Discrete transformations in the Thomson Problem

• A new upper energy limit to the general solution of the Thomson Problem is reported.
• Discrete charge transformations yield correspondences with atomic structure.
• The transformation is the difference between discrete and continuous charge systems.

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)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.

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