Two-center two-electron integrals with exponential functions

TL;DR

This paper introduces an efficient method for calculating two-center two-electron integrals with exponential functions, enabling high-precision relativistic calculations in diatomic molecules.

Contribution

It develops a novel approach to evaluate complex integrals using differential equations and recursion relations, including an analytic solution for specific cases.

Findings

Master integral derived from differential equations

Analytic expression for James-Coolidge basis

Recursion relations for arbitrary powers

Abstract

We present an efficient approach to evaluate two-center two-electron integrals with exponential functions and with an arbitrary polynomial in electron-nucleus and electron-electron distances. We show that the master integral with the single negative power of all distances can be obtained from the second order differential equation in $r$, the distance between nuclei. For particular values of nonlinear parameters corresponding to the James-Coolidge basis, we find a fully analytic expression. For integrals with arbitrary powers of all distances, we construct recursion relations which starts from the master integral. The presented approach opens a window for the high precision calculations of relativistic effects in diatomic molecules.