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

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 density-derived contact Hartree potential $V_{\mathrm{H}}\left(0\right)$. Although evaluated at the impurity site, $V_{\mathrm{H}}\left(0\right)$ is not a strictly local observable but an r-weighted moment of the full 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 linear response with nonlinear density-functional-theory reference calculations within the local-density approximation and the Perdew–Burke–Ernzerhof generalized-gradient approximation. The Estreicher–Meier LDA parameterization closely reproduces the contact Hartree potential from direct LDA and historical self-consistent results and separates its core and Friedel contributions, although its radial fit obeys only an approximate induced-density cusp. For the two ground-state local-field-factor parameterizations examined here, Kaplan–Kukkonen and Corradini–Del Sole–Onida–Palummo, the differences in the contact density and $V_{\mathrm{H}}\left(0\right)$ are negligible at the scale of the present comparisons; this does not imply a general insensitivity to local-field corrections. We then use a phase-plus-amplitude variable-phase formulation to reconstruct the full quantal density generated by constrained screened model potentials. Friedel-constrained Yukawa, hydrogenic, and Hulthén potentials satisfy the same total-screening condition but generate different contact responses, showing that the Friedel sum rule does not uniquely determine $V_{\mathrm{H}}\left(0\right)$. In the Apagyi–Nagy construction, the exact cusp is imposed on an auxiliary density ansatz used in the Poisson construction, whereas reconstructed quantum densities obey it automatically through the Coulomb singularity; neither condition fixes the complete screening cloud or its Hartree moment. The constrained model-potential results approach the nonlinear density-functional-theory references toward high density but exhibit progressive deviation with an increase in $r_s$ as the screening cloud becomes more spatially extended. In contrast, density-informed two-parameter Popović–Stott potentials, propagated through the same quantal variable-phase reconstruction of the density and Hartree potential without refitting, reproduce the corresponding historical Hartree-only and exchange–correlation-modified contact values to within $0.05\%$ relative accuracy. Taken together, these results show that the constraints used to construct a reduced potential fix only selected properties of the screening problem, not the full density response generated by the resulting potential. The Friedel sum rule constrains a global scattering quantity and the cusp constrains local short-range behavior; potentials satisfying these constraints can nevertheless generate different radial screening densities and different $V_{\mathrm{H}}\left(0\right)$. The Popović–Stott construction is distinct in additionally incorporating information from the generated screening density into the determination of the potential. The resulting one-center benchmark provides a baseline for genuine two-center screening calculations in metallic environments.