Hidden Breakdown of Linear Response: Projections of Molecular Motions in Nonequilibrium Simulations of Solvation Dynamics

The linear response (LR) approximation forms the cornerstone of nonequilibrium statistical mechanics and has found special utility in studies of solvation dynamics, in which LR implies that nonequilibrium relaxation dynamics is governed by the same molecular motions responsible for fluctuations at equilibrium. When the motions at and away from equilibrium fall in the LR regime, the equilibrium and nonequilibrium response functions are identical. However, similarity of the equilibrium and nonequilibrium solvent response functions does not guarantee that LR holds and that the underlying molecular motions are the same. In this paper, we present computer simulation studies of the removal of charge from an atomic solute in liquid tetrahydrofuran, a system for which the equilibrium and nonequilibrium solvation responses appear quite similar. We then introduce a method for projecting nonequilibrium response functions onto specific molecular motions. We find that the equilibrium relaxation is dominated by solvent rotations, whereas the nonequilibrium relaxation is much more complex, having translations dominating at early times and a delayed onset of rotations. The results imply that LR may not hold as often as is widely believed and that care should be taken when using equilibrium response functions to understand nonequilibrium solvation dynamics. Solvents are not just spectators during chemical reactions, nor are they simply a continuum in which a reaction occurs. Rather, the specific motions of individual solvent molecules directly affect the rate of electron transfer and other solutionphase chemical reactions. The study of how solvent motions couple to electronic changes in reacting solutes is known as solvation dynamics, 1 which is typically monitored via the solvation energy gap, ∆E ) E ss exc -E ss gnd , where E ss exc and E ss gnd are the solute-solvent interaction energies when the solute is in the excited and ground states, respectively. The normalized nonequilibrium solvent response function is where R denotes all of the solute and solvent positions and the overbar represents a nonequilibrium ensemble average in which the solute is promoted to the excited state at t ) 0. One of the central themes in the study of solvation dynamics is the idea of linear response (LR), which is based on the Onsager regression hypothesis. 2 In LR, the motions of the solute and solvent molecules that respond to a small perturbation are the same as those that follow naturally from a fluctuation away from equilibrium. In this limit, S(t) is identical to the equilibrium solvation time correlation function (TCF), 2 where the angled brackets denote an equilibrium ensemble average and δ∆E ) ∆E -〈∆E〉 is the equilibrium fluctuation of the energy gap (and the R dependence is repressed). The nature of LR has been explored in simulations of myriad solutesolvent systems, and most have found that, even for very large perturbations, S(t) agrees fairly well with C(t). 3 There have been a few notable exceptions, including simulations of a solute in methanol that undergoes a dipole reversal in the excited state, 4 simulations in water/methanol 5 and water/DMSO 5,6 mixtures, and simulations in water of an atomic solute that changes both size and charge in the excited state. 7 Despite these exceptions, LR is widely believed to hold, and many studies have elected to save computational resources by calculating only the equilibrium solvation TCF via eq 2 instead of computing S(t) from an ensemble of nonequilibrium trajectories, as in eq 1.