Conserving approximations: Electron gas with exchange effects

The conserving approximation includes a correct mix of the self-energy Σ(p) and the vertex function Γ(p,p+q). Only keeping the exchange self-energy Σx(p), we obtain the self-consistent equation for the vertex function Γx(p,p+q) in the closed form, which can be solved numerically. Using this vertex function, the polarization Px(q) is obtained. We evaluate the long-wavelength limit of Px(q) in the static case at both zero and nonzero temperatures. The latter case is compared with results obtained using the density-functional approach