Dielectric function of the uniform electron gas for large frequencies or wave vectors

The dielectric function of a uniform electron gas is studied on the basis of the dynamical equations which govern the response of the system to a weak external field. The deviations from the random-phase approximation caused by exchange and correlation effects are incorporated in a local-field correction which is related to the response of a certain two-particle correlation function. Using the equation of motion for the correlation function we extract the exact behavior of the local-field correction for large wave vectors or high frequencies. The high-frequency result is identical to the one obtained from the third frequency moment. For large wave vectors we find that the local-field correction tends to $\frac{2[1-g(0\left)\right]}{3}$, $g\left(0\right)\mathrm{}$ being the value of the pair distribution function at $r=0\mathrm{}$. We also recover the result of Kimball, giving a relation between the pair distribution function and its radial derivative at $r=0\mathrm{}$.