Energy dependence of the exchange-correlation kernel of time-dependent density functional theory: A simple model for solids

Abstract

Time-dependent density functional theory faces an important problem when it comes to extended systems: The long-range component of the exchange-correlation kernel ${f}_{\mathrm{xc}}$ is completely absent from local density or generalized gradient approximations, but it is believed to be present in the ``exact'' ${f}_{\mathrm{xc}}$. Several attempts have been made to solve this issue, the simplest of them being the use of a model static long-range kernel of the form $\ensuremath{-}{\ensuremath{\alpha}}^{\mathrm{static}}∕{q}^{2}$. In this paper, we propose and motivate a dynamical extension of this model of the form $\ensuremath{-}(\ensuremath{\alpha}+\ensuremath{\beta}{\ensuremath{\omega}}^{2})∕{q}^{2}$. The dynamical model is then used to calculate the dielectric function of a large variety of semiconductors and insulators. The absorption spectra of large gap insulators are remarkably improved with respect to calculations where the kernel is taken to be static. This approach is valid also for energies in the range of plasmons, and hence it yields, e.g., good electron energy loss spectra. Finally, we present some simple theoretical arguments that relate the parameters of the model to physical quantities, like the dielectric constant and the plasmon frequency.