 
                 1 
                 1 
                 0.5 
                 0.375 
               
               
                 C 2o   
                 0.42130 
                 0.42130 
                 0.42130 
                 0.56604 
                 1 
               
               
                 V e  (eV) 
                 −34.05166 
                 −32.30098 
                 −32.89066 
                 −23.32429 
                 −53.06036 
               
               
                 V p  (eV) 
                 8.45489 
                 8.48805 
                 8.54444 
                 6.94677 
                 17.94106 
               
               
                 T (eV) 
                 10.40395 
                 7.45996 
                 7.69741 
                 5.37095 
                 15.37682 
               
               
                 V m  (eV) 
                 −5.20198 
                 −3.72998 
                 −3.84870 
                 −2.68548 
                 −7.68841 
               
               
                 E(AO/HO) (eV) 
                 −7.34015 
                 −7.34015 
                 −7.34015 
                 −7.34015 
                 −14.68031 
               
               
                 ΔE H     2     MO (AO/HO) (eV) 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 E T (AO/MO) (eV) 
                 −7.34015 
                 −7.34015 
                 −7.34015 
                 −7.34015 
                 −14.68031 
               
               
                 E T (H 2 MO) (eV) 
                 −27.73495 
                 −27.42310 
                 −27.83768 
                 −21.03220 
                 −42.11120 
               
               
                 E T (atom-atom,msp 3 .AO) (eV) 
                 −3.25266 
                 −3.25266 
                 −3.25266 
                 −3.25266 
                 −6.50532 
               
               
                 E T (MO) (eV) 
                 −30.98761 
                 −30.67576 
                 −31.09034 
                 −24.28486 
                 −48.61652 
               
               
                 ω (10 15  rad/s) 
                 11.1005 
                 15.2859 
                 8.59272 
                 6.87387 
                 33.9452 
               
               
                 E K  (eV) 
                 7.30656 
                 10.06142 
                 5.65588 
                 4.52450 
                 22.34334 
               
               
                 Ē D  (eV) 
                 −0.16571 
                 −0.19250 
                 −0.14628 
                 −0.10219 
                 −0.22732 
               
               
                 Ē Kvib  (eV) 
                 0.09120 
                 0.09120 
                 0.09120 
                 0.04823 
                 0.12046 
               
               
                   
                 [14] 
                 [14] 
                 [14] 
                 [16] 
                 [17] 
               
               
                 Ē osc  (eV) 
                 −0.12011 
                 −0.14690 
                 −0.10068 
                 −0.07808 
                 −0.16709 
               
               
                 E T (Group) (eV) 
                 −31.10771 
                 −30.82266 
                 −31.19102 
                 −24.36294 
                 −48.95069 
               
               
                 E initial (c 4  AO/HO) (eV) 
                 −7.34015 
                 −7.34015 
                 −7.34015 
                 −7.34015 
                 −7.34015 
               
               
                 E initial (c 5  AO/HO) (eV) 
                 −17.42282 
                 −17.42282 
                 −17.42282 
                 −12.96764 
                 −13.61806 
               
               
                 E D (Group) (eV) 
                 6.34474 
                 6.05969 
                 6.42804 
                 4.05515 
                 7.03426 
               
               
                   
               
             
          
         
       
     
       Titanium Functional Groups and Molecules 
       [0503]    The electron configuration of titanium is [Ar]4s 2 3d 2  having the corresponding term  3 F 2 . The total energy of the state is given by the sum over the four electrons. The sum E T  (Ti,3d 4s) of experimental energies [1] of Ti, Ti + , Ti 2+ , and Ti 3+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               Ti 
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       43.2672 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       27.4917 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       13.5755 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       6.82812 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             91.16252 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.56 
                   ) 
                 
               
             
           
         
       
     
         [0000]    By considering that the central field decreases by an integer for each successive electron of the shell, the radius r 3d4s  of the Ti3d4s shell may be calculated from the Coulombic energy using Eq. (15.13): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             3 
                              
                             d 
                              
                             
                                 
                             
                              
                             4 
                              
                             s 
                           
                         
                         = 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               18 
                             
                             21 
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   Z 
                                   - 
                                   n 
                                 
                                 ) 
                               
                                
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     91.16252 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             10 
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   91.16252 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1.49248 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.57 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Z=22 for titanium. Using Eq. (15.14), the Coulombic energy E Coulomb  (Ti,3d4s) of the outer electron of the Ti3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               Ti 
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.49248 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             9.11625 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.58 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, the spin-paired 4s electrons are promoted to Ti3d4s shell as unpaired electrons. The energy for the promotion is the magnetic energy given by Eq. (15.15) at the initial radius of the 4s electrons. From Eq. (10.102) with Z=22 and n=22, the radius r 22  of Ti4s shell is 
         [0000]      r 22 =1.99261a 0   (23.59)
 
         [0000]    Using Eqs. (15.15) and (23.59), the unpairing energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                         
                           
                             2 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                                
                               2 
                             
                              
                             
                               ℏ 
                               2 
                             
                           
                           
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   22 
                                 
                                 ) 
                               
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             8 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                               μ 
                               B 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 1.99261 
                                  
                                 
                                   a 
                                   0 
                                 
                               
                               ) 
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           0.01446 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.60 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (23.58) and (23.60), the energy E(Ti,3d4s) of the outer electron of the Ti3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               Ti 
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     22 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               9.11625 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.01446 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             9.10179 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.61 
                   ) 
                 
               
             
           
         
       
     
         [0504]    Next, consider the formation of the Ti-L-bond MO of wherein each titanium atom has an Ti3d4s electron with an energy given by Eq. (23.61). The total energy of the state of each titanium atom is given by the sum over the four electrons. The sum E T (Ti Ti-L 3d4s) of energies of Ti3d4s (Eq. (23.61)), Ti + , Ti 2+ , and Ti 3+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Ti 
                                 
                                   Ti 
                                   - 
                                   L 
                                 
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               s 
                             
                             ) 
                           
                         
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       43.2672 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       27.4917 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       13.5755 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       E 
                                        
                                       
                                         ( 
                                         
                                           Ti 
                                           , 
                                           
                                             3 
                                              
                                             d 
                                              
                                             
                                                 
                                             
                                              
                                             4 
                                              
                                             s 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       43.2672 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       27.4917 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       13.5755 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       9.10179 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             93.43619 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.62 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E(Ti,3d4s) is the sum of the energy of Ti, −6.82812 eV, and the hybridization energy. 
         [0505]    The titanium HO donates an electron to each MO. Using Eq. (23.30), the radius r 3d4s  of the Ti3d4s shell calculated from the Coulombic energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             Ti 
                             - 
                             
                               L 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               s 
                             
                           
                         
                         = 
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     18 
                                   
                                   21 
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               1 
                             
                             ) 
                           
                            
                           
                             
                                
                               2 
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     93.43619 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             9 
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   93.43619 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1.31054 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.63 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (23.63), the Coulombic energy E Coulomb (Ti Ti-L ,3d4s) of the outer electron of the Ti3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 Ti 
                                 
                                   Ti 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 Ti 
                                 - 
                                 
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.31054 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             10.38180 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.64 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The only magnetic energy term is that for unpairing of the 4s electrons given by Eq. (23.60). Using Eqs. (23.32), (23.60), and (23.64), the energy E(Ti Ti-L ,3d4s) of the outer electron of the Ti3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 Ti 
                                 
                                   Ti 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   Ti 
                                   - 
                                   
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     s 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     22 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               10.38180 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.01446 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             10.36734 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.65 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (Ti-L,3d 4s), the energy change of each Ti3d4s shell with the formation of the Ti-L-bond MO is given by the difference between Eq. (23.65) and Eq. (23.61): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Ti 
                                 - 
                                 L 
                               
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   Ti 
                                   
                                     Ti 
                                     - 
                                     L 
                                   
                                 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 Ti 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               10.36734 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 9.10179 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             1.26555 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.66 
                   ) 
                 
               
             
           
         
       
     
         [0506]    The semimajor axis a solution given by Eq. (23.41) of the force balance equation, Eq. (23.39), for the σ-MO of the Ti-L-bond MO of TiL n , is given in Table 23.13 (as shown in the priority document) with the force-equation parameters Z=22, n e , and L corresponding to the orbital and spin angular momentum terms of the 3d4s HO shell. 
         [0507]    For the Ti-L functional groups, hybridization of the 4s and 3d AOs of Ti to form a single 3d4s shell forms an energy minimum, and the sharing of electrons between the Ti3d4s HO and L AO to form σ MO permits each participating orbital to decrease in radius and energy. The F AO has an energy of E(F)=−17.42282 eV, the Cl AO has an energy of E(Cl)=−12.96764 eV, the Br AO has an energy of E(Br)=−11.8138 eV, the I AO has an energy of E(I)=−10.45126 eV, the O AO has an energy of E(O)=−13.61805 eV, and the Ti3d4s HOs has an energy of E(Ti,3d4s)=−9.10179 eV (Eq. (23.61)). To meet the equipotential condition of the union of the Ti-L H 2 -type-ellipsoidal-MO with these orbitals, the hybridization factor(s), at least one of c 2  and C 2  of Eq. (15.61) for the Ti-L-bond MO given by Eq. (15.77) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               FAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Ti 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               sHO 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Ti 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               FAO 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               9.10179 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               17.42282 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.52241 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.67 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               ClAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Ti 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               sHO 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Ti 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               ClAO 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               9.10179 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               12.96764 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.70188 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.68 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               BrAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Ti 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               sHO 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               BrAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Ti 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               sHO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Ti 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               BrAO 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               9.10179 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               11.8138 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.77044 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.69 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               IAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Ti 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               sHO 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               IAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Ti 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               sHO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Ti 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               IAO 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               9.10179 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               10.45126 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.87088 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.70 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               O 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Ti 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               sHO 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Ti 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               O 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               9.10179 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               13.61805 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.66836 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.71 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Since the energy of the MO is matched to that of the Ti3d4s HO, E(AO/HO) in Eq. (15.61) is E(Ti,3d4s) given by Eq. (23.61) and twice this value for double bonds. E T (atom−atom,msp 3 .AO) of the Ti-L-bond MO is determined by considering that the bond involves an electron transfer from the titanium atom to the ligand atom to form partial ionic character in the bond as in the case of the zwitterions such as H 2 B + —F −  given in the Halido Boranes section. E T (atom−atom,msp 3 .AO) is −2.53109 eV, two times the energy of Eq. (23.66). 
         [0508]    The symbols of the functional groups of titanium coordinate compounds are given in Table 23.12. The geometrical (Eqs. (15.1-15.5) and (23.41)), intercept (Eqs. (15.31-15.32) and (15.80-15.87)), and energy (Eqs. (15.61) and (23.28-23.33)) parameters of titanium coordinate compounds are given in Tables 23.13, 23.14, and 23.15, respectively (all (as shown in the priority document). The total energy of each titanium coordinate compounds given in Table 23.16 (as shown in the priority document) was calculated as the sum over the integer multiple of each E D  (Group) of Table 23.15 (as shown in the priority document) corresponding to functional-group composition of the compound. The bond angle parameters of titanium coordinate compounds determined using Eqs. (15.88-15.117) are given in Table 23.17 (as shown in the priority document). The E T (atom−atom,msp 3 .AO) term for TiOCl 2  was calculated using Eqs. (23.30-23.33) as a linear combination of s=1 and s=2 for the energies of E(Ti,3d 4s) given by Eqs. (23.63-23.66) corresponding to a Ti—Cl single bond and a Ti═O double bond. The charge-densities of exemplary titanium coordinate compound, titanium tetrafluoride comprising the concentric shells of atoms with the outer shell bridged by one or more H 2 -type ellipsoidal MOs or joined with one or more hydrogen MOs is shown in  FIG. 40 . 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 23.12 
               
             
             
               
                   
               
               
                 The symbols of the functional groups of titanium coordinate 
               
               
                 compounds. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
               
               
                   
                 TiF group of TiF 
                 Ti—F (a) 
               
               
                   
                 TiF group of TiF 2   
                 Ti—F (b) 
               
               
                   
                 TiF group of TiF 3   
                 Ti—F (c) 
               
               
                   
                 TiF group of TiF 4   
                 Ti—F (d) 
               
               
                   
                 TiCl group of TiCl 
                 Ti—Cl (a) 
               
               
                   
                 TiCl group of TiCl 2   
                 Ti—Cl (b) 
               
               
                   
                 TiCl group of TiCl 3   
                 Ti—Cl (c) 
               
               
                   
                 TiCl group of TiCl 4   
                 Ti—Cl (d) 
               
               
                   
                 TiBr group of TiBr 
                 Ti—Br (a) 
               
               
                   
                 TiBr group of TiBr 2   
                 Ti—Br (b) 
               
               
                   
                 TiBr group of TiBr 3   
                 Ti—Br (c) 
               
               
                   
                 TiBr group of TiBr 4   
                 Ti—Br (d) 
               
               
                   
                 TiI group of TiI 
                 Ti—I (a) 
               
               
                   
                 TiI group of TiI 2   
                 Ti—I (b) 
               
               
                   
                 TiI group of TiI 3   
                 Ti—I (c) 
               
               
                   
                 TiI group of TiI 4   
                 Ti—I (d) 
               
               
                   
                 TiO group of TiO 
                 Ti—O (a) 
               
               
                   
                 TiO group of TiO 2   
                 Ti—O (b) 
               
               
                   
               
             
          
         
       
     
       Vanadium Functional Groups and Molecules 
       [0509]    The electron configuration of vanadium is [Ar]4s 2 3d 3  having the corresponding term  4 F 3/2 . The total energy of the state is given by the sum over the five electrons. The sum E T (V,3d 4s) of experimental energies [1] of V, V + , V 2+ , V 3+ , and V 4+  is 
         [0000]      E T (V,3 d 4 s )=−(65.2817 eV+46.709 eV+29.311 eV+14.618 eV+6.74619 eV)=−162.66589 eV  (23.56)
 
         [0000]    By considering that the central field decreases by an integer for each successive electron of the shell, the radius r 3d4s  of the V3d4s shell may be calculated from the Coulombic energy using Eq. (15.13): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             3 
                              
                             d 
                              
                             
                                 
                             
                              
                             4 
                              
                             s 
                           
                         
                         = 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               18 
                             
                             22 
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   Z 
                                   - 
                                   n 
                                 
                                 ) 
                               
                                
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     162.66589 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             15 
                              
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   162.66589 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1.25464 
                            
                           
                               
                           
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.72 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Z=23 for vanadium. Using Eq. (15.14), the Coulombic energy E Coulomb (V,3d 4s) of the outer electron of the V3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               V 
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.25464 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             10.844393 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.73 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, the spin-paired 4s electrons are promoted to V3d4s shell as unpaired electrons. The energy for the promotion is the magnetic energy given by Eq. (15.15) at the initial radius of the 4s electrons. From Eq. (10.102) with Z=23 and n=23, the radius r 23  of V4s shell is 
         [0000]      r 23 =2.01681a 0   (23.74)
 
         [0000]    Using Eqs. (15.15) and (23.74), the unpairing energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       ( 
                       magnetic 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         2 
                          
                         
                           πμ 
                           0 
                         
                          
                         
                           e 
                           2 
                         
                          
                         
                           ℏ 
                           2 
                         
                       
                       
                         
                           
                             m 
                             e 
                             2 
                           
                            
                           
                             ( 
                             
                               r 
                               23 
                             
                             ) 
                           
                         
                         3 
                       
                     
                     = 
                     
                       
                         
                           8 
                            
                           
                             πμ 
                             0 
                           
                            
                           
                             μ 
                             B 
                             2 
                           
                         
                         
                           
                             ( 
                             
                               2.01681 
                                
                               
                                   
                               
                                
                               
                                 a 
                                 0 
                               
                             
                             ) 
                           
                           3 
                         
                       
                       = 
                       
                         0.01395 
                          
                         
                             
                         
                          
                         eV 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.45 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (23.73) and (23.75), the energy E(V,3d4s) of the outer electron of the V3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               V 
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               - 
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 e 
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     23 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               10.844393 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.01395 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             10.83045 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.76 
                   ) 
                 
               
             
           
         
       
     
         [0510]    Next, consider the formation of the V-L-bond MO of wherein each vanadium atom has an V3d4s electron with an energy given by Eq. (23.76). The total energy of the state of each vanadium atom is given by the sum over the five electrons. The sum E T (V V-L 3d 4s) of energies of V3d4s (Eq. (23.76)), V + , V 2+ , V 3+ , and V 4+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 V 
                                 
                                   V 
                                   - 
                                   L 
                                 
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               s 
                             
                             ) 
                           
                         
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       65.2817 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       46.709 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       29.311 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       14.618 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       E 
                                        
                                       
                                         ( 
                                         
                                           V 
                                           , 
                                           
                                             3 
                                              
                                             d 
                                              
                                             
                                                 
                                             
                                              
                                             4 
                                              
                                             s 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       65.2817 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       46.709 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       29.311 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       14.618 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     10.83045 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             166.75015 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.77 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E(V,3d4s) is the sum of the energy of V, −6.74619 eV, and the hybridization energy. 
         [0511]    The vanadium HO donates an electron to each MO. Using Eq. (23.30), the radius r 3d4s  of the V3d4s shell calculated from the Coulombic energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             V 
                             - 
                             
                               L 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               s 
                             
                           
                         
                         = 
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     18 
                                   
                                   22 
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               1 
                             
                             ) 
                           
                            
                           
                             
                               e 
                               2 
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     166.75015 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             14 
                              
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   166.75015 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1.14232 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.78 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (23.78), the Coulombic energy E Coulomb (V V-L ,3d4s) of the outer electron of the V3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 V 
                                 
                                   V 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 V 
                                 - 
                                 
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.14232 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             11.91072 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.79 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The only magnetic energy term is that for unpairing of the 4s electrons given by Eq. (23.75). Using Eqs. (23.32), (23.73), and (23.79), the energy E(V V-L ,3d4s) of the outer electron of the V3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 V 
                                 
                                   V 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               - 
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   V 
                                   - 
                                   
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     s 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 e 
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     23 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               11.91072 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.01446 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             11.89678 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.80 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (V-L,3d4s), the energy change of each V3d4s shell with the formation of the V-L-bond MO is given by the difference between Eq. (23.80) and Eq. (23.76): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 V 
                                 - 
                                 L 
                               
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   V 
                                   
                                     V 
                                     - 
                                     L 
                                   
                                 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 V 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               11.89678 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 10.83045 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             1.06633 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.81 
                   ) 
                 
               
             
           
         
       
     
         [0512]    The semimajor axis a solution given by Eq. (23.41) of the force balance equation, Eq. (23.39), for the σ-MO of the V-L-bond MO of VL n  is given in Table 23.19 (as shown in the priority document) with the force-equation parameters Z=23, n e , and L corresponding to the orbital and spin angular momentum terms of the 3d4s HO shell. The semimajor axis a of carbonyl and organometallic compounds are solved using Eq. (15.51). 
         [0513]    For the V-L functional groups, hybridization of the 4s and 3d AOs of V to form a single 3d4s shell forms an energy minimum, and the sharing of electrons between the V3d4s HO and L AO to form σ MO permits each participating orbital to decrease in radius and energy. The F AO has an energy of E(F)=−17.42282 eV, the Cl AO has an energy of E(Cl)=−12.96764 eV, the C aryl 2sp 3  HO has an energy of E(C aryl ,2sp 3 )=−15.76868 eV (Eq. (14.246)), the C2sp 3  HO has an energy of E(C,2sp 3 )=−14.63489 eV (Eq. (15.25)), the N AO has an energy of E(N)=−14.53414 eV, the O AO has an energy of E(O)=−13.61805 eV, and the V3d4s HO has an energy of E Coulomb (V,3d4s)=−10.84439 eV (Eq. (23.75)) and E(V,3d4s)=−10.83045 eV (Eq. (23.76)). To meet the equipotential condition of the union of the V-L H 2 -type-ellipsoidal-MO with these orbitals, the hybridization factor(s), at least one of c 2  and C 2  of Eq. (15.61) for the V-L-bond MO given by Eq. (15.77) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                             
                         
                          
                         
                           
                             
                               C 
                               2 
                             
                              
                             
                               ( 
                               
                                 F 
                                  
                                 
                                     
                                 
                                  
                                 AO 
                                  
                                 
                                     
                                 
                                  
                                 to 
                                  
                                 
                                     
                                 
                                  
                                 V 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                                  
                                 
                                     
                                 
                                  
                                 HO 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               E 
                                
                               
                                 ( 
                                 
                                   V 
                                   , 
                                   
                                     3 
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     s 
                                   
                                 
                                 ) 
                               
