Nonequilibrium Solvation Dynamics Research Articles

Non-equilibrium solvation dynamics: results beyond linear response theory

Optically controlled non-equilibrium solvation dynamics in condensed phase have been studied extensively both experimentally and theoretically. The main drawback of linear response theory (LRT) and other existing theories is that they are unable to elucidate the dependence of non-equilibrium solvation time correlation function, S(t), on the excitation wavelength, as observed in the experiment. In order to explain the experimental observation, we first develop a kinetic equation in 1D solvation coordinate (SC) space and then we propose a new perturbative method to derive an analytical expression for S(t) for an anharmonic potential in SC space. We observe from the result of S(t) that the calculated results of the same depend strongly on the innumerable optical states through excitation wavelength, when the potential is anharmonic in SC space, indicating the breakdown of LRT. We have shown that both excitation wavelength and rotational relaxation time carry the information of micro-heterogeneity. Another significant aspect of our work is that the analytical expression obtained for S(t) corresponding to the anharmonic potential decays multi-exponentially, whereas S(t) decays single exponentially for a harmonic potential. More significantly, the calculated results for S(t) are found to be in excellent agreement with the available experimental results for the heterogeneous system.

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.

Optical controlled non-equilibrium processes: effect of excitation wavelength

Experimental results on optically controlled non-equilibrium solvation dynamics show their dependence on excitation wavelength, However, the renowned Onsager regression hypothesis and Smoluchowski equation-based approach fail to explain such dependance. Failure of this theory prompted us to develop a novel theory by projecting the dynamics in a higher-dimensional space onto a one-dimensional energy gap space, where the effect of is included explicitly. We have then derived hierarchical equations of moments in time domain for an anharmonic potential based on the new kinetic equation derived here in the energy gap space to evaluate the non-equilibrium solvation time correlation function (NSTCF). The new methodlogy presented here also removes the difficulties in perturbative approaches to evaluate the NSTCF in a condensed phase. We have also shown that the NSTCF is independent of excitation wavelength for harmonic potential but dependent on the same for anharmonic potential, which is in accord with experimental observation. However, Onsager’s regression hypothesis-based prediction for the NSTCF is independent of excitation wavelength for all kinds of potentials. More importantly, the calculated results of NSTCF is in excellent agreement with the experimental results where the Onsager regression hypothesis-based prediction fails.

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.

Non-equilibrium solvation dynamics in water-DMSO binary mixture: Composition dependence of non-linear relaxation.

Because of a larger number of intermolecular interactions and configurations available to them, aqueous binary mixtures exhibit a variety of dynamics that are not seen in pure liquids, often hard to understand or predict, and have attracted considerable interest recently. Among all such solutions, mixtures of water and dimethyl sulfoxide (DMSO) stand out for their unique role in chemistry and biology. The low DMSO concentration regime of the water-dimethyl sulfoxide (DMSO) mixture is relevant in wide ranging chemical and biological processes. Interestingly, this low concentration regime is known to display a string of yet unexplained anomalies. We probe these anomalies in atomistic simulations by studying (i) equilibrium solvation dynamics both in the ground and the excited states of the probe separately and (ii) the non-equilibrium solvation dynamics subsequent to excitation at time t = 0 and then following the solvation process. The latter needed a large number of simulations to obtain a reliable average. We carried out such studies across a large number of compositions of the water-DMSO mixture. We find that the usually employed linear response approximation breaks down at those concentrations where binary mixtures display other anomalies. The non-linearity is reflected in substantially different solvent responses in the ground and in the excited states of the solute probe indole and also in non-equilibrium solvation. The difference is maximum near 20%-35% of the DMSO concentration regime.

Optically Controlled Electron-Transfer Reaction Kinetics and Solvation Dynamics: Effect of Franck-Condon States.

Experimental results for optically controlled electron-transfer reaction kinetics (ETRK) and nonequilibrium solvation dynamics (NESD) of Coumarin 480 in DMPC vesicle show their dependence on excitation wavelength λex. However, the celebrated Marcus theory and linear-response-theory-based approaches for ETRK and NESD, respectively, predict both of the processes to be independent of λex. The above said lacuna in these theories prompted us to develop a novel theory in 1D space, where the effect of innumerable Franck-Condon states is included through λex. The present theory not only sheds light on the origin of failure of the existing theories but also gives the correct trend for the effect of λex on ETRK and NESD. More importantly, the calculated results of NESD are in excellent agreement with the experimental results for different values of λex. The new theory will therefore advance the knowledge of scientific community on the dynamics of photoinduced nonequilibrium processes.

