Efficient Criterion To Evaluate Linear Response Theory in Optical

Letter pubs.acs.org/JCTC

Efficient Criterion To Evaluate Linear Response Theory in Optical Transitions Tanping Li* School of Physics and Optoelectronic Engineering, Xidian University, Xi’an, Shaanxi 710071, People’s Republic of China ABSTRACT: 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.

L

analyzed to inspect the solvation dynamics of protein Staphylococcus nuclease (SNase). The origin of the linearity for the Stokes shift is examined, and the comparisons with the explanations in the literature are discussed. Finally in the Conclusion the results are summarized. The method provides a novel perspective on the connection between the nonequilibrium dynamics and equilibrium Gaussian fluctuations and allows development of explicit criterion to measure the appropriateness of applying linear response theory.

inear response theory plays a critical role in exploring solvation dynamics by linking the nonequilibrium relaxation of the system to its equilibrium fluctuations. It predicts that the rates of the nonequilibrium dynamics and the spontaneous regression fluctuations are identical,1 thus forming the foundation to use dynamic fluorescence Stokes shift spectroscopy to probe equilibrium hydration dynamics.2−7 The linearity of the response, owing to the small perturbation in the statistical mechanical perturbation theory,1 is revealed to result from Gaussian statistics,8−11 even though the system may very well be far out of the equilibrium.9,10 The statistical property on the ground-state surface of the chromophore has been used to approach the linear response theory in previous of studies.12−17 In more recent literature, the Stokes shift was correlated with the equilibrium statistics of the solute−solvent interactions on the perturbed surface generated by optical transitions,9,18 and the Gaussian fluctuations on excited-state surface is thus tied to the linear response for time-dependent fluorescence. To deeply understand the connection between nonequilibrium solvation dynamics and the Gaussian statistics, a formulism is presented to project the Stokes shift into the contributions of equilibrium time correlation functions of the solute−solvent interactions on the excited-state surface, in which the latter is further reformed as the time relaxations of the moments in this work. The Gaussian characteristics of the moments, especially those dominant terms, are correlated with the validity of the linear response theory. Descriptions of the above theory are presented in the Theoretical Background section. As a first application of the formulism in the Results and Discussion, molecular dynamics (MD) simulations are © 2017 American Chemical Society

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THEORETICAL BACKGROUND Stokes Shift, Moment, and Linear Response Theory. Stokes shift experiments monitor the nonequilibrium ensemble average of the solute−solvent interactions propagating on the excited-state surface upon photon probe. After the time interval t from onset of the perturbation, the solute−solvent interaction is defined as the shifted energy ΔE(t) − ⟨ΔE⟩e for each configuration. Here ΔE(t) is the difference in energy between the excited- and ground-state surface, and angular brackets ⟨···⟩e represent an equilibrium ensemble average on the excited-state 1 surface. In this context, the Boltzmann factor β = k T is B introduced as the scaling factor for the interaction energy by δΔE(t) = β(ΔE(t) − ⟨ΔE⟩e). δΔE(t) is thus a dimensionless variable with the equilibrium average ⟨δΔE⟩e = 0. Stokes shift is the nonequilibrium ensemble average of the interaction energies by ΔES(t) = ⟨δΔE(t)⟩, which relaxes and decays to zero in the long time limit. The solvation time correlation Received: November 5, 2016 Published: April 17, 2017 1867

DOI: 10.1021/acs.jctc.6b01083 J. Chem. Theory Comput. 2017, 13, 1867−1873

Letter

Journal of Chemical Theory and Computation ΔES(t ) . ΔES(0)

function is evaluated by s(t ) =

δΔE(0,t) with the initial weight Pn(δΔE). This ensemble sum of eq 4 relaxes with time and decays to zero in the long time limit and is represented by Sn(t) in eq 5. The time-dependence of Stokes shift is thus characterized by the relaxation dynamics of Sn(t) since ΔES(t) is the weighed sum over all terms of Sn(t). As δΔE exhibits Gaussian statistics on the excited-state surface, Wick’s theorem20 enables expressing the nth order correlation function ⟨δΔE(t)δΔE(0)n⟩e as being proportional to the linear order time correlation function ⟨δΔE(t)δΔE(0)⟩e when n is odd and zero when n is even.9,10 Thus, the Stokes shift arises from the contribution of only the odd-order S2n+1(t) terms in the full summation. Furthermore, the associated normalized

