JRAF: A Julia package for computation of relativistic molecular auxiliary functions

Abstract

The evaluation of relativistic molecular integrals over exponential−type spinor orbitals requires the use of relativistic auxiliary functions in prolate spheroidal coordinates, and has been recently achieved (Bağcı and Hoggan (2015) [14]). This process is used in the solution of the molecular Dirac equation for electrons moving in a Coulomb potential. A series of papers on a method for fully analytical evaluation of relativistic auxiliary functions has been published [2, 3, 4] From the perspective of computational physics, these studies demonstrate how to deal with the integrals of the product of power functions with non−integer exponents and incomplete gamma functions. The computer program package used to calculate these auxiliary functions with high accuracy is presented. It is designed using the Julia programming language and yields highly accurate results for molecular integrals over a wide range of orbital parameters and quantum numbers. Additionally, the program package facilitates the efficient calculation of the angular momentum coefficients that arise from the product of two normalized Legendre functions centered at different atomic positions, and the determination of the rotation angular functions used for both complex and real spherical harmonics. Sample calculations are performed for two−center one−electron integrals over non−integer Slater−type orbitals, and the results prove the robustness of the package. Program summaryProgram Title: JRAFCPC Library link to program files:https://doi.org/10.17632/942xsbvfdf.1Developer's repository link:https://github.com/abagciphys/JRAF.jlLicensing provisions: MITProgramming language:Julia programming languageSupplementary material: An experimental version of the computer program package written in Mathematica programming language [5].External routines/libraries:Nemo computer algebra package for the Julia programming language [6], Cuba multidimensional numerical integration using different algorithms in Julia [7].Nature of problem: Relativistic molecular auxiliary function integrals result from the expression of a two−center two−electron Coulomb energy associated with a charge density. The Coulomb energy is transformed into kinetic energy integrals using Poisson's equation and the single−center potential, considering that the Laplace expansion for the Coulomb interactions is expressed in terms of normalized non−integer Slater−type orbitals [1]. Using the resulting expression for the two−center two−electron integrals, relativistic auxiliary function integrals are derived in prolate ellipsoidal coordinates. These auxiliary functions are generalized to the entire set of physical potential operators for the Coulomb potential case.The integral of the relativistic auxiliary functions have no closed−form solutions except that their parameters are integers. As such, the analytical evaluation of these functions is challenging. They are used in the solution of the matrix form representation of the molecular Dirac−Fock self−consistent field (SCF) equation.Solution method: A criterion that considers the symmetry properties of two−center two−electron molecular integrals is initially proposed [2]. This obviates the need for the computation of incomplete and complementary incomplete gamma functions, and utilizes their sum (P+Q=1). The resulting form of the integral of the relativistic molecular auxiliary functions is expressed in terms of the convergent series representation of incomplete beta functions. Recurrence relationships are then derived for each of these sub−functions [3]. The algorithm for computation of the auxiliary functions is based on the vectorization procedure defined in [4].