Isolectronic principle and the accuracy of binding energies in the Hueckel method

In the Hkkel and other methods, binding energies are calculated by subtracting the sum of orbital electronic energies for the molecule from the sum of orbital electronic energies for the separated atoms, and not considering the internuclear repulsion. Since this last may be several orders of magnitude greater than the binding energy, reasonable results could not be obtained without an approximate cancellation with another neglected term. It is shown that such a cancellation is a consequence of the isoelectronic principle (invariance of binding energy to change in atomic number of constituent atom). Numerical examples are given. n a number of a priori and semiempirical methods, of I which the extended Huckel method2 is the best known, one calculates molecular binding energies by subtracting the electronic energy (sum of orbital contributions) of the molecule from the sum of the electronic energies of the atoms, without considering the internuclear repulsion. If we accept the argument that the parameterization in the method effectively simulates a Hartree-Fock calculation, the “electronic energies” are really sums of orbital energies. To get the true electronic energies of atom or molecule, one must subtract off in each case the interelectronic repulsion, which is being counted twice. Thus the above recipe will be valid if V” E V,,” CVeeA E AV,, (1) A where Veem and VeeA are the interelectronic repulsions (expectation values) for the molecule and for atom A, and V“ is the internuclear r e p ~ l s i o n . ~ The binding (1) Research supported by the National Science Foundation under Grant No. GP-5861; correspondence should be addressed to Chemistry Department, Syracuse University, Syracuse, N. Y. (2) R. Hoffmann and W. N. Lipscomb, J . Chem. Phys., 36, 3179 (1962); R. Hoffmann, ibid., 39, 1397 (1963). energy may be orders of magnitude smaller than VNN. Thus the error in eq 1 must be small (i.e., of the size of the binding energy itself) if reasonable binding energies are to be obtained from a wave function which is reasonable in other respects. Below, we show4 that this is in fact true in general, being a consequence of the isoelectronic principle. The proof is closely related to the derivation of a formula5 for calculating diamagnetic shieldings in molecules, also starting from the isoelectronic principle. According to this principle, two isoelectronic species have the same binding energies if they differ only by a change by unity in a nuclear charge. The example of CO us. NZ6 is perhaps the best known; one can easily find others.’ Writing ZB for the charge of nucleus B, we express this as (3) Note that it is the change in V,, from atoms to molecule which must be approximately equal to V N N , not V e e itself as has been sometimes stated. (4) J. Goodisman, Theor. Chim. Acta, in press. ( 5 ) W. H. Flygare and J. Goodisman, J . Chem. Phys., 49, 3122 (1968). (6) J. C. Slater, “Quantum Theory of Molecules and Solids,” Vol. I, (7) J. Berkowitz, J . Chem. Phys., 30, 858 (1959); also cf. ref 5 . McGraw-Hill Book Co., Inc., New York, N. Y., 1963, p 134. Journal of the American Chemical Society 1 91:24 November 19, 1969 b (BE) = 0 bZB We define the binding energy (a positive quantity) as (3) A where Eem is the molecular electronic energy E,* the electronic energy of atom A and VNN the internuclear repulsion Here, F and TA are the kinetic energies (expectation values) for the molecule and for atom A, V,," and VeeA the interelectronic repulsion energies for the molecule and for atom A, and V A ~ " and V A , ~ the expectation values of