A comparison of the Laplace and the alternative multipole expansion series for the Coulomb potential

Abstract

Multipole expansion is a powerful technique used in many-body physics to solve dynamical problems involving correlated interactions between constituent particles. The Laplace multipole expansion series of the Coulomb potential is well established in literature. We compare it with our recently developed perturbative and analytical alternative multipole expansion series of the Coulomb potential. In our working, we confirm that both expansion series are complete but quite different in the basis functions used. The analytical expansion, being the infinite limit of the perturbative expansion, is confirmed to be equivalent to the Laplace multipole expansion of the Coulomb potential. In terms of performance, the perturbative alternative multipole expansion series yields the lower bound while the Laplace and the analytical alternative multipole expansion series yield an upper bound of the expected results. The results show that only a finite number of terms in the series expansion of the basis functions for the perturbative alternative multipole expansion series are necessary for converged and accurate results. Our findings are likely to be useful in the perturbative treatment of the Coulomb potential in electronic structure calculations as well as in celestial mechanics