Correlation Energy of an Electron Gas at High Density

TLDR

The correlation energy per particle of an electron gas, ε_c, is defined in rydbergs and depends on the dimensionless density parameter r_s, where r_s^{-3} is proportional to electron density. The authors compute ε_c by summing the most divergent terms of the perturbation series under the integral sign, employing a technique analogous to Feynman’s field‑theory methods to obtain a convergent result. For high‑density (small r_s) electron gases, ε_c is found to follow ε_c = 0.0622 ln r_s + C + O(r_s), with an exact expression for the constant C presented for the first time and a method for determining the next r_s correction.

Abstract

The quantity ${\ensuremath{\epsilon}}_{c}$ is defined as the correlation energy per particle of an electron gas expressed in rydbergs. It is a function of the conventional dimensionless parameter ${r}_{s}$, where ${{r}_{s}}^{\ensuremath{-}3}$ is proportional to the electron density. Here ${\ensuremath{\epsilon}}_{c}$ is computed for small values of ${r}_{s}$ (high density) and found to be given by ${\ensuremath{\epsilon}}_{c}=A\mathrm{ln}{r}_{s}+C+O({r}_{s})$. The value of $A$ is found to be 0.0622, a result that could be deduced from previous work of Wigner, Macke, and Pines. An exact formula for the constant $C$ is given here for the first time; earlier workers had made only approximate calculations of $C$. Further, it is shown how the next correction in ${r}_{s}$ can be computed. The method is based on summing the most highly divergent terms of the perturbation series under the integral sign to give a convergent result. The summation is performed by a technique similar to Feynman's methods in field theory.