A New Expression for the Coulomb Potential Corresponding to the Product of Two Exponential Functions Based on the Properties of the Integral Representations of the Bessel Functions

Authors

  • C. J. Alturria Lanzardo Departamento de Matemática, Facultad de Ciencias Exactas, Físico-Químicas y Naturales, Universidad Nacional de Río Cuarto https://orcid.org/0000-0002-0177-1531
  • J. E. Pérez Departamento de Física, Facultad de Ciencias Exactas, Físico-Químicas y Naturales, Universidad Nacional de Río Cuarto
  • M. L. Tardivo Departamento de Computación, Facultad de Ciencias Exactas, Físico-Químicas y Naturales, Universidad Nacional de Río Cuarto
  • G. Fraschetti Departamento de Computación, Facultad de Ciencias Exactas, Físico-Químicas y Naturales, Universidad Nacional de Río Cuarto
  • J. C. Cesco Instituto de Matemática Aplicada de San Luis (CONICET-UNSL)

DOI:

https://doi.org/10.5540/tcam.2022.024.01.00001

Keywords:

Bessel functions, non-rational functions, integral representation, improper integrals, oscillating integrand, Exponential Type Functions, Coulomb Potential

Abstract

The calculation of the Coulomb Potential corresponding to the product of two Exponential Type Functions, inherently has numerical challenges that must be resolved. In order to address these problems, in this paper it is presented a new partition of the Coulomb Potential. The proposed partition involves two terms. One of the terms is a one-dimensional integral, which allows geometrical and statistical interpretations. The other term is proportional to a Modified Bessel Function and it is obtained from a two-step procedure. As a first step, a Non-Rational Function is used for approximating one of the two integrals involved. Then, the remaining improper integral can be identified with an integral representation of an appropriate Modified Bessel Function. The existence of such a Non-Rational Approximant is proved and its numerical performance is shown through some examples.

References

J. E. Pérez, O. E. Taurian, A. Bouferguene, and P. E. Hoggan, “On a transformation for the electrostatic potential, generated by the product of two 1s Slater type orbitals, giving an efficient expression,” Advances in Quantum Chemistry, vol. 67, pp. 65–71, 2013.

C. J. Alturria. Lanzardo, J. Pérez, M. L. Tardivo, G. Fraschetti, and J. C. Cesco, “Potencial de coulomb para funciones 1s de Slater expresado en términos de funciones de Bessel,” in Matemática Aplicada Computacional e Industrial, vol. 8, pp. 81–84, ASAMACI, 2021.

I. Stegun and M. Abramowitz, Handbook of mathematical functions and tables, Bessel function of fractional order. National Bureau of Standards, App Math. Series, Dover, 1972.

J. C. Cesco, J. E. Pérez, C. C. Denner, G. O. Giubergia, and A. E. Rosso,

“Rational Approximants to Evaluate four-center Electron Repulsion ntegrals for 1s Hydrogen Slater Type Functions,” Applied Numerical Mathematics, vol. 55, pp. 179–190, 2005.

C. J. Alturria Lanzardo, J. Pérez, and J. Cesco, “Aproximación de integrales impropias cuyo integrando tiene una parte oscilante,” in Matemática Aplicada Computacional e Industrial, vol. 7, pp. 181–184, ASAMACI, 2019.

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Published

2023-03-14

How to Cite

Alturria Lanzardo, C. J., Pérez, J. E., Tardivo, M. L., Fraschetti, G., & Cesco, J. C. (2023). A New Expression for the Coulomb Potential Corresponding to the Product of Two Exponential Functions Based on the Properties of the Integral Representations of the Bessel Functions. Trends in Computational and Applied Mathematics, 24(1), 1–7. https://doi.org/10.5540/tcam.2022.024.01.00001

Issue

Vol. 24 No. 1 (2023)

Section

Original Article

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