Alternatively, the Stokes

shift can be rearranged as the equilibrium accumulation ΔES(t ) =

⟨δ ΔE(t )e δΔE⟩e ⟨e δΔE⟩e

on the excited-state surface.9,19 Upon

Taylor expanding eδΔE, ΔES(t) can be expressed as the weighted sum of the time correlation function ⟨δΔE(t) δΔE(0)n⟩e of the central variable δΔE. The corresponding weights will be discussed later. The equilibrium time correlation function ⟨δΔE(t)δΔE(0)n⟩e is further converted into a time relaxation process, which is illustrated and defined as Sn(t) in the following context. A representative example of the calculation of the time correlation functions is illustrated in Figure 1. The time zero energies were sampled over a long

time correlation functions (NTCFs) c 2n + 1(t ) =

S2n + 1(t ) S2n + 1(0)

comply

with the equality in eq 6 indicating the identical decay rates of S2n+1(t). c 2n + 1(t ) = c1(t )

(6)

The above mathematical expressions naturally lead to the equality in eq 7 between the solvation time correlation function s(t) and the linear order NTCF, thus validating the linear response theory. s(t ) = c1(t )

The odd orders of S2n+1(t) compose the Stokes shift and characterize the Gaussian statistics of the interaction energies δΔE and are defined as moments. This terminology is in analogy to the usage in the early investigations on the Raman and infrared spectroscopy.21−23 In the following, these moments will be further investigated as well as their contributions to the Stokes shift. Decompositions of Stokes Shift with Moments. Under the circumstances of Gaussian statistics, the excited-state

Figure 1. Illustrative example of how the time correlation functions are accumulated from a long equilibrium trajectory on the excited-state surface. For a given value of δΔE(0), multiple time origins are sampled and subsequently propagate into δΔEi(t) upon the time interval t.

equilibrium trajectory on the excited-state surface, where the total number of the points is N. For a given value of δΔE(0), the number of occurrences sampled over the trajectory is NδΔE(0). For each δΔE(0), the propagated energy after the time interval t is represented as δΔEi(t), where i varies from 1 to NδΔE(0). In eq 1, the nth order time correlation function is accumulated by enumerating all possible values of δΔE(0). ⟨δ ΔE(t )δ ΔE(0)n ⟩e = =

1 N

∑ ∑ δΔEi(t )δΔE(0)n δ ΔE(0)

∑

probability distribution takes the form Pe(δΔE) =

NδΔE(0) N

δ ΔE(0)

weight functions Pn(δ ΔE) = δ ΔEn

2πσ 2

e−δΔE

2 /2σ 2

2πσ 2

introduced in eq 4.

(2)

∫ dδΔE(0)δΔE(0, t )δΔE(0)n Pe(δΔE(0))

(3)

=

∫ dδΔEδΔE(0, t )Pn(δΔE)

(4)

In eq 2, the quantity δ ΔE(0, t ) =

2 /2σ 2

Figure 2 shows examples of these the odd- and even-ordered weight functions. In contrast to the symmetric distribution of the even-order weight functions, the odd-order weight functions exhibit the antisymmetric distributions. For the weight function of the moment 2n + 1, two stationary points

(1)

=

= Sn(t )

e−δΔE

with the variance σ, leading to an explicit expression of the

i

δ ΔE(0, t )δ ΔE(0)n

(7)

(5) 1 NδΔE(0)

∑i δ ΔEi(t ) is the

average of δΔEi(t) propagating from the same δΔE(0) as sampled from equilibrium fluctuations. The summation is expressed as an integration (eq 3) using the excited-state equilibrium probability distribution Pe(δΔE(0)). In eq 4, a modulated distribution Pn(δΔE) = δΔEnPe(δΔE) is proposed by combining factor δΔEn and Pe(δΔE), which plays the role of a weight function for the integration. The time correlation function ⟨δΔE(t)δΔE(0)n⟩e can thus be referred to as the time evolution of a weighted summation over the ensemble

