Static Response and Local Field Factor of the Electron Gas

We have evaluated the density-density static response of the electron gas at zero temperature and in the metallic regime by diffusion Monte Carlo. The computed local field factor $G\left(q\right)$ smoothly interpolates between the asymptotic small and large $q$ behavior, with a crossover around ${2q}_F$. In fact, $G\left(q\right)$ appears to be almost completely given by its asymptotes, being accurately reproduced by the local density approximation to density functional theory for $q\lesssim {2q}_F$. We give a simple formula to reproduce $G\left(q\right)$ at relevant values of the wave vector.