Methods for accurate calculations of multi-center integrals of squared Coulomb potentials for lower bounds to energy levels of molecular systems

TL;DR

This paper develops new methods for calculating multi-center integrals of squared Coulomb potentials with Slater-type orbitals, enabling more accurate lower bounds for molecular energy levels.

Contribution

It introduces exact analytic and semi-analytic expressions for these integrals, which were previously unavailable, using advanced mathematical techniques.

Findings

Derived exact analytic expressions for fundamental integrals.

Provided numerical results demonstrating the methods' accuracy.

Compared new methods with existing approaches, showing improved precision.

Abstract

In this paper methods for calculations of multi-center integrals of squared Coulomb potentials and Slater-type orbitals (STO) are derived. These integrals are necessary for accurate lower bounds to energy levels of molecular systems. All multi-center integrals are reduced to fundamental integrals using the Gaunt coefficients and translation of STO. When the potential is the usual Coulomb potential, using the Laplace expansion or the Neumann expansion of the potential the integrals can be calculated. However, for the squared Coulomb potentials such expansions are not known. For the fundamental one-center and two-center integrals with squared Coulomb potentials, by methods free from such expansions exact analytic expressions and expressions by one-dimensional integrals of analytic functions are derived. The methods mainly rely on the integration in ellipsoidal coordinates, the Fourier transform, Hobson's theorem and expansion of differential operators by simple ones suitable for the calculation. Numerical results by these expressions are given and compared.