Figure 2. Examples of odd- (black) and even- (cyan) order weight functions originating from the Gaussian statistics of δΔE. The heights of the curves were scaled for visual clarity. The arrows indicate the optimal configurations of the odd-order weight function. 1868


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Journal of Chemical Theory and Computation

and 2, which arise from the distinct structural alignment of a salt-bridge type residue pair E129-K133 to the tryptophan.25 The local structure around W140 including this salt bridge is shown in Figure 3A. The equilibrium probability distribution of

±σ 2n + 1 are found (Figure 2). It is noted that higher order moments lead to further separation of their optimal configurations away from the origin. As illustrated by eq 4, these values of δΔE are critical on the relaxation dynamics of the S2n+1(t) by the prevailing initial weights among the ensemble and denoted as the optimal configurations. The accurate accumulation of the resulting NCTF requires the sufficient sampling of configurations with interaction energies at or near the associated optimal configurations, which makes the evaluations of the high orders c2n+1(t) difficult due to the rare sampling over those values of δΔE. Now let us revisit the Stokes shift propagating of the initial probability Pg(δΔE), namely the equilibrium distribution of δΔE on the ground-state surface. The distribution function is expanded into a sum of the Gaussian-type weight functions by Pg(δ ΔE) =

∑ fn Pn(δΔE)

(8)

f n is the nth coefficient and can be solved for by Taylor expanding on both sides and matching corresponding terms. This treatment is especially efficient with the occurrence of the multiple potential energy basins on the ground-state surface and enables investigation of the relaxation dynamics of individual isomers. Eq 8 enables the Stokes shift as being the sum of the moments with the selecting weights f 2n+1, in which each moment S2n+1(t) initiates the odd-order weight functions P2n+1(δΔE). The above relationship is depicted in eq 9 as ΔES(t ) =

∑ f2n+ 1 S2n+ 1(t )

Figure 3. A. Local structure of protein SNase around indole of W140 (yellow). Charged residues K133 and E129 form a salt bridge within 5 Å of indole. B. Probability distributions of δΔE computed from molecular simulations of SNase of the excited-state surface (black) and two isomers (red and green) on the ground-state surface.

(9)

where the even-order weight functions contribute null. Since the total energy shift of the Stokes shift is ΔES(0) = Σf 2n+1S2n+1(0) and S2n+1(0) could be performed analytically using eq 3, the contribution of the moment 2n + 1 to the Stokes shift is given in eq 10 f2n + 1 S2n + 1(0) = f2n + 1

∫ dδΔEδΔE2n+2 e

the energy variable δΔE for the excited- and both ground-state isomers is shown in Figure 3B. The excited-state distribution

−δ ΔE 2 /2σ 2

2πσ 2

exhibits a Gaussian form by Pe(δΔE) =

(10)

e−δΔE

2 /2σ 2

2πσ 2

, where the

variance σ is fitted as 3.8. As discussed earlier, the scaling factor for the interaction energies was chosen such that δΔE(t) and σ are dimensionless quantities. The corresponding distribution for the ground-state isomers was fitted with a shifted Gaussian

As Stokes shift is decomposed into all moment terms with specific weights, those orders with the most significant contributions are defined as dominant moments. The validity of the linear response theory therefore relies on whether those dominant terms are in compliance with the properties of Gaussian statistics, namely the equality c2n+1(t) = c1(t) between the associated NTCFs and the linear order one.

