Hückel method
The Hückel method or Hückel molecular orbital theory, proposed by Erich Hückel in 1930, is a simple method for calculating molecular orbitals as linear combinations of atomic orbitals. The theory predicts the molecular orbitals for π-electrons in π-delocalized molecules, such as ethylene, benzene, butadiene, and pyridine. It provides the theoretical basis for Hückel's rule that cyclic, planar molecules or ions with formula_1 π-electrons are aromatic. It was later extended to conjugated molecules such as pyridine, pyrrole and furan that contain atoms other than carbon and hydrogen (heteroatoms). A more dramatic extension of the method  to include σ-electrons, known as the extended Hückel method (EHM), was developed by Roald Hoffmann. The extended Hückel method gives some degree of quantitative accuracy for organic molecules in general (not just planar systems) and was used to provide computational justification for the Woodward–Hoffmann rules. To distinguish the original approach from Hoffmann's extension, the Hückel method is also known as the simple Hückel method (SHM). In spite of its simplicity, the Hückel method in its original form makes qualitatively accurate and chemically useful predictions for many common molecules and is therefore a powerful and widely taught educational  tool. It is described in many introductory quantum chemistry and physical organic chemistry textbooks, and organic chemists in particular still routinely apply Hückel theory to obtain a very approximate, back-of-the-envelope understanding of π-bonding. Hückel characteristics. The method has several characteristics: Hückel results. Results for simple molecules and general results for cyclic and linear systems. The results for a few simple molecules are tabulated below: The theory predicts two energy levels for ethylene with its two π electrons filling the low-energy HOMO and the high energy LUMO remaining empty. In butadiene the 4 π-electrons occupy  2 low energy molecular orbitals, out of a total of 4, and for benzene 6 energy levels are predicted, two of them degenerate. For linear and cyclic systems (with "N" atoms), general solutions exist: The energy levels for cyclic systems can be predicted using the mnemonic (named after the American chemist ). A circle centered at α with radius 2β is inscribed with a regular "N-"gon with one vertex pointing down; the "y"-coordinate of the vertices of the polygon then represent the orbital energies of the ["N"]annulene/annulenyl system. Related mnemonics exists for linear and Möbius systems. The values of α  and β. The value of α is the energy of an electron in a 2p orbital, relative to an unbound electron at infinity. This quantity is negative, since the electron is stabilized by being electrostatically bound to the positively charged nucleus. For carbon this value is known to be approximately –11.4 eV. Since Hückel theory is generally only interested in energies relative to a reference localized system, the value of α is often immaterial and can be set to zero without affecting any conclusions. Roughly speaking, β physically represents the energy of stabilization experienced by an electron allowed to delocalize  in a π molecular orbital formed from the 2p orbitals of adjacent atoms, compared to being localized in an isolated 2p atomic orbital. As such, it is also a negative number, although it is often spoken of in terms of its absolute value. The value for |β| in Hückel theory is roughly constant for structurally similar compounds, but not surprisingly, structurally dissimilar compounds will give very different values for |β|. For example, using the π bond energy of ethylene (65 kcal/mole) and comparing the energy of a doubly-occupied π orbital (2α + 2β) with the energy of electrons  in two isolated p orbitals (2α), a value of |β| = 32.5 kcal/mole can be inferred. On the other hand, using the resonance energy of benzene (36 kcal/mole, derived from heats of hydrogenation) and comparing benzene (6α + 8β) with a hypothetical "non-aromatic 1,3,5-cyclohexatriene" (6α + 6β), a much smaller value of |β| = 18 kcal/mole emerges. These differences are not surprising, given the substantially shorter bond length of ethylene (1.33 Å) compared to benzene (1.40 Å). The shorter distance between the interacting p orbitals accounts for the greater energy of interaction, which is reflected  by a higher value of |β|. Nevertheless, heat of hydrogenation measurements of various polycyclic aromatic hydrocarbons like naphthalene and anthracene all imply values of |β| between 17 and 20 kcal/mol. However, even for the same compound, the correct assignment of |β| can be controversial. For instance, it is argued that the resonance energy measured experimentally via heats of hydrogenation is diminished by the distortions in bond lengths that must take place going from the single and double bonds of "non-aromatic 1,3,5-cyclohexatriene" to the delocalized bonds of benzene. Taking this distortion energy into account, the value of |β|  for delocalization without geometric change (called the "vertical resonance energy") for benzene is found to be around 37 kcal/mole. On the other hand, experimental measurements of electronic spectra have given a value of |β| (called the "spectroscopic resonance energy") as high as 3 eV (~70 kcal/mole) for benzene. Given these subtleties, qualifications, and ambiguities, Hückel theory should not be called upon to provide accurate quantitative predictions – only semi-quantitative or qualitative trends and comparisons are reliable and robust. Other successful predictions. With this caveat in mind, many predictions of the theory have been experimentally verified: Application in Optical  Activity Analysis. The analysis of the optical activity of a molecule depends to a certain extent on the study of its chiral characteristics. However, for achiral molecules applying pesudoscalars to simplify the calculations of optical avtivity cannot be achieved due to the lack of spatial average. Instead of traditional chiroptical solution measurements, Hückel theory helps focus on oriented π systems by separating from σ electrons especially in the planar, formula_8-symmetric cases. Transition dipole moments derived by multiplying each wavefunction of individual planar molecule one by one, contribute to the directions of the most optical activity, where sit at  the bisectors of two orthogonal ones. Despite of the zero value for the trace of the tensor, cis-butadiene shows considerable offdiagonal component which was computated as the first optical avtivity evaluation of achiral molecule. By using 3,5-dimethylene-1-cyclopentene as an example. Transition electric dipole, magnetic dipole and electric quadrupole moments interactions result in optical rotation(OR), which can be described by both tensor components and chemical geometries. The in phase overlap of two molecular orbitals yield negative charge while depleting charge out of phase. The movement can be interpreted quantitatively by corresponding π and π* orbitals coefficients. Delocalization energy , π-bond orders, and π-electron populations. The delocalization energy, π-bond orders, and π-electron population are chemically significant parameters that can be gleaned from the orbital energies and coefficients that are the direct outputs of Hückel theory. These are quantities strictly derived from theory, as opposed to measurable physical properties, though they correlate with measurable qualitative and quantitative properties of the chemical species. Delocalization energy is defined as the difference in energy between that of the most stable localized Lewis structure and the energy of the molecule computed from Hückel theory orbital energies and occupancies. Since all energies  are relative, we set formula_9 without loss of generality to simplify discussion. The energy of the localized structure is then set to be 2β for every two-electron localized π-bond. The Hückel energy of the molecule is formula_10, where the sum is over all Hückel orbitals, formula_11 is the occupancy of orbital "i", set to be 2 for doubly-occupied orbitals, 1 for singly-occupied orbitals, and 0 for unoccupied orbitals, and formula_12 is the energy of orbital "i". Thus, the delocalization energy, conventionally a positive number, is defined as In the case of benzene, the occupied orbitals have  energies (again setting formula_9) 2β, β, and β. This gives the Hückel energy of benzene as formula_15. Each Kekulé structure of benzene has three double bonds, so the localized structure is assigned an energy of formula_16. The delocalization energy, measured in units of formula_17, is then formula_18. The π-bond orders derived from Hückel theory are defined using the orbital coefficients of the Hückel MOs. The π-bond order between atoms "j" and "k" is defined as where formula_11 is again the orbital occupancy of orbital "i" and formula_21 and formula_22 are the coefficients on atoms "j" and "k", respectively , for orbital "i". For benzene, the three occupied MOs, expressed as linear combinations of AOs formula_23, are: Perhaps surprisingly, the π-bond order formula gives a bond order of for the bond between carbons 1 and 2. The resulting total (σ + π) bond order of formula_31 is the same between any other pair of adjacent carbon atoms. This is more than the naive π-bond order of formula_32 (for a total bond order of formula_33) that one might guess when simply considering the Kekulé structures and the usual definition of bond order in valence bond theory. The Hückel definition  of bond order attempts to quantify any additional stabilization that the system enjoys resulting from delocalization. In a sense, the Hückel bond order suggests that there are four π-bonds in benzene instead of the three that are implied by the Kekulé-type Lewis structures. The "extra" bond is attributed to the additional stabilization that results from the aromaticity of the benzene molecule. (This is only one of several definitions for non-integral bond orders, and other definitions will lead to different values that fall between 1 and 2.) The π-electron