Modified uniform density exchange-approximation for finite electron systems

The goal of this project is to rework an exchange-approximation based upon the uniform electron gas and determine whether additional accuracy can be achieved for a finite number of electrons. We found that the ratio of exact exchange energy to the original Kohn-Sham expression for the exchange energy is a simple fraction that depends on the number of electrons in atoms. Alternatively, there are several ways to represent the ratio in terms of integrals involving the uniform density. After presenting our derivation, we use similar formulae to investigate whether a single empirical parameter can correctly provide the exact exchange energy, with or without self-interaction corrections, for atoms, and both finite and infinite uniform density systems. In addition, we will discuss how these expressions vary when the Fermi sphere is replaced by Brillouin zones with arbitrary aspect ratios. This study further examines the kinetic and exchange energies of a system of M electrons confined in different three-dimensional k-space Brillouin zones under a constant real-space volume V=L³. Three box configurations are considered: (a₁, b₁, c₁) = (L, L, L), (a₂, b₂, c₂) = (L/2, L/2, 4L), and (a₃, b₃, c₃) = (L/2, L, 2L). For each, the total kinetic energy is obtained by summing over discrete k-states up to the Fermi level. As a benchmark, kinetic energy is also determined for the same number of electrons distributed within a Fermi sphere of equivalent volume. Exchange energy is evaluated for both box and spherical configurations using a uniform-electron-gas approximation. Results show that k-space geometry influences both kinetic and exchange contributions. Deviations from spherical symmetry cause systematic variations in exchange energy, revealing limitations of standard uniform-gas approximations. These findings enhance understanding of many-electron behavior in anisotropic systems and support refinement of exchange-correlation functionals in density functional theory. The modified free-electron-gas, exchange-only model is applied to noble-gas atoms.