expression Pgi(δΔE) = e

−(δ ΔE − Xi)2 /2σi 2

2πσi 2

, where i = 1, 2 is the index of

isomer 1 and isomer 2. The corresponding variances are σ1 = 3.0 and σ2 = 2.7 with centers X1 = 11.3 and X2 = 16.1. Given the Gaussian form of the distribution Pe(δΔE) on the excited-state surface, the resulting weight functions Pn(δΔE) were computed, which enables the decomposition of the total Stokes shifts into the contributions of the associated moments. The optimal configurations associated with the weight functions of the moments are derived straightforwardly. It is ±σ 2n + 1 , which are ±3.8, ±6.7, ±8.6, ±10.2, ±11.6 for the orders 1, 3, 5, 7, 9. For isomer 1, the equilibrium probability distribution Pg1(δΔE) was expanded as the sum of the weight functions using the relationship Pg1(δΔE) = Σf nPn(δΔE) (Figure 4A), where the coefficients f n were solved. It noted that the negative contributions of the odd order P2n+1(δΔE) are canceled by the even order functions. By applying eq 10, the contributions of moments to the total Stokes shift, namely f 2n+1S2n+1(0), are thus derived and plotted in Figure 4B. The dominant ones arise from order 7, 9, and 11 with the values of 2.8, 4.0, and 3.2 out of the total energy shift 11.3. The maximal

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RESULTS AND DISCUSSION Weight Function, Optimal Configuration, and Dominant Moments. Molecular dynamics simulations were employed to reveal the role of the excited-state moments resulting from excited-state Gaussian statistics on the linear response theory of the fluorescence Stokes shift. To explore the mathematical relations discussed in the previous section, MD trajectories of the protein SNase were analyzed, which contains a single inherent tryptophan W140 used as the photon probe and has been extensively studied using the ultrafast fluorescence techniques.5,10,24,25 A more specific account of the simulation details was discussed in earlier work.10,25 The equilibrium simulations were performed to probe the statistical property of the central variable δΔE on both the ground- and excited-state surface of the chromophore. Structural transitions were observed in the ground-state simulations, leading to dynamic hopping between potential energy basins defining isomers 1 1869


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Figure 4. A. Probability distribution Pg1(δΔE) of isomer 1 (red) is expanded into the weight functions by fnPn(δΔE), where the contributions from order 7, 8, 9, and 11 are presented. The configuration with the maximal probability in the red curve corresponds to the optimal configuration of the 9th moment, as indicated by the red dashed line. B. The total Stokes shift of isomer 1 is decomposed into the contributions from individual moments by applying eq 10. The maximal contribution arises from the 9th moment; all even-order time correlation functions contribute null by definition resulting from Gaussian statistics.

Figure 5. Comparison of time correlation functions. Top panel: the almost identical relaxation rates of the different order moments indicate the appropriateness of Gaussian statistics of δΔE on the excited-state surface; bottom panel: linear order NTCF (black) shows excellent agreement with the nonequilibrium solvation time correlation function sg1 (red) of isomer 1 compared to observed deviations for sg2 (green) of isomer 2. It is noted that the dynamics by the ground-state linear response approach (orange) of isomer 1 differs from the nonequilibrium solvation dynamics sg1.

clearly indicates that the fluctuations of δΔE on the excitedstate surface, at least up to the first 9 moments, obey Gaussian statistics as displayed in eq 6. This is consistent with the observation that the probability distribution Pe(δΔE) exhibits a Gaussian profile at least up to the optimal configurations for the first 9 moments. The insufficient sampling over the optimal configurations of higher order moments makes the calculations of their NTCFs difficult. Thus, the statistical property of δΔE was cautiously classified as that of Gaussian type since the higher orders NTCF have not been evaluated. Additionally, two sets of nonequilibrium MD simulations with the time length of 200 ps (ps) were performed on the excited-state surface to calculate the Stokes shifts, where the initial configurations were sampled from the structural fluctuations of isomer 1 and isomer 2 structures. The solvation time correlation functions were obtained separately and are compared with the linear order NTCF to check the validity of the linear response theory. For isomer 1, the solvation time correlation function sg1(t) shows excellent agreement with the NTCF c1 (as well as c3, c5, c7, and c9). This is attributed to the equivalent relaxation rates between the moments of the linear order and those higher ones dominating the Stokes shift, although the NTCFs of some pertinent orders (such as moment 11) have not been obtained. As a comparison, the normalized time correlation function