                             
                             
                               E 
                                
                               
                                 ( 
                                 
                                   F 
                                    
                                   
                                       
                                   
                                    
                                   AO 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               10.83045 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               17.42282 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                           0.62162 
                         
                          
                         
                             
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.82 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                             
                         
                          
                         
                           
                             
                               C 
                               2 
                             
                              
                             
                               ( 
                               
                                 Cl 
                                  
                                 
                                     
                                 
                                  
                                 AO 
                                  
                                 
                                     
                                 
                                  
                                 to 
                                  
                                 
                                     
                                 
                                  
                                 V 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                                  
                                 
                                     
                                 
                                  
                                 HO 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               E 
                                
                               
                                 ( 
                                 
                                   V 
                                   , 
                                   
                                     3 
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     s 
                                   
                                 
                                 ) 
                               
                             
                             
                               E 
                                
                               
                                 ( 
                                 
                                   Cl 
                                    
                                   
                                       
                                   
                                    
                                   AO 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               10.83045 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               12.96764 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.83519 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.83 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                             
                         
                          
                         
                           
                             
                               C 
                               2 
                             
                              
                             
                               ( 
                               
                                 C 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                   sp 
                                   3 
                                 
                                  
                                 HO 
                                  
                                 
                                     
                                 
                                  
                                 to 
                                  
                                 
                                     
                                 
                                  
                                 V 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                   s 
                                    
                                   HO 
                                 
                               
                               ) 
                             
                           
                           = 
                             
                            
                           
                             
                               
                                 
                                   E 
                                   Coulomb 
                                 
                                  
                                 
                                   ( 
                                   
                                     V 
                                     , 
                                     
                                       3 
                                        
                                       d 
                                        
                                       
                                           
                                       
                                        
                                       4 
                                        
                                       s 
                                     
                                   
                                   ) 
                                 
                               
                               
                                 E 
                                  
                                 
                                   ( 
                                   
                                     C 
                                     , 
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         sp 
                                         3 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                              
                             
                               c 
                               2 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           ( 
                           
                             C 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                               sp 
                               3 
                             
                              
                             HO 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 - 
                                 10.84439 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             
                               
                                 - 
                                 14.63489 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                           
                            
                           
                             ( 
                             0.91771 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.68002 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.84 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       c 
                       2 
                     
                      
                     
                       ( 
                       
                         
                           C 
                           aryl 
                         
                          
                         
                             
                         
                          
                         2 
                          
                         
                           sp 
                           3 
                         
                          
                         HO 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         V 
                          
                         
                             
                         
                          
                         3 
                          
                         d 
                          
                         
                             
                         
                          
                         4 
                          
                         s 
                          
                         
                             
                         
                          
                         HO 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         c 
                         2 
                       
                        
                       
                         ( 
                         
                           
                             C 
                             aryl 
                           
                            
                           
                               
                           
                            
                           2 
                            
                           
                             sp 
                             3 
                           
                            
                           HO 
                            
                           
                               
                           
                            
                           to 
                            
                           
                               
                           
                            
                           V 
                            
                           
                               
                           
                            
                           3 
                            
                           d 
                            
                           
                               
                           
                            
                           4 
                            
                           
                             s 
                              
                             HO 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               V 
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 C 
                                 aryl 
                               
                               , 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                       
                       = 
                       
                         
                           
                             
                               - 
                               10.84439 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               15.76868 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                         = 
                         0.68772 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.85 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                             
                         
                          
                         
                           
                             
                               c 
                               2 
                             
                              
                             
                               ( 
                               
                                 N 
                                  
                                 
                                     
                                 
                                  
                                 AO 
                                  
                                 
                                     
                                 
                                  
                                 to 
                                  
                                 
                                     
                                 
                                  
                                 V 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                   s 
                                    
                                   HO 
                                 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               C 
                               2 
                             
                              
                             
                               ( 
                               
                                 N 
                                  
                                 
                                     
                                 
                                  
                                 AO 
                                  
                                 
                                     
                                 
                                  
                                 to 
                                  
                                 
                                     
                                 
                                  
                                 V 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                   s 
                                    
                                   HO 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 V 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               NAO 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               10.83045 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               14.53414 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.74517 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.86 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                             
                         
                          
                         
                           
                             
                               c 
                               2 
                             
                              
                             
                               ( 
                               
                                 O 
                                  
                                 
                                     
                                 
                                  
                                 to 
                                  
                                 
                                     
                                 
                                  
                                 V 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                   s 
                                    
                                   HO 
                                 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               E 
                                
                               
                                 ( 
                                 
                                   V 
                                   , 
                                   
                                     3 
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     s 
                                   
                                 
                                 ) 
                               
                             
                             
                               E 
                                
                               
                                 ( 
                                 O 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               10.83045 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               13.61805 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.79530 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.87 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Eqs. (15.76), (15.79), and (13.430) were used in Eq. (23.84). Since the energy of the MO is matched to that of the V3d4s HO of coordinate compounds, E(A0/HO) in Eq. (15.61) is E(V,3d4s) given by Eq. (23.76) and twice this value for double bonds. For carbonyls and organometallics, the energy of the MO is matched to that of the Coulomb energy of the V3d4s HO such that E(AO/HO) in Eq. (15.61) is E Coulomb (V,3d 4s) given by Eq. (23.73). E T (atom−atom,msp 3 .AO) of the V-L-bond MO is determined by considering that the bond involves an electron transfer from the vanadium atom to the ligand atom to form partial ionic character in the bond as in the case of the zwitterions such as H 2 B + —F −  given in the Halido Boranes section. For coordinate compounds, E T (atom−atom,msp 3 .AO) is −2.53109 eV, two times the energy of Eq. (23.81). For carbonyl and organometallic compounds, E T (atom−atom,msp 3 .AO) is −1.65376 eV and −2.26759 eV, respectively. The former is based on the energy match between the V3d4s HO and the C2sp 3  HO of a carbonyl group and is given by the linear combination of −0.72457 eV (Eq. (14.151)) and −0.92918 eV (Eq. (14.513)), respectively. The latter is equivalent to that of ethylene and the aryl group, −2.26759 eV, given by Eq. (14.247). The C═O functional group of carbonyls is equivalent to that of formic acid given in Carboxylic Acids section except that Ē Kvib  corresponds to that of a metal carbonyl and E T (AO/HO) of Eq. (15.47) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               AO 
                               / 
                               HO 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             Δ 
                           
                            
                           
                               
                           
                            
                           
                             
                               E 
                               
                                 
                                   H 
                                   2 
                                 
                                  
                                 MO 
                               
                             
                              
                             
                               ( 
                               
                                 AO 
                                 / 
                                 HO 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   - 
                                   14.63489 
                                 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               - 
                               
                                 3.58557 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           18.22046 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.88 
                   ) 
                 
               
             
           
         
       
     
         [0000]    wherein the additional E(AO/HO)=−14.63489 eV (Eq. (15.25)) component corresponds to the donation of both unpaired electrons of the C2sp 3  HO of the carbonyl group to the metal-carbonyl bond. The benzene groups of organometallic, V(C 6 H 6 ) 2  are equivalent to those given in the Aromatic and Heterocyclic Compounds section.
 
The symbols of the functional groups of vanadium coordinate compounds are given in Table 23.18. The geometrical (Eqs. (15.1-15.5) and (23.41)), intercept (Eqs. (15.31-15.32) and (15.80-15.87)), and energy (Eqs. (15.61) and (23.28-23.33)) parameters of vanadium coordinate compounds are given in Tables 23.19, 23.20, and 23.21, respectively (all as shown in the priority document). The total energy of each vanadium coordinate compounds given in Table 23.22 (as shown in the priority document) was calculated as the sum over the integer multiple of each E D  (Group) of Table 23.21 (as shown in the priority document) corresponding to functional-group composition of the compound. The bond angle parameters of vanadium coordinate compounds determined using Eqs. (15.88-15.117) are given in Table 23.23 (as shown in the priority document). The E T (atom−atom,msp 3 .AO) term for VOCl 3  was calculated using Eqs. (23.30-23.33) with s=1 for the energies of E(V,3d4s) given by Eqs. (23.78-23.81). The charge-densities of exemplary vanadium carbonyl and organometallic compounds, vanadium hexacarbonyl (V (CO) 6 ) and dibenzene vanadium (V(C 6 H 6 ) 2 ), respectively, comprising the concentric shells of atoms with the outer shell bridged by one or more H 2 -type ellipsoidal MOs or joined with one or more hydrogen MOs are shown in  FIGS. 41 and 42 .
 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 23.18 
               
             
             
               
                   
               
               
                 The symbols of the functional groups of vanadium coordinate 
               
               
                 compounds. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
               
               
                   
                 VF group of VF 5   
                 V—F 
               
               
                   
                 VCl group of VCl 4   
                 V—Cl 
               
               
                   
                 VN group of VN 
                 V—N 
               
               
                   
                 VO group of VO and VO 2   
                 V—O 
               
               
                   
                 VCO group of V(CO) 6   
                 V—CO 
               
               
                   
                 C═O 
                 C═O 
               
               
                   
                 VC aryl  group of V(C 6 H 6 ) 2   
                 V—C 6 H 6   
               
               
                   
                 CC (aromatic bond) 
                 C 3e ═C 
               
               
                   
                 CH (aromatic) 
                 CH 
               
               
                   
               
             
          
         
       
     
       Chromium Functional Groups and Molecules 
       [0514]    The electron configuration of chromium is [Ar]4s 1 3d 5  having the corresponding term  7 S 3 . The total energy of the state is given by the sum over the six electrons. The sum E T (Cr,3d4s) of experimental energies [1] of Cr, Cr + , Cr 2+ , Cr 3+ , Cr 4+ , and Cr 5+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             3 
                              
                             d 
                              
                             
                                 
                             
                              
                             4 
                              
                             s 
                           
                         
                         = 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               18 
                             
                             23 
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   Z 
                                   - 
                                   n 
                                 
                                 ) 
                               
                                
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     263.46711 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             21 
                              
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   263.46711 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1.08447 
                            
                           
                               
                           
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.90 
                   ) 
                 
               
             
           
         
       
     
         [0000]    By considering that the central field decreases by an integer for each successive electron of the shell, the radius r 3d4s  of the Cr3d4s shell may be calculated from the Coulombic energy using Eq. (15.13): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               Cr 
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       90.6349 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       69.46 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       49.16 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       30.96 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       16.4857 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       6.76651 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             263.46711 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.89 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Z=24 for chromium. Using Eq. (15.14), the Coulombic energy E Coulomb (Cr,3d45) of the outer electron of the Cr3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               Cr 
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.08447 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             12.546053 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.91 
                   ) 
                 
               
             
           
         
       
     
         [0515]    Next, consider the formation of the Cr-L-bond MO of wherein each chromium atom has an Cr3d4s electron with an energy given by Eq. (23.91). The total energy of the state of each chromium atom is given by the sum over the six electrons. The sum E T (Cr Cr-L 3d4s) of energies of Cr3d4s (Eq. (23.91)), Cr + , Cr 2+ , Cr 3+ , Cr 4+ , and Cr 5+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Cr 
                                 
                                   Cr 
                                   - 
                                   L 
                                 
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               s 
                             
                             ) 
                           
                         
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       90.6349 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       69.46 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       49.16 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       30.96 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       16.4857 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       
                                         E 
                                         Coulomb 
                                       
                                        
                                       
                                         ( 
                                         
                                           Cr 
                                           , 
                                           
                                             3 
                                              
                                             d 
                                              
                                             
                                                 
                                             
                                              
                                             4 
                                              
                                             s 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       90.6349 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       69.46 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       49.16 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       30.96 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       16.4857 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       12.546053 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             269.24665 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.92 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E(Cr,3d4s) is the sum of the energy of Cr, −6.76651 eV, and the hybridization energy. 
         [0516]    The chromium HO donates an electron to each MO. Using Eq. (23.30), the radius r 3d4s  of the Cr3d4s shell calculated from the Coulombic energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             Cr 
                             - 
                             
                               L 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               s 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     18 
                                   
                                   23 
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               1 
                             
                             ) 
                           
                            
                           
                             
                               e 
                               2 
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     269.24665 
                                      
                                     
                                         
                                     
                                      
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     V 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               20 
                                
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     269.24665 
                                      
                                     
                                       
                                           
                                       
                                        
                                       
                                           
                                       
                                     
                                      
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     V 
                                   
                                   ) 
                                 
                               
                             
                           
                           = 
                           
                             1.01066 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.93 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (23.93), the Coulombic energy E Coulomb (Cr Cr-L ,3 d 4s) of the outer electron of the Cr3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 Cr 
                                 
                                   Cr 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 Cr 
                                 - 
                                 
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.01066 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             13.46233 
                           
                            
                           
                               
                           
                            
                           e 
                            
                           
                               
                           
                            
                           V 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.94 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (Cr-L,3d4s), the energy change of each Cr3d4s shell with the formation of the Cr-L-bond MO is given by the difference between Eq. (23.94) and Eq. (23.91): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Cr 
                                 - 
                                 L 
                               
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   Cr 
                                   
                                     Cr 
                                     - 
                                     L 
                                   
                                 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 Cr 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               13.46233 
                             
                              
                             
                                 
                             
                              
                             e 
                              
                             
                                 
                             
                              
                             V 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 12.546053 
                               
                                
                               
                                   
                               
                                
                               e 
                                
                               
                                   
                               
                                
                               V 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             0.91628 
                           
                            
                           
                               
                           
                            
                           e 
                            
                           
                               
                           
                            
                           V 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.95 
                   ) 
                 
               
             
           
         
       
     
         [0517]    The semimajor axis a solution given by Eq. (23.41) of the force balance equation, Eq. (23.39), for the σ-MO of the Cr-L-bond MO of CrL n , is given in Table 23.25 (as shown in the priority document) with the force-equation parameters Z=24, n e , and L corresponding to the orbital and spin angular momentum terms of the 3d4s HO shell. The semimajor axis a of carbonyl and organometallic compounds are solved using Eq. (15.51). 
         [0518]    For the Cr-L functional groups, hybridization of the 4s and 3d AOs of Cr to form a single 3d4s shell forms an energy minimum, and the sharing of electrons between the Cr3d4s HO and L AO to form σ MO permits each participating orbital to decrease in radius and energy. The F AO has an energy of E(F)=−17.42282 eV, the Cl AO has an energy of E(Cl)=−12.96764 eV, the C aryl 2sp 3  HO has an energy of E(C aryl ,2sp 3 )=−15.76868 eV (Eq. (14.246)), the C2sp 3  HO has an energy of E(C,2sp 3 )=−14.63489 eV (Eq. (15.25)), the O AO has an energy of E(O)=−13.61805 eV, and the Cr3d4s HO has an energy of E Coulomb (Cr,3d4s)=−12.54605 eV (Eq. (23.91)). To meet the equipotential condition of the union of the Cr-L H 2 -type-ellipsoidal-MO with these orbitals, the hybridization factor(s), at least one of c 2  and C 2  of Eq. (15.61) for the Cr-L-bond MO given by Eq. (15.77) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               FAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Cr 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                   
                               
                                
                               s 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               FAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Cr 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               
                                 d 
                                  
                                 4 
                               
                                
                               
                                   
                               
                                
                               s 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               E 
                               Coulomb 
                             
                              
                             
                               ( 
                               
                                 Cr 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               FAO 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 - 
                                 12.54605 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             
                               
                                 - 
                                 17.42282 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                           
                           = 
                             
                            
                           0.72009 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.96 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               ClAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Cr 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               ClAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Cr 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               E 
                               Coulomb 
                             
                              
                             
                               ( 
                               
                                 Cr 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               ClAO 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 - 
                                 12.54605 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             
                               
                                 - 
                                 12.96764 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                           
                           = 
                           0.96749 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.97 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               C 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Cr 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               C 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                 
                                     
                                 
                                  
                                 
                                     
                                 
                               
                                
                               Cr 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               E 
                               Coulomb 
                             
                              
                             
                               ( 
                               
                                 Cr 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 C 
                                 , 
                                 
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 - 
                                 12.54605 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             
                               
                                 - 
                                 14.63489 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                           
                           = 
                           0.85727 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.98 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 C 
                                 aryl 
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Cr 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               E 
                               Coulomb 
                             
                              
                             
                               ( 
                               
                                 Cr 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   C 
                                   aryl 
                                 
                                 , 
                                 
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 - 
                                 12.54605 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             
                               
                                 - 
                                 15.76868 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                           
                           = 
                           0.79563 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.99 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               O 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Cr 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               O 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Cr 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               E 
                               Coulomb 
                             
                              
                             
                               ( 
                               
                                 Cr 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               O 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 - 
                                 12.54605 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             
                               
                                 - 
                                 13.61805 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                           
                           = 
                           0.92128 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.100 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Since the energy of the MO is matched to that of the V Coulomb 3d 4s HO, E(AO/HO) in Eq. (15.61) is E Coulomb (Cr,3d4s) given by Eq. (23.91) and twice this value for double bonds. E T (atom−atom,msp 3 .AO) of the Cr-L-bond MO is determined by considering that the bond involves an electron transfer from the chromium atom to the ligand atom to form partial ionic character in the bond as in the case of the zwitterions such as H 2 B + —F −  given in the Halido Boranes section. For coordinate compounds, E T (atom−atom,msp 3 .AO) is −1.83256 eV, two times the energy of Eq. (23.95). For carbonyl and organometallic compounds, E T (atom−atom,msp 3 .AO) is −1.44915 eV (Eq. (14.151)), and the C═O functional group of carbonyls is equivalent to that of vanadium carbonyls. The benzene and substituted benzene groups of organometallics are equivalent to those given in the Aromatic and Heterocyclic Compounds section. 
         [0519]    The symbols of the functional groups of chromium coordinate compounds are given in Table 23.24. The corresponding designation of the structure of the (CH 3 ) 3  C 6 H 3  group of Cr((CH 3 ) 3 C 6 H 3 ) 2  is equivalent to that of toluene shown in  FIG. 43 . The geometrical (Eqs. (15.1-15.5) and (23.41)), intercept (Eqs. (15.31-15.32) and (15.80-15.87)), and energy (Eqs. (15.61) and (23.28-23.33)) parameters of chromium coordinate compounds are given in Tables 23.25, 23.26, and 23.27, respectively (all as shown in the priority document). The total energy of each chromium coordinate compounds given in Table 23.28 (as shown in the priority document) was calculated as the sum over the integer multiple of each E D  (Group) of Table 23.27 (as shown in the priority document) corresponding to functional-group composition of the compound. The bond angle parameters of chromium coordinate compounds determined using Eqs. (15.88-15.117) are given in Table 23.29 (as shown in the priority document). The E T (atom−atom,msp 3 .AO) term for CrOCl 3  was calculated using Eqs. (23.30-23.33) with s=1 for the energies of E Coulomb (Cr,3d4s) given by Eqs. (23.93-23.95). The charge-densities of exemplary chromium carbonyl and organometallic compounds, chromium hexacarbonyl (Cr (CO) 6 ) and di-(1,2,4-trimethylbenzene) chromium (Cr ((CH 3 ) 3  C 6 H 3 ) 2 ), respectively, comprising the concentric shells of atoms with the outer shell bridged by one or more H 2 -type ellipsoidal MOs or joined with one or more hydrogen MOs are shown in  FIGS. 44 and 45 . 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 23.24 
               
             
             
               
                   
               
               
                 The symbols of the functional groups of chromium coordinate 
               
               
                 compounds. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
               
               
                   
                 CrF group of CrF 2   
                 Cr—F 
               
               
                   
                 CrCl group of CrCl 2   
                 Cr—Cl 
               
               
                   
                 CrO group of CrO 
                 Cr—O (a) 
               
               
                   
                 CrO group of CrO 2   
                 Cr—O (b) 
               
               
                   
                 CrO group of CrO 3   
                 Cr—O (c) 
               
               
                   
                 CrCO group of Cr(CO) 6   
                 Cr—CO 
               
               
                   
                 C═O 
                 C═O 
               
               
                   
                 CrC aryl  group of Cr(C 6 H 6 ) 2  and 
                 Cr—C 6 H 6   
               
               
                   
                 Cr((CH 3 ) 3 C 6 H 3 ) 2   
                   
               
               
                   
                 CC (aromatic bond) 
                 C 3e ═C 
               
               
                   
                 CH (aromatic) 
                 CH 
               
               
                   
                 C a —C b  (CH 3  to aromatic bond) 
                 C—C 
               
               
                   
                 CH 3  group 
                 C—H (CH 3 ) 
               
               
                   
               
             
          
         
       
     