population is calculated in a very similar way  to the bond order using the orbital coefficients of the Hückel MOs. The π-electron population on atom "j" is defined as The associated Hückel Coulomb charge is defined as formula_35, where formula_36 is the number of π-electrons contributed by a neutral, sp-hybridized atom "j" (we always have formula_37 for carbon). For carbon 1 on benzene, this yields a π-electron population of Since each carbon atom contributes one π-electron to the molecule, this gives a Coulomb charge of 0 for carbon 1 (and all other carbon atoms), as expected. In the cases of benzyl cation and  benzyl anion shown above, Mathematics behind the Hückel method. The mathematics of the Hückel method is based on the Ritz method. In short, given a basis set of "n" normalized atomic orbitals formula_43, an "ansatz" molecular orbital formula_44 is written down, with normalization constant "N" and coefficients formula_45 which are to be determined. In other words, we are assuming that the molecular orbital (MO) can be written as a linear combination of atomic orbitals, a conceptually intuitive and convenient approximation (the linear combination of atomic orbitals or LCAO approximation). The variational theorem states that given an eigenvalue problem formula_46 with  smallest eigenvalue formula_47 and corresponding wavefunction formula_48, any normalized trial wavefunction formula_49 (i.e., formula_50 holds) will satisfy with equality holding if and only if formula_52. Thus, by minimizing formula_53 with respect to coefficients formula_45 for normalized trial wavefunctions formula_55, we obtain a closer approximation of the true ground-state wavefunction and its energy. To start, we apply the normalization condition to the "ansatz" and expand to get an expression for "N" in terms of the formula_45. Then, we substitute the "ansatz" into the expression for "E" and expand, yielding In the remainder of the derivation, we will assume that the  atomic orbitals are real. (For the simple case of the Hückel theory, they will be the 2p"" orbitals on carbon.) Thus, formula_61, and because the Hamiltonian operator is hermitian, formula_62. Setting formula_63 for formula_64 to minimize "E" and collecting terms, we obtain a system of "n" simultaneous equations When formula_66, formula_67 and formula_68 are called the overlap and resonance (or exchange) integrals, respectively, while formula_69 is called the Coulomb integral, and formula_70 simply expresses that fact that the formula_23 are normalized. The "n" × "n" matrices formula_72 and formula_73 are known as the overlap and Hamiltonian matrices, respectively. By a  well-known result from linear algebra, nontrivial solutions formula_74 to the above system of linear equations can only exist if the coefficient matrix formula_75 is singular. Hence, formula_76 must have a value such that the determinant of the coefficient matrix vanishes: This determinant expression is known as the secular determinant and gives rise to a generalized eigenvalue problem. The variational theorem guarantees that the lowest value of formula_76 that gives rise to a nontrivial (that is, not all zero) solution vector formula_79 represents the best LCAO approximation of the energy of the most stable π orbital; higher values of formula_76  with nontrivial solution vectors represent reasonable estimates of the energies of the remaining π orbitals. The Hückel method makes a few further simplifying assumptions concerning the values of the formula_67 and formula_68. In particular, it is first assumed that distinct formula_23 have zero overlap. Together with the assumption that formula_23 are normalized, this means that the overlap matrix is the "n" × "n" identity matrix: formula_85. Solving for "E" in (*) then reduces to finding the eigenvalues of the Hamiltonian matrix. Second, in the simplest case of a planar, unsaturated hydrocarbon, the Hamiltonian matrix formula_86 is parameterized in the following way : To summarize, we are assuming that: (1) the energy of an electron in an isolated C(2p"") orbital is formula_88; (2) the energy of interaction between C(2p"") orbitals on adjacent carbons "i" and "j" (i.e., "i" and "j" are connected by a σ-bond) is formula_89; (3) orbitals on carbons not joined in this way are assumed not to interact, so "formula_90" for nonadjacent "i" and "j"; and, as mentioned above, (4) the spatial overlap of electron density between different orbitals, represented by non-diagonal elements of the overlap matrix, is ignored by setting formula_91, "even when the orbitals are adjacent . This neglect of orbital overlap is an especially severe approximation. In actuality, orbital overlap is a prerequisite for orbital interaction, and it is impossible to have formula_89 while formula_93. For typical bond distances (1.40 Å) as might be found in benzene, for example, the true value of the overlap for C(2p") orbitals on adjacent atoms i" and "j" is about formula_94; even larger values are found when the bond distance is shorter (e.g., formula_95 ethylene). A