Nonlinear Static Screening of Positive Charges in an Electron Gas: Contact Hartree Energy

Abstract

Electron screening of positive charges in metals is most strongly nonlinear in the static near-field regime. We revisit screening of a static single protonic charge in a homogeneous electron gas, focusing on the induced density and the contact Hartree energy $U_{\text{H}}(0)$. Although evaluated at the impurity position, $U_{\text{H}}(0)$ is not purely local: our formulation makes it explicit as a nonlocal quantity set by a radial moment of the full induced density, applicable to both linear-response and nonlinear density-functional-theory (DFT) descriptions. We compare Thomas--Fermi, Lindhard/random-phase-approximation, and local-field-corrected dielectric models with nonlinear DFT benchmarks. The Estreicher--Meier local-density-approximation (LDA) parametrization reproduces the contact Hartree energy from our direct LDA calculations and the self-consistent results of Almbladh \emph{et al.} [\href{https://doi.org/10.1103/PhysRevB.14.2250}{Phys. Rev. B \textbf{14}, 2250 (1976)}]. This validates the unified $U_{\text{H}}(0)$ implementation, separates the hydrogenic density profile from non-negligible Friedel oscillations, and provides a compact nonlinear reference for linear-response theory. Testing modern local-field factors, the Corradini--Del Sole--Onida--Palummo and Kaplan--Kukkonen parametrizations yield indistinguishable contact screening despite differing near $q\simeq 2k_F$. We also analyze Yukawa, hydrogenic, and Hulthén screened Coulomb potentials via a variable-phase scattering formulation constrained by the Friedel sum rule; these give a useful phase-shift representation of static screening but cannot alone reproduce the nonlinear DFT contact Hartree energy. The results establish a one-center nonlinear screening benchmark for proton impurities in jellium and clarify the baseline needed before treating two-center screening relevant to low-energy fusion in condensed matter.