J. Phys. Chem. B 2008, 112, 369-377
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Anomalously Slow Solvent Structural Relaxation Accompanying High-Energy Rotational Relaxation† Guohua Tao and Richard M. Stratt* Department of Chemistry, Brown UniVersity, ProVidence, Rhode Island 02912 ReceiVed: July 19, 2007; In Final Form: August 31, 2007
Experimental and theoretical work on the relaxation of rapidly rotating solutes in liquids have pointed out a number of striking features. Even in rapidly relaxing solvents, the relaxation proceeds quite slowly, exhibiting a manifestly nonlinear response that depends explicitly on the initial rotational energy. In this paper, we show how the long-time behavior, in particular, stems from a strong coupling of solute orientation to local solvent geometry. This coupling creates a rotational friction that decreases sharply with rotational energy, allowing for the protracted survival of not only high-angular-momentum rotational states but the cavity-like low-friction solvent geometries. We show, further, that the slow dynamics is dynamically heterogeneous. The distribution of excited rotors is marked by a distinct population of slowly relaxing hot rotational states. This population can be traced directly to the small subset of liquid configurations that happen to have low rotational friction values at the instant at which the rapid rotation started, indicating an unusual failure of the normally chaotic environment of a liquid to randomize initial conditions.
I. Introduction The energy relaxation of a molecule dissolved in a liquid typically follows whatever course the solvent sets for it. As one would expect from linear response theory, the normal, equilibrium, fluctuations of the solvent are usually what determine the rates and even the mechanisms of energy dissipation from a solute.1,2 So, at least in situations where linear response holds, one has a very direct interpretation of the phenomenon of friction on a molecular level.3-9 However, in a series of recent papers, we noted that there was a spectroscopically accessible example of solute energy relaxation, high-energy rotational relaxation, in which linear response not only fails, it fails in a fashion that allows us to trace the molecular origins of that failure.10,11 This paper is a continuation of our study of the microscopics behind this relaxation, but, now, we turn our attention to the complementary question this example raises: How is it that a solute can end up controlling the dynamics of the surrounding solvent? When a very rapidly rotating molecule is created in a liquid, linear response actually provides an accurate description of the initial rotational energy loss, but within a fraction of a rotational period, the rotor suddenly begins to relax much more slowly.10,11 What happens is that the rotor expels some of the innermost part of the first solvent shell, creating a kind of cavity around itself. This event diminishes the molecular friction, thus allowing the solute to keep rotating for times much longer than one would have expected. In our previous work, we examined the behavior of this rotational friction in some detail, pointing out how the evolution of the solvent structure leads to parallel changes in the friction and how it does so in a way that depends strongly on the initial rotational energy and the intrinsic time scales of the solvent.11 One question we never answered, though, was why it takes so long for the solvent structure itself to revert back to equilibrium. It is this topic that motivates the present paper. †
Part of the “James T. (Casey) Hynes Festschrift”. * To whom correspondence should be addressed.
The existence of a fundamental link between failures of linear response and the slow evolution of solvent geometry was observed fairly early on.12 Solvation dynamics involving solvent hydrogen-bond breaking and net solute size changes have both produced clear instances in which linear response expectations were not met.13-15 The means by which the solute causes the solvent to relax slowly in these situations, however, seems to have received less attention. An interesting clue to this long-time behavior comes from a recent study of rotational relaxation by Gelin and Kosov.16 Following up on Gordon’s suggestion that there were physical mechanisms that could lead to slower rotational relaxation rates at higher rotational energies,17 Gelin and Kosov noted that, at a Fokker-Planck level, one could generate a slow response in this problem just by invoking an angular-momentum-dependent rotational friction. What we will show is that this explanation is in fact germane to our slow relaxation and that the molecular origins of this angular-velocity-dependent friction come from the same