contribution arises from the 9th moment, which is consistent with the observation that the maximal probability energy (11.3) of the Pg1(δΔE) approximately corresponds to the optimal configuration of moment 9 (11.6). The first linear order term only contributes to 0.02, and moments 15 and 17 are minor negative contributions. All even orders contribute null to the Stokes shift by definition as a result of Gaussian statistics. As compared with isomer 1 (Figure 3B), the equilibrium distribution of δΔE for isomer 2 is located further away from the origin. The decomposition of Pg2(δΔE) using eq 8 makes the main contributions from the higher order weight functions. For example, the δΔE with the maximal probability (16.1) is about the same as the optimal configuration (16.0) of moment 17, where the samplings of these configurations in excited-state MD simulations are rare. Since the structural fluctuations of isomer 2 mostly visit the regions of configuration space sampled with low probability via the excited-state simulations, the associated statistical properties of δΔE are vague, and the protocol of applying the Gaussian moments on the Stokes shift may not be applicable. Gaussian Statistics and Linear Response Theory. The linear response theory of the fluorescence Stokes shift is examined by comparing the relaxation rates of the moments with those of the nonequilibrium process for both ground-state isomers. First, the normalized time correlation function of the moments on the excited-state surface was calculated. The c2n+1 correlation functions were accumulated over the equilibrium simulations on the excited-state surface for the moment order 1, 3, 5, 7, and 9. As shown in Figure 5, all the time correlation functions exhibit very similar relaxation dynamics. This result

c g (t ) = 1

⟨δ ΔE(t )δ ΔE(0)⟩g

1

⟨δ ΔE(0)δ ΔE(0)⟩g

is also accumulated over the ground-

1

state equilibrium fluctuations of isomer 1, which represents the ground-state linear response approach. The resulting decay rates are clearly different from those of the nonequilibrium Stokes shift (bottom panel of Figure 5). For isomer 2, 1870


Letter

Journal of Chemical Theory and Computation deviations are observed between c1 and the solvation time correlation function sg2(t). As discussed earlier, the Stokes shift of isomer 2 mainly arises from the contributions of the dominant moments with the high orders. Although the relaxation rates c2n+1 of those moments have not been derived by the equilibrium simulations on the excited-state surface, it is still speculated that the Gaussian profile of the excited-state distribution Pe (δΔE) distorts near or at the optimal configurations of those moments. Correspondingly, Gaussian statistics are corrupted for the dominant moments, leading to the resulting discrepancy between the nonequilibrium dynamics and the relaxation of the linear order moment. While the Gaussian characteristics of the dominant moments are linked with the linear response of the solvent in isomer 1, the explanations of the linearity in the literature are also examined. The equilibrium statistics of the energy variable on the ground-state surface have been correlated with Stokes shift in plenty of studies.12−17 However, the comparison in Figure 5 clearly shows that the ground-state linear response approach is incapable of accurately describing the nonequilibrium solvation dynamics. This agrees with the observations by Laird and Thompson.9,18 By examining the nanoconfined solvents, they found that the ground-state linear response results could differ substantially from those of nonequilibrium simulations and the excited-state linear response approach.18 The linearization between the high order time correlations and the linear one of the excited state was put forward as the criterion on the linear response theory.9,18 In this work, the identical relaxation rates c2n+1(t) = c1(t) between the dominant moments and the linear order are tied with the appropriateness of the linear response theory. This criterion, although closely correlated with the one by Laird and Thompson, presents a straightforward interpretation on the mechanism of the linear response theory. The dominant moments further bring in an implementation for the associated checking process. On the other hand, the time evolutions of the probability distributions of δΔE upon the optical transitions are also examined for isomer 1 by analyzing the associated MD trajectories. Several snapshots of the energy distribution during the nonequilibrium relaxation are displayed in Figure 6. Fitting with a shifted Gaussian function, the variances of the profiles change with time and are not stationary. For example, the variance of the initial distribution 3.0 (±0.10) and the final distribution 3.8 (±0.06) obviously differ with each other. This disagreement should not deem to be the violation with the results of Tachiya,26 and the associated discussions appear in the previous publication.25 This nonstationary character does not correlate with the validity of the linear response theory in the case of isomer 1, although it was assigned to explain the discrepancy between the dynamics of the nonequilibrium relaxation and the equilibrium fluctuations in the literature.8

