Exact static exchange benchmark for the uniform electron gas: Dynamic-kernel tests and exchange-correlation cancellation near 2kF

Carl A. Kukkonen

doi:10.5281/zenodo.21497366

paper

Exact static exchange benchmark for the uniform electron gas: Dynamic-kernel tests and exchange-correlation cancellation near 2kF

arXiv:2607.20403 · doi:10.5281/zenodo.21497366

Abstract

The static local field factors of the uniform electron gas are known accurately from quantum Monte Carlo (QMC) calculations, but two of their most striking features have lacked a mechanism: $G_{+}$ grows almost exactly as $q^{2}$ up to nearly $2 k_{F}$ , and the violent $2 k_{F}$ physics of the Fermi surface leaves only a faint trace in it. Combining the textbook first-order local-field relation with the exact static exchange polarizability gives a compact exact benchmark: the universal exchange-only local field factor $G_{x} (q, 0) = - I (Q) Q^{2} / L (Q)^{2}$ , $Q = q /2 k_{F}$ , with exact values $(1/4) (q / k_{F})^{2}$ at small $q$ , $I ¨ ^{2} /6$ at $2 k_{F}$ -- apparently never before stated as a local-field value -- and $1/3$ at large $q$ , giving the exchange-only spin-averaged contact value $g (0) = 1/2$ of the unpolarized gas (BDL 1977). $G_{x}$ rises to a finite maximum of 1.9924 at $1.9685 k_{F}$ and passes through $I ¨ ^{2} /6$ at exactly $2 k_{F}$ with a slope diverging as $ln^{3} ∣ q - 2 k_{F} ∣$ . Against this benchmark, the 1980 BDL tabulation is accurate to about one percent below $1.5 k_{F}$ , while the 2024 McLachlan-variational kernel of Nazarov and Silkin is two-percent accurate only below $0.5 k_{F}$ , is a factor of 1.74 low at $2 k_{F}$ , and its large-q limit is inconsistent with the contact constraint. A spin-resolved decomposition of QMC local field factors about the exact baseline shows that, within the resolution and smoothness assumptions of the QMC-based representations, both features follow from a strong exchange-correlation cancellation: at the QMC point closest to $2 k_{F}$ ( $q = 2.01 k_{F}$ , $r_{s} = 2$ ), correlation cancels 0.43(7) of the exact exchange value 1.33. Computed consistently at each wavevector, the static exchange-kernel thresholds derived from the closed form reproduce, to 0.2%, the finite-wavevector charge threshold $r_{s} \approx 10.6$ near $q = 1.94 k_{F}$ obtained numerically in 1980, and recovered again in 2014.

17 pages, 4 figures, 1 table