Calculation of the Gaunt coefficients over real spherical harmonics

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 Schrödinger 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.