solute-solvent coupling that causes the slow solvent dynamics. We will also emphasize that this connection creates a pronounced dynamical heterogeneity that one could easily miss in a Fokker-Planck treatment. The instantaneous solvent environments that lead to slow solute relaxation are precisely the ones that lead to slow solvent relaxation. The remainder of this paper will be organized as follows: Section II summarizes our model and some of the numerical methods we use. Section III starts our analysis by presenting a solute-centered perspective on the slow relaxation, looking, in particular at the unusual probability distribution of solute energies that occurs in this problem. Section IV follows up by examining the coupling between the solute and solvent dynamics, emphasizing the role that dynamical heterogeneity plays in the slow dynamics. We conclude in section V with a few remarks. II. Models and Numerical Methods Although the experiments that prompted these studies were carried out with photochemically generated CN radicals in water
10.1021/jp075664a CCC: $40.75 © 2008 American Chemical Society Published on Web 11/02/2007
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and in alcohols,10,18 the essential dynamical features, including, somewhat surprisingly, the slow solvent relaxations, seem to have been captured by classical molecular dynamics calculations on a diatomic molecule dissolved in an atomic solvent.10,11 We therefore continue to use the same simple model system we have been using: a single diatomic CN molecule in the presence of a liquid of Ar atoms. The CN bond length is held fixed at 1.268 Å (corresponding to a rotational constant B ) 1.621 cm-1), and the CN/Ar and Ar/Ar interaction are taken as sums of pair potentials. The CN/Ar pair potential was derived from an ab initio potential surface19 that we fit to a five-site potential of the form 5
u(rja) )
∑
a)1
ua(rja), ua(r) ) cae-bar -
da r6
(2.1)
where a labels the sites on the CN and rja is the distance between the ath CN site and the jth Ar atom. The locations of the sites and the (ba, ca, da) potential parameters are given in our previous paper.11 The Ar/Ar pair potential is of the Lennard-Jones form
u(r) ) 4
[(σr ) - (σr ) ], 12
6
( ) 119.8 K, σ ) 3.405 Å) (2.2)
All calculations were carried out in the liquid phase (reduced Ar number density Fσ3 ) 0.800 and reduced temperature kBT/ ) 1.00) with periodic boundary conditions. Both equilibrium and nonequilibrium simulations were performed with a time step of 2.16 fs using the velocity-Verlet algorithm for center-of-mass translation and the leapfrog algorithm for CN rotation.20 After melting the initial lattice configuration, a minimum of 151 ps was allowed for system equilibration after the final temperature rescaling. Equilibrium quantities were averaged over data sampled every 10 time steps. The equilibrium energy fluctuation correlation functions, for example, were averaged over 2.5 × 106 configurations. Nonequilibrium simulations were carried out with initial phase-space points selected every 100 time steps. For each set of initial conditions, the CN angular velocity was rescaled to the desired initial value and the subsequent dynamics was followed by computing appropriate correlation and distribution functions. Nonequilibrium quantities such as the rotational energy response function and the rotational energy distribution function were all averaged over 5 × 105 initial conditions. Our simulations were checked for finite-size effects (most notably for the effects of system heating following nonequilibrium relaxation), by comparing 106 Ar atom results with those from systems with 254 and 498 Ar atoms. There were no significant differences observed in the energy relaxation profiles as a function of system size, so except where otherwise noted, all of the molecular dynamics results we present here are based on 106 Ar atom simulations.21 Our numerical solutions of the rotational-state-population master equation used a fourth-order Runge-Kutta method22 with a time step of 0.002 τ (with τ the characteristic time scale defined in the next section. Checks run using a time step of 0.001 τ yielded no noticeable change in the results. III. Solute’s Perspective on the Slow Relaxation A. Exact Dynamics. The most obvious way to monitor the relaxation in our liquid is to follow the time evolution of the solute’s rotational kinetic energy EROT. Figure 1 shows the
Figure 1. Equilibrium, CE(t), and nonequilibrium, SE(t), rotational energy relaxation as a function of time, t. The curves are all derived from molecular dynamics results for a CN solute in liquid Ar. The nonequilibrium relaxations are shown for initial rotational kinetic energies EROT(0) ) 4, 8, 12, 16, 20, and 24kBT.
