Normal electric modes at small polyhedral particles

The electric potential of a normal mode at a dielectric or metallic polyhedron is expanded in terms of spherical multipoles. By inducing surface charges each multipole induces secondary multipoles. It is shown that all matrix elements of the resultant secular problem can be calculated analytically, which considerably improves convergence of the eigenvalues of the normal modes. The total shift of eigenvalues caused by multipoles of equal degree is proved to be independent of the shape of the polyhedron under consideration. Simple recurrence relations for separating the secular problem in the case of tetrahedral or cubic symmetry are derived and explicit eigenvalue schemes for some standard polyhedrons are given.