Projections of quantum observables onto classical degrees of freedom in mixed quantum-classical simulations: Understanding linear response failure for the photoexcited hydrated electron

We present a general analytic method for understanding how specific motions of a classical bath influence the dynamics of quantum-mechanical observables in mixed quantum-classical molecular dynamics simulations. We apply our method and develop expressions for the special case of quantum solvation, allowing us to examine how specific classical solvent motions couple to the equilibrium energy fluctuations and nonequilibrium energy relaxation of a quantum-mechanical solute. As a first application of our formalism, we investigate the motions of classical water underlying the equilibrium and nonequilibrium excited-state solvent response functions of the hydrated electron; the results allow us to explain why the linear response approximation fails for this system.