Discrete Transformations in the Thomson Problem

Abstract

A significantly lower upper limit to minimum energy solutions of the electrostatic Thomson Problem is reported. A point charge is introduced to the origin of each $N$-charge solution. This raises the total energy by $N$ as an upper limit to each $(N+1)$-charge solution. Minimization of energy to $U(N+1)$ is well fit with $-0.5518(3/2)\sqrt N+1/2$ for up to $N=500$. The energy distribution due to this displacement exhibits correspondences with shell-filling behavior in atomic systems. This work may aid development of more efficient and innovative numerical search algorithms to obtain $N$-charge configurations having global energy minima and yield new insights to atomic structure.