A Hartree-Fock Analysis of the Finite Jellium Model

Abstract

A Hartree--Fock analysis of the ground-state electronic structure of the finite spherical jellium model is carried out for systems containing up to $520$ electrons in a positive background field with densities ranging from $10^{-3}$ to $1$. The study focuses on quantifying the effects of confinement on the local-density models of the exchange and kinetic energies used in orbital-free density-based quantum computation methods. Significant discrepancies are observed between the energies obtained from the Hartree--Fock approximation and those predicted by the local density approximation (LDA) and the Thomas--Fermi model (TF) evaluated at the computed electron densities, both in the inner region and on the surface of the system. To reconcile these differences, refined expressions for the local one-electron energy densities, parametrized by the system's size and background charge density, are proposed. These models are also compared with commonly used gradient-based energy functionals.

Figures (9)

• Figure 1: Jellium potential $V$ and five representative one-electron radial wave functions $u_{nl}$ for the system of radius $R = 5$ and background charge $Q = 15$. Horizontal dashed lines indicate corresponding energy levels. Vertical dotted line marks the system radius $R$.
• Figure 2: Dependence of the one-electron energy levels on the radius of the nanoparticle. (a) Positive background charge is fixed at $Q = 1$. (b) Positive background charge density is fixed at $n_\text{I} = 10^{-2}$.
• Figure 3: Evolution of the one-electron Hartree energies $\epsilon_{nl}$ as a function of electron number $N$ for the six lowest-energy orbital configurations, $n_\text{I} = 10^{-2}$.
• Figure 4: Radial electron density distributions $n(r)$, normalized by the background charge density $n_{\text{I}}$, as functions of the normalized radius $\bar{r} = r/R$. (a) Background charge density $n_{\text{I}} = 10^{-2}$, $N$ corresponds to three different systems with fully filled orbitals. (b) Comparison of normalized radial electron densities of systems with fully filled orbitals with $N = 486$ electrons, evaluated at different background charge densities.
• Figure 5: Dependence of the one-electron exchange energy density $\varepsilon_\text{x}(r)$ on the electron density $n(r)$ for $n_\text{I} = 10^{-2}$ together with the functions $f$ and $g$ given by Eqs. (\ref{['eq:f']}), (\ref{['eq:g']}) for parameters $A = 0.748$, $B = 4.10$, $C = 4.60$, and $\beta = 8.88\times10^{-2}$.