Convergence of the Laplace and the alternative multipole expansion approximation series for the Coulomb potential

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 its convergence with our recently developed perturbative and analytical alternative multipole expansion series of the Coulomb potential. In our working, we confirm that the Laplace and the analytical alternative multipole expansion series are equivalent as expected. In terms of performance, the perturbative alternative multipole expansion series underapproximate the expected results to some extent while the Laplace and the analytical alternative multipole expansion series yield results which are relatively accurate but oscillatory in nature even with a higher number of angular momentum terms employed. As a practical example, we have evaluated the Slater double integrals for two-electron systems using the multipole expansion techniques and a mean field approximation. The estimated results show that only spherical interactions are dominant while the higher-order interactions are negligible. To highlight the discrepancy in the application of each of the formulations of the multipole expansion series for the electron-electron interaction potential, an estimation of the non-relativistic groundstate energies of some helium-like systems, evaluated using the spherical approximation of the multipole potential, is provided. Our findings are likely to be useful in the treatment of the Coulomb potential in electronic structure calculations as well as in celestial mechanics.