ON THE COULOMB ENERGY OF A FINITE-TEMPERATURE ELECTRON-GAS

We rederive the Coulomb expansion of the electron gas average energy a t finite temperature, starting from scratch, i.e., using only the fram ework of the grand canonical ensemble and not the finite-T Green's fun ction formalism. We recover the analytical expressions of the exchange and correlation energy in both the high- T and the T = 0 limits. We e xplicitly show the origin of the crossover of the correlation energy l eading term from e4 In e2 at zero temperature to e3 at finite T We als o discuss the relative importance of exchange and correlation in both limits.