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

Abstract: We derive an exact reduction of the screened second-order exchange (SOSEX) energy in the uniform electron gas to a triple integral for a specific class of single-pole screened interaction. The reduction proceeds by rescaling the frequency variable to factorize the propagator denominators, applying a Fourier decomposition to separate the two particle-hole blocks, and finally performing a change of integration variables that brings the geometric structure into a tractable form. The reduction to a one-variable integral kernel is possible if and only if the screened interaction belongs to a one-pole class characterized by a single momentum-independent frequency scale~$μ$, which we call the reduction-compatible single-pole (RC-SP) model. The RC-SP model does not approximate plasmon dispersions in real materials, but provides an exactly reducible reference model for analyzing dynamically screened exchange, and gives natural basis elements for approximating more general one-pole screening. We analyze the $μ$-dependence of the SOSEX energy asymptotically at both small and large~$μ$ and establish the leading behaviors at the theorem level. Under a power-law mapping from~$μ$ to the density parameter~$r_s$, this asymptotic structure constrains the analytic form of the screened-exchange correction in $r_s$-space, providing a diagrammatically justified basis for beyond-RPA functional construction. Direct numerical integration of the reduced representation confirms the asymptotic behaviors quantitatively.