Correlated exponential functions in high precision calculations for diatomic molecules

TL;DR

This paper develops and analyzes methods for high-precision calculation of two-electron integrals over explicitly correlated exponential functions, crucial for accurate diatomic molecule modeling.

Contribution

It introduces a compact integral representation and recurrence relations for calculating two-center two-electron integrals with arbitrary powers, enhancing computational accuracy.

Findings

A compact one-dimensional integral representation was derived.

Recurrence relations enable calculation of integrals with arbitrary powers.

An alternative Taylor series approach is viable when recurrences are unstable.

Abstract

Various properties of the general two-center two-electron integral over the explicitly correlated exponential function are analyzed for the potential use in high precision calculations for diatomic molecules. A compact one dimensional integral representation is found, which is suited for the numerical evaluation. Together with recurrence relations, it makes possible the calculation of the two-center two-electron integral with arbitrary powers of electron distances. Alternative approach via the Taylor series in the internuclear distance is also investigated. Although numerically slower, it can be used in cases when recurrences lose stability. Separate analysis is devoted to molecular integrals with integer powers of interelectronic distances $r_{12}$ and the vanishing corresponding nonlinear parameter. Several methods of their evaluation are proposed.