Calculation of the Gaunt coefficients over real spherical harmonics

Yukcu S. A., Atalay Ş., Yukcu N., Öztekin E.

CANADIAN JOURNAL OF PHYSICS, 2024 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Publication Date: 2024
  • Doi Number: 10.1139/cjp-2024-0161
  • Journal Name: CANADIAN JOURNAL OF PHYSICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Applied Science & Technology Source, Chemical Abstracts Core, Communication Abstracts, Compendex, Computer & Applied Sciences, Environment Index, Metadex, zbMATH, Civil Engineering Abstracts
  • Ondokuz Mayıs University Affiliated: Yes

Abstract

The Gaunt coefficients are numerical coefficients that describe the interaction between two different angular momentum quantum states. In the quantum physics, this interaction affects the energy levels and structures of atoms and molecules. The calculation of Gaunt coefficients plays a significant role in molecular integral evaluations, which emerge from the solving Schr & ouml;dinger equation. This work presents the analytical evaluation methods for the Gaunt coefficients using real spherical harmonics and associated Legendre functions. In this paper, the general integral form of the Gaunt coefficients is written in two main parts as polar and azimuthal integrals. The new analytical relations and symmetry properties are generated for these integrals. The analytical expressions are obtained in terms of the beta functions, and these coefficients are calculated for different atomic orbitals of angular momentum using the Mathematica programming language. The obtained numerical results are compared with the values in the literature and we found completely agreement to 15 digits.