Time-dependent Hartree-Fock dielectric function for the uniform electron gas

The time-dependent Hartree-Fock (TDHF) formula for the dielectric function, which was derived by the authors in a previous paper, is here adapted to the uniform electron gas. The TDHF formula is put into a form that enables one to apply the analytic tetrahedron method. Convergence studies show that a large number of tetrahedrons are required to accurately compute the first or random-phase approximation (RPA) term in the TDHF susceptibility, while a smaller number are required for the second or exchange term. Hartree-Fock (HF) energies and an $r_s$ value of 1.74 were used to compute both terms in the TDHF susceptibility as a function of wave vector $\overrightarrow{\mathrm{q}}$. The second or exchange term was found to be of the same order of magnitude as the first or RPA term. The full TDHF susceptibility was found to differ substantially from the RPA susceptibility computed with Hartree energies. Thus exchange, which is contained in both the HF energies and the exchange term itself, was found to have a major effect on the susceptibility. It was found that the TDHF susceptibility has an absolute maximum at a nonzero value of $\overrightarrow{\mathrm{q}}$, indicating that when exchange effects are included in the susceptibility, the ground state of the uniform electron gas contains a spin density wave.