In the multi-center molecular integrals, Gaunt coefficients have an important role for numerical calculations. If we want to find correct results from molecular interactions, then we have to calculate accurately their molecular integrals which contain Gaunt coefficients. These coefficients and atomic spherical harmonics have a remarkable relationship. Namely, analytical productions of spherical harmonics generate these coefficients. In this paper, we present some best-known definitions for Gaunt coefficients, and later for them, we find new representations with binomial coefficients. Finally, we will give other mathematical details and numerical results for these coefficients to emphasize their quantum mechanical importance.