A Bi−directional method for evaluating integrals involving higher transcendental functions. HyperRAF: A Julia package for new hyper−radial functions

Abstract

The electron repulsion integrals over Slater−type orbitals with non−integer principal quantum numbers are investigated. These integrals are useful in both non−relativistic and relativistic calculations of many−electron systems. They involve hyper−geometric functions that are practically difficult to compute. Relationships free from hyper−geometric functions for expectation values of Coulomb potential (r21−1) are derived. These relationships are new and show that the complication coming from two−range nature of Laplace expansion for the Coulomb potential is removed. This is achieved by utilizing auxiliary functions represented in finite power series. They serve as essential components in deriving straightforward recurrence relationships for electron repulsion integrals. In the context of computing the expectation values of potentials with arbitrary power, the methodology presented here for evaluation of these integrals forms the initial condition. It is also adapted to multi−center integrals. Program summary: Program Title: HyperRAF CPC Library link to program files: https://doi.org/10.17632/6pbv2y7s42.1 Developer's repository link: https://github.com/abagciphys/HyperRAF.git Licensing provisions: MIT Programming language: Julia Programming Language [1] Supplementary material: An exploratory variant of the software program written in the Mathematica Programming Language [2]. External routines/libraries: Nemo, a computer algebra package for the Julia programming language [3], JRAF, a Julia package for computation of relativistic molecular auxiliary functions [4]. Nature of problem: Definite integrals involving higher transcendental functions given by, fmn1(a,b,x)=xm−1e−bxΓ[n,ax], and fmn2(a,b,x)=xm−1e−bxγ[n,ax] are frequently encountered in atomic physics, with the electron repulsion integral being a notable illustration. For exclusive solutions to these integrals, one can refer to Erdélyi's [5] or Gradshteyn and Ryzhik's [6] books. The solutions are obtained by using the series representation of incomplete gamma functions. The result is hyper−geometric functions of the form F12[1,b,c;z], where b=m+n, c=m+1, z=b/(a+b) for fmn1(a,b,x) and c=n+1, z=a/(a+b) for fmn2(a,b,x), respectively. Due to the non−trivial structure of infinite series that are used to define them, the computation for hyper−geometric poses challenges. Convergence of their series strictly depends on the values of parameters. Computational issues such as cancellation or round−off error emerge. Solution method: This research introduces novel bi−directional hyper−radial functions that are used to establish fresh recurrence relationships for electron repulsion integrals, eliminating the dependence on hyper−geometric functions. Dual functionality is inherent in the hyper−radial functions as they offer alternative solutions for definite integrals involving higher transcendental functions. Additionally, they transform the representation of the hyper−geometric functions into finite power series. References: [1] J. Bezanson, A. Edelman, S. Karpinski, V.B. Shah, Julia: a fresh approach to numerical computing, SIAM Rev. 59 (1) (2017) 65–98, https://doi.org/10.1137/141000671. [2] https://www.wolfram.com/mathematica/. [3] C. Fieker, W. Hart, T. Hofmann, F. Johansson, Nemo/Hecke: computer algebra and number theory packages for the Julia programming language, in: Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation, New York, USA, 2017, pp. 157–164, https://doi.org/10.1145/3087604.3087611. [4] A. Bağcı, JRAF: a Julia package for computation of relativistic molecular auxiliary functions, Comput. Phys. Commun. 273 (2022) 108276, https://doi.org/10.1016/j.cpc.2021.108276. [5] H. Bateman, A. Erdélyi, Higher Transcendental Functions. Vol. II, McGraw Hill, New York, 1954, pp. 308–309. [6] I.S. Gradshteyn, I.M. Ryzhik, 6−7−Definite integrals of special functions, in: A. Jeffrey, D. Zwillinger (Eds.), 8th ed., Table of Integrals, Series, and Products, Academic Press, Amsterdam, 2014, pp. 665. https://doi.org/10.1016/B978-0-12-384933-5.00006-0. © 2023 Elsevier B.V.