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 VH(0). Although evaluated at the impurity site, VH(0) 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 VH(0) 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 VH(0). 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 rs 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 VH(0). 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.