  We formulate a rigorous method for calculating a nonadiabatic
(frequency-dependent) exchange-correlation (XC) kernel required for correct
description of both equilibrium and nonequilibrium properties of strongly
correlated systems within Time-Dependent Density Functional Theory (TDDFT). To
do so we use the expression for charge susceptibility provided by Dynamical
Mean Field Theory (DMFT) for the effective multi-orbital Hubbard Model. We
tested our formalism by applying it to the one-band Hubbard model: our
nonadiabatic kernel leads to a significant modification of the excitation
spectrum, shifting the peak that appears in adiabatic (simplified) solutions
and disclosing a new one, in agreement with the DMFT solution. We also used our
method to track the nonequilibrium charge-density response of a multi-orbital
perovskite Mott insulator, YTiO3, to a perturbation by a femtosecond (fs) laser
pulse. The results were quite different from those provided by the
corresponding adiabatic formalism. These initial investigations indicate that
electron-electron correlations and nonadiabatic features can significantly
affect the spectrum and nonequilibrium properties of strongly correlated
systems.
