Screened second-order exchange in the uniform electron gas: exact reduction, a single-pole reference model and asymptotic analysis

Abstract

The uniform electron gas underlies the local-density approximation of density-functional theory. At high density the random-phase approximation (RPA) captures long-range screening but misses short-range, exchange-type correlation, leaving a systematic error in the correlation energy. A natural correction is the screened second-order exchange (SOSEX): the second-order exchange diagram with one interaction line replaced by a frequency-dependent screened interaction. We evaluate this contribution in a model where it can be treated analytically. The reduction rests on exact multiplicative separation of the momentum variable $q$ and the scaled frequency $z=ξ/q$. Within the single-pole class, this separation selects a momentum-independent screening scale $Λ$; we call the resulting model the reduction-compatible single-pole (RC-SP) model, in which the contribution becomes a one-dimensional integral in $Λ$. The quantity obtained is not the full correlation energy of the Coulomb system, but the contribution of the screened-second-order-exchange diagram. As $Λ\to\infty$ the screened interaction returns to the bare Coulomb one and the contribution reduces to the bare second-order exchange; we evaluate this limit in closed form and find agreement with the Onsager-Mittag-Stephen value, a nontrivial check that also fixes an absolute energy scale. We analyze the $Λ$-dependence by the Mellin-Barnes method, obtaining small- and large-$Lambda$ asymptotics. Numerics confirm these and show a non-monotone dependence: over an intermediate range the normalized contribution exceeds its static-limit value. Mapping $Λ$ to the density parameter $r_s$ yields specific functional forms that, following from the diagram structure rather than empirical choice, can guide modeling of screened-exchange contributions beyond the RPA.