       Manganese Functional Groups and Molecules 
       [0520]    The electron configuration of manganese is [Ar]4s 2 3d 5  having the corresponding term  6 S 5/2 . The total energy of the state is given by the sum over the seven electrons. The sum E T (Mn,3d4s) of experimental energies [1] of Mn, Mn + , Mn 2+ , Mn 3+ , Mn 4+ , Mn 5+ , and Mn 6+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               Mn 
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       119.203 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       95.6 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       72.4 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       51.2 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       33.668 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       15.6400 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       14.22133 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             401.93233 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.101 
                   ) 
                 
               
             
           
         
       
     
         [0000]    By considering that the central field decreases by an integer for each successive electron of the shell, the radius r 3d4s  of the Mn3d4s shell may be calculated from the Coulombic energy using Eq. (15.13): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             3 
                              
                             d 
                              
                             
                                 
                             
                              
                             4 
                              
                             s 
                           
                         
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               18 
                             
                             24 
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   Z 
                                   - 
                                   n 
                                 
                                 ) 
                               
                                
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     395.14502 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             28 
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   395.14502 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.96411 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.102 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Z=25 for manganese. Using Eq. (15.14), the Coulombic energy E Coulomb (Mn,3d4s) of the outer electron of the Mn3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               Mn 
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 s 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             0.96411 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             14.112322 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.103 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, the spin-paired 4s electrons are promoted to Mn3d4s shell as unpaired electrons. The energy for the promotion is the magnetic energy given by Eq. (15.15) at the initial radius of the 4s electrons. From Eq. (10.102) with Z=25 and n=25, the radius r 25  of Mn4s shell is 
         [0000]      r 25 =1.83021a 0   (23.104)
 
         [0000]    Using Eqs. (15.15) and (23.104), the unpairing energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               4 
                                
                               
                                   
                               
                                
                               s 
                             
                           
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             2 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                               e 
                               2 
                             
                              
                             
                               ℏ 
                               2 
                             
                           
                           
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   25 
                                 
                                 ) 
                               
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             8 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                               μ 
                               B 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 1.83021 
                                  
                                 
                                   a 
                                   0 
                                 
                               
                               ) 
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.01866 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.105 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The electrons from the 4s and 3d shells successively fill unoccupied HOs until the HO shell is filled with unpaired electrons, then the electrons pair per HO. In the case of the Mn3d4s shell having seven electrons and six orbitals, one set of electrons is paired. Using Eqs. (15.15) and (23.102), the paring energy is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                   
                               
                                
                               s 
                             
                           
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           - 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 e 
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     
                                       3 
                                        
                                       d 
                                        
                                       
                                           
                                       
                                        
                                       4 
                                        
                                       s 
                                     
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           - 
                           
                             
                               8 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 μ 
                                 B 
                                 2 
                               
                             
                             
                               
                                 ( 
                                 
                                   0.96411 
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             0.12767 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.105 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, after Eq. (23.28), the energy E(Mn,3d4s) of the outer electron of the Mn3d4s shell is given by adding the magnetic energy of unpairing the 4s electrons (Eq. (23.105)) and paring of one set of Mn3d4s electrons (Eq. (23.106)) to E Coulomb (Mn,3d 4s) (Eq. (23.103)): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               Mn 
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 e 
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 r 
                                 
                                   4 
                                    
                                   s 
                                 
                                 3 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 pairs 
                               
                             
                              
                             
                               
                                 2 
                                  
                                 
                                   πμ 
                                   0 
                                 
                                  
                                 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ℏ 
                                   2 
                                 
                               
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   r 
                                   
                                     3 
                                      
                                     d 
                                   
                                   3 
                                 
                               
                             
                           
                           - 
                           
                             
                               ∑ 
                               
                                 HO 
                                  
                                 
                                     
                                 
                                  
                                 pairs 
                               
                             
                              
                             
                               
                                 2 
                                  
                                 
                                   πμ 
                                   0 
                                 
                                  
                                 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ℏ 
                                   2 
                                 
                               
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   r 
                                   
                                     3 
                                      
                                     
                                         
                                     
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     s 
                                   
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               14.112322 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.01866 
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           0.12767 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             14.22133 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.107 
                   ) 
                 
               
             
           
         
       
     
         [0521]    Next, consider the formation of the Mn-L-bond MO of wherein each manganese atom has an Mn3d 4s electron with an energy given by Eq. (23.107). The total energy of the state of each manganese atom is given by the sum over the seven electrons. The sum E T (Mn Mn-L 3d4s) of energies of Mn3d 4s (Eq. (23.107)), Mn + , Mn 2+ , Mn 3+ , Mn 4+ , Mn 5+ , and Mn 6+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Mn 
                                 
                                   Mn 
                                   - 
                                   L 
                                 
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               s 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       119.203 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       95.6 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       72.4 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       51.2 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             33.668 
                                              
                                             
                                                 
                                             
                                              
                                             eV 
                                           
                                           + 
                                           
                                             15.6400 
                                              
                                             
                                                 
                                             
                                              
                                             eV 
                                           
                                           + 
                                         
                                       
                                     
                                     
                                       
                                         
                                           E 
                                            
                                           
                                             ( 
                                             
                                               Mn 
                                               , 
                                               
                                                 3 
                                                  
                                                 d 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 4 
                                                  
                                                 s 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       119.203 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       95.6 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       72.4 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       51.2 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             33.668 
                                              
                                             
                                                 
                                             
                                              
                                             eV 
                                           
                                           + 
                                           
                                             15.6400 
                                              
                                             
                                                 
                                             
                                              
                                             eV 
                                           
                                           + 
                                         
                                       
                                     
                                     
                                       
                                         
                                           14.22133 
                                            
                                           
                                               
                                           
                                            
                                           eV 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             401.93233 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.108 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E(Mn, 3d 4s) is the sum of the energy of Mn, −7.43402 eV, and the hybridization energy. 
         [0522]    The manganese HO donates an electron to each MO. Using Eq. (23.30), the radius r 3d4s  of the Mn3d4s shell calculated from the Coulombic energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             Mn 
                             - 
                             
                               L 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               s 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     18 
                                   
                                   24 
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               1 
                             
                             ) 
                           
                            
                           
                             
                               e 
                               2 
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     401 
                                      
                                     .93233 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             27 
                              
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   401.93233 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.91398 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.109 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (23.109), the Coulombic energy E Coulomb (Mn Mn-L ,3d 4s) of the outer electron of the Mn3d 4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 Mn 
                                 
                                   Mn 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 Mn 
                                 - 
                                 
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             0.91398 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             14.88638 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.110 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The magnetic energy terms are those for unpairing of the 4s electrons (Eq. (23.105)) and paring one set of Mn3d4s electrons (Eq. (23.106)). Using Eqs. (23.32), (23.105), (23.106), and (23.110), the energy E(Mn Mn-L , 3d4s) of the outer electron of the Mn3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 Mn 
                                 
                                   Mn 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   Mn 
                                   - 
                                   
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     s 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 e 
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     25 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             2 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                               e 
                               2 
                             
                              
                             
                               ℏ 
                               2 
                             
                           
                           
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   
                                     3 
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     
                                         
                                     
                                      
                                     s 
                                   
                                 
                                 ) 
                               
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               14.88638 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.01866 
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           0.12767 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             14.99539 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.111 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (Mn-L,3d4s), the energy change of each Mn3d4s shell with the formation of the Mn-L-bond MO is given by the difference between Eq. (23.111) and Eq. (23.107): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Mn 
                                 - 
                                 L 
                               
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   Mn 
                                   
                                     Mn 
                                     - 
                                     L 
                                   
                                 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 Mn 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               14.99539 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 14.22133 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             0.77406 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.112 
                   ) 
                 
               
             
           
         
       
     
         [0523]    The semimajor axis a solution given by Eq. (23.41) of the force balance equation, Eq. (23.39), for the σ-MO of the Mn-L-bond MO of MnL n , is given in Table 23.31 with the force-equation parameters Z=25, n e , and L corresponding to the orbital and spin angular momentum terms of the 3d4s HO shell. The semimajor axis a of carbonyl and organometallic compounds are solved using Eq. (15.51). 
         [0524]    For the Mn-L functional groups, hybridization of the 4s and 3d AOs of Mn to form a single 3d4s shell forms an energy minimum, and the sharing of electrons between the Mn3d4s HO and L AO to form σ MO permits each participating orbital to decrease in radius and energy. The F AO has an energy of E(F)=−17.42282 eV, the Cl AO has an energy of E(Cl)=−12.96764 eV, the C2sp 3  HO has an energy of E(C,2sp 3 )=−14.63489 eV (Eq. (15.25)), the Coulomb energy of Mn3d4s HO is E Coulomb (Mn,3d4s)=−14.11232 eV (Eq. (23.103)), the Mn3d4s HO has an energy of E(Mn,3d4s)=−14.22133 eV (Eq. (23.107)), and 13.605804 eV is the magnitude of the Coulombic energy between the electron and proton of H (Eq. (1.243)). To meet the equipotential condition of the union of the Mn-L H 2 -type-ellipsoidal-MO with these orbitals, the hybridization factor(s), at least one of c 2  and C 2  of Eq. (15.61) for the Mn-L-bond MO given by Eq. (15.77) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               F 
                                
                               
                                   
                               
                                
                               AO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Mn 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                   
                               
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Mn 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 F 
                                  
                                 
                                     
                                 
                                  
                                 AO 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               14.22133 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               17.42282 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.81625 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.113 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               Cl 
                                
                               
                                   
                               
                                
                               AO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Mn 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                   
                               
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Cl 
                                  
                                 
                                     
                                 
                                  
                                 AO 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 Mn 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               12.96764 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               14.22133 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.91184 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.114 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                             
                         
                          
                         
                           
                             
                               c 
                               2 
                             
                              
                             
                               ( 
                               
                                 C 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                                  
                                 HO 
                                  
                                 
                                     
                                 
                                  
                                 toMn 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 
                                   s 
                                    
                                   HO 
                                 
                               
                               ) 
                             
                           
                           = 
                             
                            
                           
                             
                               
                                 E 
                                 Coulomb 
                               
                                
                               
                                 ( 
                                 
                                   Mn 
                                   , 
                                   
                                     3 
                                      
                                     
                                         
                                     
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     
                                         
                                     
                                      
                                     s 
                                   
                                 
                                 ) 
                               
                             
                             
                               E 
                                
                               
                                 ( 
                                 
                                   C 
                                   , 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       sp 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               C 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 - 
                                 14.11232 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             
                               
                                 - 
                                 14.63489 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                           
                            
                             
                            
                           
                             ( 
                             0.91771 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.88495 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.115 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             
                               C 
                               2 
                             
                              
                             
                               ( 
                               
                                 Mn 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 
                                   s 
                                    
                                   HO 
                                 
                                  
                                 
                                     
                                 
                                  
                                 to 
                                  
                                 
                                     
                                 
                                  
                                 Mn 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 
                                   s 
                                    
                                   HO 
                                 
                               
                               ) 
                             
                           
                           = 
                             
                            
                           
                             
                               E 
                                
                               
                                 ( 
                                 H 
                                 ) 
                               
                             
                             
                               
                                 E 
                                 Coulomb 
                               
                                
                               
                                 ( 
                                 
                                   Mn 
                                   , 
                                   
                                     3 
                                      
                                     
                                         
                                     
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     
                                         
                                     
                                      
                                     s 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               
                                 - 
                                 13.605804 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             
                               
                                 - 
                                 14.11232 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           0.96411 
                         
                       
                     
                   
                    
                   
                     
 
                   
                 
               
               
                 
                   ( 
                   23.116 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Eqs. (15.76), (15.79), and (13.430) were used in Eq. (23.115) and Eq. (15.71) was used in Eq. (23.116). Since the energy of the MO is matched to that of the Mn3d4s HO in coordinate compounds, E(AO/HO) in Eq. (15.61) is E(Mn,3d4s) given by Eq. (23.107) and E(AO/HO) in Eq. (15.61) of carbonyl compounds is E Coulomb (Mn,3d4s) given by Eq. (23.103). E T (atom−atom,msp 3 .AO) of the Mn-L-bond MO is determined by considering that the bond involves an electron transfer from the manganese atom to the ligand atom to form partial ionic character in the bond as in the case of the zwitterions such as H 2 B + —F −  given in the Halido Boranes section. For the coordinate compounds, E T (atom−atom,msp 3 .AO) is −1.54812 eV, two times the energy of Eq. (23.112). For the Mn—CO bonds of carbonyl compounds, E T (atom−atom,msp 3 .AO) is −1.44915 eV (Eq. (14.151)), and the C═O functional group of carbonyls is equivalent to that of vanadium carbonyls. 
         [0525]    The symbols of the functional groups of manganese coordinate compounds are given in Table 23.30. The geometrical (Eqs. (15.1-15.5) and (23.41)), intercept (Eqs. (15.31-15.32) and (15.80-15.87)), and energy (Eqs. (15.61) and (23.28-23.33)) parameters of manganese coordinate compounds are given in Tables 23.31, 23.32 (as shown in the priority document), and 23.33, respectively. The total energy of each manganese coordinate compounds given in Table 23.34 (as shown in the priority document) was calculated as the sum over the integer multiple of each E D  (Group) of Table 23.33 corresponding to functional-group composition of the compound. The charge-densities of exemplary manganese carbonyl compound, dimanganese decacarbonyl (Mn 2  (CO) 10 ) comprising the concentric shells of atoms with the outer shell bridged by one or more H 2 -type ellipsoidal MOs or joined with one or more hydrogen MOs is shown in  FIG. 46 . 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 23.30 
               
             
             
               
                   
               
               
                 The symbols of the functional groups of manganese 
               
               
                 coordinate compounds. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
               
               
                   
                 MnF group of MnF 
                 Mn—F 
               
               
                   
                 MnCl group of MnCl 
                 Mn—Cl 
               
               
                   
                 MnCO group of Mn 2 (CO) 10   
                 Mn—CO 
               
               
                   
                 MnMn group of Mn 2 (CO) 10   
                 Mn—Mn 
               
               
                   
                 C═0 
                 C═0 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 23.31 
               
             
             
               
                   
               
               
                 The geometrical bond parameters of manganese 
               
               
                 coordinate compounds and experimental values. 
               
             
          
           
               
                   
                 Mn—F 
                 Mn—Cl 
                 Mn—CO 
                 Mn—Mn 
                 C═O 
               
               
                 Parameter 
                 Group 
                 Group 
                 Group 
                 Group 
                 Group 
               
               
                   
               
               
                 n e   
                 2 
                 3 
                   
                 5 
                   
               
               
                   
               
               
                 L 
                 
                   
                     
                       
                         2 
                         + 
                         
                           4 
                            
                           
                             
                               3 
                               4 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         4 
                         + 
                         
                           6 
                            
                           
                             
                               3 
                               4 
                             
                           
                         
                       
                     
                   
                 
                   
                 
                   
                     
                       
                         3 
                          
                         
                           
                             3 
                             4 
                           
                         
                       
                     
                   
                 
                   
               
               
                   
               
               
                 a (a 0 ) 
                 2.21856 
                 2.86785 
                 2.23676 
                 3.60392 
                 1.184842 
               
               
                 c′ (a 0 ) 
                 1.64864 
                 2.04780 
                 1.72695 
                 2.73426 
                 1.08850  
               
               
                 Bond Length 
                 1.74484 
                 2.16729 
                 1.82772 
                 2.89382 
                 1.15202  
               
               
                 2c′ (Å) 
                   
                   
                   
                   
                   
               
               
                 Exp. Bond 
                 1.729 [45] 
                 2.202 [15] 
                 1.830 [46] 
                 2.923 [46] 
                 1.151 [29, 46] 
               
               
                 Length 
                 (MnF 2 ) 
                 (MnCl 2 ) 
                 (Mn 2 (CO) 10 ) 
                 (Mn 2 (CO) 10 ) 
                 (Mn 2 (CO) 10 ) 
               
               
                 (Å) 
                   
                   
                   
                   
                   
               
               
                 b, c (a 0 ) 
                 1.48459 
                 2.00775 
                 1.42153 
                 2.34778 
                 0.46798  
               
               
                 e 
                 0.74311 
                 0.71405 
                 0.77208 
                 0.75869 
                 0.91869  
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 23.33 
               
             
             
               
                   
               
               
                 The energy parameters (eV) of functional groups of 
               
               
                 manganese coordinate compounds. 
               
             
          
           
               
                   
                 Mn—F 
                 Mn—Cl 
                 Mn—CO 
                 Mn—Mn 
                 C═O 
               
               
                 Parameters 
                 Group 
                 Group 
                 Group 
                 Group 
                 Group 
               
               
                   
               
             
          
           
               
                 f 1   
                 1 
                 1 
                 1 
                 1 
                 1 
               
               
                 n 1   
                 1 
                 1 
                 1 
                 1 
                 2 
               
               
                 n 2   
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 n 3   
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 C 1   
                 0.5 
                 0.375 
                 0.375 
                 0.25 
                 0.5 
               
               
                 C 2   
                 0.81625 
                 0.91184 
                 1 
                 0.96411 
                 1 
               
               
                 c 1   
                 1 
                 1 
                 1 
                 1 
                 1 
               
               
                 c 2   
                 1 
                 1 
                 0.88495 
                 1 
                 0.85395 
               
               
                 c 3   
                 0 
                 0 
                 0 
                 0 
                 2 
               
               
                 c 4   
                 1 
                 1 
                 2 
                 2 
                 4 
               
               
                 c 5   
                 1 
                 1 
                 0 
                 0 
                 0 
               
               
                 C 1o   
                 0.5 
                 0.375 
                 0.375 
                 0.25 
                 0.5 
               
               
                 C 2o   
                 0.81625 
                 0.91184 
                 1 
                 0.96411 
                 1 
               
               
                 V e  (eV) 
                 −31.60440 
                 −23.79675 
                 −28.59791 
                 −19.76726 
                 −134.96850 
               
               
                 V p  (eV) 
                 8.25276 
                 6.64412 
                 7.87853 
                 4.97605 
                 24.99908 
               
               
                 T (eV) 
                 7.12272 
                 4.14889 
                 6.39271 
                 2.74246 
                 56.95634 
               
               
                 V m  (eV) 
                 −3.56136 
                 −2.07445 
                 −3.19636 
                 −1.37123 
                 −28.47817 
               
               
                 E(AO/HO) (eV) 
                 −14.22133 
                 −14.22133 
                 −14.11232 
                 −14.11232 
                 0 
               
               
                 ΔE H     2     MO (AO/HO) (eV) 
                 0 
                 0 
                 0 
                 0 
                 −18.22046 
               
               
                 E T (AO/HO) (eV) 
                 −14.22133 
                 −14.22133 
                 −14.11232 
                 −14.11232 
                 18.22046 
               
               
                 E T (H 2 MO) (eV) 
                 −34.01162 
                 −29.29952 
                 −31.63535 
                 −27.53231 
                 −63.27080 
               
               
                 E T (atom-atom,msp 3 .AO) (eV) 
                 −1.54812 
                 −1.54812 
                 −1.44915 
                 −1.54005 
                 −3.58557 
               
               
                 E T (MO) (eV) 
                 −35.55974 
                 −30.84764 
                 −33.08452 
                 −29.07235 
                 −66.85630 
               
               
                 ω (10 15  rad/s) 
                 7.99232 
                 4.97768 
                 7.56783 
                 2.96657 
                 22.6662 
               
               
                 E K  (eV) 
                 5.26068 
                 3.27640 
                 4.98128 
                 1.95265 
                 14.91930 
               
               
                 Ē D  (eV) 
                 −0.16136 
                 −0.11046 
                 −0.14608 
                 −0.08037 
                 −0.25544 
               
               
                 Ē Kvib  (eV) 
                 0.07672 
                 0.04772 
                 0.04749 
                 0.01537 
                 0.24962 
               
               
                   
                 [47] 
                 [47] 
                 [29] 
                 [48] 
                 [29] 
               
               
                 Ē osc  (eV) 
                 −0.12299 
                 −0.08660 
                 −0.12234 
                 −0.07268 
                 −0.13063 
               
               
                 E mag  (eV) 
                 0.12767 
                 0.12767 
                 0.14803 
                 0.12767 
                 0.11441 
               
               
                 E T (Group) (eV) 
                 −35.68273 
                 −30.93425 
                 −33.20686 
                 −29.14504 
                 −67.11757 
               
               
                 E initial (c 4  AO/HO) (eV) 
                 −14.22133 
                 −14.22133 
                 −14.63489 
                 −14.11232 
                 −14.63489 
               
               
                 E initial (c 5  AO/HO) (eV) 
                 −17.42282 
                 −12.96764 
                 0 
                 0 
                 0 
               
