Theoretical Insights on Nonlinear Response Theory of Fluorescence Spectroscopy in Liquids

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.