Calculation of two-centre two-electron integrals over Slater-type orbitals revisited. I. Coulomb and hybrid integrals

TL;DR

This paper develops and compares two numerical methods for calculating two-centre Coulomb and hybrid integrals over Slater-type orbitals, ensuring stability and efficiency for complex quantum chemistry calculations.

Contribution

It introduces general formulae and two stable numerical schemes for evaluating complex integrals over STOs without restrictions on quantum numbers.

Findings

Numerical methods are stable and efficient for high angular momentum integrals.

Transformations and recursive schemes effectively evaluate integrals with minimal computational overhead.

The methods are applicable over a wide range of nonlinear parameters.

Abstract

In this paper, which constitutes the first part of the series, we consider calculation of two-centre Coulomb and hybrid integrals over Slater-type orbitals (STOs). General formulae for these integrals are derived with no restrictions on the values of the quantum numbers and nonlinear parameters. Direct integration over the coordinates of one of the electrons leaves us with the set of overlap-like integrals which are evaluated by using two distinct methods. The first one is based on the transformation to the ellipsoidal coordinates system and the second utilises a recursive scheme for consecutive increase of the angular momenta in the integrand. In both methods simple one-dimensional numerical integrations are used in order to avoid severe digital erosion connected with the straightforward use of the alternative analytical formulae. It is discussed that the numerical integration does not introduce a large computational overhead since the integrands are well-behaved functions, calculated recursively with decent speed. Special attention is paid to the numerical stability of the algorithms. Applicability of the resulting scheme over a large range of the nonlinear parameters is tested on examples of the most difficult integrals appearing in the actual calculations including at most 7i-type functions (l=6).