For any static interaction the self-energy and vertex corrections to the electron-gas polarizability combine exactly into one integrand free of the Coulomb singularity

Abstract

The first-order corrections to the polarizability of the uniform electron gas are a self-energy insertion and a vertex correction. For the Coulomb interaction the coefficient of q^2 in each diverges logarithmically at small wave vector, and the two divergences have opposite signs and cancel in their sum. This Note shows that for any interaction that depends only on the momentum it transfers, the two integrands combine exactly into one before any integration is done. At zero temperature the combined integrand has no Coulomb singularity and no chemical-potential counterterm. For screened interactions, which must be integrated numerically, it should be computed in place of the two diagrams. The Note gives the derivation and its physical reason, the one term added at finite temperature, and checks for bare, Yukawa and screened interactions. With a frequency-dependent interaction the combination is no longer exact.