General expressions for reducing the Slater-Koster linear combination of atomic orbitals integrals to the two-center approximation

Abstract

This paper contains a derivation of general expressions for reducing Slater-Koster integrals ${E}_{\mathrm{ij}}(l,m,n)$ to the two-center approximation for orbitals $i$ (and $j$) having arbitrary angular momentum quantum numbers; $l$,$m$,$n$ describe the location of one center relative to the other. This has allowed the author to check the entries in Table I of the Slater-Koster paper and provides a means for extending these entries to include states of higher angular momenta. The derived general expressions are especially useful for machine calculations when a large number of particular expressions are required to be evaluated. As a collateral result the missing factors in the energy matrix elements along the $\ensuremath{\Delta}$ line for the face-centered cubic structure given by Slater and Koster and reviewed by Reitz have been noted.