Exchange and the Dielectric Screening Function

Using many-body techniques, we evaluate the three dielectric screening functions of a free-electron gas including exchange self-energy and exchange ladder bubble diagrams. The three dielectric functions are ${\epsilon }_{\mathrm{kt}}(\kappa , \omega )$, ${\epsilon }_{\mathrm{tt}}(\kappa , \omega )$, and ${\epsilon }_{\mathrm{kk}}$, ($\kappa , \omega$), appropriate for screening the interaction between an electron and a test charge, between a pair of test charges, and between a pair of electrons. An approximation for integrals of the screened exchange interaction previously used by Hubbard and by us enables us to evaluate the dielectric screening functions for all values of wave vector $\kappa$ and frequency $\omega$. The form of ${\epsilon }_{\mathrm{kt}}$ and ${\epsilon }_{\mathrm{tt}}$ is similar but not identical to that obtained previously using self-consistent-field techniques.