Energy of a finite three-dimensional electron gas of spinless electrons

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We study a finite three-dimensional electron gas system consisting of an arbitrary number of electrons embedded in a finite cubic domain. The electrons are treated as spinless particles implying that the system under consideration represents a fully spin-polarized Fermi quantum phase of electrons. The cubic region is uniformly filled with a positive background that ensures overall charge neutrality. We apply a Hartree-Fock approach that starts with a Slater determinant wave function of normalized plane wave orbitals. The treatment enables us to obtain the energy per particle of the finite system at any given number of electrons. The potential (exchange) energy is conveniently obtained by simplifying the calculation of the ensuing two-particle integrals over the finite cubic domain in terms of expressions that involve compact analytic auxiliary functions. Results are provided for both the kinetic and potential energy per particle for various numbers of electrons. It is shown that the kinetic and the potential energy per particle converge towards their bulk thermodynamic limit values in a non-monotonic way as a function of the number of particles. The results derived may apply to finite systems of delocalized electrons in alkali metal nanoclusters in which the positive ionic core is approximated as a cubic jellium region.

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