Analytical evaluation of relativistic molecular integrals. I. Auxiliary functions

TL;DR

This paper discusses improved methods for computing relativistic molecular auxiliary functions, enabling accurate and unrestricted evaluation of integrals for molecules with Slater-type orbitals, applicable in both relativistic and non-relativistic theories.

Contribution

It introduces two solution methods that replace ill-conditioned series with convergent series and extends the domain of convergence for accurate integral computation.

Findings

Highly accurate integral results achieved

No restrictions on quantum numbers or orbital parameters

Methods applicable to relativistic and non-relativistic theories

Abstract

The auxiliary functions provide efficient computation of integrals arising at the self-consistent field (SCF) level for molecules using Slater-type bases. This applies both in relativistic and non-relativistic electronic structure theory. The relativistic molecular auxiliary functions derived in our previous paper [Phys. Rev. E 91, 023303 (2015)] are discussed here in detail. Two solution methods are proposed in the present study. The ill-conditioned binomial series representation formulae first, are replaced by convergent series representation for incomplete beta functions then, they are improved by inserting an extra parameter used to extend the domain of convergence. Highly accurate results can be achieved for integrals by the procedures discussed in the present study which also places no restrictions on quantum numbers in all ranges of orbital parameters. The difficulty of obtaining analytical relations associated with using non-integer Slater-type orbitals which are non-analytic in the sense of complex analysis at r=0 is therefore, eliminated.