  Multipole representation is proposed for the anisotropic Coulomb interactions
in solids. Any local interactions can be expressed as the product of two
multipole operators, and the interaction parameters are systematically
classified based on the point group symmetry. The form of the multipole
interactions are restricted not only by the symmetry and Hermiticity but also
by the spatial structure of the interaction, which is closely related to the
presence or absence of the odd-rank multipoles. As an exemplary demonstration,
the screened Coulomb interaction for SrVO$_3$ is considered, where only a few
parameters are necessary for its description. By comparing it with the
unscreened version, the totally symmetric $A_1$ representation is found to be
strongly suppressed, but the $A_1$ component still gives a dominant
contribution for the anisotropic part of the interaction. The anisotropic
interactions are also applied to the localized two $f$-electron wave functions,
which give the same-order contribution as the one-body level splitting
estimated by the band structure calculation.
