Unified Treatment for Arbitrary-rank Cartesian Electric and Magnetic Multipole Moment Operators

In this study, the theory of cartesian electric and magnetic multipole moments is extended in a unified way. The general analytical expressions for distinct components of arbitrary rank cartesian electric and magnetic multipole moment operators are derived as linear combination of corresponding spherical operators, which can be used as interconversion between cartesian and spherical electric and magnetic multipole moment tensors. The transformation properties, such as translation and rotation of cartesian electric and magnetic multipole moments are given in a very simple general form. The relationship between distinct and linearly independent components of cartesian multipole moment tensors in system of linear symmetry is also presented. The formulae obtained in this paper can be utilized to calculate the interaction energies between charge distributions.