Figure 6. Time evolution of the energy distribution of δΔE during the nonequilibrium relaxation of isomer 1. The curves correspond to the t = 0 equilibrium distribution (red) of isomer 1, distributions after the system has relaxed for 5 ps (purple), 200 ps (orange), and infinite time, namely the excited-state equilibrium distribution (black). The time zero and time infinity curves are fitted with Pg1 (δΔE) = 2 /2σ 2 1

e−(δΔE − X1)

2πσ12

and Pe(δΔE) =

e−δΔE

2 /2σ 2

2πσ 2

, resulting in the variance σ1 and

σ of 3.0 (±0.10) and 3.8 (±0.06). To obtain the error bars in the brackets, the energy distributions were sampled over different time blocks of the isomer 1 and excited-state equilibrium trajectories. The error bars were estimated by the block averaging.

interactions.8,9,31 In this approach, the introduction of moments presents a quantitative description of Gaussian statistics on the excited-state surface and thus accounts for the origin of the linearity. The protocol described herein suggests an appropriate strategy on the application of linear responses theory: whether the dominant moments obey the Gaussian statistics with the equality c2n+1(t) = c1(t) is the key to determine whether the nonequilibrium dynamics accurately tracks the equilibrium regression rates. On the other hand, the Gaussian statistics on the excited-state surface should be emphasized,9,19,32,33 which has not gained enough attention in most of the current usage. These observations originate from the Onsager regression hypothesis1,34,35 and are enhanced by the Gaussian fluctuations in the macromolecular system. As the solute is immersed into a macroscopic number of solvent molecules (including protein side chains), Gaussian statistics arises from central limit theorem.1,36 However, the corruptions were indeed observed. Nonlinearity exhibits with the appearance of the gross change on the environmental solvation structures.32,37−42 In a system containing a neutral sodium atom (in the form of a cation and a electron) solvated in liquid tetrahydrofuran, the distinct solvent arrangements of the initial states (induced by an electron shifting) affect the rates of the relaxations.37 The inherent mechanism is analogous to the observations in this work. Variations of Gaussian fluctuations are introduced by different solvent structures leading to the failure of the linear response theory. Another ubiquitous issue is the variation of the protein conformations reported by MD simulations. In the theoretical investigation of SNase using a different force field,24 the conformational heterogeneity exhibited in this study was not reported. The disagreement may arise from the length of the simulation time or even the force field itself. Longer groundstate simulation may recover the parabolic free energy surface. However, the perspective on the connection between the nonequilibrium dynamics and equilibrium Gaussian fluctuations is anticipated to hold under the circumstances of both the homogeneous and heterogeneous conformations.

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CONCLUSION The linear response theory has been applied extensively to investigate hydration dynamics not only because of its convenience in evaluating the fluorescence Stokes shift but also for its bridge between the spontaneous equilibrium fluctuations and the experimentally accessible measurements, 12,14,19,27−29 where the connection between the autocorrelation of fluctuations and the magnitude of Stokes shift was specifically investigated.30 The linearity often relies on the assumption of the Gaussian fluctuations of solute−solvent 1871


