Investigating Real- and Reciprocal- Space Effects on Kinetic and Exchange Energies of Many-Electron Systems

The goal of this project is to rework an exchange-approximation based upon the free-electron gas and determine whether additional accuracy can be achieved. This study examines the total kinetic and exchange energies of a system of M electrons confined within different three-dimensional k-space Brillouin zones under the constraint of a constant real-space volume V=L³. Three box configurations are considered, with side lengths (a₁,b₁,c₁)= (L,L,L), (a₂,b₂,c₂) =(L/2,L/2,4L), (a₃,b₃,c₃)=(L/2,L,2L), For each configuration, the total kinetic energy is calculated by summing over discrete k-states occupied up to the Fermi level. As the first benchmark, the kinetic energy is determined for the same number of electrons M distributed within a Fermi sphere of equivalent volume. Furthermore, the exchange energy is evaluated for both the box and spherical configurations using an approximation appropriate for a uniform electron gas. The results highlight how the geometry of the occupied region in k-space influences both the kinetic and exchange energy contributions. In particular, deviations from spherical symmetry result in systematic variations in exchange energy, revealing limitations of standard uniform-gas approximations. These findings improve our understanding of many-electron behavior in anisotropic systems and provide insight relevant to refining exchange-correlation functionals in density functional theory. The free electron gas model thus serves as a foundational tool, not only simplifying complex quantum systems but enabling the development of more accurate models for realistic materials under varying geometric and symmetry constraints. Use of this modified free-electron-gas exchange-only system is applied to the nobel-gas atoms.