Difference in Coulomb Electrostatic Energy for Localized versus Delocalized Electrons and Electron Pairs—Exact Results Based on Cubic Charge Distributions

Abstract

Wigner showed that a sufficiently thin electron gas will condense into a crystal of localized electrons. Here, we show, using a model based on cubic charge distributions that gives exact results, that the Coulomb repulsion energy of localized charge distributions is lower than that of delocalized distributions in spite of the fact that the total overall charge distribution is the same. Assuming a simple cubic geometry, we obtain an explicit result for the energy reduction. This reduction results from the exclusion of self-interactions of the electrons. The corresponding results for electron pairs are also discussed.