               
                 E D (Group) (eV) 
                 4.03858 
                 3.74528 
                 3.93708 
                 0.92039 
                 8.34918 
               
               
                   
               
             
          
         
       
     
       Iron Functional Groups and Molecules 
       [0526]    The electron configuration of iron is [Ar]4s 2 3 d 6  having the corresponding term  5 D 4 . The total energy of the state is given by the sum over the eight electrons. The sum E T (Fe,3d4s) of experimental energies [1] of Fe, Fe + , Fe 2+ , Fe 3+ , Fe 4+ , Fe 5+ , Fe 6+ , and Fe 7+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               Fe 
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       151.06 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       124.98 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       99.1 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       75.0 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       54.8 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       30.652 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       16.1877 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       7.9024 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             559.68210 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.117 
                   ) 
                 
               
             
           
         
       
     
         [0000]    By considering that the central field decreases by an integer for each successive electron of the shell, the radius r 3d4s  of the Fe3d4s shell may be calculated from the Coulombic energy using Eq. (15.13): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             3 
                              
                             
                                 
                             
                              
                             d 
                              
                             
                                 
                             
                              
                             4 
                              
                             
                                 
                             
                              
                             s 
                           
                         
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               18 
                             
                             25 
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   Z 
                                   - 
                                   n 
                                 
                                 ) 
                               
                                
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     559 
                                      
                                     .68210 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             36 
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   559 
                                    
                                   .68210 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.87516 
                            
                           
                               
                           
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.118 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Z=26 for iron. Using Eq. (15.14), the Coulombic energy E Coulomb (Fe,3d4s) of the outer electron of the Fe3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               Fe 
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 s 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             
                               πɛ 
                               0 
                             
                              
                             0.87516 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             15.546725 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.119 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, the spin-paired 4s electrons and the one set of paired 3d electrons are promoted to Fe3d4s shell as initially unpaired electrons. The energies for the promotions are given by Eq. (15.15) at the initial radii of the 4s and 3d electrons. From Eq. (10.102) with Z=26 and n=26, the radius r 26  of Fe4s shell is 
         [0000]      r 26 =1.72173a 0   (23.120)
 
         [0000]    and with Z=26 and n=24, the radius r 24  of Fe3d shell is 
         [0000]      r 24 =1.33164a 0   (23.121)
 
         [0000]    Using Eqs. (15.15), (23.120), and (23.121), the unpairing energies are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               4 
                                
                               
                                   
                               
                                
                               s 
                             
                           
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               πμ 
                               0 
                             
                              
                             
                                
                               2 
                             
                              
                             
                               ℏ 
                               2 
                             
                           
                           
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   26 
                                 
                                 ) 
                               
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             8 
                              
                             
                                 
                             
                              
                             
                               πμ 
                               0 
                             
                              
                             
                               μ 
                               B 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 1.72173 
                                  
                                 
                                     
                                 
                                  
                                 
                                   a 
                                   0 
                                 
                               
                               ) 
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.02242 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.122 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               3 
                                
                               
                                   
                               
                                
                               d 
                             
                           
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               πμ 
                               0 
                             
                              
                             
                                
                               2 
                             
                              
                             
                               ℏ 
                               2 
                             
                           
                           
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   24 
                                 
                                 ) 
                               
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             8 
                              
                             
                                 
                             
                              
                             
                               πμ 
                               0 
                             
                              
                             
                               μ 
                               B 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 1.33164 
                                  
                                 
                                     
                                 
                                  
                                 
                                   a 
                                   0 
                                 
                               
                               ) 
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.04845 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.123 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The electrons from the 4s and 3d shells successively fill unoccupied HOs until the HO shell is filled with unpaired electrons, then the electrons pair per HO. In the case of the Fe3d4s shell having eight electrons and six orbitals, two sets of electrons are paired. Using Eqs. (15.15) and (23.118), the paring energy is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               3 
                                
                               
                                   
                               
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                   
                               
                                
                               s 
                             
                           
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           - 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     
                                       3 
                                        
                                       
                                           
                                       
                                        
                                       d 
                                        
                                       
                                           
                                       
                                        
                                       4 
                                        
                                       
                                           
                                       
                                        
                                       s 
                                     
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           - 
                           
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 μ 
                                 B 
                                 2 
                               
                             
                             
                               
                                 ( 
                                 
                                   0.87516 
                                    
                                   
                                       
                                   
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.17069 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.124 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, after Eq. (23.28), the energy E(Fe,3d4s) of the outer electron of the Fe3d4s shell is given by adding the magnetic energies of unpairing the 4s (Eq. (23.122)) and 3d electrons (Eq. (23.123)) and paring of two sets of Fe3d4s electrons (Eq. (23.124)) to E Coulomb (Fe,3d4s) (Eq. (23.119)): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               Fe 
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 r 
                                 
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                                 3 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 pairs 
                               
                             
                              
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   πμ 
                                   0 
                                 
                                  
                                 
                                    
                                   2 
                                 
                                  
                                 
                                   ℏ 
                                   2 
                                 
                               
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   r 
                                   
                                     3 
                                      
                                     
                                         
                                     
                                      
                                     d 
                                   
                                   3 
                                 
                               
                             
                           
                           - 
                           
                             
                               ∑ 
                               
                                 HO 
                                  
                                 
                                     
                                 
                                  
                                 pairs 
                               
                             
                              
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   πμ 
                                   0 
                                 
                                  
                                 
                                    
                                   2 
                                 
                                  
                                 
                                   ℏ 
                                   2 
                                 
                               
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   r 
                                   
                                     3 
                                      
                                     
                                         
                                     
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     
                                         
                                     
                                      
                                     s 
                                   
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               15.546725 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.02242 
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             0.04845 
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             2 
                              
                             
                               ( 
                               
                                 0.17069 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             15.81724 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.125 
                   ) 
                 
               
             
           
         
       
     
         [0527]    Next, consider the formation of the Fe-L-bond MO of wherein each iron atom has an Fe3d4s electron with an energy given by Eq. (23.125). The total energy of the state of each iron atom is given by the sum over the eight electrons. The sum E T (Fe Fe-L 3d4s) of energies of Fe3d4s (Eq. (23.125)), Fe + , Fe 2+ , Fe 3+ , Fe 4+ , Fe 5+ , Fe 6+ , and Fe 7+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Fe 
                                 Fe—L 
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                   
                               
                                
                               s 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       151.06 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       124.98 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       99.1 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       75.0 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       54.8 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       30.652 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       16.1877 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     E 
                                      
                                     
                                       ( 
                                       
                                         Fe 
                                         , 
                                         
                                           3 
                                            
                                           
                                               
                                           
                                            
                                           d 
                                            
                                           
                                               
                                           
                                            
                                           4 
                                            
                                           
                                               
                                           
                                            
                                           s 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       151.06 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       124.98 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       99.1 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       75.0 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       54.8 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       30.652 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       16.1877 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     15.81724 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             567.59694 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.126 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E(Fe, 3d 4s) is the sum of the energy of Fe, −7.9024 eV, and the hybridization energy. 
         [0528]    The iron HO donates an electron to each MO. Using Eq. (23.30), the radius r 3d4s  of the Fe3d4s shell calculated from the Coulombic energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             Fe—L 
                              
                             
                                 
                             
                              
                             3 
                              
                             
                                 
                             
                              
                             d 
                              
                             
                                 
                             
                              
                             4 
                              
                             
                                 
                             
                              
                             s 
                           
                         
                         = 
                           
                          
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     18 
                                   
                                   25 
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               1 
                             
                             ) 
                           
                            
                           
                             
                                
                               2 
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     567 
                                      
                                     .59694 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               35 
                                
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     567 
                                      
                                     .59694 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                           = 
                           
                             0.83898 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.127 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (23.127), the Coulombic energy E Coulomb (Fe Fe-L ,3d4s) of the outer electron of the Fe3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 Fe 
                                 Fe—L 
                               
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 Fe—L 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 s 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             
                               πɛ 
                               0 
                             
                              
                             0.83898 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             16.21706 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.128 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The magnetic energy terms are those for unpairing of the 4s and 3d electrons (Eqs. (23.122) and (23.123), respectively) and paring two sets of Fe3d4s electrons (Eq. (23.124)). Using Eqs. (23.32), (23.128) and (23.122-23.124), the energy E(Fe Fe-L ,3d4s) of the outer electron of the Fe3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 Fe 
                                 Fe—L 
                               
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   Fe—L 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     26 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     24 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     
                                       3 
                                        
                                       
                                           
                                       
                                        
                                       d 
                                        
                                       
                                           
                                       
                                        
                                       4 
                                        
                                       
                                           
                                       
                                        
                                       s 
                                     
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               16.21706 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.02242 
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             0.04845 
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             2 
                              
                             
                               ( 
                               
                                 0.17069 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             16.48757 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.129 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (Fe-L,3d4s), the energy change of each Fe3d4s shell with the formation of the Fe-L-bond MO is given by the difference between Eq. (23.129) and Eq. (23.125): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                           ( 
                           
                             
                               Fe 
                                
                               —L 
                             
                             , 
                             
                               3 
                                
                               
                                   
                               
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                   
                               
                                
                               s 
                             
                           
                           ) 
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   Fe 
                                   Fe—L 
                                 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 Fe 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               16.48757 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 15.81724 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             0.67033 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.130 
                   ) 
                 
               
             
           
         
       
     
         [0529]    The semimajor axis a solution given by Eq. (23.41) of the force balance equation, Eq. (23.39), for the σ-MO of the Fe-L-bond MO of FeL n  is given in Table 23.36 (as shown in the priority document) with the force-equation parameters Z=26, n e , and L corresponding to the orbital and spin angular momentum terms of the 3d4s HO shell. The semimajor axis a of carbonyl and organometallic compounds are solved using Eq. (15.51). 
         [0530]    For the Fe-L functional groups, hybridization of the 4s and 3d AOs of Fe to form a single 3d4s shell forms an energy minimum, and the sharing of electrons between the Fe3d4s HO and L AO to form σ MO permits each participating orbital to decrease in radius and energy. The F AO has an energy of E(F)=−17.42282 eV, the Cl AO has an energy of E(Cl)=−12.96764 eV, the C aryl 2sp 3  HO has an energy of E(C aryl 2sp 3 )=−15.76868 eV (Eq. (14.246)), the C2sp 3  HO has an energy of E(C,2sp 3 )=−14.63489 eV (Eq. (15.25)), the O AO has an energy of E(O)=−13.61805 eV, the Coulomb energy of Fe3d4s HO is E Coulomb (Fe,3d4s)=−15.546725 eV (Eq. (23.119)), and the Fe3d4s HO has an energy of E(Fe,3d4s)=−15.81724 eV (Eq. (23.125)). To meet the equipotential condition of the union of the Fe-L H 2 -type-ellipsoidal-MO with these orbitals, the hybridization factor(s), at least one of c 2  and C 2  of Eq. (15.61) for the Fe-L-bond MO given by Eq. (15.77) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                             
                         
                          
                         
                           
                             
                               c 
                               2 
                             
                              
                             
                               ( 
                               
                                 F 
                                  
                                 
                                     
                                 
                                  
                                 AO 
                                  
                                 
                                     
                                 
                                  
                                 to 
                                  
                                 
                                     
                                 
                                  
                                 Fe 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 sHO 
                               
                               ) 
                             
                           
                           = 
                             
                            
                           
                             
                               C 
                               2 
                             
                              
                             
                               ( 
                               
                                 F 
                                  
                                 
                                     
                                 
                                  
                                 AO 
                                  
                                 
                                     
                                 
                                  
                                 to 
                                  
                                 
                                     
                                 
                                  
                                 Fe 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 sHO 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Fe 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 F 
                                  
                                 
                                     
                                 
                                  
                                 AO 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               15.81724 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               17.42282 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.90785 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.131 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               Cl 
                                
                               
                                   
                               
                                
                               AO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Fe 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                   
                               
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               Cl 
                                
                               
                                   
                               
                                
                               AO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Fe 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                   
                               
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Cl 
                                  
                                 
                                     
                                 
                                  
                                 AO 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 Fe 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               12.96764 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               15.81724 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.81984 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.132 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               C 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Fe 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                   
                               
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 C 
                                 , 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             
                               E 
                               Coulomb 
                             
                              
                             
                               ( 
                               
                                 Fe 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               C 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 - 
                                 14.63489 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             
                               
                                 - 
                                 15.54673 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                           
                            
                           
                             ( 
                             0.91771 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.86389 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.133 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 C 
                                 aryl 
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Fe 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                   
                               
                                
                               sHO 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       C 
                                       aryl 
                                     
                                      
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       sp 
                                       3 
                                     
                                      
                                     HO 
                                      
                                     
                                         
                                     
                                      
                                     to 
                                   
                                 
                               
                               
                                 
                                   
                                     Fe 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                      
                                     
                                         
                                     
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     
                                         
                                     
                                      
                                     
                                       s 
                                        
                                       HO 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 C 
                                 , 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             
                               E 
                               Coulomb 
                             
                              
                             
                               ( 
                               
                                 Fe 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 C 
                                 aryl 
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               14.63489 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               15.54673 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           ( 
                           0.85252 
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.80252 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.134 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                             
                         
                          
                         
                           
                             
                               c 
                               2 
                             
                              
                             
                               ( 
                               
                                 O 
                                  
                                 
                                     
                                 
                                  
                                 to 
                                  
                                 
                                     
                                 
                                  
                                 Fe 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 
                                   s 
                                    
                                   HO 
                                 
                               
                               ) 
                             
                           
                           = 
                             
                            
                           
                             
                               C 
                               2 
                             
                              
                             
                               ( 
                               
                                 O 
                                  
                                 
                                     
                                 
                                  
                                 to 
                                  
                                 
                                     
                                 
                                  
                                 Fe 
                                  
                                 
                                     
                                 
                                  
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 
                                   s 
                                    
                                   HO 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               O 
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 Fe 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               13.61805 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               15.81724 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.86096 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.135 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Eqs. (15.76), (15.79), and (13.430) were used in Eq. (23.133) and Eqs. (15.76), (15.79), and (14.417) were used in Eq. (23.134). Since the energy of the MO is matched to that of the Fe3d4s HO in coordinate compounds, E(AO/HO) in Eq. (15.61) is E(Fe,3d4s) given by Eq. (23.125) and E(AO/HO) in Eq. (15.61) of carbonyl and organometallic compounds is E Coulomb (Fe,3d4s) given by Eq. (23.119). E T (atom−atom,msp 3 .AO) of the Fe-L-bond MO is determined by considering that the bond involves an electron transfer from the iron atom to the ligand atom to form partial ionic character in the bond as in the case of the zwitterions such as H 2 B + —F −  given in the Halido Boranes section. For the coordinate compounds, E T (atom−atom,msp 3 .AO) is −1.34066 eV, two times the energy of Eq. (23.130). For the Fe—C bonds of carbonyl and organometallic compounds, E T (atom−atom,msp 3 .AO) is −1.44915 eV (Eq. (14.151)), and the C═O functional group of carbonyls is equivalent to that of vanadium carbonyls. The aromatic cyclopentadienyl moieties of organometallic Fe(C 5 H 5 ) 2  comprise 
         [0000]    
       
         
           
             C 
              
             
               = 
               
                 3 
                  
                 
                     
                 
                  
                 e 
               
             
              
             C 
           
         
       
     
         [0000]    and CH functional groups that are equivalent to those given in the Aromatic and Heterocyclic Compounds section. 
         [0531]    The symbols of the functional groups of iron coordinate compounds are given in Table 23.35. The geometrical (Eqs. (15.1-15.5) and (23.41)), intercept (Eqs. (15.31-15.32) and (15.80-15.87)), and energy (Eqs. (15.61) and (23.28-23.33)) parameters of iron coordinate compounds are given in Tables 23.36, 23.37, and 23.38, respectively (all as shown in the priority document). The total energy of each iron coordinate compounds given in Table 23.39 (as shown in the priority document) was calculated as the sum over the integer multiple of each E D  (Group) of Table 23.38 (as shown in the priority document) corresponding to functional-group composition of the compound. The charge-densities of exemplary iron carbonyl and organometallic compounds, iron pentacarbonyl (Fe (CO) 5 ) and bis-cylopentadienyl iron or ferrocene (Fe (C 5 H 5 ) 2 ) comprising the concentric shells of atoms with the outer shell bridged by one or more H 2 -type ellipsoidal MOs or joined with one or more hydrogen MOs are shown in  FIG. 47  as  48 , respectively. 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 23.35 
               
             
             
               
                   
               
               
                 The symbols of the functional groups of iron coordinate compounds. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
               
               
                   
                 FeF group of FeF 
                 Fe—F (a) 
               
               
                   
                 FeF 2  group of FeF 2   
                 Fe—F (b) 
               
               
                   
                 FeF 3  group of FeF 3   
                 Fe—F (c) 
               
               
                   
                 FeCl group of FeCl 
                 Fe—Cl (a) 
               
               
                   
                 FeCl 2  group of FeCl 2   
                 Fe—Cl (b) 
               
               
                   
                 FeCl 3  group of FeCl 3   
                 Fe—Cl (c) 
               
               
                   
                 FeO group of FeO 
                 Fe—O 
               
               
                   
                 FeCO group of Fe(CO) 5   
                 Fe—CO 
               
               
                   
                 C═O 
                 C═O 
               
               
                   
                 FeC aryl  group of Fe(C 5 H 5 ) 2   
                 Fe—C 5 H 5   
               
               
                   
                 CC (aromatic bond) 
                 C 3e ═C 
               
               
                   
                 CH (aromatic) 
                 CH 
               
               
                   
               
             
          
         
       
     
       Cobalt Functional Groups and Molecules 
       [0532]    The electron configuration of cobalt is [Ar]4s 2 3d 7  having the corresponding term  4 F 9/2 . The total energy of the state is given by the sum over the nine electrons. The sum E T (Co,3d 4s) of experimental energies [1] of Co, Co + , Co 2+ , Co 3+ , Co 4+ , Co 5+ , Co 6+ , Co 7+ , and Co 8+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               Co 
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       186.13 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       157.8 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       128.9 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       102.0 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       79.5 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       51.3 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       33.50 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       17.084 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     7.88101 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             764.09501 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.136 
                   ) 
                 
               
             
           
         
       
     
         [0000]    By considering that the central field decreases by an integer for each successive electron of the shell, the radius r 3d4s  of the Co3d4s shell may be calculated from the Coulombic energy using Eq. (15.13): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             3 
                              
                             d 
                              
                             
                                 
                             
                              
                             4 
                              
                             s 
                           
                         
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               18 
                             
                             26 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   Z 
                                   - 
                                   n 
                                 
                                 ) 
                               
                                
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     764.09501 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             45 
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   764.09501 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.80129 
                            
                           
                               
                           
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.137 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Z=27 for cobalt. Using Eq. (15.14), the Coulombic energy E Coulomb (Co,3d4s) of the outer electron of the Co3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               Co 
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             0.80129 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             16.979889 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.138 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, the spin-paired 4s electrons and the two sets of paired 3d electrons are promoted to Co3d4s shell as initially unpaired electrons. The energies for the promotions are given by Eq. (15.15) at the initial radii of the 4s and 3d electrons. From Eq. (10.102) with Z=27 and n=27, the radius r 27  of Co4s shell is 
         [0000]      r 27 =1.72640a 0   (23.139)
 
         [0000]    and with Z=27 and n=25, the radius r 25  of Co3d shell is 
         [0000]      r 25 =1.21843a 0   (23.140)
 
         [0000]    Using Eqs. (15.15), (23.139), and (23.140), the unpairing energies are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               4 
                                
                               s 
                             
                           
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                         
                           
                             2 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                                
                               2 
                             
                              
                             
                               ℏ 
                               2 
                             
                           
                           
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   27 
                                 
                                 ) 
                               
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             8 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                               μ 
                               B 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 1.72640 
                                  
                                 
                                     
                                 
                                  
                                 
                                   a 
                                   0 
                                 
                               
                               ) 
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           0.02224 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.141 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               3 
                                
                               d 
                             
                           
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                         
                           
                             2 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                                
                               2 
                             
                              
                             
                               ℏ 
                               2 
                             
                           
                           
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   25 
                                 
                                 ) 
                               
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             8 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                               μ 
                               B 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 1.21843 
                                  
                                 
                                     
                                 
                                  
                                 
                                   a 
                                   0 
                                 
                               
                               ) 
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           0.06325 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.142 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The electrons from the 4s and 3d shells successively fill unoccupied HOs until the HO shell is filled with unpaired electrons, then the electrons pair per HO. In the case of the Co3d4s shell having nine electrons and six orbitals, three sets of electrons are paired. Using Eqs. (15.15) and (23.137), the paring energy is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               s 
                             
