The construction of the maxwell representation for a cylindrically symmetric spherical harmonic of arbitrary order

Abstract

The general structure of the sets of characteristic directions in the Maxwell invariant representation (vide infra) of spherical harmonics which have the symmetry of those defined by the moments of a cylindrically symmetric distribution has been determined. These results, valid for arbitrary orders, have been used to derive an algorithm for the construction of the characteristic directions. The calculations in the new algorithm are significantly simpler than those in the algorithm for the general spherical harmonic and in test calculations comparing the two algorithms the new one gave more accurate sets of characteristic directions. Previous work has established the convenience of the invariant representation for: (i) the calculation of the interaction energy of either a finite set or a crystal lattice of charge distributions; (ii) the construction of algorithms for problems which require partial derivatives of the electric field (e.g., mutual torques or induced multipoles and their contribution to the net electrostatic energy).