Benchmark values for molecular three-center integrals arising in the Dirac equation

TL;DR

This paper evaluates the accuracy of new computational methods for three-center nuclear attraction integrals in relativistic molecular calculations, extending previous two-center integral work and exploring integrals without quantum number restrictions.

Contribution

It introduces new molecular auxiliary functions for three-center integrals and assesses their accuracy across all quantum numbers and orbital parameters.

Findings

New auxiliary functions enable accurate three-center integral evaluation.

The methods work across all quantum numbers and orbital parameters.

First comprehensive study of unrestricted three-center integrals in relativistic context.

Abstract

The authors in their previous papers obtained compact, arbitrarily accurate expressions for two-center one- and two-electron relativistic molecular integrals expressed over Slater-type orbitals. In this present study, the accuracy limits of given expressions is examined for three-center nuclear attraction integrals, which are the first integral set do not have analytically closed form relations. They are expressed through new molecular auxiliary functions obtained via Neumann expansion of Coulomb interaction. The numerical global adaptive method is used to evaluate these integrals for arbitrarily values of orbital parameters, quantum numbers. Several methods, such as Laplace expansion of Coulomb interaction, single-center expansion, Fourier transformation method, have been performed in order to evaluate these integrals considering the values of principal quantum numbers in the set of positive integer numbers. This is the first attempts to study the three-center integrals without any restrictions on quantum numbers and in all ranges of orbital parameters.