                           
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                         
                           - 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     
                                       3 
                                        
                                       d 
                                        
                                       
                                           
                                       
                                        
                                       4 
                                        
                                       s 
                                     
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           - 
                           
                             
                               8 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 μ 
                                 B 
                                 2 
                               
                             
                             
                               
                                 ( 
                                 
                                   0.80129 
                                    
                                   
                                       
                                   
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             0.22238 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.143 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, after Eq. (23.28), the energy E(Co,3d4s) of the outer electron of the Co3d4s shell is given by adding the magnetic energies of unpairing the 4s (Eq. (23.141)) and 3d electrons (Eq. (23.142)) and paring of three sets of Co3d4s electrons (Eq. (23.143)) to E Coulomb (Co,3d 4s) (Eq. (23.138)): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               Co 
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 r 
                                 
                                   4 
                                    
                                   s 
                                 
                                 3 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 pairs 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 2 
                                  
                                 
                                   πμ 
                                   0 
                                 
                                  
                                 
                                    
                                   2 
                                 
                                  
                                 
                                   ℏ 
                                   2 
                                 
                               
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   r 
                                   
                                     3 
                                      
                                     d 
                                   
                                   3 
                                 
                               
                             
                           
                           - 
                           
                             
                               ∑ 
                               
                                 HO 
                                  
                                 
                                     
                                 
                                  
                                 pairs 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 2 
                                  
                                 
                                   πμ 
                                   0 
                                 
                                  
                                 
                                    
                                   2 
                                 
                                  
                                 
                                   ℏ 
                                   2 
                                 
                               
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   r 
                                   
                                     3 
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     s 
                                   
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               16.979889 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.02224 
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             2 
                              
                             
                               ( 
                               
                                 0.06325 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                           - 
                           
                             3 
                              
                             
                               ( 
                               
                                 0.22238 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             17.49830 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.144 
                   ) 
                 
               
             
           
         
       
     
         [0533]    Next, consider the formation of the Co-L-bond MO of wherein each cobalt atom has an Co3d4s electron with an energy given by Eq. (23.144). The total energy of the state of each cobalt atom is given by the sum over the nine electrons. The sum E T (CO Co-L 3d 4s) of energies of Co3d4s (Eq. (23.144)), Co + , Co 2+ , Co 3+ , Co 4+ , Co 5+ , Co 6+ , Co 7+ , and Co 8+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Co 
                                 
                                   Co 
                                   - 
                                   L 
                                 
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               s 
                             
                             ) 
                           
                         
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       186.13 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       157.8 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       128.9 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       102.0 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       79.5 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       51.3 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       33.50 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       17.084 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       E 
                                        
                                       
                                         ( 
                                         
                                           Co 
                                           , 
                                           
                                             3 
                                              
                                             d 
                                              
                                             
                                                 
                                             
                                              
                                             4 
                                              
                                             s 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       186.13 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       157.8 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       128.9 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       102.0 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       79.5 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       51.3 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       33.50 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       17.084 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       17.49830 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             773.71230 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.145 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E(Co,3d4s) is the sum of the energy of Co, −7.88101 eV, and the hybridization energy. 
         [0534]    The cobalt HO donates an electron to each MO. Using Eq. (23.30), the radius r 3d4s  of the Co3d4s shell calculated from the Coulombic energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             Co 
                             - 
                             
                               L 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               s 
                             
                           
                         
                         = 
                           
                          
                         
                           ( 
                           
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   18 
                                 
                                 26 
                               
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   Z 
                                   - 
                                   n 
                                 
                                 ) 
                               
                             
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                              
                             2 
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   773.71230 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             44 
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   773.71230 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.77374 
                            
                           
                               
                           
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.146 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (23.146), the Coulombic energy E Coulomb (CO Co-L ,3d 4s) of the outer electron of the Co3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 Co 
                                 
                                   Co 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 Co 
                                 - 
                                 
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             0.77374 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             17.58437 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.147 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The magnetic energy terms are those for unpairing of the 4s and 3d electrons (Eqs. (23.141) and (23.142), respectively) and paring three sets of Co3d4s electrons (Eq. (23.143)). Using Eqs. (23.32), (23.147) and (23.141-23.143), the energy E(CO Co-L ,3d4s) of the outer electron of the Co3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 Co 
                                 
                                   Co 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   Fe 
                                   - 
                                   
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     s 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     27 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     25 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                           - 
                           
                             3 
                              
                             
                               
                                 2 
                                  
                                 
                                   πμ 
                                   0 
                                 
                                  
                                 
                                    
                                   2 
                                 
                                  
                                 
                                   ℏ 
                                   2 
                                 
                               
                               
                                 
                                   
                                     m 
                                     e 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       r 
                                       
                                         3 
                                          
                                         d 
                                          
                                         
                                             
                                         
                                          
                                         4 
                                          
                                         s 
                                       
                                     
                                     ) 
                                   
                                 
                                 3 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               17.58437 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.02224 
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             2 
                              
                             
                               ( 
                               
                                 0.06325 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                           - 
                           
                             3 
                              
                             
                               ( 
                               
                                 0.22238 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             18.10278 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.148 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (Co-L, 3d 4s), the energy change of each Co3d4s shell with the formation of the Co-L-bond MO is given by the difference between Eq. (23.148) and Eq. (23.144): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Co 
                                 - 
                                 L 
                               
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   Co 
                                   
                                     Co 
                                     - 
                                     L 
                                   
                                 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 Co 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               18.10278 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 17.49830 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             0.60448 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.149 
                   ) 
                 
               
             
           
         
       
     
         [0535]    The semimajor axis a solution given by Eq. (23.41) of the force balance equation, Eq. (23.39), for the σ-MO of the Co-L-bond MO of CoL n  is given in Table 23.41 (as shown in the priority document) with the force-equation parameters Z=27, n e  and L corresponding to the orbital and spin angular momentum terms of the 3d4s HO shell. The semimajor axis a of carbonyl and organometallic compounds are solved using Eq. (15.51). 
         [0536]    For the Co-L functional groups, hybridization of the 4s and 3d AOs of Co to form a single 3d4s shell forms an energy minimum, and the sharing of electrons between the Co3d4s HO and L AO to form σ MO permits each participating orbital to decrease in radius and energy. The F AO has an energy of E(F)=−17.42282 eV, the Cl AO has an energy of E(Cl)=−12.96764 eV, the C2sp 3  HO has an energy of E(C,2sp 3 )=−14.63489 eV (Eq. (15.25)), the Coulomb energy of Co3d4s HO is E Coulomb (CO 3 3d 4s)=−16.979889 eV (Eq. (23.138)), 13.605804 eV is the magnitude of the Coulombic energy between the electron and proton of H (Eq. (1.243)), and the Co3d4s HO has an energy of E(Co,3d4s)=−17.49830 eV (Eq. (23.144)). To meet the equipotential condition of the union of the Co-L H 2 -type-ellipsoidal-MO with these orbitals, the hybridization factor(s), at least one of c 2  and C 2  of Eq. (15.61) for the Co-L-bond MO given by Eq. (15.77) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               FAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               
                                 Co 
                                  
                                 3d 
                               
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               FAO 
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 Co 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               17.42282 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               17.49830 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.99569 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.150 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               ClAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Co 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               ClAO 
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 Co 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               12.96764 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               17.49830 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.74108 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.151 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               C 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                                
                               
                                 
                                     
                                 
                                  
                                 
                                     
                                 
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Co3 
                                
                               
                                   
                               
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 C 
                                 , 
                                 
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             
                               E 
                               Coulomb 
                             
                              
                             
                               ( 
                               
                                 Co 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               C 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               14.63489 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               16.97989 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           ( 
                           0.91771 
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.79097 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.152 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               HAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Co 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               HAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Co 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               H 
                               ) 
                             
                           
                           
                             
                               E 
                               Coulomb 
                             
                              
                             
                               ( 
                               
                                 Co 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               13.605804 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               16.97989 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.80129 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.153 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Eqs. (15.76), (15.79), and (13.430) were used in Eq. (23.152) and Eq. (15.71) was used in Eq. (23.153). Since the energy of the MO is matched to that of the Co3d4s HO in coordinate compounds, E(AO/HO) in Eq. (15.61) is E(Co,3d4s) given by Eq. (23.144) and E(AO/HO) in Eq. (15.61) of carbonyl compounds is E Coulomb (Co,3d 4s) given by Eq. (23.138). E T (atom−atom,msp 3 .AO) of the Co-L-bond MO is determined by considering that the bond involves an electron transfer from the cobalt atom to the ligand atom to form partial ionic character in the bond as in the case of the zwitterions such as H 2 B + —F −  given in the Halido Boranes section. For the coordinate compounds, E T (atom−atom,msp 3 .AO) is −1.20896 eV, two times the energy of Eq. (23.149). For the Co—C bonds of carbonyl compounds, E T (atom−atom,msp 3 .AO) is −1.13379 eV (Eq. (14.247)), and the C═O functional group of carbonyls is equivalent to that of vanadium carbonyls. 
         [0537]    The symbols of the functional groups of cobalt coordinate compounds are given in Table 23.40. The geometrical (Eqs. (15.1-15.5) and (23.41)), intercept (Eqs. (15.31-15.32) and (15.80-15.87)), and energy (Eqs. (15.61) and (23.28-23.33)) parameters of cobalt coordinate compounds are given in Tables 23.41, 23.42, and 23.43, respectively (all as shown in the priority document). The total energy of each cobalt coordinate compounds given in Table 23.44 (as shown in the priority document) was calculated as the sum over the integer multiple of each E D  (Group) of Table 23.43 (as shown in the priority document) corresponding to functional-group composition of the compound. The charge-densities of exemplary cobalt carbonyl compound, cobalt tetracarbonyl hydride (CoH(CO) 4  comprising the concentric shells of atoms with the outer shell bridged by one or more H 2 -type ellipsoidal MOs or joined with one or more hydrogen MOs is shown in  FIG. 49 . 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 23.40 
               
             
             
               
                   
               
               
                 The symbols of the functional groups of cobalt coordinate compounds. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
               
               
                   
                 CoF 2  group of CoF 2   
                 Co—F 
               
               
                   
                 CoCl group of CoCl 
                 Co—Cl (a) 
               
               
                   
                 CoCl 2  group of CoCl 2   
                 Co—Cl (b) 
               
               
                   
                 CoCl 3  group of CoCl 3   
                 Co—Cl (c) 
               
               
                   
                 CoH group of CoH(CO) 4   
                 Co—H 
               
               
                   
                 CoCO group of CoH(CO) 4   
                 Co—CO 
               
               
                   
                 C═O 
                 C═O 
               
               
                   
               
             
          
         
       
     
       Nickel Functional Groups and Molecules 
       [0538]    The electron configuration of nickel is [Ar]4s 2 3d 8  having the corresponding term  3 F 4 . The total energy of the state is given by the sum over the ten electrons. The sum E T (Ni,3d4s) of experimental energies [1] of Ni, Ni + , Ni 2+ , Ni 3+ , Ni 4+ , Ni 5+ , Ni 6+ , Ni 7+ , Ni 8+ , and Ni 9+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               Ni 
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       224.6 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       193 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       162 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       133 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       108 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       76.06 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       54.9 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       35.19 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       18.16884 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     7.6398 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             1012.55864 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.154 
                   ) 
                 
               
             
           
         
       
     
         [0000]    By considering that the central field decreases by an integer for each successive electron of the shell, the radius r 3d4s  of the Ni3d4s shell may be calculated from the Coulombic energy using Eq. (15.13): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             3 
                              
                             d 
                              
                             
                                 
                             
                              
                             4 
                              
                             s 
                           
                         
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               18 
                             
                             27 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   Z 
                                   - 
                                   n 
                                 
                                 ) 
                               
                                
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     1012.55864 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             55 
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   1012.55864 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.73904 
                            
                           
                               
                           
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.155 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Z=28 for nickel. Using Eq. (15.14), the Coulombic energy E Coulomb  (Ni,3d4s) of the outer electron of the Ni3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               Ni 
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             0.73904 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             18.410157 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.156 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, the spin-paired 4s electrons and the three sets of paired 3d electrons are promoted to Ni3d4s shell as initially unpaired electrons. The energies for the promotions are given by Eq. (15.15) at the initial radii of the 4s and 3d electrons. From Eq. (10.102) with Z=28 and n=28, the radius r 28  of Ni4s shell is 
         [0000]      r 28 =1.78091a 0   (23.157)
 
         [0000]    and with Z=28 and n=26, the radius r 26  of Ni3d shell is 
         [0000]      r 26 =1.15992a 0   (23.158)
 
         [0000]    Using Eqs. (15.15), (23.157), and (23.158), the unpairing energies are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               4 
                                
                               s 
                             
                           
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                         
                           
                             2 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                                
                               2 
                             
                              
                             
                               ℏ 
                               2 
                             
                           
                           
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   28 
                                 
                                 ) 
                               
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             8 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                               μ 
                               B 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 1.78091 
                                  
                                 
                                     
                                 
                                  
                                 
                                   a 
                                   0 
                                 
                               
                               ) 
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           0.02026 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.159 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               3 
                                
                               d 
                             
                           
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                         
                           
                             2 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                                
                               2 
                             
                              
                             
                               ℏ 
                               2 
                             
                           
                           
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   26 
                                 
                                 ) 
                               
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             8 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                               μ 
                               B 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 1.15992 
                                  
                                 
                                     
                                 
                                  
                                 
                                   a 
                                   0 
                                 
                               
                               ) 
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           0.07331 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.160 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The electrons from the 4s and 3d shells successively fill unoccupied HOs until the HO shell is filled with unpaired electrons, then the electrons pair per HO. In the case of the Ni3d4s shell having ten electrons and six orbitals, four sets of electrons are paired. Using Eqs. (15.15) and (23.155), the paring energy is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               3 
                                
                               d4s 
                             
                           
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           - 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     
                                       3 
                                        
                                       d 
                                        
                                       
                                           
                                       
                                        
                                       4 
                                        
                                       s 
                                     
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           - 
                           
                             
                               8 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 μ 
                                 B 
                                 2 
                               
                             
                             
                               
                                 ( 
                                 
                                   0.73904 
                                    
                                   
                                       
                                   
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             0.28344 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.161 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, after Eq. (23.28), the energy E(Ni,3d4s) of the outer electron of the Ni3d4s shell is given by adding the magnetic energies of unpairing the 4s (Eq. (23.159)) and 3d electrons (Eq. (23.160)) and paring of four sets of Ni3d4s electrons (Eq. (23.161)) to E Coulomb (Ni,3d 4s) (Eq. (23.156)): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               Ni 
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 r 
                                 
                                   4 
                                    
                                   s 
                                 
                                 3 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 pairs 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 2 
                                  
                                 
                                   πμ 
                                   0 
                                 
                                  
                                 
                                    
                                   2 
                                 
                                  
                                 
                                   ℏ 
                                   2 
                                 
                               
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   r 
                                   
                                     3 
                                      
                                     d 
                                   
                                   3 
                                 
                               
                             
                           
                           - 
                           
                             
                               ∑ 
                               
                                 HO 
                                  
                                 
                                     
                                 
                                  
                                 pairs 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 2 
                                  
                                 
                                   πμ 
                                   0 
                                 
                                  
                                 
                                    
                                   2 
                                 
                                  
                                 
                                   ℏ 
                                   2 
                                 
                               
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   r 
                                   
                                     3 
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     s 
                                   
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               18.410157 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.02026 
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             3 
                              
                             
                               ( 
                               
                                 0.07331 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                           - 
                           
                             4 
                              
                             
                               ( 
                               
                                 0.28344 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             19.30374 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.162 
                   ) 
                 
               
             
           
         
       
     
         [0539]    Next, consider the formation of the Ni-L-bond MO of wherein each nickel atom has an Ni3d4s electron with an energy given by Eq. (23.162). The total energy of the state of each nickel atom is given by the sum over the ten electrons. The sum E T  (Ni Ni-L 3d4s) of energies of Ni3d4s (Eq. (23.162)), Ni + , Ni 2+ , Ni 3+ , Ni 4+ , Ni 5+ , Ni 6+ , Ni 7+ , Ni 8+ , and Ni 9+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Ni 
                                 
                                   Ni 
                                   - 
                                   L 
                                 
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               s 
                             
                             ) 
                           
                         
                         = 
                         
                           - 
                             
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       224.6 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       193 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       162 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       133 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       108 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       76.06 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       54.9 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       35.19 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       18.16884 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       E 
                                        
                                       
                                         ( 
                                         
                                           Ni 
                                           , 
                                           
                                             3 
                                              
                                             d 
                                              
                                             
                                                 
                                             
                                              
                                             4 
                                              
                                             s 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           - 
                             
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       224.6 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       193 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       162 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       133 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       108 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       76.06 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       54.9 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       35.19 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       18.16884 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       19.30374 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             1024.22258 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.163 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E(Ni,3d4s) is the sum of the energy of Ni, −7.6398 eV, and the hybridization energy. 
         [0540]    The nickel HO donates an electron to each MO. Using Eq. (23.30), the radius r 3d4s  of the Ni3d4s shell calculated from the Coulombic energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             Ni 
                             = 
                             
                               L 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               s 
                             
                           
                         
                         = 
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     18 
                                   
                                   27 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               1 
                             
                             ) 
                           
                            
                           
                             
                                
                               2 
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     1024.22258 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             54 
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   1024.22258 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           0.71734 
                            
                           
                               
                           
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.164 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (23.164), the Coulombic energy E Coulomb (Ni Ni-L ,3d 4s) of the outer electron of the Ni3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 Ni 
                                 
                                   Ni 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 Ni 
                                 - 
                                 
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             0.71734 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             18.96708 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.165 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The magnetic energy terms are those for unpairing of the 4s and 3d electrons (Eqs. (23.159) and (23.160), respectively) and paring four sets of Ni3d4s electrons (Eq. (23.161)). Using Eqs. (23.32), (23.165) and (23.159-23.161), the energy E(Ni Ni-L ,3d4s) of the outer electron of the Ni3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 Ni 
                                 
                                   Ni 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   Ni 
                                   - 
                                   
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                      
                                     
                                         
                                     
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     s 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     28 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             3 
                              
                             
                               
                                 2 
                                  
                                 
                                   πμ 
                                   0 
                                 
                                  
                                 
                                    
                                   2 
                                 
                                  
                                 
                                   ℏ 
                                   2 
                                 
                               
                               
                                 
                                   
                                     m 
                                     e 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       r 
                                       26 
                                     
                                     ) 
                                   
                                 
                                 3 
                               
                             
                           
                           - 
                           
                             4 
                              
                             
                               
                                 2 
                                  
                                 
                                   πμ 
                                   0 
                                 
                                  
                                 
                                    
                                   2 
                                 
                                  
                                 
                                   ℏ 
                                   2 
                                 
                               
                               
                                 
                                   
                                     m 
                                     e 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       r 
                                       
                                         3 
                                          
                                         d 
                                          
                                         
                                             
                                         
                                          
                                         4 
                                          
                                         s 
                                       
                                     
                                     ) 
                                   
                                 
                                 3 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               18.96708 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.02026 
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             3 
                              
                             
                               ( 
                               
                                 0.07331 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                           - 
                           
                             4 
                              
                             
                               ( 
                               
                                 0.28344 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             19.86066 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.166 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (Ni-L,3d4s), the energy change of each Ni3d4s shell with the formation of the Ni-L-bond MO is given by the difference between Eq. (23.166) and Eq. (23.162): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Ni 
                                 - 
                                 L 
                               
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   Ni 
                                   
                                     Ni 
                                     - 
                                     L 
                                   
                                 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 Ni 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               19.86066 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 19.30374 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             0.55693 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.167 
                   ) 
                 
               
             
           
         
       
     
         [0541]    The semimajor axis a solution given by Eq. (23.41) of the force balance equation, Eq. (23.39), for the σ-MO of the Ni-L-bond MO of NiL n , is given in Table 23.46 (as shown in the priority document) with the force-equation parameters Z=28, n e , and L corresponding to the orbital and spin angular momentum terms of the 3d4s HO shell. The semimajor axis a of carbonyl and organometallic compounds are solved using Eq. (15.51). 
         [0542]    For the Ni-L functional groups, hybridization of the 4s and 3d AOs of Ni to form a single 3d4s shell forms an energy minimum, and the sharing of electrons between the Ni3d4s HO and L AO to form σ MO permits each participating orbital to decrease in radius and energy. The Cl AO has an energy of E(Cl)=−12.96764 eV, the C aryl ,2sp 3  HO has an energy of E(C aryl ,2sp 3 )=−15.76868 eV (Eq. (14.246)), the C2sp 3  HO has an energy of E(C,2sp 3 )=−14.63489 eV (Eq. (15.25)), the Coulomb energy of Ni3d4s HO is E Coulomb (Ni,3d4s)=−18.41016 eV (Eq. (23.156)), and the Ni3d4s HO has an energy of E(Ni,3d4s)=−19.30374 eV (Eq. (23.162)). To meet the equipotential condition of the union of the Ni-L H 2 -type-ellipsoidal-MO with these orbitals, the hybridization factor(s), at least one of c 2  and C 2  of Eq. (15.61) for the Ni-L-bond MO given by Eq. (15.77) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               ClAO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Ni 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                   
                               
