Exchange-correlation kernel in time-dependent density functional theory derived from many-body theory

We derive a simplified Bethe-Salpeter equation for calculating optical absorption based on the assumption of a local electron-hole interaction. The original four-point equation for the kernel is reduced to a two-point one. A connection to the exchange-correlation kernel in time-dependent density functional theory can be established. The resulting f(xc) is found to be -W/2 where W contains only the short-range (local) part of the Coulomb screened interaction. This simple approximation was successfully applied to optical absorption spectra of some excitonic crystals, reproducing not only the continuum excitons but also the bound ones.