Single-centre expansion of Gaussian basis functions and the angular decomposition of their overlap integrals

Single-centre partial-wave expansions are derived for several Gaussian-type functions: simple, solid harmonic and spherical Gaussians. Single-centre expansions for the most commonly used Cartesian Gaussians are obtained by expanding these functions in spherical Gaussians. Transformation matrices for expanding Cartesian in spherical Gaussians are given for s, p, d and f type functions. The single-centre expansions are used to calculate the partial-wave decomposition of overlap integrals for all Gaussian-type functions specified. The formulae given are suitable for fast numerical computation and were tested with programs developed for this purpose.