major consequence of having nonzero overlap integrals is the fact that, compared to non-interacting isolated orbitals, bonding orbitals are not energetically stabilized by  nearly as much as antibonding orbitals are destabilized. The orbital energies derived from the Hückel treatment do not account for this asymmetry ("see" Hückel solution for ethylene (below) "for details"). The eigenvalues of formula_96 are the Hückel molecular orbital energies formula_97, expressed in terms of formula_98 and formula_99, while the eigenvectors are the Hückel MOs formula_100, expressed as linear combinations of the atomic orbitals formula_23. Using the expression for the normalization constant "N" and the fact that formula_85, we can find the normalized MOs by incorporating the additional condition The Hückel MOs are thus uniquely determined when eigenvalues are all  distinct. When an eigenvalue is degenerate (two or more of the formula_12 are equal), the eigenspace corresponding to the degenerate energy level has dimension greater than 1, and the normalized MOs at that energy level are then "not" uniquely determined. When that happens, further assumptions pertaining to the coefficients of the degenerate orbitals (usually ones that make the MOs orthogonal and mathematically convenient) have to be made in order to generate a concrete set of molecular orbital functions. If the substance is a planar, unsaturated hydrocarbon, the coefficients of the MOs can be found without appeal to empirical parameters, while  orbital energies are given in terms of only formula_98 and formula_99. On the other hand, for systems containing heteroatoms, such as pyridine or formaldehyde, values of correction constants formula_107 and formula_108 have to be specified for the atoms and bonds in question, and formula_98 and formula_99 in (**) are replaced by formula_111 and formula_112, respectively. Hückel solution for ethylene in detail. In the Hückel treatment for ethylene, we write the Hückel MOs formula_113 as a linear combination of the atomic orbitals (2p orbitals) on each of the carbon atoms: Applying the result obtained by the Ritz method, we have the system  of equations where: (Since 2p"" atomic orbital can be expressed as a pure real function, the * representing complex conjugation can be dropped.) The Hückel method assumes that all overlap integrals (including the normalization integrals) equal the Kronecker delta, formula_118, all Coulomb integrals formula_119 are equal, and the resonance integral formula_120 is nonzero when the atoms "i" and "j" are bonded. Using the standard Hückel variable names, we set The Hamiltonian matrix is The matrix equation that needs to be solved is then or, dividing by formula_99, Setting formula_129, we obtain This homogeneous system of equations has nontrivial solutions for formula_131  (solutions besides the physically meaningless formula_132) iff the matrix is singular and the determinant is zero: Solving for formula_134, Since formula_137, the energy levels are The coefficients can then be found by expanding (***): Since the matrix is singular, the two equations are linearly dependent, and the solution set is not uniquely determined until we apply the normalization condition. We can only solve for formula_142 in terms of formula_143: After normalization with formula_146, the numerical values of formula_143 and formula_142 can be found: Finally, the Hückel molecular orbitals are The constant β in the energy term is negative; therefore, formula_152 with  formula_153 is the lower energy corresponding to the HOMO energy and formula_154 with formula_155 is the LUMO energy. If, contrary to the Hückel treatment, a positive value for formula_156 were included, the energies would instead be while the corresponding orbitals would take the form An important consequence of setting formula_159 is that the bonding (in-phase) combination is always stabilized to a lesser extent than the antibonding (out-of-phase) combination is destabilized, relative to the energy of the free 2p orbital. Thus, in general, 2-center 4-electron interactions, where both the bonding and antibonding orbitals are occupied, are  destabilizing overall. This asymmetry is ignored by Hückel theory. In general, for the orbital energies derived from Hückel theory, the sum of stabilization energies for the bonding orbitals is equal to the sum of destabilization energies for the antibonding orbitals, as in the simplest case of ethylene shown here and the case of butadiene shown below. Hückel solution for 1,3-butadiene. The Hückel MO theory treatment of 1,3-butadiene is largely analogous to the treatment of ethylene, shown in detail above, though we must now find the eigenvalues and eigenvectors of a 4 × 4 Hamiltonian matrix. We first write  the molecular orbital formula_113 as a linear combination of the four atomic orbitals formula_23 (carbon 2p orbitals) with coefficients formula_45: The Hamiltonian matrix is In the same way, we write the secular equations in matrix form as which leads to and The orbitals are given by