Long-wavelength limit and the logarithmic singularities in the dielectric function with dynamical exchange effects

The dielectric function with dynamical exchange effects, derived earlier, was obtained by applying the Hartree-Fock decoupling in the equation of motion for the Wigner distribution function. A "local-field correction" $G(q,\omega )$ was obtained with a variational technique. In the present paper, the decoupled equation of motion is solved exactly in the long-wavelength limit, showing that the variational approach correctly describes dynamical exchange effects in the dielectric function to order $q^2$. Furthermore, it is shown that the logarithmic singularities, obtained earlier, might be due to the particular choice of the trial Wigner distribution function. A more general trial function eliminates these singularities at the boundaries of the particle-hole continuum, but otherwise does not appreciably alter the behavior of the dielectric function.