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(18) Laird, B. B.; Thompson, W. H. Time-dependent fluorescence in nanoconfined solvents: linear-response approximations and Gaussian statistics. J. Chem. Phys. 2011, 135, 084511. (19) Carter, E. A.; Hynes, J. T. Solvation Dynamics for an Ion-Pair in a Polar-Solvent - Time-Dependent Fluorescence and Photochemical Charge-Transfer. J. Chem. Phys. 1991, 94, 5961−5979. (20) Wick, G. C. The Evaluation of the Collision Matrix. Phys. Rev. 1950, 80, 268−272. (21) Gordon, R. G. Molecular Motion and the Moment Analysis of Molecular Spectra in Condensed Phases. I. Dipole-Allowed Spectra. J. Chem. Phys. 1963, 39, 2788−2797. (22) Gordon, R. G. Molecular Motion and the Moment Analysis of Molecular Spectra. II. The Rotational Raman Effect. J. Chem. Phys. 1964, 40, 1973−1985. (23) Gordon, R. G. Molecular Motion and the Moment Analysis of Molecular Spectra. III. Infrared Spectra. J. Chem. Phys. 1964, 41, 1819−1829. (24) Scott, J. N.; Callis, P. R. Insensitivity of Tryptophan Fluorescence to Local Charge Mutations. J. Phys. Chem. B 2013, 117, 9598−9605. (25) Li, T. P. Validity of Linear Response Theory for TimeDependent Fluorescence in Staphylococcus Nuclease. J. Phys. Chem. B 2014, 118, 12952−12959. (26) Tachiya, M. Relation between the Electron-Transfer Rate and the Free-Energy Change of Reaction. J. Phys. Chem. 1989, 93, 7050− 7052. (27) Stratt, R. M.; Maroncelli, M. Nonreactive dynamics in solution: The emerging molecular view of solvation dynamics and vibrational relaxation. J. Phys. Chem. 1996, 100, 12981−12996. (28) Maroncelli, M.; Fleming, G. R. Computer-Simulation of the Dynamics of Aqueous Solvation. J. Chem. Phys. 1988, 89, 5044−5069. (29) Schwartz, B. J.; Rossky, P. J. Aqueous Solvation Dynamics with a Quantum-Mechanical Solute - Computer-Simulation Studies of the Photoexcited Hydrated Electron. J. Chem. Phys. 1994, 101, 6902− 6916. (30) Loring, R. F. Statistical Mechanical Calculation of Inhomogeneously Broadened Absorption-Line Shapes in Solution. J. Phys. Chem. 1990, 94, 513−515. (31) Vuilleumier, R.; Tay, K. A.; Jeanmairet, G.; Borgis, D.; Boutin, A. Extension of Marcus Picture for Electron Transfer Reactions with Large Solvation Changes. J. Am. Chem. Soc. 2012, 134, 2067−2074. (32) Tao, G. H.; Stratt, R. M. The molecular origins of nonlinear response in solute energy relaxation: The example of high-energy rotational relaxation. J. Chem. Phys. 2006, 125 (125), 114501. (33) Maroncelli, M. Computer-Simulations of Solvation Dynamics in Acetonitrile. J. Chem. Phys. 1991, 94, 2084−2103. (34) Onsager, L. Reciprocal relations in irreversible processes. II. Phys. Rev. 1931, 38, 2265−2279. (35) Bernard, W.; Callen, H. B. Irreversible Thermodynamics of Nonlinear Processes and Noise in Driven Systems. Rev. Mod. Phys. 1959, 31, 1017−1044. (36) Khinchin, A. I. Mathematical Foundations of Statistical Mechanics; Dover Publications: New York, 1949. (37) Bragg, A. E.; Cavanagh, M. C.; Schwartz, B. J. Linear response breakdown in solvation dynamics induced by atomic electron-transfer reactions. Science 2008, 321, 1817−1822. (38) Stratt, R. M. Chemistry - Nonlinear thinking about molecular energy transfer. Science 2008, 321, 1789−1790. (39) Moskun, A. C.; Jailaubekov, A. E.; Bradforth, S. E.; Tao, G. H.; Stratt, R. M. Rotational coherence and a sudden breakdown in linear response seen in room-temperature liquids. Science 2006, 311, 1907− 1911. (40) Skaf, M. S.; Ladanyi, B. M. Molecular dynamics simulation of solvation dynamics in methanol-water mixtures. J. Phys. Chem. 1996, 100, 18258−18268. (41) Turi, L.; Minary, P.; Rossky, P. J. Non-linear response and hydrogen bond dynamics for electron solvation in methanol. Chem. Phys. Lett. 2000, 316, 465−470.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Tanping Li: 0000-0002-3729-2639 Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS The author thanks Professor Sherwin Singer and Dr. Christopher Knight for the stimulating discussions. REFERENCES