Solvation dynamics in water confined within layered manganese dioxide

The confined environment presented by layered transition metal oxides is conducive to a variety of chemical reactions. Despite intense interest in these materials, little is known regarding the microscopic details relevant to their catalytic activity. We characterize aspects of the dynamics governing a redox reaction in the interlayer environment between manganese dioxide sheets. The nonequilibrium solvation dynamics surrounding charge transfer between an ion and the surface are highly non-linear and exhibit long-time relaxation that is governed by collective dynamics. These dynamics are rationalized in terms of structural rearrangements, allowing connections to be made to more complex reactions in these materials.

On the validity of linear response approximations regarding the solvation dynamics of polyatomic solutes

The time-dependent fluorescence of a chromophore can be calculated from either nonequilibrium simulations, or, as long as linear response theory holds true, from equilibrium solvent fluctuations in the ground or excited state if the perturbation inflicted by the chromophore is small. The assumption of Gaussian statistics, in contrast, links the nonequilibrium dynamics to solvent fluctuations solely in the excited state, as long as the energy gap distribution is Gaussian throughout the process. The validity of linear response theories on the ground and excited state surface as well as Gaussian statistics is thoroughly tested in this study by calculating the time-dependent Stokes shift of different benzene-like solutes. The effect of the size of change in partial charges of the solute, the multipolar order of charge distribution, the direction of change, as well as the influence of different solvents on the validity of linear response theory is examined by simulating 54 different systems. Calculation of the Gaussian character of the energy distribution in equilibrium, as well as the time-evolution of the peak width in the nonequilibrium simulation sheds light on the validity of Gaussian statistics in a nonstationary regime. We observed that a large intermediate broadening of the width of the energy distribution correlates with a failure of correlation functions to describe the nonequilibrium event. These results are accompanied by analysis of higher order correlation functions, as well as the structure of the solvents water, acetonitrile and methanol around the solute, to yield a comprehensive view, as well as general guidelines, on when and why equilibrium solvent fluctuations can correctly depict solvation dynamics.

Extended hydrodynamic approach to quantum-classical nonequilibrium evolution. II. Application to nonpolar solvation

The mixed quantum-classical formulation derived in our companion paper [D. Bousquet, K. H. Hughes, D. Micha, and I. Burghardt, J. Chem. Phys. 134, 064116 (2011)], which is based upon a hydrodynamic representation of the classical sector, is applied to nonequilibrium nonpolar solvation dynamics as exemplified by the solvation of the electronically excited NO molecule in a rare gas environment. Derived from a partition of the Hamiltonian into a primary (quantum) part and a secondary (classical) part the hydrodynamic equations are formulated for multi-quantum states and result in explicit equations of motion for populations and coherences. The hierarchy of hydrodynamic equations is truncated by the following approximate closure schemes: Gauss-Hermite closure, dynamical density functional theory approximation, and a generalized Maxwellian closure. A comparison of the dynamics using these three closure methods showed that the suitability of a particular closure scheme was dependent on the initial conditions and the nonequilibrium character of the dynamics.

Solution reaction space Hamiltonian based on an electrostatic potential representation of solvent dynamics

Quantum chemical solvation models usually rely on the equilibrium solvation condition and is thus not immediately applicable to the study of nonequilibrium solvation dynamics, particularly those associated with chemical reactions. Here we address this problem by considering an effective Hamiltonian for solution-phase reactions based on an electrostatic potential (ESP) representation of solvent dynamics. In this approach a general ESP field of solvent is employed as collective solvent coordinate, and an effective Hamiltonian is constructed by treating both solute geometry and solvent ESP as dynamical variables. A harmonic bath is then attached onto the ESP variables in order to account for the stochastic nature of solvent dynamics. As an illustration we apply the above method to the proton transfer of a substituted phenol-amine complex in a polar solvent. The effective Hamiltonian is constructed by means of the reference interaction site model self-consistent field method (i.e., a type of quantum chemical solvation model), and a mixed quantum/classical simulation is performed in the space of solute geometry and solvent ESP. The results suggest that important dynamical features of proton transfer in solution can be captured by the present approach, including spontaneous fluctuations of solvent ESP that drives the proton from reactant to product potential wells.

Signature of Nonadiabatic Transitions on the Pump−Probe Infrared Spectra of a Hydrogen-Bonded Complex Dissolved in a Polar Solvent: A Computational Study

The mixed quantum-classical Liouville equation provides a unified and self-consistent platform for modeling the spectral signatures of nonequilibrium solvation dynamics, non-Condon effects, and nonadiabatic transitions. These features are demonstrated in this paper in the context of the pump-probe infrared spectra of the hydrogen stretch in a moderately strong hydrogen-bonded complex dissolved in a polar solvent. Particular emphasis is put on incorporating nonadiabatic transitions and accounting for their unique spectral signature.

