Minimum energy configurations of classical charges: Large N asymptotics

TL;DR

This paper analyzes the large N asymptotic behavior of minimum energy configurations of electrons around atomic nuclei, explicitly determining the electron distribution and screening effects through Gamma-convergence techniques.

Contribution

It provides an explicit characterization of the asymptotic electron distribution in large N limits, revealing the screening phenomenon in multi-nuclear systems.

Findings

Electron distribution explicitly determined asymptotically.

Number of electrons around each nucleus matches nuclear charge.

Screening effect confirmed in large N limit.

Abstract

We study minimum energy configurations of $N$ particles in $\R^3$ of charge -1 (`electrons') in the potential of $M$ particles of charges $Z_\alpha>0$ (`atomic nuclei'). In a suitable large-N limit, we determine the asymptotic electron distribution explicitly, showing in particular that the number of electrons surrounding each nucleus is asymptotic to the nuclear charge ("screening"). The proof proceeds by establishing, via Gamma-convergence, a coarse-grained variational principle for the limit distribution, which can be solved explicitly.