Leading-order behavior of the correlation energy in the uniform electron gas

We show that, in the high-density limit, restricted Møller-Plesset (RMP) perturbation theory yields $E_{\text{RMP}}^{(2)} = π^{-2}(1-\ln 2) \ln r_s + O(r_s^0)$ for the correlation energy per electron in the uniform electron gas, where $r_s$ is the Seitz radius. This contradicts an earlier derivation which yielded $E_{\text{RMP}}^{(2)} = O(\ln|\ln r_s|)$. The reason for the discrepancy is explained.

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