  Coulomb matrix elements are needed in all studies in solid-state theory that
are based on Hubbard-type multi-orbital models. Due to symmetries, the matrix
elements are not independent. We determine a set of independent Coulomb
parameters for a $d$-shell and a $f$-shell and all point groups with up to $16$
elements ($O_h$, $O$, $T_d$, $T_h$, $D_{6h}$, and $D_{4h}$). Furthermore, we
express all other matrix elements as a function of the independent Coulomb
parameters. Apart from the solution of the general point-group problem we
investigate in detail the spherical approximation and first-order corrections
to the spherical approximation.
