Algebraic Solution of Gaunt Coefficients via the Angular Momentum Ladder Operators

Abstract

In this study, Gaunt coefficients, frequently encountered in quantum mechanical calculations of atomic and molecular structures, have been algebraically derived. Firstly, the Gaunt coefficient, equal to the integral over the solid angle of the product of three spherical harmonics, is written in terms of angular momentum ladder operators. Subsequently, raising or lowering operators are applied to spherical harmonics, and the obtained integrals are solved using the recurrence and orthogonality relations of spherical harmonics. As a result, algebraic expressions for Gaunt coefficients are obtained in terms of quantum numbers.