Correlation Energy of an Electron Gas at High Density

The quantity ${\epsilon }_c$ 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 ${r_s}^{-3\mathrm{}}$ is proportional to the electron density. Here ${\epsilon }_c$ is computed for small values of $r_s$ (high density) and found to be given by ${\epsilon }_c=A\ln r_s+C+O\left(r_s\right)$. 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.