Convergence of the multipole expansion of the polarization interaction

TL;DR

This paper demonstrates that the multipole expansion of the polarization potential for a confined hydrogen atom converges absolutely outside the confinement radius, contrasting with its known asymptotic divergence in unconfined systems.

Contribution

It shows that confinement can lead to absolute convergence of the multipole expansion, providing new insights into polarization interactions in confined atomic systems.

Findings

Multipole expansion of confined hydrogen atom's polarization potential converges absolutely outside the confinement radius.

Likely similar convergence behavior for dispersion interactions between confined atoms at large separations.

Contrasts with the known divergence of multipole expansion in unconfined systems.

Abstract

The multipole expansion of the polarization interaction between a charged particle and an electrically charged particle has long been known to be asymptotic in nature, i.e. the multiple expansion diverges at any finite distance from the atom. However, it is shown that the multipole expansion of the polarization potential of a confined hydrogen atom is absolutely convergent at a distance outside the atoms confinement radius. It is likely that the multipole expansion of the dispersion interaction of two confined atoms will also be absolutely convergent provided the internuclear separation of the two atoms is sufficiency large to exclude any overlap between the electron charge clouds of the two atoms.