Calculation of two-centre two-electron integrals over Slater-type orbitals revisited. II. Neumann expansion of the exchange integrals

TL;DR

This paper develops new stable and efficient methods for calculating two-centre two-electron exchange integrals over Slater-type orbitals using Neumann expansion, differential equations, series expansions, and improved numerical integration techniques.

Contribution

It introduces novel differential equations and series expansions that enhance the stability and accuracy of exchange integral calculations over a broad parameter range.

Findings

New differential equations for basic integrals.

Series expansions for small and large parameter values.

A stable algorithm combining multiple computational methods.

Abstract

In this paper we consider calculation of two-centre exchange integrals over Slater-type orbitals (STOs). We apply the Neumann expansion of the Coulomb interaction potential and consider calculation of all basic quantities which appear in the resulting expression. Analytical closed-form equations for all auxiliary quantities have already been known but they suffer from large digital erosion when some of the parameters are large or small. We derive two differential equations which are obeyed by the most difficult basic integrals. Taking them as a starting point, useful series expansions for small parameter values or asymptotic expansions for large parameter values are systematically derived. The resulting novel expansions replace the corresponding analytical expressions when the latter introduce significant cancellations. Additionally, we reconsider numerical integration of some necessary quantities and present a new way to calculate the integrand with a controlled precision. All proposed methods are combined to lead to a general, stable algorithm. We perform extensive numerical tests of the introduced expressions to verify their validity and usefulness. Advances reported here provide methodology to compute two-electron exchange integrals over STOs for a broad range of the nonlinear parameters and large angular momenta.