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

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.

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.

Efficient Criterion To Evaluate Linear Response Theory in Optical Transitions

The role of the Gaussian statistics on the solvation dynamics upon the photon excitation of the chromophore is deeply explored. The linear response theory for the fluorescence Stokes shift is investigated. An analytical formulism is presented to recast Stokes shift into the contributions of the equilibrium time correlation functions of the solute-solvent interactions on the excited-state surface, and the latter is further reformed and depicted by the time relaxation of the moment. As the first application of the formulism in the molecular dynamics simulations, it is verified that the efficiency of the linear response theory relies on the Gaussian characteristics of the dominant moments in terms of the Stokes shift, which is identified by the same relaxation dynamics between those moments and the linear order one. The comparisons between the above observations on the linearity of Stokes shift and the explanations in the literature are discussed. The key finding is the development of explicit criterion to measure the appropriateness of applying linear response theory.