Spherical to Cartesian Coordinates Transformation for Solid Harmonics Revisited

Spherical Harmonic Gaussian type orbitals and Slater functions can be expressed using spherical coordinates or a linear combinations of the appropriate Cartesian functions. General expressions for the transformation coefficients between the two representations are provided. Values for the transformation coefficients are tabulated up to the quantum number $\mathrm{\ell }=10$. The formula is applied to construct the Hartree potential by an arbitrary-order multipole expansion.