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               ClAO 
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 Ni 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               12.96764 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               19.30374 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.67177 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.168 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               C 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Ni 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 C 
                                 , 
                                 
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             
                               E 
                               Coulomb 
                             
                              
                             
                               ( 
                               
                                 Ni 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               C 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 - 
                                 14.63489 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             
                               
                                 - 
                                 18.41016 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                           
                            
                           
                             ( 
                             0.91771 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.72952 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.169 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 C 
                                 aryl 
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Ni 
                                
                               
                                   
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                 s 
                                  
                                 HO 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 C 
                                 , 
                                 
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             
                               E 
                               Coulomb 
                             
                              
                             
                               ( 
                               
                                 Ni 
                                 , 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 C 
                                 aryl 
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               14.63489 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               18.41016 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           ( 
                           0.85252 
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.67770 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.170 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Eqs. (15.76), (15.79), and (13.430) were used in Eq. (23.169) and Eqs. (15.76), (15.79), and (14.417) were used in Eq. (23.170). Since the energy of the MO is matched to that of the Ni3d4s HO in coordinate compounds, E(AO/HO) in Eq. (15.61) is E(Ni,3d4s) given by Eq. (23.162) and E(AO/HO) in Eq. (15.61) of carbonyl compounds and organometallics is E Coulomb (Ni,3d4s) given by Eq. (23.156). E T (atom−atom,msp 3 .AO) of the Ni-L-bond MO is determined by considering that the bond involves an electron transfer from the nickel atom to the ligand atom to form partial ionic character in the bond as in the case of the zwitterions such as H 2 B + —F −  given in the Halido Boranes section. For the coordinate compounds, E T (atom−atom,msp 3 .AO) is −1.11386 eV, two times the energy of Eq. (23.167). For the Ni—C bonds of carbonyl compound, Ni (CO) 4  and organometallic, nickelocene, E T (atom−atom,msp 3 .AO) is −1.85837 eV (two times Eq. (14.513)) and −0.92918 eV (Eq. (14.513)), respectively. The C═O functional group of Ni(CO) 4  is equivalent to that of vanadium carbonyls. The aromatic cyclopentadienyl moieties of organometallic Ni (C 5 H 5 ) 2  comprise C 3e ═C and CH functional groups that are equivalent to those given in the Aromatic and Heterocyclic Compounds section. 
         [0543]    The symbols of the functional groups of nickel coordinate compounds are given in Table 23.45. The geometrical (Eqs. (15.1-15.5) and (23.41)), intercept (Eqs. (15.31-15.32) and (15.80-15.87)), and energy (Eqs. (15.61) and (23.28-23.33)) parameters of nickel coordinate compounds are given in Tables 23.46, 23.47, and 23.48, respectively (all as shown in the priority document). The total energy of each nickel coordinate compounds given in Table 23.49 was calculated as the sum over the integer multiple of each E D  (Group) of Table 23.48 (as shown in the priority document) corresponding to functional-group composition of the compound. The charge-densities of exemplary nickel carbonyl and organometallic compounds, nickel tetracarbonyl (Ni (CO) 4 ) and bis-cylopentadienyl nickel or nickelocene (Ni (C 5 H 5 ) 2 ) comprising the concentric shells of atoms with the outer shell bridged by one or more H 2 -type ellipsoidal MOs or joined with one or more hydrogen MOs are shown in  FIGS. 50 and 51 , respectively. 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 23.45 
               
             
             
               
                   
               
               
                 The symbols of the functional groups of nickel coordinate compounds. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
               
               
                   
                 NiCl group of NiCl 
                 Ni—Cl (a) 
               
               
                   
                 NiCl 2  group of NiCl 2   
                 Ni—Cl (b) 
               
               
                   
                 NiCO group of Ni(CO) 4   
                 Ni—CO 
               
               
                   
                 C═O 
                 C═O 
               
               
                   
                 NiC aryl  group of Ni(C 5 H 5 ) 2   
                 Ni—C 5 H 5   
               
               
                   
                 CC (aromatic bond) 
                 C 3e ═C 
               
               
                   
                 CH (aromatic) 
                 CH 
               
               
                   
               
             
          
         
       
     
       Copper Functional Groups and Molecules 
       [0544]    The electron configuration of copper is [Ar]4s′3d 10  having the corresponding term  2 S 1/2 . The single outer 4s [61] electron having an energy of −7.72638 eV [1] forms a single bond to give an electron configuration with filled 3d and 4s shells. Additional bonding of copper is possible involving a double bond or two single bonds by the hybridization of the 3d and 4s shells to form a Cu3d4s shell and the donation of an electron per bond. The total energy of the copper  2 S 1/2  state is given by the sum over the eleven electrons. The sum E T (Cu, 3d4s) of experimental energies [1] of Cu, Cu + , Cu 2+ , Cu 3+ , Cu 4+ , Cu 5+ , Cu 6+ , Cu 7+ , Cu 8+ , Cu 9+ , and Cu 10+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               Cu 
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       265.3 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       232 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       199 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       166 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       139 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       103 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       79.8 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       57.38 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       36.841 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       20.2924 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       7.72638 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             1306.33978 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.171 
                   ) 
                 
               
             
           
         
       
     
         [0000]    By considering that the central field decreases by an integer for each successive electron of the shell, the radius r 3d4s  of the Cu3d4s shell may be calculated from the Coulombic energy using Eq. (15.13): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             3 
                              
                             d 
                              
                             
                                 
                             
                              
                             4 
                              
                             s 
                           
                         
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               18 
                             
                             28 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   Z 
                                   - 
                                   n 
                                 
                                 ) 
                               
                                
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     1306.33978 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             66 
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   1306.33978 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.68740 
                            
                           
                               
                           
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.172 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Z=29 for copper. Using Eq. (15.14), the Coulombic energy E Coulomb (Cu, 3d4s) of the outer electron of the Cu3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               Cu 
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             0.68740 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             19.793027 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.173 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, the unpaired 4s electron and five sets of spin-paired 3d electrons are promoted to Cu3d4s shell as initially unpaired electrons. The energies for the promotions of the initially paired electrons are given by Eq. (15.15) at the initial radius of the 3d electrons. From Eq. (10.102) with Z=29 and n=28, the radius r 28  of Cu3d shell is 
         [0000]      r 28 =1.34098a 0   (23.174)
 
         [0000]    Using Eqs. (15.15), and (23.174), the unpairing energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               3 
                                
                               d 
                             
                           
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             2 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                                
                               2 
                             
                              
                             
                               ℏ 
                               2 
                             
                           
                           
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   28 
                                 
                                 ) 
                               
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             8 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                               μ 
                               B 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 1.34098 
                                  
                                 
                                     
                                 
                                  
                                 
                                   a 
                                   0 
                                 
                               
                               ) 
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.04745 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.175 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The electrons from the 4s and 3d shells successively fill unoccupied HOs until the HO shell is filled with unpaired electrons, then the electrons pair per HO. In the case of the Cu3d4s shell having eleven electrons and six orbitals, five sets of electrons are paired. Using Eqs. (15.15) and (23.172), the paring energy is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               s 
                             
                           
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           - 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     
                                       3 
                                        
                                       d 
                                        
                                       
                                           
                                       
                                        
                                       4 
                                        
                                       s 
                                     
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           - 
                           
                             
                               8 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 μ 
                                 B 
                                 2 
                               
                             
                             
                               
                                 ( 
                                 
                                   0.68740 
                                    
                                   
                                       
                                   
                                    
                                   
                                     a 
                                     0 
                                   
                                 
                                 ) 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             0.35223 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.176 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, after Eq. (23.28), the energy E(Cu,3d4s) of the outer electron of the Cu3d4s shell is given by adding the magnetic energies of unpairing five sets of 3d electrons (Eq. (23.175)) and paring of five sets of Cu3d4s electrons (Eq. (23.176)) to E Coulomb (Cu,3d4s) (Eq. (23.173)): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               Cu 
                               , 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   3 
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   s 
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 r 
                                 
                                   4 
                                    
                                   s 
                                 
                                 3 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 3 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 pairs 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 2 
                                  
                                 
                                   πμ 
                                   0 
                                 
                                  
                                 
                                    
                                   2 
                                 
                                  
                                 
                                   ℏ 
                                   2 
                                 
                               
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   r 
                                   
                                     3 
                                      
                                     d 
                                   
                                   3 
                                 
                               
                             
                           
                           - 
                           
                             
                               ∑ 
                               
                                 HO 
                                  
                                 
                                     
                                 
                                  
                                 pairs 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 2 
                                  
                                 
                                   πμ 
                                   0 
                                 
                                  
                                 
                                    
                                   2 
                                 
                                  
                                 
                                   ℏ 
                                   2 
                                 
                               
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   r 
                                   
                                     3 
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     s 
                                   
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               19.793027 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0 
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             5 
                              
                             
                               ( 
                               
                                 0.04745 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                           - 
                           
                             5 
                              
                             
                               ( 
                               
                                 0.35223 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             21.31697 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.177 
                   ) 
                 
               
             
           
         
       
     
         [0545]    Next, consider the formation of the Cu-L-bond MO of wherein each copper atom has an Cu3d4s electron with an energy given by Eq. (23.177). The total energy of the state of each copper atom is given by the sum over the eleven electrons. The sum E T (C Cu-L  3d4s) of energies of Cu3d4s (Eq. (23.177)), Cu + , Cu 2+ , Cu 3+ , Cu 4+ , Cu 5+ , Cu 6+ , Cu 7+ , Cu 8+ , Cu 9+ , and Cu 10+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Cu 
                                 
                                   Cu 
                                   - 
                                   L 
                                 
                               
                                
                               3 
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               s 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       265.3 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       232 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       199 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       166 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       139 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       103 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       79.8 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       57.38 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       36.841 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       20.2924 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       E 
                                        
                                       
                                         ( 
                                         
                                           Cu 
                                           , 
                                           
                                             3 
                                              
                                             d 
                                              
                                             
                                                 
                                             
                                              
                                             4 
                                              
                                             s 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       265.3 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       232 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       199 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       166 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       139 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       103 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       79.8 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       57.38 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       36.841 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       20.2924 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       21.31697 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             1319.93037 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.178 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E(Cu,3d4s) is the sum of the energy of Cu, −7.72638 eV, and the hybridization energy. 
         [0546]    The copper HO donates an electron to each MO. Using Eq. (23.30), the radius r 3d4s  of the Cu3d4s shell calculated from the Coulombic energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             Cu 
                             - 
                             
                               L 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                   
                               
                                
                               s 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     18 
                                   
                                   28 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               1 
                             
                             ) 
                           
                            
                           
                             
                                
                               2 
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     1319.93037 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             65 
                              
                             
                                 
                             
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   1319.93037 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.67002 
                            
                           
                               
                           
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.179 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (23.179), the Coulombic energy E Coulomb (C Cu-L ,3d 4s) of the outer electron of the Cu3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 Cu 
                                 
                                   Cu 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 Cu 
                                 - 
                                 
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             0.67002 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             20.30662 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.180 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The magnetic energy terms are those for unpairing of the five sets of 3d electrons (Eq. (23.175)) and paring of five sets of Cu3d4s electrons (Eq. (23.176)). Using Eqs. (23.32), (23.180), and (23.175-23.176), the energy E(Cu Cu-L ,3d4s) of the outer electron of the Cu3d4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 Cu 
                                 
                                   Cu 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   Cu 
                                   - 
                                   
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                      
                                     
                                         
                                     
                                      
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     
                                         
                                     
                                      
                                     s 
                                   
                                 
                               
                             
                           
                           + 
                           
                             0 
                              
                             
                               
                                 2 
                                  
                                 
                                   πμ 
                                   0 
                                 
                                  
                                 
                                    
                                   2 
                                 
                                  
                                 
                                   ℏ 
                                   2 
                                 
                               
                               
                                 
                                   
                                     m 
                                     e 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       r 
                                       29 
                                     
                                     ) 
                                   
                                 
                                 3 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             5 
                              
                             
                               
                                 2 
                                  
                                 
                                   πμ 
                                   0 
                                 
                                  
                                 
                                    
                                   2 
                                 
                                  
                                 
                                   ℏ 
                                   2 
                                 
                               
                               
                                 
                                   
                                     m 
                                     e 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       r 
                                       28 
                                     
                                     ) 
                                   
                                 
                                 3 
                               
                             
                           
                           - 
                           
                             5 
                              
                             
                               
                                 2 
                                  
                                 
                                   πμ 
                                   0 
                                 
                                  
                                 
                                    
                                   2 
                                 
                                  
                                 
                                   ℏ 
                                   2 
                                 
                               
                               
                                 
                                   
                                     m 
                                     e 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       r 
                                       
                                         3 
                                          
                                         
                                             
                                         
                                          
                                         d 
                                          
                                         
                                             
                                         
                                          
                                         4 
                                          
                                         
                                             
                                         
                                          
                                         s 
                                       
                                     
                                     ) 
                                   
                                 
                                 3 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               20.30662 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0 
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             5 
                              
                             
                               ( 
                               
                                 0.04745 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                           - 
                           
                             5 
                              
                             
                               ( 
                               
                                 0.35223 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             21.83056 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.181 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (Cu-L,3d4s), the energy change of each Cu3d4s shell with the formation of the Cu-L-bond MO is given by the difference between Eq. (23.181) and Eq. (23.177): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Cu 
                                 - 
                                 L 
                               
                               , 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   Cu 
                                   
                                     Cu 
                                     - 
                                     L 
                                   
                                 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 Cu 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               21.83056 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 21.31697 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             0.51359 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.182 
                   ) 
                 
               
             
           
         
       
     
         [0547]    The semimajor axis a solution given by Eq. (23.41) of the force balance equation, Eq. (23.39), for the σ-MO of the Cu-L-bond MO of CuL n  is given in Table 23.51 with the force-equation parameters Z=29, n e  and L corresponding to the orbital and spin angular momentum terms of the 3d4s HO shell. 
         [0548]    For the Cu-L functional groups, hybridization of the 4s and 3d AOs of Cu to form a single 3d4s shell forms an energy minimum, and the sharing of electrons between the Cu3d4s HO and L AO to form σ MO permits each participating orbital to decrease in radius and energy. The F AO has an energy of E(F)=−17.42282 eV, the Cl AO has an energy of E(Cl)=−12.96764 eV, the O AO has an energy of E(O)=−13.61805 eV, the Cu AO has an energy of E(Cu)=−7.72638 eV, and the Cu3d4s HO has an energy of E(Cu, 3d4s)=−21.31697 eV (Eq. (23.177)). To meet the equipotential condition of the union of the Cu-L H 2 -type-ellipsoidal-MO with these orbitals, the hybridization factor(s), at least one of c 2  and C 2  of Eq. (15.61) for the Cu-L-bond MO given by Eq. (15.77) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               F 
                                
                               
                                   
                               
                                
                               A 
                                
                               
                                   
                               
                                
                               O 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               CuAO 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               CuAO 
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 F 
                                  
                                 
                                     
                                 
                                  
                                 A 
                                  
                                 
                                     
                                 
                                  
                                 O 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               7.72638 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               17.42282 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.44346 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.183 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             
                               c 
                               2 
                             
                              
                             
                               ( 
                               
                                 ClAO 
                                  
                                 
                                     
                                 
                                  
                                 to 
                                  
                                 
                                     
                                 
                                  
                                 CuAO 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               C 
                               2 
                             
                              
                             
                               ( 
                               
                                 ClAO 
                                  
                                 
                                     
                                 
                                  
                                 to 
                                  
                                 
                                     
                                 
                                  
                                 CuAO 
                               
                               ) 
                             
                           
                         
                          
                         
                             
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               CuAO 
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               ClAO 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               7.72638 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               12.96764 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.59582 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.184 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               F 
                                
                               
                                   
                               
                                
                               A 
                                
                               
                                   
                               
                                
                               O 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Cu 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                   
                               
                                
                               sHO 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 F 
                                  
                                 
                                     
                                 
                                  
                                 A 
                                  
                                 
                                     
                                 
                                  
                                 O 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 Cu 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               17.42282 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               21.31697 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.81732 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.185 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               O 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Cu 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                   
                               
                                
                               d 
                                
                               
                                   
                               
                                
                               4 
                                
                               
                                   
                               
                                
                               sHO 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             E 
                              
                             
                               ( 
                               O 
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 Cu 
                                 , 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               13.61805 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               21.31697 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.63884 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.186 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Since the energy of the MO is matched to that of the Cu3d 4s HO in coordinate compounds, E(AO/HO) in Eq. (15.61) is E(Cu, 3d 4s) given by Eq. (23.177) and twice this value for double bonds. E T (atom−atom,msp 3 .AO) of the Cu-L-bond MO is determined by considering that the bond involves an electron transfer from the copper atom to the ligand atom to form partial ionic character in the bond as in the case of the zwitterions such as H 2 B + —F −  given in the Halido Boranes section. For the two-bond coordinate compounds, E T (atom−atom,msp 3 .AO) is −1.02719 eV, two times the energy of Eq. (23.182). 
         [0549]    The symbols of the functional groups of copper coordinate compounds are given in Table 23.50. The geometrical (Eqs. (15.1-15.5) and (23.41)), intercept (Eqs. (15.31-15.32) and (15.80-15.87)), and energy (Eqs. (15.61) and (23.28-23.33)) parameters of copper coordinate compounds are given in Tables 23.51, 23.52, and 23.53 (all as shown in the priority document), respectively. The total energy of each copper coordinate compounds given in Table 23.54 was calculated as the sum over the integer multiple of each E D  (Group) of Table 23.53 (as shown in the priority document) corresponding to functional-group composition of the compound. The charge-densities of exemplary copper coordinate compounds, copper chloride (CuCl) and copper dichloride (CuCl 2 ) comprising the concentric shells of atoms with the outer shell bridged by one or more H 2 -type ellipsoidal MOs or joined with one or more hydrogen MOs are shown in  FIGS. 52 and 53 , respectively. 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 23.50 
               
             
             
               
                   
               
               
                 The symbols of the functional groups of copper coordinate 
               
               
                 compounds. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
               
               
                   
                 CuF group of CuF 
                 Cu—F (a) 
               
               
                   
                 CuF 2  group of CuF 2   
                 Cu—F (b) 
               
               
                   
                 CuCl group of CuCl 
                 Cu—Cl 
               
               
                   
                 CuO group of CuO 
                 Cu—O 
               
               
                   
               
             
          
         
       
     
       Zinc Functional Groups and Molecules 
       [0550]    The electron configuration of zinc is [Ar]4s 2 3d 10  having the corresponding term  1 S 0 . The two outer 4s [61] electrons having energies of −9.394199 eV and −17.96439 eV [1] hybridize to form a single shell comprising two HOs. Each HO donates an electron to any single bond that participates in bonding with the HO such that two single bonds with ligands are possible to achieve a filled, spin-paired outer electron shell. Then, the total energy of the  1 S 0  state of the bonding zinc atom is given by the sum over the two electrons. The sum E T (Zn,4sHO) of experimental energies [1] of Zn, and Zn + , is 
         [0000]      E T (Zn,4 s HO)=−(17.96439 eV+9.394199 eV)=−27.35859 eV  (23.187)
 
         [0000]    By considering that the central field decreases by an integer for each successive electron of the shell, the radius r 4sHO  of the Zn4s HO shell may be calculated from the Coulombic energy using Eq. (15.13): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             4 
                              
                             
                                 
                             
                              
                             sHO 
                           
                         
                         = 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               28 
                             
                             29 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   Z 
                                   - 
                                   n 
                                 
                                 ) 
                               
                                
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     27 
                                      
                                     .35859 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             3 
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   27 
                                    
                                   .35859 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1.49194 
                            
                           
                               
                           
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.188 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Z=30 for zinc. Using Eq. (15.14), the Coulombic energy E Coulomb (Zn,4sHO) of the outer electron of the Zn4s shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               Zn 
                               , 
                               
