Energy in a finite two-dimensional spinless electron gas

We study the properties of a finite two-dimensional electron gas system in the Hartree-Fock approximation. We obtain exact analytical expressions for the energy in a finite two-dimensional fully spin-polarized (spinless) system of electrons interacting with a Coulomb potential immersed in a finite square region uniformly filled with a neutralizing positive charge. The difficult two-electron integrals over the finite square domain are reduced to simple compact expressions involving analytic auxiliary functions. We provide results for the potential energy of systems with a finite number of electrons and show how the energy slowly converges towards its thermodynamic limit bulk value.