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arxiv: 2606.23065 · v1 · pith:DN26I6XJnew · submitted 2026-06-22 · ❄️ cond-mat.other

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

M. Sherafati , G. Rodway-Gant , A. X. Chen This is my paper

Pith reviewed 2026-06-26 06:07 UTC · model grok-4.3

classification ❄️ cond-mat.other
keywords nonlinear screeninghomogeneous electron gascontact Hartree energylocal density approximationproton impurityFriedel oscillationslocal field factors
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0 comments X

The pith

The Estreicher-Meier LDA parametrization reproduces the contact Hartree energy for a proton in jellium, matching direct LDA calculations and Almbladh et al. results.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies static screening of a single proton by electrons in a uniform gas, where nonlinearity is strongest near the impurity. It formulates the contact Hartree energy as a radial integral over the full induced density rather than a purely local quantity. Comparisons of dielectric models against nonlinear DFT show that one LDA parametrization recovers the benchmark value from both new direct calculations and 1976 self-consistent work. This match confirms a consistent implementation across linear and nonlinear regimes and isolates the hydrogenic density component from Friedel oscillations. The study also finds that certain modern local-field factors produce nearly identical contact energies while simple screened potentials fall short.

Core claim

The Estreicher--Meier local-density-approximation parametrization reproduces the contact Hartree energy from direct LDA calculations and the self-consistent results of Almbladh et al. This agreement validates the unified formulation of the contact Hartree energy as a nonlocal radial moment of the induced density, separates the hydrogenic density profile from non-negligible Friedel oscillations, and supplies a compact nonlinear reference value against which linear-response models can be tested.

What carries the argument

The contact Hartree energy U_H(0), defined as a radial moment of the full induced density around the proton impurity.

If this is right

  • The unified U_H(0) expression applies equally to linear-response dielectric functions and nonlinear DFT.
  • Corradini-Del Sole-Onida-Palummo and Kaplan-Kukkonen local-field factors produce indistinguishable contact screening values.
  • Yukawa, hydrogenic, and Hulthén potentials constrained by the Friedel sum rule cannot reproduce the nonlinear DFT contact Hartree energy.
  • The benchmark cleanly separates the hydrogenic density component from Friedel oscillations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The one-center nonlinear benchmark supplies the reference needed before treating two-center screening problems such as low-energy fusion.
  • Band-structure and core-electron effects in real metals may shift the near-field screening relative to the jellium prediction.
  • The variable-phase scattering representation of screened potentials offers a phase-shift view that could be calibrated directly against the DFT contact energy.

Load-bearing premise

The homogeneous electron gas model together with the chosen exchange-correlation functional accurately describes the nonlinear induced density close to the proton.

What would settle it

A direct computation or measurement of the induced density or contact Hartree energy near a proton embedded in a real crystalline metal that deviates by more than a few percent from the jellium LDA benchmark.

Figures

Figures reproduced from arXiv: 2606.23065 by A. X. Chen, G. Rodway-Gant, M. Sherafati.

Figure 1
Figure 1. Figure 1: FIG. 1. Short-range dominance of the electrostatic KS [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Screening functions [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Static density local-field factor [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Failure of linear response in the contact region of [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Contact-density breakdown of linear response. The [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Linear-response contact Hartree energy [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Contact Hartree potential energy as a function of [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Partial-wave phase shifts for the three screened [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 2
Figure 2. Figure 2: Although all three potentials share the same [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Cumulative partial-wave Friedel sum for a pro [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Self-consistent screening parameters and the cor [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. (Top Panel) Contact Hartree potential energy [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Induced electron charge density ∆ [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
read the original 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 Hulth\'en 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript claims that a unified, nonlocal definition of the contact Hartree energy U_H(0) as the radial moment 4π ∫ n_ind(r) r dr of the induced density allows consistent comparison of linear-response dielectric models (Thomas-Fermi, Lindhard/RPA, local-field corrected) with nonlinear LDA-DFT calculations for a static proton in jellium. It reports that the Estreicher-Meier LDA parametrization reproduces U_H(0) from the authors' direct LDA runs and from Almbladh et al. (1976), separates the hydrogenic profile from Friedel oscillations, finds that Corradini-Del Sole-Onida-Palummo and Kaplan-Kukkonen local-field factors give indistinguishable contact screening, and shows that Yukawa/hydrogenic/Hulthén potentials constrained by the Friedel sum rule cannot alone recover the nonlinear DFT value. The work positions the results as a one-center benchmark for jellium screening.

Significance. If the reported numerical agreement holds, the paper supplies a reproducible, explicitly nonlocal benchmark for nonlinear static screening of a point charge in jellium that cleanly separates local and oscillatory contributions. Credit is due for the direct, side-by-side comparison against the Almbladh et al. self-consistent results and for testing modern local-field factors on the same observable. The formulation is useful as a reference before extending to two-center problems; the stress-test concern about transferability to real metals with band structure does not land because the manuscript confines its claims to the jellium model.

minor comments (2)
  1. The abstract states that the Estreicher-Meier parametrization 'reproduces' the contact energy from direct LDA and Almbladh et al.; the main text should tabulate the numerical U_H(0) values (with any stated convergence tolerances) so that the degree of agreement is immediately verifiable.
  2. Notation for the induced density n_ind(r) and the moment definition of U_H(0) is introduced in the abstract; a short dedicated subsection or equation block early in the methods would improve readability for readers comparing linear-response and DFT routes.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines U_H(0) explicitly as the nonlocal radial moment 4π ∫ n_ind(r) r dr of the full induced density, independent of any model. It then performs consistency checks by comparing this quantity across Thomas-Fermi, Lindhard, local-field-corrected dielectrics, direct LDA calculations, the Estreicher-Meier parametrization, and the external 1976 Almbladh et al. benchmark. No load-bearing step reduces a prediction to a fitted input, self-citation, or definitional equivalence; the reproduction is an external validation inside the jellium LDA framework. The derivation chain is self-contained against independent benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review limits visibility into explicit free parameters or invented entities; the work relies on standard jellium assumptions and existing dielectric models without introducing new postulates.

axioms (2)
  • domain assumption Homogeneous electron gas (jellium) model accurately represents screening near a proton impurity
    Invoked throughout the abstract as the physical setting for all calculations and comparisons.
  • standard math Friedel sum rule constrains the phase shifts of screened potentials
    Used to analyze Yukawa, hydrogenic, and Hulthén potentials.

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