Efficient Criterion To Evaluate Linear Response Theory in Optical Transitions

Explicit Projection of Stokes Shifts onto Solvent Motion in an Aqueous Liquid and Linear Response Theory

We investigate the molecular origin of the fluorescence Stokes shift in an aqueous liquid. By examining the speed of energy change, the solvation response function is explicitly projected onto the translational and rotational motions of water molecules for both nonequilibrium relaxation and equilibrium fluctuations. Molecular dynamics simulations of a tryptophan solution show that these two processes have highly consistent dynamics, not only for the total response function but also for the decomposed components in terms of specific molecular movements. We found that the rotational mode governs the relaxation of the Stokes shift, whereas the translational mode contributes non-negligibly with slower dynamics. This consistency implies the similarity of the underlying translational and rotational movements of water molecules as the system is far away from and at equilibrium, supporting the validity of the linear response theory at the molecular level. The decomposition methodology is also applicable to a rigid solvent.

Solvation Dynamics in Water. 4. On the Initial Regime of Solvation Relaxation

It is shown, by means of numerical and analytic work, that initial molecular momenta play little significant role in the initial fast solvation relaxation that follows electronic excitation of, and charge creation for, a standard model system of a solute in water. Instead, the nonequilibrium dynamics are predominantly described by noninertial "steering" by the torques directly generated by the newly created charge distribution. It is this process that largely overcomes inertia and drives the relaxation dynamics on a time scale of a few tens of femtoseconds in the key initial regime of the dynamics. These results are discussed in the context of commonly employed descriptions such as inertial, Gaussian, and underdamped dynamical behavior.

To unravel the connection between the non-equilibrium and equilibrium solvation dynamics of tryptophan: success and failure of the linear response theory of fluorescence Stokes shift

The connections between the non-equilibrium solvation dynamics upon optical transitions and the system's equilibrium fluctuations are explored in aqueous liquid. Linear response theory correlates time-dependent fluorescence with the equilibrium time correlation functions. In the previous work [T. Li, J. Chem. Theory Comput., 2017, 13, 1867], Stokes shift was explicitly decomposed into the contributions of various order time correlation functions on the excited state surface. Gaussian fluctuations of the solute-solvent interactions validate linear response theory. Correspondingly, the deviation of the Gaussian statistics causes the inefficiency of linear response evaluation. The above mechanism is thoroughly tested in this study. By employing molecular simulations, multiple non-equilibrium processes, not necessarily initiated from the ground state equilibrium minimum, were examined for tryptophan. Both the success and failure of linear response theory are found for this simple system and the mechanism is analyzed. These observations, assisted by the width dynamics, the initial state linear response approach, and the variation of the solvation structures, integrally verify the virtue of the excited state Gaussian statistics on the dynamics of Stokes shift.

Linear Response Theory for Decomposition Energies of Stokes Shift in Proteins

In aqueous solution, fluorescence Stokes shift experiments monitor the relaxation of the solute-solvent interactions upon photon excitation of the solute chromophore. Linear response (LR) theory expects the identical dynamics between the Stokes shift and the system's spontaneous fluctuations. Whether this identity guarantees similar dynamics between the nonequilibrated and equilibrium processes for the decomposition energy of the Stokes shift is the main focus of this study. In our previous work [Li, T. J. Chem. Theory Comput. 2017, 13, 1867-1873], Stokes shift is properly correlated with various order time-correlation functions. As a continuation, its decomposition energy from the subsystem is further represented as the full summation of all of the cross-time correlation functions between the decomposition energy and the total solute-solvent interactions. Gaussian statistics of the total solute-solvent interactions ensure the same decay rates among the odd orders not only for the time-correlation functions but also for the cross-time correlation functions, validating the LR of the Stokes shift and the decompositions, respectively. The above mechanism is verified by molecular dynamics simulations in the protein Staphylococcus nuclease and is robust even as the decomposed energy associated with an individual residue exhibits typical non-Gaussian properties. Further examinations reveal the consistent molecular motions for a specific residue over the nonequilibrium and equilibrium processes, which are responsible for the nonequilibrium dynamics of the associated decomposed energy. Our results show the appropriateness of LR on finer molecular scales.

Theoretical Insights on Nonlinear Response Theory of Fluorescence Spectroscopy in Liquids

Correlations between the nonequilibrium solvation dynamics upon the photon excitation of the chromophore and a system's equilibrium fluctuations are deeply studied. As the linear response of the solvent has been linked with Gaussian statistics of the energy fluctuations in the literature, we specifically explore the cases beyond the regime of the linear response theory due to deviation from Gaussian fluctuations. As a continuation of our previous work, an analytical formalism is presented to project the energy shift with various order moments, where the non-Gaussian statistics arise from the overlap of the energy basins on the perturbed potential energy surface. It is shown that the nonequilibrium dynamics still correlate with the spontaneous regressions at equilibrium and are controlled by the decay rates of those higher order components with the prevailing contributions to the energy shift. Molecular dynamics simulations were performed in the protein Staphylococcus nuclease, in which even the dynamics of the high order moments are available. The results further verify the above relationship. Our scheme is used to evaluate Stokes shift using the information on non-Gaussian statistics at equilibrium, thus presenting a broad picture on the correlation between the nonequilibrium process and equilibrium properties in liquids.