Wave-vector dependence of the exchange contribution to the electron-gas response functions: An analytic derivation

We present an analytic procedure for evaluating the wave-vector (q) dependence of the lowest-order exchange contribution to the density and spin-density response functions for the homogeneous electron gas. The two types of contributing diagrams are calculated separately by different methods. The simpler one based on the lowest-order self-energy insertion can be integrated directly. To obtain the analytic form of the more complex ‘‘vertex correction’’ diagram a differential equation is derived from the original integral representation and then integrated. The derivative of the result has a (ln‖q-2${\mathit{k}}_{\mathit{F}}$‖${)}^3$ divergence at q=2${\mathit{k}}_{\mathit{F}}$, which is stronger than that of the Lindhard function. Some consequences of this singularity are discussed, e.g., the asymptotic structure of the statically screened potential of an impurity in a metal and the density fluctuation induced by it. Furthermore, from the low-q expansion of the result we obtain higher-order gradient corrections to the exchange energy functional within linear response.