Analytical evaluation of relativistic molecular integrals. I. Auxiliary functions

Abstract

The Slater-type orbital basis with non-integer principal quantum numbers is a physically and mathematically motivated choice for molecular electronic structure calculations in both non-relativistic and relativistic theory. The non-analyticity of these orbitals at r = 0, however, requires analytical relations for multi-center integrals to be derived. This is nearly insurmountable. Previous papers by present authors eliminated this difficulty. Highly accurate results can be achieved by the procedure described in these papers, which place no restrictions on quantum numbers in all ranges of orbital parameters. The purpose of this work is to investigate computational aspects of the formulae given in the previous paper. It is to present a method which helps to increase computational efficiency. In terms of the processing time, evaluation of integrals over Slater-type orbitals with non-integer principal quantum numbers are competitive with those over Slater-type orbitals with integer principal quantum numbers.Keywords Slater-type orbitals · Multi-center integrals · Auxiliary functions 1 IntroductionIn the first paper [1] of aforementioned series, history and importance of usage the auxiliary function method was summarized. Applications in molecular calculations were briefly

Analytical evaluation of relativistic molecular integrals. II: Method of computation for molecular auxiliary functions involved

“…The derived relationships in our previous papers [99,100] for molecular auxiliary functions while p 3 = 0 can be recapitulated in the present paper. The Eq.…”

“…For p 3 = 0 we have the following relationship for the auxiliary functions given in the left hand sidde of Eq. (16) with q = 0 [99],…”

“…Note that, the analytical evaluation of these functions involve some challenges namely, power functions with non-integer exponents, incomplete gamma functions and their multiplications have no explicit closed-form relations. In following the definition and the origin of these new molecular auxiliary functions re-visited, an analytical method based on a recurrence strategy which is based on the criterion defined in our previous papers of the series, developed for computation of them [99,100].…”

JRAF: A Julia Package for Computation of the Relativistic Molecular Auxiliary Functions

“…The above procedure by the author improved to a general set of functions where the parameters are free from any restriction [14]. A symmetry feature for two−center two−electron molecular integrals that pointed out and identified by a criterion in [24,25], eliminates the necessity for immediate expansion for incomplete gamma functions or using the relations for the normalized complementary incomplete and the normalized incomplete gamma functions as [20,22] cause some restrictions for the set of functions to be used.…”

“…A computer program based on the analytical methods of computation, described in the previous series of papers [24,26,27] by the author for relativistic molecular auxiliary functions, is presented here. It is unique inasmuch as it is the only computer program code that enables highly accurate calculation of the molecular integrals involving power functions with non−integer exponents.…”