Signatures of Nonequilibrium Solvation Dynamics on Multidimensional Spectra

Multidimensional electronic and vibrational spectroscopies have established themselves over the last decade as uniquely detailed probes of intramolecular structure and dynamics. However, these spectroscopies can also provide powerful tools for probing solute-solvent interactions and the solvation dynamics that they give rise to. To this end, it should be noted that multidimensional spectra can be expressed in terms of optical response functions that differ with respect to the chromophore's quantum state during the various time intervals separating light-matter interactions. The dynamics of the photoinactive degrees of freedom during those time intervals (that is, between pulses) is dictated by potential energy surfaces that depend on the corresponding state of the chromophore. One therefore expects the system to hop between potential surfaces in a manner dictated by the optical response functions. Thus, the corresponding spectra should reflect the system's dynamics during the resulting sequence of nonequilibrium solvation processes. However, the interpretation of multidimensional spectra is often based on the assumption that they reflect the equilibrium dynamics of the photoinactive degrees of freedom on the potential surface that corresponds to the chromophore's ground state. In this Account, we present a systematic analysis of the signature of nonequilibrium solvation dynamics on multidimensional spectra and the ability of various computational methods to capture it. The analysis is performed in the context of the following three model systems: (A) a two-state chromophore with shifted harmonic potential surfaces that differ in frequency, (B) a two-state atomic chromophore in an atomic liquid, and (C) the hydrogen stretch of a moderately strong hydrogen-bonded complex in a dipolar liquid. The following computational methods are employed and compared: (1) exact quantum dynamics (model A only), (2) the semiclassical forward-backward initial value representation (FB-IVR) method (models A and B only), (3) the linearized semiclassical (LSC) method (all three models), and (4) the standard ground-state equilibrium dynamics approach (all three models). The results demonstrate how multidimensional spectra can be used to probe nonequilibrium solvation dynamics in real time and with an unprecedented level of detail. We also show that, unlike the standard method, the LSC and FB-IVR methods can accurately capture the signature of solvation dynamics on the spectra. Our results also suggest that LSC and FB-IVR yield similar results in the presence of rapid dephasing, which is typical in complex condensed-phase systems. This observation gives credence to the use of the LSC method for modeling spectra in complex systems for which an exact or even FB-IVR-based calculation is prohibitively expensive.

Effects of Solute Electronic Polarizability on Solvation in a Room-Temperature Ionic Liquid

The effects of solute polarizability on solvation and solute transport in the room-temperature ionic liquid 1-ethyl-3-methylimidazolium hexafluorophosphate (EMI+PF(6)-) are investigated via molecular dynamics simulations. A valence-bond description is employed to account for the instantaneous adjustment of the solute electronic charge distribution to the fluctuating solvent environment. It is found that the ultrafast inertial component of solvation dynamics becomes slower as the solute polarizability grows. Moreover, its contribution to overall solvent relaxation becomes reduced with increasing polarizability, especially in the case of nonequilibrium solvation dynamics. Overall, the inclusion of the solute electronic polarizability in the simulations improves the agreement with time-dependent Stokes shift measurements.

A molecular dynamics computer simulation study of room-temperature ionic liquids. II. Equilibrium and nonequilibrium solvation dynamics

The molecular dynamics (MD) simulation study of solvation structure and free energetics in 1-ethyl-3-methylimidazolium chloride and 1-ethyl-3-methylimidazolium hexafluorophosphate using a probe solute in the preceding article [Y. Shim, M. Y. Choi and H. J. Kim, J. Chem. Phys. 122, 044510 (2005)] is extended to investigate dynamic properties of these liquids. Solvent fluctuation dynamics near equilibrium are studied via MD and associated time-dependent friction is analyzed via the generalized Langevin equation. Nonequilibrium solvent relaxation following an instantaneous change in the solute charge distribution and accompanying solvent structure reorganization are also investigated. Both equilibrium and nonequilibrium solvation dynamics are characterized by at least two vastly different time scales--a subpicosecond inertial regime followed by a slow diffusive regime. Solvent regions contributing to the subpicosecond nonequilibrium relaxation are found to vary significantly with initial solvation configurations, especially near the solute. If the solvent density near the solute is sufficiently high at the outset of the relaxation, subpicosecond dynamics are mainly governed by the motions of a few ions close to the solute. By contrast, in the case of a low local density, solvent ions located not only close to but also relatively far from the solute participate in the subpicosecond relaxation. Despite this difference, linear response holds reasonably well in both ionic liquids.

Dynamics of an Excess Electron at Metal/Polar Interfaces

The low-temperature equilibrium and nonequilibrium solvation dynamics of an excess electron in bulk and at a surface interface have been characterized using mixed quantum/classical simulation methods. A methanol bath was modeled using classical molecular dynamics, whereas the properties of an excess electron were calculated from a wave function based approach. The temperature dependence of the bulk response has been found to be minimal, which may indicate that large scale hydrogen bond breaking and diffusive motion may not play important roles in the solvation dynamics. The equilibrium dynamics have also been compared to nonequilibrium simulations of charge injection into the neat bath. An excess electron at a methanol/Pt(100) surface interface has been shown to become solvated by the bath yet is less bound by a factor of ∼2 compared to the bulk. The solvent response function also displays interesting differences when compared to the low temperature bulk glass. These results also reveal many of the microscopic properties of the solvated interfacial electrons that have recently been observed from ultrafast two photon photoelectron spectroscopy studies.

