THE VALENCE-BOND METHOD FOR LATTICE ELECTRON STUDIES

$^{1}$ L. F. Mattheiss, Phys. Rev. 123, 1209 (1961). $^{2}$ J. W. Moskowitz, J. Chem. Phys. 37, 677 (1962).Author Institution: Department of Chemistry, Massachusetts Institute of TechnologyIn a model lattice with one electron per atom the band theory gives an adequate description for the electron behavior only at small lattice spacings. At very large separations the lowest state corresponds to electrons localized on individual atoms. The transition between these limiting cases is of great interest to the theory of solids and of conjugated molecules. Most critical studies on this problem have been restricted to finite lattices of hydrogen atoms as the model systems, with most attention fixed to date on the 2N-membered rings. The hexagonal $H_{6}$ ring has been studied completely; a full configuration interaction study was made by $Mattheiss^{1}$ and the ``alternant molecular orbital'' study by $Moskowitz^{2}$. Mattheiss' calculation may be regarded as the ``exact'' results by which other methods may be evaluated. The valence-bond method has never been applied to problems of lattices because of the large number of canonical singlets which are in full resonance. On the other hand it is physically realistic, emphasizing localization and correlation of nearest neighbor electrons. We have employed the valence-bond method as a partial description of the wave function for model lattices, working out the energy expressions using a) orthogonal atomic orbitals, b) ionic character in each pair bond (the same in every bond), c) ``nearest neighbor bonds only''. For the 2N-membered rings this restricts the VB set to the $Kekul\'{e}$ structures. In order to represent true delocalization of electrons, dominant at short distances, the lowest molecular orbital wave function was also included. With such a simple three-term wave function, with one variable parameter (ionic character in the VB terms) we obtained lower energies for the ground singlet in $H_{6}$ than any other calculation except that of Mattheiss. The method has significant advantages in comparison with the ``alternant m. o.'' method, and is not identical to it. We believe it brings some rather helpful physical insight into the treatment of the lattice problem. Extension of the method to other model systems and possible implications of our results to date are discussed