The exact universal exchange-only local field factor of the uniform electron gas and its near-cancellation by parallel-spin correlation near 2kF

Abstract

The exchange-only local field factor Gx(q,omega) of the uniform electron gas is exact and universal, a single curve at every density when q is measured in units of kF and omega in units of the Fermi energy. This paper assembles the known pieces of the static Gx(q,0) into one expression, evaluates it exactly at every wavevector, and releases it as a short Python module. It rises as (q/kF)^2/4, has a large peak of 1.99 just below 2kF, passes through the exact value pi^2/6 at 2kF, and tends to 1/3. Used alone in static linear response it predicts a spin density wave at rs = 4.96 and a charge density wave at rs = 10.62. Quantum Monte Carlo shows the gas has neither. Exchange enters the measured density and spin factors G+ and G- identically, so their difference, twice the antiparallel-spin factor, contains no exchange at all and is measured to be small and smooth through 2kF. Whatever removed the exchange peak is therefore in the parallel-spin channel, where everything beyond first-order exchange is large, negative, and mirrors the exchange peak. In the measured G+ the net removal is 40 to 48 percent of the peak at rs = 1 to 10. The stability of the metallic electron gas against exchange-driven density waves is a parallel-spin effect, and the decomposition shows where microscopic calculations of each channel should go.