Evaluation of the Gaunt Coefficients by Using Recurrence Relations for Spherical Harmonics

Year 2023, Volume: 18 Issue: 3, 213 - 222, 23.11.2023

Selda Özay

https://doi.org/10.29233/sdufeffd.1312399

Abstract

The Gaunt coefficient is one of the important coefficients to be known for calculating molecular integrals in quantum theory of coupling of three angular momenta. Generally, these coefficients are calculated analytically by using the properties of the associated Legendre polynomials. In this study, Gaunt coefficients were calculated algebraically by using the recurrence relations and orthogonality conditions of spherical harmonics and different mathematical expressions were obtained from known analytical expressions for Gaunt coefficients in terms of factorial functions or binomial coefficients. By using the program written in the Mathematica programming language, both the analytical expressions and the algebraic expressions were calculated, and the numerical results obtained were compared. Numerical results are in quite agreement with the literature and each other.

Keywords

Clebsch – Gordan Coefficients, Vector Coupling Constant, Gaunt Coefficients

References

• G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed., London, Academic Press, 1029 pages, 2001.
• D. A. Varshalovich, A. N. Moskalev and V. K. Khersonskii, Quantum Theory of Angular Momentum, Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols, Singapore, World Scientific Publishing Co. Pte. Ltd, 514 pages, 1988.
• T. Shimpuku, “General Theory and Numerical Tables of Clebsch-Gordan Coefficients”, Progress of Theoretical Physics Supplement, 13, 1-136, 1960.
• A. Bandzaitis and A. Yutsis, “Once more on the formulas for the Clebsch-Gordan coefficients”, Lietuvos Fiz. Rinkinys, 4, 45-49, 1964.
• V. A. Fock, “New deduction of the vector model”, JETP 10, 383-393, 1940.
• S. D. Majumdar, “The Clebsch-Gordan coefficients”, Progress of Theoretical Physics, 20(6), 798-803, 1958.
• G. Racah, “Theory of complex spectra. II”, Physical Review, 62(9), 438-462, 1942.
• A. R. Edmonds, Angular Momentum in Quantum Mechanics, 2nd ed., New Jersey, Princeton University Press, 146 pages, 1960.
• M. E. Rose, Elementary Theory of Angular Momentum, New York, John Wiley&Sons, 248 pages, 1957.
• R. N. Zare, Angular Momentum, Understanding Spatial Aspects in Chemistry and Physics, New York, John Wiley&Sons, 349 pages, 1988.
• E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, Cambridge, Cambridge University Press, 441 pages, 1959.
• A. R. Edmonds, Angular Momenta in Quantum Mechanics, CERN 55-26, Geneva, 82 pages, 1955.
• J. A. Gaunt, “The triplets of Helium”, Philosophical Transactions of the Royal Society of London Series A, 228, 151-196, 1929.
• O. R. Cruzan, “Translational addition theorems for spherical vector wave functions”, Quarterly Applied Mathematics, 20, 33-40, 1962.
• Y. L. Xu, “Fast evaluation of Gaunt coefficients”, Mathematics of Computation, 65(216), 1601-1612, 1996.
• E. J. Weniger and E. O. Steinborn, “Programs for the coupling of spherical harmonics, Computer Physics Communications, 25, 149-157, 1982.
• J. Rasch and A. C. H. Yu, “Efficient storage scheme for precalculated Wigner 3j, 6j and Gaunt coefficients”, Journal on Scientific Computing, 25(4), 1416-1428, 2004.
• H. H. Homeier and E. O. Steinborn, “Some properties of the coupling coefficients of real spherical harmonics and their relation to Gaunt coefficients”, Journal of Molecular Structure (Theochem), 368, 31-37, 1996.
• S. A. Yükçü, N. Yükçü and E. Öztekin, “New representations for Gaunt coefficients”, Chemical Physics Letters, 735 (UNSP 136769), 1-3, 2019.
• E. P. Wigner, Group Theory, New York, Academic Press, 372 pages, 1959.
• E. L. Akim and A. A. Levin, “Generating Function for Clebsch-Gordan Coefficients”, Doklady Akademii Nauk USSR, 138, 503-505, 1961.
• S. Wolfram, A system for doing mathematics by computer, 2nd ed., United Kingdom, Addison Wesley, 261 pages, 1998.
• D. Sebilleau, “On the computation of the integrated products of three spherical harmonics”, Journal of Physics A: Mathematical and General Physics, 31(34), 7157-7168, 1998.