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 sHO 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 sHO 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.49194 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             9.119530 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.189 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, the spin-paired 4s AO electrons are promoted to Zn4s HO shell as unpaired electrons. The energy for the promotion is given by Eq. (15.15) at the initial radius of the 4s electrons. From Eq. (10.102) with Z=30 and n=30, the radius r 30  of Zn4s AO shell is 
         [0000]      r 30 =1.44832a 0   (23.190)
 
         [0000]    Using Eqs. (15.15) and (23.190), the unpairing energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             
                               4 
                                
                               
                                   
                               
                                
                               s 
                             
                           
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                         
                           
                             2 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                                
                               2 
                             
                              
                             
                               ℏ 
                               2 
                             
                           
                           
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   30 
                                 
                                 ) 
                               
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             8 
                              
                             
                               πμ 
                               o 
                             
                              
                             
                               μ 
                               B 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 1.44832 
                                  
                                 
                                     
                                 
                                  
                                 
                                   a 
                                   0 
                                 
                               
                               ) 
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           0.03766 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.191 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (23.189) and (23.191), the energy E(Zn, 4sHO) of the outer electron of the Zn4s HO shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               Zn 
                               , 
                               
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 sHO 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   sHO 
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     30 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               9.119530 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.03766 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             9.08187 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.192 
                   ) 
                 
               
             
           
         
       
     
         [0551]    Next, consider the formation of the Zn-L-bond MO wherein each zinc atom has a Zn4sHO electron with an energy given by Eq. (23.192). The total energy of the state of each zinc atom is given by the sum over the two electrons. The sum E T (Zn Zn-L  4sHO) of energies of Zn4sHO (Eq. (23.192)) and Zn +  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Zn 
                                 
                                   Zn 
                                   - 
                                   L 
                                 
                               
                                
                               4 
                                
                               
                                   
                               
                                
                               sHO 
                             
                             ) 
                           
                         
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 17.96439 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               + 
                               
                                 E 
                                  
                                 
                                   ( 
                                   
                                     Zn 
                                     , 
                                     
                                       4 
                                        
                                       
                                           
                                       
                                        
                                       sHO 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           - 
                           
                             ( 
                             
                               
                                 17.96439 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               + 
                               
                                 9.08187 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             27.04626 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.193 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E(Zn,4 s HO) is the sum of the energy of Zn, −9.394199 eV eV, and the hybridization energy. 
         [0552]    The zinc HO donates an electron to each MO. Using Eq. (23.30), the radius r 4sHO  of the Zn4sHO shell calculated from the Coulombic energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             Zn 
                             - 
                             
                               L 
                                
                               
                                   
                               
                                
                               4 
                                
                               sHO 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     28 
                                   
                                   29 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               1 
                             
                             ) 
                           
                            
                           
                             
                                
                               2 
                             
                             
                               8 
                                
                               
                                 
                                   πɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                      
                                      
                                     
                                         
                                     
                                      
                                     27.04626 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   27.04626 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           1.00611 
                            
                           
                               
                           
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.194 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (23.194), the Coulombic energy E Coulomb (Zn Zn-L ,4sHO) of the outer electron of the Zn4sHO shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 Zn 
                                 
                                   Zn 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 sHO 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 Zn 
                                 - 
                                 
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   sHO 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                                
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.00611 
                              
                             
                                 
                             
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             13.52313 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.195 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, the spin-paired 2s electrons are promoted to Zn4sHO shell as unpaired electrons. The energy for the promotion is the magnetic energy given by Eq. (23.191). Using Eqs. (23.195) and (23.191), the energy E(Zn Zn-L ,4s HO) of the outer electron of the Zn4sHO shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 Zn 
                                 
                                   Zn 
                                   - 
                                   L 
                                 
                               
                                
                               4 
                                
                               
                                   
                               
                                
                               sHO 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               - 
                               
                                  
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   Zn 
                                   - 
                                   
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     4 
                                      
                                     
                                         
                                     
                                      
                                     sHO 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                  
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     30 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               13.52313 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.03766 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             13.48547 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.196 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (Zn-L, 4s HO), the energy change of each Zn4sHO shell with the formation of the Zn-L-bond MO is given by the difference between Eq. (23.196) and Eq. (23.192): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Zn 
                                 - 
                                 L 
                               
                               , 
                               
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 sHO 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   Zn 
                                   
                                     Zn 
                                     - 
                                     L 
                                   
                                 
                                 , 
                                 
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   sHO 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 Zn 
                                 , 
                                 
                                   4 
                                    
                                   
                                       
                                   
                                    
                                   sHO 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               13.48547 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                           
                             ( 
                             
                               
                                 - 
                                 9.08187 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             4.40360 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.197 
                   ) 
                 
               
             
           
         
       
     
         [0553]    The semimajor axis a solution given by Eq. (23.41) of the force balance equation, Eq. (23.39), for the σ-MO of the Zn-L-bond MO of ZnL n , is given in Table 23.56 (as shown in the priority document) with the force-equation parameters Z=30, n e  and L corresponding to the orbital and spin angular momentum terms of the 4s HO shell. The semimajor axis a of organometallic compounds are solved using Eq. (15.51). 
         [0554]    For the Zn-L functional groups, hybridization of the 4s AOs of Zn to form a single 4s HO shell forms an energy minimum, and the sharing of electrons between the Zn4s HO and L AO to form σ MO permits each participating orbital to decrease in radius and energy. The Cl AO has an energy of E(Cl)=−12.96764 eV, the C2sp 3  HO has an energy of E(C,2sp 3 )=−14.63489 eV (Eq. (15.25)), the Coulomb energy of the Zn4s HO is E Coulomb (Zn,4sHO)=−9.119530 eV (Eq. (23.189)), and the Zn4s HO has an energy of E(Zn, 4sHO)=−9.08187 eV (Eq. (23.192)). To meet the equipotential condition of the union of the Zn-L H 2 -type-ellipsoidal-MO with these orbitals, the hybridization factor(s), at least one of c 2  and C 2  of Eq. (15.61) for the Zn-L-bond MO given by Eq. (15.77) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                             
                         
                          
                         
                           
                             
                               C 
                               2 
                             
                              
                             
                               ( 
                               
                                 ClAO 
                                  
                                 
                                     
                                 
                                  
                                 to 
                                  
                                 
                                     
                                 
                                  
                                 Zn 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 sHO 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               E 
                                
                               
                                 ( 
                                 
                                   Zn 
                                   , 
                                   
                                     34 
                                      
                                     
                                         
                                     
                                      
                                     sHO 
                                   
                                 
                                 ) 
                               
                             
                             
                               E 
                                
                               
                                 ( 
                                 ClAO 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               - 
                               9.08187 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               12.96764 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.70035 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.198 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             
                               c 
                               2 
                             
                              
                             
                               ( 
                               
                                 C 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                                  
                                 HO 
                                  
                                 
                                     
                                 
                                  
                                 to 
                                  
                                 
                                     
                                 
                                  
                                 Zn 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 sHO 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               C 
                               2 
                             
                              
                             
                               ( 
                               
                                 C 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                                  
                                 HO 
                                  
                                 
                                     
                                 
                                  
                                 to 
                                  
                                 
                                     
                                 
                                  
                                 Zn 
                                  
                                 
                                     
                                 
                                  
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 sHO 
                               
                               ) 
                             
                           
                         
                          
                         
                             
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 E 
                                 Coulomb 
                               
                                
                               
                                 ( 
                                 
                                   Zn 
                                   , 
                                   
                                     4 
                                      
                                     
                                         
                                     
                                      
                                     sHO 
                                   
                                 
                                 ) 
                               
                             
                             
                               E 
                                
                               
                                 ( 
                                 
                                   C 
                                   , 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       sp 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                            
                           
                             
                               c 
                               2 
                             
                              
                             
                               ( 
                               
                                 C 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sp 
                                   3 
                                 
                                  
                                 HO 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 - 
                                 9.11953 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             
                               
                                 - 
                                 14.63489 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                           
                            
                           
                             ( 
                             0.91771 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         0.57186 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.199 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Eqs. (15.76), (15.79), and (13.430) were used in Eq. (23.199). Since the energy of the MO is matched to that of the Zn4sHO in coordinate compounds, E(AO/HO) in Eq. (15.61) is E(Zn,4sHO) given by Eq. (23.192) and E(Zn,4s HO) for organometallics is E Coulomb (Zn, 4sHO) given by Eq. (23.189). E T  (atom−atom,msp 3 .AO) of the Zn-L-bond MO is determined by considering that the bond involves an electron transfer from the zinc atom to the ligand atom to form partial ionic character in the bond as in the case of the zwitterions such as H 2 B + —F −  given in the Halido Boranes section. For the coordinate compounds, E T (atom−atom,msp 3 .AO) is −8.80720 eV, two times the energy of Eq. (23.197). 
         [0555]    The symbols of the functional groups of zinc coordinate compounds are given in Table 23.55. The geometrical (Eqs. (15.1-15.5) and (23.41)), intercept (Eqs. (15.31-15.32) and (15.80-15.87)), and energy (Eqs. (15.61) and (23.28-23.33)) parameters of zinc coordinate compounds are given in Tables 23.56, 23.57, and 23.58 (all as shown in the priority document), respectively (all as shown in the priority document). The total energy of each zinc coordinate compounds given in Table 23.59 (as shown in the priority document) was calculated as the sum over the integer multiple of each E D (Group) of Table 23.58 (as shown in the priority document) corresponding to functional-group composition of the compound. The charge-densities of exemplary zinc coordinate and organometallic compounds, zinc chloride (ZnCl) and di-n-butylzinc (Zn(C 4 H 9 ) 2 ) comprising the concentric shells of atoms with the outer shell bridged by one or more H 2 -type ellipsoidal MOs or joined with one or more hydrogen MOs are shown in  FIG. 54  as  55 , respectively. 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 23.55 
               
             
             
               
                   
               
               
                 The symbols of the functional groups of zinc coordinate compounds. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
               
               
                   
                 ZnCl group of ZnCl 
                 Zn—Cl (a) 
               
               
                   
                 ZnCl 2  group of ZnCl 2   
                 Zn—Cl (b) 
               
               
                   
                 ZnC alkyl  group of RZnR′ 
                 Zn—C 
               
               
                   
                 CH 3  group 
                 C—H (CH 3 ) 
               
               
                   
                 CH 2  group 
                 C—H (CH 2 ) 
               
               
                   
                 CC bond (n-C) 
                 C—C 
               
               
                   
               
             
          
         
       
     
       Tin Functional Groups and Molecules 
       [0556]    As in the cases of carbon and tin, the bonding in the tin atom involves four sp 3  hybridized orbitals formed from the 5p and 5s electrons of the outer shells. Sn—X X=halide,oxide, Sn—H, and Sn—Sn bonds form between Sn5sp 3  HOs and between a halide or oxide AO, a H1s AO, and a Sn5sp 3  HO, respectively to yield tin halides and oxides, stannanes, and distannes, respectively. The geometrical parameters of each Sn—X X=halide,oxide, Sn—H, and Sn—Sn functional group is solved from the force balance equation of the electrons of the corresponding σ-MO and the relationships between the prolate spheroidal axes. Then, the sum of the energies of the H 2 -type ellipsoidal MOs is matched to that of the Sn5sp 3  shell as in the case of the corresponding carbon and tin molecules. As in the case of the transition metals, the energy of each functional group is determined for the effect of the electron density donation from the each participating Sn5sp 3  HO and AO to the corresponding MO that maximizes the bond energy. 
         [0557]    The branched-chain alkyl stannanes and distannes, Sn m C n H 2(m+n)+2 , comprise at least a terminal methyl group (CH 3 ) and at least one Sn bound by a carbon-tin single bond comprising a C—Sn group, and may comprise methylene (CH 2 ), methylyne (CH), C—C, SnH n=1,2,3 , and Sn—Sn functional groups. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C—C bonds can be identified. The n-alkane C—C bond is the same as that of straight-chain alkanes. In addition, the C—C bonds within isopropyl ((CH 3 ) 2  CH) and t-butyl ((CH 3 ) 3 C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C—C bonds comprise functional groups. 
         [0558]    The Sn electron configuration is [Kr]5s 2 4d 10 5p 2 , and the orbital arrangement is 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       5 
                        
                       p 
                        
                       
                           
                       
                        
                       state 
                     
                      
                     
                       
 
                     
                      
                     
                       
                         
                           
                             ↑ 
                             1 
                           
                         
                         
                           
                             ↑ 
                             0 
                           
                         
                         
                           
                             
                                 
                             
                             
                               - 
                               1 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.200 
                   ) 
                 
               
             
           
         
       
     
         [0000]    corresponding to the ground state  3 P 0 . The energy of the carbon 5p shell is the negative of the ionization energy of the tin atom [1] given by 
         [0000]      E(Sn,5 p shell )=−E(ionization;Sn)=−7.34392 eV  (23.201)
 
         [0000]    The energy of tin is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243), but the atomic orbital may hybridize in order to achieve a bond at an energy minimum. After Eq. (13.422), the Sn5s atomic orbital (AO) combines with the Sn5p AOs to form a single Sn5sp 3  hybridized orbital (HO) with the orbital arrangement 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       5 
                        
                       
                         sp 
                         3 
                       
                        
                       
                           
                       
                        
                       state 
                     
                      
                     
                       
 
                     
                      
                     
                       
                         
                           
                             ↑ 
                             
                               0 
                               , 
                               0 
                             
                           
                         
                         
                           
                             ↑ 
                             
                               1 
                               , 
                               
                                 - 
                                 1 
                               
                             
                           
                         
                         
                           
                             ↑ 
                             
                               1 
                               , 
                               0 
                             
                           
                         
                         
                           
                             ↑ 
                             
                               1 
                               , 
                               1 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.202 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the quantum numbers (l, m l ) are below each electron. The total energy of the state is given by the sum over the four electrons. The sum E T (Sn, 4sp 3 ) of experimental energies [1] of Sn, Sn + , Sn 2+ , and Sn 3+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               Sn 
                               , 
                               
                                 5 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             40.73502 
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             30.50260 
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             14.6322 
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             7.34392 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           93.21374 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.203 
                   ) 
                 
               
             
           
         
       
     
         [0000]    By considering that the central field decreases by an integer for each successive electron of the shell, the radius r 5sp     3    of the Sn5sp 3  shell may be calculated from the Coulombic energy using Eq. (15.13): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             5 
                              
                             
                               sp 
                               3 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               46 
                             
                             49 
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   Z 
                                   - 
                                   n 
                                 
                                 ) 
                               
                                
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               π 
                                
                               
                                   
                               
                                
                               
                                 
                                   ɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     93.21374 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             10 
                              
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               
                                 ɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   93.21374 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           1.45964 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.204 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Z=50 for tin. Using Eq. (15.14), the Coulombic energy E Coulomb (Sn, 5sp 3 ) of the outer electron of the Sn5sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               Sn 
                               , 
                               
                                 5 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 5 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             1.45964 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             9.321374 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.205 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, the spin-paired 5s electrons are promoted to Sn5sp 3  shell as unpaired electrons. The energy for the promotion is the magnetic energy given by Eq. (15.15) at the initial radius of the 5s electrons. From Eq. (10.255) with Z=50, the radius r 48  of Sn5s shell is 
         [0000]      r 48 =1.33816a 0   (23.206)
 
         [0000]    Using Eqs. (15.15) and (23.206), the unpairing energy is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             magnetic 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             2 
                              
                             
                               πμ 
                               0 
                             
                              
                             
                               e 
                               2 
                             
                              
                             
                               ℏ 
                               2 
                             
                           
                           
                             
                               
                                 m 
                                 e 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   48 
                                 
                                 ) 
                               
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             8 
                              
                             
                               πμ 
                               o 
                             
                              
                             
                               μ 
                               B 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 1.33816 
                                  
                                 
                                   a 
                                   0 
                                 
                               
                               ) 
                             
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           0.04775 
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.207 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (23.203) and (23.207), the energy E(Sn,5sp 3 ) of the outer electron of the Sn5sp3 shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               Sn 
                               , 
                               
                                 5 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   5 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 e 
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     48 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               9.321374 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.04775 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             9.27363 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.208 
                   ) 
                 
               
             
           
         
       
     
         [0559]    Next, consider the formation of the Sn-L-bond MO of tin compounds wherein L is a ligand including tin and each tin atom has a Sn5sp 3  electron with an energy given by Eq. (23.208). The total energy of the state of each tin atom is given by the sum over the four electrons. The sum E T (Sn Sn-L ,5sp 3 ) of energies of Sn5sp 3  (Eq. (23.208)), Sn + , Sn 2+ , and Sn 3+  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Sn 
                                 
                                   Sn 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 5 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       40.73502 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       30.50260 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       14.6322 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       E 
                                        
                                       
                                         ( 
                                         
                                           Sn 
                                           , 
                                           
                                             5 
                                              
                                             
                                               sp 
                                               3 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       40.73502 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       30.50560 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       14.6322 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     + 
                                     
                                       9.27363 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             95.14345 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.209 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E(Sn,5sp 3 ) is the sum of the energy of Sn, −7.34392 eV, and the hybridization energy. 
         [0560]    A minimum energy is achieved while matching the potential, kinetic, and orbital energy relationships given in the Hydroxyl Radical (OH) section with the donation of electron density from the participating Sn5sp 3  HO to each Sn-L-bond MO. As in the case of acetylene given in the Acetylene Molecule section, the energy of each Sn-L functional group is determined for the effect of the charge donation. For example, as in the case of the Si—Si-bond MO given in the Alkyl Silanes and Disilanes section, the sharing of electrons between two Sn5sp 3  HOs to form a Sn—Sn-bond MO permits each participating orbital to decrease in size and energy. In order to further satisfy the potential, kinetic, and orbital energy relationships, each Sn5sp 3  HO donates an excess of 25% of its electron density to the Sn—Sn-bond MO to form an energy minimum. By considering this electron redistribution in the distannane molecule as well as the fact that the central field decreases by an integer for each successive electron of the shell, in general terms, the radius r Sn-L5s     3   , of the Sn5sp 3  shell may be calculated from the Coulombic energy using Eq. (15.18): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             Sn 
                             - 
                             
                               L 
                                
                               
                                   
                               
                                
                               5 
                                
                               
                                 sp 
                                 3 
                               
                             
                           
                         
                         = 
                           
                          
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     46 
                                   
                                   49 
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               0.25 
                             
                             ) 
                           
                            
                           
                             
                               e 
                               2 
                             
                             
                               8 
                                
                               π 
                                
                               
                                   
                               
                                
                               
                                 
                                   ɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     95.14345 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             9.75 
                              
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   95.14345 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           1.39428 
                            
                           
                             a 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.210 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (23.210), the Coulombic energy E Coulomb (Sn Sn-L ,5sp 3 ) of the outer electron of the Sn5sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 Sn 
                                 
                                   Sn 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 5 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 Sn 
                                 - 
                                 
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   5 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               ɛ 
                               0 
                             
                              
                             1.39428 
                              
                             
                               a 
                               0 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             9.75830 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.211 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, the spin-paired 5s electrons are promoted to Sn5sp 3  shell as unpaired electrons. The energy for the promotion is the magnetic energy given by Eq. (23.207). Using Eqs. (23.207) and (23.211), the energy E(Sn Sn-L ,5sp 3 ) of the outer electron of the Si3sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 Sn 
                                 
                                   Sn 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 5 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               π 
                                
                               
                                   
                               
                                
                               
                                 ɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   Sn 
                                   - 
                                   
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     5 
                                      
                                     
                                       sp 
                                       3 
                                     
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 e 
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     48 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               9.75830 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.04775 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             9.71056 
                           
                            
                           
                               
                           
                            
                           eV 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.212 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (Sn-L,5sp 3 ), the energy change of each Sn5sp 3  shell with the formation of the Sn-L-bond MO is given by the difference between Eq. (23.212) and Eq. (23.208): 
         [0000]      E T (Sn-L,5 sp   3 )=E(Sn Sn-L ,5 sp   3 )−E(Sn,5 sp   3 )=−0.43693 eV  (23.213)
 
         [0561]    Next, consider the formation of the Si-L-bond MO of additional functional groups wherein each tin atom contributes a Sn5sp 3  electron having the sum E T (Sn Sn-L ,5sp 3 ) of energies of Sn5sp 3  (Eq. (23.208)), Sn + , Sn 2+ , and Sn 3+  given by Eq. (23.209). Each Sn-L-bond MO of each functional group Si-L forms with the sharing of electrons between a Sn5sp 3  HO and a AO or HO of L, and the donation of electron density from the Sn5sp 3  HO to the Sn-L-bond MO permits the participating orbitals to decrease in size and energy. In order to further satisfy the potential, kinetic, and orbital energy relationships while forming an energy minimum, the permitted values of the excess fractional charge of its electron density that the Sn5sp 3  HO donates to the Si-L-bond MO given by Eq. (15.18) is (0.25); s=1,2,3,4 and linear combinations thereof. By considering this electron redistribution in the tin molecule as well as the fact that the central field decreases by an integer for each successive electron of the shell, the radius r Sn-L5sp     3    of the Sn5sp 3  shell may be calculated from the Coulombic energy using Eq. (15.18): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             Sn 
                             - 
                             
