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Asymptotic expansion of two-electron integrals and its application to Coulomb and exchange lattice sums in metallic, semimetallic, and nonmetallic crystals.

A simple, easily implemented, accurate, and efficient approximation of long-range electron-electron-repulsion and electron-nucleus-attraction integrals is proposed. It replaces each product of two atomic-orbital (AO) basis functions of an electron by a point charge centered at the midpoint of the two AO's. The magnitude of the point charge is equal to the overlap integral of the two AO's. Each integral is then rapidly evaluated in the direct algorithm as a Coulomb interaction between two point charges. This scheme is implemented in ab initio Hartree-Fock crystalline orbital theory and tested for one-, two-, and three-dimensional solids of metallic, semimetallic, and nonmetallic electronic structures, in which the lattice sums of the direct Coulomb and/or exchange interactions are expected to be slowly convergent. It is shown that this approximation reduces operation and/or memory costs by up to an order of magnitude to achieve converged lattice sums, although the scaling (size dependence) of operation cost is unchanged. An improved criterion for truncating the exchange lattice sum is also proposed.