# Floating Wigner crystal with no boundary charge fluctuations

## TL;DR

This paper introduces a modified trial state for the classical Homogeneous Electron Gas that suppresses boundary charge fluctuations, demonstrating the equivalence of three different ground state energy definitions.

## Contribution

It proposes a novel boundary treatment in the trial state for Jellium, unifying various definitions of the ground state energy.

## Key findings

- Boundary charge fluctuations are effectively suppressed.
- The three definitions of the ground state energy are shown to coincide.
- The modified trial state improves understanding of boundary effects in Jellium.

## Abstract

We modify the "floating crystal" trial state for the classical Homogeneous Electron Gas (also known as Jellium), in order to suppress the boundary charge fluctuations that are known to lead to a macroscopic increase of the energy. The argument is to melt a thin layer of the crystal close to the boundary and consequently replace it by an incompressible fluid. With the aid of this trial state we show that three different definitions of the ground state energy of Jellium coincide. In the first point of view the electrons are placed in a neutralizing uniform background. In the second definition there is no background but the electrons are submitted to the constraint that their density is constant, as is appropriate in Density Functional Theory. Finally, in the third system each electron interacts with a periodic image of itself, that is, periodic boundary conditions are imposed on the interaction potential.