Four-center Integral of a Dipolar Two-electron Potential Between s-type GTO's

TL;DR

This paper presents a method to compute four-center integrals of a dipolar two-electron potential between s-type Gaussian orbitals by reducing them to two-center products and expressing the integrals with special functions.

Contribution

It introduces a reduction technique for two-electron integrals involving dipolar potentials using Boys' contraction and hypergeometric functions.

Findings

Derived explicit formulas for four-center integrals

Reduced complex integrals to two-center products

Expressed integrals in terms of confluent hypergeometric functions

Abstract

We reduce two-electron 4-center products of Cartesian Gaussian Type Orbitals with Boys' contraction to 2-center products of the form psi_alpha(r_i-A) psi_beta(r_j-B), and compute the 6-dimensional integral over d^3r_i d^3r_j over these with the effective potential V_{ij} = (r_i-r_j) . r_j / |r_i-r_j|^3 in terms of Shavitt's confluent hypergeometric functions.