Approximate Screening Functions in Metals

A variational principle is used to solve an approximate integral equation for the screening functions of the electron gas. It is shown that the simplest possible trial function gives a solution to the equation for the dielectric constant which is exact in both the limits of small and large momentum transfers. The results are compared with other calculations. It is shown that the approximation developed recently by Kleinman is quite good in the static large-k limit, but otherwise incorrect. The dielectric constant derived from the variational calculation is used to derive an expression for the ground-state energy; this expression is similar in its essential features to the interpolation schemes of Hubbard and of Nozi\`eres and Pines, even though the Hubbard approximation considerably underestimates the exchange enhancement of the vertex function at large k. Finally, it is suggested that similar variational principles may have other uses, as in the paramagnon problem.