(1) Chandler, D. Introduction to Modern Statistical Mechanics; Oxford University Press: New York, 1987. (2) Pal, S. K.; Peon, J.; Zewail, A. H. Biological water at the protein surface: Dynamical solvation probed directly with femtosecond resolution. Proc. Natl. Acad. Sci. U. S. A. 2002, 99, 1763−1768. (3) Zhong, D. P.; Pal, S. K.; Zhang, D. Q.; Chan, S. I.; Zewail, A. H. Femtosecond dynamics of rubredoxin: Tryptophan solvation and resonance energy transfer in the protein. Proc. Natl. Acad. Sci. U. S. A. 2002, 99, 13−18. (4) Peon, J.; Pal, S. K.; Zewail, A. H. Hydration at the surface of the protein Monellin: Dynamics with femtosecond resolution. Proc. Natl. Acad. Sci. U. S. A. 2002, 99, 10964−10969. (5) Qiu, W. H.; Kao, Y. T.; Zhang, L. Y.; Yang, Y.; Wang, L. J.; Stites, W. E.; Zhong, D. P.; Zewail, A. H. Protein surface hydration mapped by site-specific mutations. Proc. Natl. Acad. Sci. U. S. A. 2006, 103, 13979−13984. (6) Qiu, W. H.; Zhang, L. Y.; Okobiah, O.; Yang, Y.; Wang, L. J.; Zhong, D. P.; Zewail, A. H. Ultrafast solvation dynamics of human serum albumin: Correlations with conformational transitions and siteselected recognition. J. Phys. Chem. B 2006, 110, 10540−10549. (7) Zhang, L. Y.; Wang, L. J.; Kao, Y. T.; Qiu, W. H.; Yang, Y.; Okobiah, O.; Zhong, D. P. Mapping hydration dynamics around a protein surface. Proc. Natl. Acad. Sci. U. S. A. 2007, 104, 18461−18466. (8) Geissler, P. L.; Chandler, D. Importance sampling and theory of nonequilibrium solvation dynamics in water. J. Chem. Phys. 2000, 113, 9759−9765. (9) Laird, B. B.; Thompson, W. H. On the connection between Gaussian statistics and excited-state linear response for time-dependent fluorescence. J. Chem. Phys. 2007, 126, 211104. (10) Li, T. P.; Kumar, R. Role of excited state solvent fluctuations on time-dependent fluorescence Stokes shift. J. Chem. Phys. 2015, 143, 174501. (11) Simonson, T. Gaussian fluctuations and linear response in an electron transfer protein. Proc. Natl. Acad. Sci. U. S. A. 2002, 99, 6544− 6549. (12) Li, T. P.; Hassanali, A. A. P.; Kao, Y. T.; Zhong, D. P.; Singer, S. J. Hydration dynamics and time scales of coupled water-protein fluctuations. J. Am. Chem. Soc. 2007, 129, 3376−3382. (13) Li, T. P.; Hassanali, A. A.; Singer, S. J. Origin of Slow Relaxation Following Photoexcitation of W7 in Myoglobin and the Dynamics of Its Hydration Layer. J. Phys. Chem. B 2008, 112, 16121−16134. (14) Nilsson, L.; Halle, B. Molecular origin of time-dependent fluorescence shifts in proteins. Proc. Natl. Acad. Sci. U. S. A. 2005, 102, 13867−13872. (15) Furse, K. E.; Corcelli, S. A. The dynamics of water at DNA interfaces: Computational studies of Hoechst 33258 bound to DNA. J. Am. Chem. Soc. 2008, 130, 13103−13109. (16) Golosov, A. A.; Karplus, M. Probing polar solvation dynamics in proteins: A molecular dynamics simulation analysis. J. Phys. Chem. B 2007, 111, 1482−1490. (17) Pal, N.; Shweta, H.; Singh, M. K.; Verma, S. D.; Sen, S. PowerLaw Solvation Dynamics in G-Quadruplex DNA: Role of Hydration Dynamics on Ligand Solvation inside DNA. J. Phys. Chem. Lett. 2015, 6, 1754−1760. 1872


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Journal of Chemical Theory and Computation (42) Bedard-Hearn, M. J.; Larsen, R. E.; Schwartz, B. J. Projections of quantum observables onto classical degrees of freedom in mixed quantum-classical simulations: understanding linear response failure for the photoexcited hydrated electron. Phys. Rev. Lett. 2006, 97, 130403.

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