Frequency Dependence of the Exact Exchange-Correlation Kernel of Time-Dependent Density-Functional Theory

We present a scheme for numerically reconstructing the exact and the Kohn-Sham time-dependent density-density response functions, and from their inverse the numerical representation of the exact frequency-dependent exchange-correlation kernel f(xc) of time-dependent Kohn-Sham density-functional theory. We investigate the exact f(xc) in detail for a system that is known to show strong memory effects. The reconstructed kernel fulfills the appropriate sum rule. Using it in linear response calculations, we show how the exact f(xc), due to its frequency dependence, yields the exact excitation energies, including the ones of double excitation character.