Variational Solution of Vertex Equation and Dielectric Function of an Interacting-Electron Gas

It is shown that the static dielectric function computed from a variational solution of the appropriate vertex function obeys the compressibility sum rule and reaches the well-established limit for large $\overrightarrow{\mathrm{k}}$. The vertex equation contains the self-energy contributions to the one-electron states as well as the exchange contributions. A comparison of the results with a recent scheme due to Toigo and Woodruff shows that the moment-conserving approximation used by these authors is correct for small wave vectors and incorrect for large wave vectors in contrast to the variational method.