Exact dynamical exchange-correlation kernel of a weakly inhomogeneous electron gas

The dynamical exchange-correlation kernel fxc of a nonuniform electron gas is an essential input for the time-dependent density-functional theory of electronic systems. The long-wavelength behavior of this kernel is known to be of the form fxc=α/q2 where q is the wave vector and α is a frequency-dependent coefficient. We show that in the limit of weak nonuniformity the coefficient α has a simple and exact expression in terms of the ground-state density and the frequency-dependent kernel of a uniform electron gas at the average density. We present an approximate evaluation of this expression for Si and discuss its implications for the theory of excitonic effects.