Nonlinear Static Screening of Positive Charges in an Electron Gas: Contact Hartree Energy
Electron screening of positive charges in metals is most strongly nonlinear in the static near-field regime. We revisit the static screening of a proton embedded in a homogeneous electron gas, focusing on the induced electron density and the contact Hartree energy $U_{\rm H}(0)$. Although evaluated at the impurity site, $U_{\rm H}(0)$ is a nonlocal radial moment of the induced density in a formulation applicable to both linear-response and nonlinear density-functional descriptions. We compare Thomas--Fermi, random-phase-approximation, and local-field-corrected dielectric screening with nonlinear density-functional-theory benchmarks. The Estreicher--Meier local-density-approximation parametrization closely reproduces the contact Hartree energies from our direct LDA calculations and from Almbladh \emph{et al.} [\href{https://doi.org/10.1103/PhysRevB.14.2250}{Phys. Rev. B \textbf{14}, 2250 (1976)}], separating hydrogenic core and Friedel-oscillation contributions. The contact energy and on-top density are nearly insensitive to the choice between modern quantum-Monte-Carlo-consistent local-field factors. We then analyze Yukawa, hydrogenic, and Hulth\'en screened Coulomb potentials using a variable-phase formulation constrained by the Friedel sum rule. These model potentials provide a useful phase-shift representation of static screening, but a single Friedel constraint does not determine the nonlinear contact Hartree energy quantitatively. The results establish a one-center nonlinear screening benchmark for protons in jellium and a baseline for future two-center screening calculations in metallic environments.