                               L 
                                
                               
                                   
                               
                                
                               5 
                                
                               
                                 sp 
                                 3 
                               
                             
                           
                         
                         = 
                           
                          
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     46 
                                   
                                   49 
                                 
                                  
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     n 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 s 
                                  
                                 
                                   ( 
                                   0.25 
                                   ) 
                                 
                               
                             
                             ) 
                           
                            
                           
                             
                               e 
                               2 
                             
                             
                               8 
                                
                               π 
                                
                               
                                   
                               
                                
                               
                                 
                                   ɛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     e 
                                      
                                     
                                         
                                     
                                      
                                     95.14345 
                                      
                                     
                                         
                                     
                                      
                                     eV 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ( 
                               
                                 10 
                                 - 
                                 
                                   s 
                                    
                                   
                                     ( 
                                     0.25 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                              
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               
                                 ɛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   95.14345 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.214 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.19) and (23.214), the Coulombic energy E Coulomb (Sn Sn-L ,5sp 3 ) of the outer electron of the Sn5sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             Coulomb 
                           
                            
                           
                             ( 
                             
                               
                                 Sn 
                                 
                                   Sn 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 5 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               r 
                               
                                 Sn 
                                 - 
                                 
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   5 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                               e 
                               2 
                             
                           
                           
                             8 
                              
                             
                               πɛ 
                               0 
                             
                              
                             
                               
                                 
                                   ( 
                                   
                                     10 
                                     - 
                                     
                                       s 
                                        
                                       
                                         ( 
                                         0.25 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   e 
                                   2 
                                 
                               
                               
                                 8 
                                  
                                 
                                   
                                     πɛ 
                                     0 
                                   
                                    
                                   
                                     ( 
                                     
                                       e 
                                        
                                       
                                           
                                       
                                        
                                       95.14345 
                                        
                                       
                                           
                                       
                                        
                                       eV 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             95.14345 
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             ( 
                             
                               10 
                               - 
                               
                                 s 
                                  
                                 
                                   ( 
                                   0.25 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.215 
                   ) 
                 
               
             
           
         
       
     
         [0000]    During hybridization, the spin-paired 5s electrons are promoted to Sn5sp 3  shell as unpaired electrons. The energy for the promotion is the magnetic energy given by Eq. (23.207). Using Eqs. (23.207) and (23.215), the energy E(Sn Sn-L , 5 sp 3 ) of the outer electron of the Si3sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             ( 
                             
                               
                                 Sn 
                                 
                                   Sn 
                                   - 
                                   L 
                                 
                               
                               , 
                               
                                 5 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                 e 
                                 2 
                               
                             
                             
                               8 
                                
                               
                                 πɛ 
                                 0 
                               
                                
                               
                                 r 
                                 
                                   Sn 
                                   - 
                                   
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     5 
                                      
                                     
                                       sp 
                                       3 
                                     
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               2 
                                
                               
                                 πμ 
                                 0 
                               
                                
                               
                                 e 
                                 2 
                               
                                
                               
                                 ℏ 
                                 2 
                               
                             
                             
                               
                                 
                                   m 
                                   e 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     48 
                                   
                                   ) 
                                 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               95.14345 
                                
                               
                                   
                               
                                
                               eV 
                             
                             
                               ( 
                               
                                 10 
                                 - 
                                 
                                   s 
                                    
                                   
                                     ( 
                                     0.25 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             0.04775 
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.216 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, E T (Sn-L,5sp 3 ), the energy change of each Sn5sp 3  shell with the formation of the Sn-L-bond MO is given by the difference between Eq. (23.216) and Eq. (23.208): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 Sn 
                                 - 
                                 L 
                               
                               , 
                               
                                 5 
                                  
                                 
                                   sp 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   Sn 
                                   
                                     Sn 
                                     - 
                                     L 
                                   
                                 
                                 , 
                                 
                                   5 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             E 
                              
                             
                               ( 
                               
                                 Sn 
                                 , 
                                 
                                   5 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                 95.14345 
                                 
                                   ( 
                                   
                                     10 
                                     - 
                                     
                                       s 
                                        
                                       
                                         ( 
                                         0.25 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           + 
                           
                             0.04775 
                              
                             
                                 
                             
                              
                             eV 
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           ( 
                           
                             
                               - 
                               9.27363 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.217 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eq. (15.28) for the case that the energy matching and energy minimum conditions of the MOs in the tin molecule are met by a linear combination of values of s (s 1  and s 2 ) in Eqs. (23.214-23.217), the energy E(Sn Sn-L ,5sp 3 ) of the outer electron of the Si3sp 3  shell is 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       ( 
                       
                         
                           Sn 
                           
                             Sn 
                             - 
                             L 
                           
                         
                         , 
                         
                           5 
                            
                           
                             sp 
                             3 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               
                                 95.14345 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               
                                 ( 
                                 
                                   10 
                                   - 
                                   
                                     
                                       s 
                                       1 
                                     
                                      
                                     
                                       ( 
                                       0.25 
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 95.14345 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               
                                 ( 
                                 
                                   10 
                                   - 
                                   
                                     
                                       s 
                                       2 
                                     
                                      
                                     
                                       ( 
                                       0.25 
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             2 
                              
                             
                               ( 
                               
                                 0.04775 
                                  
                                 
                                     
                                 
                                  
                                 eV 
                               
                               ) 
                             
                           
                         
                       
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   23.218 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using Eqs. (15.13) and (23.218), the radius corresponding to Eq. (23.218) is: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           r 
                           
                             5 
                              
                             
                               sp 
                               3 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             e 
                             2 
                           
                           
                             8 
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               ɛ 
                               0 
                             
                              
                             
                               E 
                                
                               
                                 ( 
                                 
                                   
                                     Sn 
                                     
                                       Sn 
                                       - 
                                       L 
                                     
                                   
                                   , 
                                   
                                     5 
                                      
                                     
                                       sp 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             e 
                             2 
                           
                           
                             8 
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               
                                 ɛ 
                                 0 
                               
                               ( 
                               
                                 e 
                                 ( 
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               
                                                 95.14345 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 eV 
                                               
                                               
                                                 ( 
                                                 
                                                   10 
                                                   - 
                                                   
                                                     
                                                       s 
                                                       1 
                                                     
                                                      
                                                     
                                                       ( 
                                                       0.25 
                                                       ) 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                             + 
                                             
                                               
                                                 95.14345 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 eV 
                                               
                                               
                                                 ( 
                                                 
                                                   10 
                                                   - 
                                                   
                                                     
                                                       s 
                                                       2 
                                                     
                                                      
                                                     
                                                       ( 
                                                       0.25 
                                                       ) 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                       
                                         
                                           
                                             2 
                                              
                                             
                                               ( 
                                               
                                                 0.04775 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 eV 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                     
                                     + 
                                   
                                   2 
                                 
                                 ) 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.219 
                   ) 
                 
               
             
           
         
       
     
         [0000]    E T (Sn-L, 5sp 3 ), the energy change of each Sn5sp 3  shell with the formation of the Sn-L-bond MO is given by the difference between Eq. (23.219) and Eq. (23.208): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       T 
                     
                      
                     
                       ( 
                       
                         
                           Sn 
                           - 
                           L 
                         
                         , 
                         
                           5 
                            
                           
                               
                           
                            
                           
                             sp 
                             3 
                           
                         
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       E 
                        
                       
                         ( 
                         
                           
                             Sn 
                             
                               Sn 
                               - 
                               L 
                             
                           
                           , 
                           
                             5 
                              
                             
                               sp 
                               3 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       E 
                        
                       
                         ( 
                         
                           Sn 
                           , 
                           
                             5 
                              
                             
                                 
                             
                              
                             
                               sp 
                               3 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           
                             
                               
                                 
                                   95.14345 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 
                                   ( 
                                   
                                     10 
                                     - 
                                     
                                       
                                         s 
                                         1 
                                       
                                        
                                       
                                         ( 
                                         0.25 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   95.14345 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 
                                   ( 
                                   
                                     10 
                                     - 
                                     
                                       
                                         s 
                                         2 
                                       
                                        
                                       
                                         ( 
                                         0.25 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               2 
                                
                               
                                 ( 
                                 
                                   0.04775 
                                    
                                   
                                       
                                   
                                    
                                   eV 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       2 
                     
                     - 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     ( 
                     
                       
                         - 
                         9.27363 
                       
                        
                       
                           
                       
                        
                       eV 
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    E T (Sn-L,5sp 3 ) is also given by Eq. (15.29). Bonding parameters for Sn-L-bond MO of tin functional groups due to charge donation from the HO to the MO are given in Table 23.60. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 23.60 
               
             
             
               
                   
               
               
                 The values of r Sn5sp     3   , E Coulomb  (Sn Sn-L ,5sp 3 ), and E(Sn Sn-L ,5sp 3 ) and the 
               
               
                 resulting E T  (Sn-L,5sp 3 ) of the MO due to charge donation from 
               
               
                 the HO to the MO. 
               
             
          
           
               
                 MO 
                   
                   
                   
                 E Coulomb (Sn Sn-L ,5sp 3 ) 
                 E(Sn Sn-L ,5sp 3 ) 
                   
               
               
                 Bond 
                   
                   
                 r Sn5sp     3    (a 0 ) 
                 (eV) 
                 (eV) 
                 E T (Sn-L,5sp 3 ) 
               
               
                 Type 
                 s1 
                 s2 
                 Final 
                 Final 
                 Final 
                 (eV) 
               
               
                   
               
             
          
           
               
                 0 
                 0 
                 0 
                 1.45964 
                 −9.321374 
                 −9.27363 
                 0 
               
               
                 I 
                 1 
                 0 
                 1.39428 
                 −9.75830 
                 −9.71056 
                 −0.43693 
               
               
                 II 
                 2 
                 0 
                 1.35853 
                 −10.01510 
                 −9.96735 
                 −0.69373 
               
               
                 III 
                 3 
                 0 
                 1.32278 
                 −10.28578 
                 −10.23803 
                 −0.96440 
               
               
                 IV 
                 4 
                 0 
                 1.28703 
                 −10.57149 
                 −10.52375 
                 −1.25012 
               
               
                 I + II 
                 1 
                 2 
                 1.37617 
                 −9.88670 
                 −9.83895 
                 −0.56533 
               
               
                 II + III 
                 2 
                 3 
                 1.34042 
                 −10.15044 
                 −10.10269 
                 −0.82906 
               
               
                   
               
             
          
         
       
     
         [0562]    The semimajor axis a solution given by Eq. (23.41) of the force balance equation, Eq. (23.39), for the σ-MO of the Sn-L-bond MO of SnL n  is given in Table 23.62 (as shown in the priority document) with the force-equation parameters Z=50, n e , and L corresponding to the orbital and spin angular momentum terms of the 4s HO shell. The semimajor axis a of organometallic compounds, stannanes and distannes, are solved using Eq. (15.51). 
         [0563]    For the Sn-L functional groups, hybridization of the 5p and 5s AOs of Sn to form a single Sn5sp 3  HO shell forms an energy minimum, and the sharing of electrons between the Sn5sp 3  HO and L AO to form σ MO permits each participating orbital to decrease in radius and energy. The Cl AO has an energy of E(Cl)=−12.96764 eV, the Br AO has an energy of E(Br)=−11.8138 eV, the I AO has an energy of E(I)=−10.45126 eV, the O AO has an energy of E(O)=−13.61805 eV, the C2sp 3  HO has an energy of E(C,2sp 3 )=−14.63489 eV (Eq. (15.25)), 13.605804 eV is the magnitude of the Coulombic energy between the electron and proton of H (Eq. (1.243)), the Coulomb energy of the Sn5sp 3  HO is E Coulomb (Sn,5sp 3  HO=−9.32137 eV (Eq. (23.205)), and the Sn5sp 3  HO has an energy of E(Sn,5s, HO)=−9.27363 eV (Eq. (23.208)). To meet the equipotential condition of the union of the Sn-L H 2 -type-ellipsoidal-MO with these orbitals, the hybridization factor(s), at least one of c 2  and C 2  of Eq. (15.61) for the Sn-L-bond MO given by Eq. (15.77) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               Cl 
                                
                               
                                   
                               
                                
                               A 
                                
                               
                                   
                               
                                
                               O 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Sn 
                                
                               
                                   
                               
                                
                               5 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               Cl 
                                
                               
                                   
                               
                                
                               A 
                                
                               
                                   
                               
                                
                               O 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Sn 
                                
                               
                                   
                               
                                
                               5 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Sn 
                                 , 
                                 
                                   5 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 Cl 
                                  
                                 
                                     
                                 
                                  
                                 A 
                                  
                                 
                                     
                                 
                                  
                                 O 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               9.27363 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               12.96764 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.71514 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.221 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               Br 
                                
                               
                                   
                               
                                
                               A 
                                
                               
                                   
                               
                                
                               O 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Sn 
                                
                               
                                   
                               
                                
                               5 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Sn 
                                 , 
                                 
                                   5 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 Br 
                                  
                                 
                                     
                                 
                                  
                                 A 
                                  
                                 
                                     
                                 
                                  
                                 O 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               9.27363 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               11.8138 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.78498 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.222 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               I 
                                
                               
                                   
                               
                                
                               A 
                                
                               
                                   
                               
                                
                               O 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Sn 
                                
                               
                                   
                               
                                
                               5 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Sn 
                                 , 
                                 
                                   Sn 
                                    
                                   
                                       
                                   
                                    
                                   5 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 I 
                                  
                                 
                                     
                                 
                                  
                                 A 
                                  
                                 
                                     
                                 
                                  
                                 O 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               9.27363 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               10.45126 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.88732 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.223 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               O 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Sn 
                                
                               
                                   
                               
                                
                               5 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               O 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Sn 
                                
                               
                                   
                               
                                
                               5 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Sn 
                                 , 
                                 
                                   5 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               O 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               9.27363 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               13.61805 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.68098 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.224 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               H 
                                
                               
                                   
                               
                                
                               A 
                                
                               
                                   
                               
                                
                               O 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Sn 
                                
                               
                                   
                               
                                
                               5 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               E 
                               Coulomb 
                             
                              
                             
                               ( 
                               
                                 Sn 
                                 , 
                                 
                                   5 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               H 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               9.32137 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               13.605804 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.68510 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.225 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             C 
                             2 
                           
                            
                           
                             ( 
                             
                               C 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Sn 
                                
                               
                                   
                               
                                
                               5 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 Sn 
                                 , 
                                 
                                   5 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sp 
                                     3 
                                   
                                    
                                   HO 
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               
                                 C 
                                 , 
                                 
                                   2 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               C 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 - 
                                 9.27363 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                             
                               
                                 - 
                                 14.63489 
                               
                                
                               
                                   
                               
                                
                               eV 
                             
                           
                            
                           
                             ( 
                             0.91771 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.58152 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.226 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             2 
                           
                            
                           
                             ( 
                             
                               Sn 
                                
                               
                                   
                               
                                
                               5 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               Sn 
                                
                               
                                   
                               
                                
                               5 
                                
                               
                                 sp 
                                 3 
                               
                                
                               HO 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               E 
                               Coulomb 
                             
                              
                             
                               ( 
                               
                                 Sn 
                                 , 
                                 
                                     
                                 
                                  
                                 
                                   5 
                                    
                                   
                                     sp 
                                     3 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             E 
                              
                             
                               ( 
                               H 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               9.32137 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                           
                             
                               - 
                               13.605804 
                             
                              
                             
                                 
                             
                              
                             eV 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0.68510 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23.227 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Eq. (15.71) was used in Eqs. (23.225) and (23.227) and Eqs. (15.76), (15.79), and (13.430) were used in Eq. (23.226). Since the energy of the MO is matched to that of the Sn5sp 3  HO, E(AO/HO) in Eq. (15.61) is E(Sn,5sp 3  HO) given by Eq. (23.208) for single bonds and twice this value for double bonds. E T (atom−atom,msp 3 .A 0) of the Sn-L-bond MO is determined by considering that the bond involves up to an electron transfer from the tin atom to the ligand atom to form partial ionic character in the bond as in the case of the zwitterions such as H 2 B + —F −  given in the Halido Boranes section. For the tin compounds, E T (atom−atom,msp 3 .AO) is that which forms an energy minimum for the hybridization and other bond parameter. The general values of Table 23.60 are given by Eqs. (23.217) and (23.220), and the specific values for the tin functional groups are given in Table 23. 64. 
         [0564]    The symbols of the functional groups of tin compounds are given in Table 23.61. The geometrical (Eqs. (15.1-15.5) and (23.41)), intercept (Eqs. (15.31-15.32) and (15.80-15.87)), and energy (Eqs. (15.61) and (23.28-23.33)) parameters of tin compounds are given in Tables 23.62, 23.63, and 23.64, respectively (all as shown in the priority document). The total energy of each tin compounds given in Table 23.65 (as shown in the priority document) was calculated as the sum over the integer multiple of each E D  (Group) of Table 23.64 (as shown in the priority document) corresponding to functional-group composition of the compound. The bond angle parameters of tin compounds determined using Eqs. (15.88-15.117) are given in Table 23.66. The E T (atom−atom,msp 3 .AO) term for SnCl 4  was calculated using Eqs. (23.214-23.217) with s=1 for the energies of E(Sn,5sp 3 ). The charge-densities of exemplary tin coordinate and organometallic compounds, tin tetrachloride (5 nCl 4 ) and hexaphenyldistannane ((C 6 H 5 ) 3  SnSn(C 6 H 5 ) 3 ) comprising the concentric shells of atoms with the outer shell bridged by one or more H 2 -type ellipsoidal MOs or joined with one or more hydrogen MOs are shown in  FIG. 56  as  57 , respectively. 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 23.61 
               
             
             
               
                   
               
               
                 The symbols of functional groups of tin compounds. 
               
             
          
           
               
                   
                 Functional Group 
                 Group Symbol 
               
               
                   
                   
               
               
                   
                 SnCl group 
                 Sn—Cl 
               
               
                   
                 SnBr group 
                 Sn—Br 
               
               
                   
                 SnI group 
                 Sn—I 
               
               
                   
                 SnO group 
                 Sn—O 
               
               
                   
                 SnH group 
                 Sn—H 
               
               
                   
                 SnC group 
                 Sn—C 
               
               
                   
                 SnSn group 
                 Sn—Sn 
               
               
                   
                 CH 3  group 
                 C—H (CH 3 ) 
               
               
                   
                 CH 2  alkyl group 
                 C—H (CH 2 ) (i) 
               
               
                   
                 CH alkyl 
                 C—H (i) 
               
               
                   
                 CC bond (n-C) 
                 C—C (a) 
               
               
                   
                 CC bond (iso-C) 
                 C—C (b) 
               
               
                   
                 CC bond (tert-C) 
                 C—C (c) 
               
               
                   
                 CC (iso to iso-C) 
                 C—C (d) 
               
               
                   
                 CC (t to t-C) 
                 C—C (e) 
               
               
                   
                 CC (t to iso-C) 
                 C—C (f) 
               
               
                   
                 CC double bond 
                 C═C 
               
               
                   
                 C vinyl single bond to —C(C)═C 
                 C—C (i) 
               
               
                   
                 C vinyl single bond to —C(H)═C 
                 C—C (ii) 
               
               
                   
                 C vinyl single bond to —C(C)═CH 2   
                 C—C (iii) 
               
               
                   
                 CH 2  alkenyl group 
                 C—H (CH 2 ) (ii) 
               
               
                   
                   
               
               
                   
                 CC (aromatic bond) 
                 
                   
                     
                       
                         C 
                          
                         
                           
                             — 
                             — 
                           
                           
                               
                           
                           
                             3 
                              
                             e 
                           
                         
                          
                         C 
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                 CH (aromatic) 
                 CH (ii) 
               
               
                   
                 C a —C b  (CH 3  to aromatic bond) 
                 C—C (iv) 
               
               
                   
                 C—C(O) 
                 C—C(O) 
               
               
                   
                 C═O (aryl carboxylic acid) 
                 C═O 
               
               
                   
                 (O)C—O 
                 C—O 
               
               
                   
                 OH group 
                 OH 
               
               
                   
                   
               
             
          
         
       
     
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