Systematic Extensions of the Local-Orbitals Method

In prior publications, the Adams---Gilbert local-orbitals method has been used in the limit of small overlaps. In this paper, these methods are extended to systems of higher or even very great overlap. The self-consistent equations are derived to second order in overlap. Rules for classifying integrals in powers of overlap are presented. Methods for solving the local-orbitals equations exactly are presented and exact solutions are obtained for several solid-state and molecular cases. First order in overlap solutions are obtained for some cases and the first-order solutions are compared with the exact solutions.