Joint time-dependent density-functional theory for excited states of electronic systems in solution

We present a joint time-dependent density-functional theory for the description of solute-solvent systems in time-dependent external potentials. Starting with the exact quantum-mechanical action functional for both electrons and nuclei, we systematically eliminate solvent degrees of freedom and thus arrive at coarse-grained action functionals that retain the highly accurate ab initio description for the solute and are, in principle, exact. This procedure allows us to examine approximations underlying popular embedding theories for excited states. Finally, we introduce an approximate action functional for the solute-water system and compute the solvatochromic shift of the lowest singlet excited state of formaldehyde in aqueous solution, which is in good agreement with experimental findings.