Hartree–Fock energy of a finite two-dimensional electron gas system in a jellium background

We adopt a Hartree-Fock approach and calculate the energy of a finite two-dimensional electron gas system confined to a region that is treated as a positive jellium background. The electrons are considered fully spin-polarized (spinless) and interact with a Coulomb potential. The calculation of the exact potential energy of electrons in a finite square jellium domain is very challenging since the mathematical expressions depend on each component of particle's position and not the radial distance from the center of the domain. In order to address this issue we introduce an approximation to the problem. We assess the quality of this approximation and discuss instances where its use is not only desirable, but also fairly accurate. The results give a correct picture of how the energy of the finite system evolves towards the bulk value as the size of the system increases.