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

TL;DR

This paper introduces JRAF, a Julia package for accurate computation of relativistic molecular auxiliary functions essential for solving the molecular Dirac equation, including tools for angular momentum and spherical harmonics calculations.

Contribution

The paper presents a new Julia-based computational package that efficiently evaluates relativistic auxiliary functions and related integrals with high accuracy for molecular quantum chemistry.

Findings

The package computes two-center one-electron integrals over non-integer Slater-type orbitals.

It provides tools for angular momentum coefficients and spherical harmonics calculations.

Sample results demonstrate robustness and high accuracy of the package.

Abstract

Evaluation of relativistic molecular integrals over exponential-type spinor orbitals require using the relativistic auxiliary functions in prolate spheroidal coordinates. They have derived recently by the author [Physical Review E 91, 023303 (2015)]. They are used in solution of the molecular Dirac equation for electrons moving around Coulomb potential. A series of papers on a method for fully analytical evaluation of relativistic auxiliary functions in following, published [2, 3, 4]. From the computational physics point of view, these works also demonstrate how to deal with the integrals involve product of power functions with non-integer exponents and incomplete gamma functions. The computer program package to calculate these auxiliary functions in high accuracy is presented. It is designed in Julia programming language. It is capable of yielding highly accurate results for molecular integrals over a wide range of orbital parameters and quantum numbers. Additionally, the program package provides numerous tools such as efficient calculation for the angular momentum coefficients arising in the product of two normalized associated Legendre functions centered on different atomic positions, 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. The results prove that the package is robust.