Understanding Nonequilibrium Solute and Solvent Motions through Molecular Projections: Computer Simulations of Solvation Dynamics in Liquid Tetrahydrofuran (THF)

In this paper, we investigate the solvation dynamics of the weakly polar organic solvent tetrahydrofuran (THF) via classical molecular dynamics simulation. We find that the relaxation dynamics of all of the rotational and translational degrees of freedom of neat THF occur on similar time scales and have similar power spectra, making it impossible to use spectral density analysis to discern which specific molecular motions are involved in solvation. Instead, we probe the molecular origins of solvation dynamics using a nonequilibrium projection formalism that we originally outlined in M. J. Bedard-Hearn et al., J. Phys. Chem. A 2003, 107 (24), 4773. Here, we expand this formalism and use it to study the nonequilibrium solvation dynamics for a model reaction in THF in which a charge is removed from an anionic Lennard-Jones (LJ) solute, leaving behind a smaller neutral atom. The solute parameters are chosen to model the photodetachment of an electron from a sodium anion, Na- → Na0, to compare to the results of ultrafast spectroscopic experiments of this reaction being performed in our lab. We are able to explain the hidden breakdown of linear response for this system that we uncovered in our previous work in terms of the dynamical properties of the neat liquid and the structural properties of the solutions. In particular, our nonequilibrium projection analysis shows that four distinct solvation mechanisms are operative: (1) a rapid relaxation (t ≤ 700 fs) caused by longitudinal translational motions that dramatically change the local solvation structure; (2) a slower relaxation (t > 700 fs) caused by diffusive longitudinal translational motions that completes the transformation of the long-range solute−solvent packing; (3) an early-time relaxation (t < 500 fs) caused by solvent rotational motions that destabilizes the (unoccupied) anion ground state via Coulomb interactions; (4) a longer time process (t ≥ 500 fs) caused by solvent rotational motions that increases the solvation energy gap. This last process results from the LJ interaction component of the solvent rotational motions, as the nonequilibrium dynamics bring smaller THF−oxygen sites closer to the excited solute in place of larger THF−methylene sites. Overall, the simulations indicate that nonequilibrium solvation dynamics involves cooperative motions of both the solute and solvent and consists of multiple competing relaxation processes that can affect the solvation energy gap in opposite directions.

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.

Importance sampling and theory of nonequilibrium solvation dynamics in water

We have devised a novel importance sampling method for nonequilibrium processes. Like transition path sampling, the method employs a Monte Carlo procedure to confine or bias the search through trajectory space. In this way, molecular dynamics trajectories consistent with the nonequilibrium dynamics of interest are generated efficiently. Using results of this sampling, we demonstrate that statistics of the energy gap between a solute’s electronic states are Gaussian throughout the dynamics of nonequilibrium solvation in water. However, these statistics do change in time, reflecting linear response that is nonstationary. Discrepancies observed between the dynamics of nonequilibrium relaxation and of equilibrium fluctuations are thus explained. We analyze a simple Gaussian field theory that accounts for this nonstationary response.

Study of the van der Zwan–Hynes model for dipole isomerization reaction rates from the viewpoint of critical phenomena

We study the van der Zwan–Hynes model of nonequilibrium solvation dynamics for reactions in polar solvents. In this model those authors showed the equivalence of a multidimensional reaction coordinate picture to a simple one-dimensional description. They identified several distinct regimes and successfully described these in molecular terms. They found that in the regime of long solvent response times, the reaction rate had singular aspects which could be described in terms of simple power laws. There are two sets of exponents depending on whether the solvent is underdamped or overdamped. In this paper, we show that this behavior can be readily understood by mapping this problem onto critical phenomena. We find that the two sets of exponents correspond to two types of critical behaviors. There is also a regime when both types of critical behaviors are present and one can see the crossover effects between the two by varying the parameters. Studying the resulting crossover scaling function gives new insights into this problem and reveals its rich behavior.

Effects of non-equilibrium solvation dynamics on barrierless electronic relaxation in solution

The effects of non-equilibrium solvation dynamics on zero-barrier photoisomerization reactions are studied theoretically. A theory based on general considerations is presented. A two-state model for the time dependence of the sink function is introduced and solved analytically for the survival probability of the excited state. Several interesting non-equilibrium solvation-induced effects which may be experimentally observable are predicted.