Analysis of atomic integrals involving explicit correlation factors for the three-electron case. I. Connection to the hypergeometric function $$_{3}{{\varvec{F}}}_{2}$$

Abstract

This work considers the solution of atomic three-electron integrals that involve explicit inter-electronic separation factors. The traditional Sack expansion is replaced by an alternative expansion, which avoids the breakup of the radial integrals into a number of factors depending on the lesser and greater of the inter-electronic separation distances. The present approach avoids the N! increase in the number of required auxiliary functions, where N is the number of electrons. The new approach leads to additional infinite summations, but these summations either converge very quickly in a serial calculation, or can be effectively dealt with using the massively parallel architecture of a graphics processing unit. The new auxiliary functions that arise are discussed in detail.