Please use this identifier to cite or link to this item: https://hdl.handle.net/11499/56069
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dc.contributor.authorBağcı, A.-
dc.contributor.authorAucar, G.A.-
dc.date.accessioned2023-12-16T06:44:48Z-
dc.date.available2023-12-16T06:44:48Z-
dc.date.issued2024-
dc.identifier.issn0010-4655-
dc.identifier.urihttp://dx.doi.org/10.1016/j.cpc.2023.108990-
dc.identifier.urihttps://hdl.handle.net/11499/56069-
dc.description.abstractThe 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.en_US
dc.description.sponsorshipDI-2023-2493-APN-GRH#CONICET; Consejo Nacional de Investigaciones Científicas y Técnicas, CONICET; Universidad Nacional del Nordeste, UNNEen_US
dc.description.sponsorshipOne of the authors A.B. acknowledges funding for a postdoctoral research fellowship from CONICET-IMIT Instituto de Modelado e Innovación Tecnológica (Consejo Nacional de Investigaciones Científicas y Técnicas - Universidad Nacional del Nordeste ), with Grant Number DI-2023-2493-APN-GRH#CONICET . The authors would like to express their gratitude to the anonymous reviewers for their constructive suggestions and thank them for their valuable comments to improve the quality of this paper.en_US
dc.description.sponsorshipThe authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Ali Bagci reports financial support was provided by Instituto de Modelado e Innovación Tecnológica (Consejo Nacional de Investigaciones Científicas y Técnicas - Universidad Nacional del Nordeste)-IMIT (Grant Number: DI-2023-2493-APN-GRH#CONICET).One of the authors A.B. acknowledges funding for a postdoctoral research fellowship from CONICET-IMIT Instituto de Modelado e Innovación Tecnológica (Consejo Nacional de Investigaciones Científicas y Técnicas - Universidad Nacional del Nordeste), with Grant Number DI-2023-2493-APN-GRH#CONICET. The authors would like to express their gratitude to the anonymous reviewers for their constructive suggestions and thank them for their valuable comments to improve the quality of this paper.en_US
dc.language.isoenen_US
dc.publisherElsevier B.V.en_US
dc.relation.ispartofComputer Physics Communicationsen_US
dc.rightsinfo:eu-repo/semantics/closedAccessen_US
dc.subjectBi−directional methoden_US
dc.subjectCoulomb potentialen_US
dc.subjectHyper−radial functionsen_US
dc.subjectLaplace expansionen_US
dc.subjectNon−integer Slater−type orbitalsen_US
dc.subjectAlgebraen_US
dc.subjectComputer programming languagesen_US
dc.subjectElectric fieldsen_US
dc.subjectElectronsen_US
dc.subjectFunction evaluationen_US
dc.subjectLaplace transformsen_US
dc.subjectQuantum theoryen_US
dc.subjectBi-directionalen_US
dc.subjectBi−directional methoden_US
dc.subjectCoulomb potentialen_US
dc.subjectGeometric functionsen_US
dc.subjectHigher transcendental functionsen_US
dc.subjectHyper−radial functionen_US
dc.subjectLaplace expansionsen_US
dc.subjectNon−integer slater−type orbitalen_US
dc.subjectRadial functionsen_US
dc.subjectSlater-type orbitalsen_US
dc.subjectQuantum chemistryen_US
dc.titleA Bi−directional method for evaluating integrals involving higher transcendental functions. HyperRAF: A Julia package for new hyper−radial functionsen_US
dc.typeArticleen_US
dc.identifier.volume295en_US
dc.departmentPamukkale Universityen_US
dc.identifier.doi10.1016/j.cpc.2023.108990-
dc.relation.publicationcategoryMakale - Uluslararası Hakemli Dergi - Kurum Öğretim Elemanıen_US
dc.authorscopusid56337698800-
dc.authorscopusid6602580499-
dc.identifier.scopus2-s2.0-85176286062en_US
dc.identifier.wosWOS:001111374000001en_US
dc.institutionauthor-
item.languageiso639-1en-
item.openairetypeArticle-
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item.fulltextNo Fulltext-
item.openairecristypehttp://purl.org/coar/resource_type/c_18cf-
crisitem.author.dept17.03. Physics-
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