nonequilibrium rotational energy response function
SE(t) )
E h ROT(t) - E h ROT(∞) E h ROT(0) - E h ROT(∞)
(3.1)
for a variety of different initial rotational energies EROT(0). The overbars here denote nonequilibrium averages over the rest of the system’s initial conditions. If the system obeyed linear response theory perfectly, the initial preparation would not matter. All of these curves would be identical, and they would all match the decay of rotational energy fluctuations δEROT predicted by the equilibrium correlation function
CE(t) )
〈δEROT(0) δEROT(t)〉
, δEROT ) EROT - 〈EROT〉
〈δEROT(0) δEROT(0)〉
(3.2)
with the angular brackets representing equilibrium averages.11 At modest initial rotational energies (EROT(0) ) 333 cm-1 ) 4 kBT), the relaxation is quite rapid, following a course almost exactly along the lines predicted by linear response theory. However, higher initial energies lead to a considerable slowing down of the relaxation. Interestingly, even the long-time decay rates τE-1 (the asymptotic slopes of the log plots) decrease with increasing initial rotational energy, a rather curious occurrence given that the relaxation appears to be largely complete at long times. Over the interval during which SE(t) declines from 0.1 to 0.01, for example, the 333 cm-1 (4kBT), 1665 cm-1 (20kBT), and 1998 cm-1 (24kBT) results have exponential decay constants τE ) 0.72, 1.5, and 2.1 ps, respectively, despite the fact that the average rotational energy in all three drops under the 4kBT mark in this regime. A more detailed look at the process reveals that focusing on the average relaxation is somewhat misleading. Figures 2 and 3 show the time evolution of the rotational energy probability
Slow Solvent Structural Relaxation
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Figure 2. Molecular dynamics results for PJ(t), the probability distribution of classical rotational states J at a series of times, for CN in liquid Ar. The normalized probability distributions are divided by (2J + 1), in order to allow us to follow both the high and low angular momentum states, with the corresponding equilibrium distribution indicated by the dashed curve at t ) 0. The initial rotational energy here is EROT(0) ) 333 cm-1, (J(0) ) 14.3).
Figure 3. Molecular dynamics results for PJ(t), the probability distribution of classical rotational states J for CN in liquid Ar (as in Figure 2), but here for an initial rotational energy, EROT(0) ) 1998 cm-1, (J(0) ) 35.1).
distributions for the EROT(0) ) 333 cm-1and 1998 cm-1 cases. The former shows the initial delta-function rotational energy distribution simply broadening and shifting until it reaches the equilibrium distribution. The higher energy example, though, exhibits a markedly different behavior. The probability distribution now becomes bimodal with both a residual hot population and a cooler, near-equilibrium, component coexisting for a considerable period of time. Just how long a period this coexistence persists for is hard to see in Figure 3, but some sense of its remarkable longevity can be seen in Figure 4, where we plot the deviation of the nonequilibrium energy-weighted probability distribution of rotational states J from its (classical) equilibrium equivalent
EJ∆PJ(t) t E clJ [PJ(t) - Peq-cl ], J 2 cl ) (2βBJ)e-βE J , E J ) BJ (3.3) Peq-cl J cl
looking, in particular, at what remains after t ) 10.8 ps has elapsed since an initial 1998 cm-1 rotational excitation. (Here β ) (kBT)-1 and kB is Boltzmann’s constant.)
Figure 4. Energy-weighted difference between the nonequilibrium and equilibrium probability densities of rotational states at long times. The results shown here are from a molecular dynamics simulation of CN in liquid Ar monitored 10.8 ps after being launched with an initial rotational energy EROT(0) ) 1998 cm-1 corresponding to the arrow.
This level of excitation corresponds to an initial J ) J(0) ≈ 35, which eventually has to relax to an equilibrium distribution peaked at J ≈ 9. However, despite the rather lengthy EclJ Peq-cl J time delay shown in the figure, there is a clear indication of a residual hot population centered around J ≈ 30. The system evidently retains at least some memory of the initial excitation for a substantial period. B. Master Equation Predictions for the Rotational State Populations. The first question one should probably ask is whether this bimodal behavior is unexpected; is it, for example, anything more than a consequence of our, somewhat artificial, microcanonical initial rotational conditions? Since the long-time behavior is what is at issue here, we should be able to pursue this point just by constructing a master equation description of the time evolution of the rotational state populations.23 If we call PJ(t) the nonequilibrium average probability of having total angular momentum quantum number J at time t, we can write down a set of kinetic equations for these probabilities subject to the conditions that all of the population resides initially in the J ) J(0) energy level, that the asymptotic distribution matches the quantum mechanical equilibrium distribution Peq-qm , and that probability is conserved at all J times
PJ(0) ) δJ,J(0)
(3.4a)
) (2J + 1)e-βBJ(J+1)/qrot (3.4b) PJ(t f ∞) ) Peq-qm J ∞
∑ PJ(t) ) 1
(3.4c)
J)0
where qrot, the rotational partition function, is defined implicitly by the normalization condition, eq 3.4c. Suppose, for definiteness, we take the influence of the solvent to have the leading symmetry of our heteronuclear diatomic solute.24 That is, suppose we assume the solvent perturbation is of the form
V(θ, φ) ) V0 cos θ
(3.5)
where the angles θ and φ are defined with respect to the instantaneous bond axis of the diatomic. Then the Fermi’sgolden-rule predictions for the level-to-level rate constants kJfJ′ limit our transitions to J f J ( 1
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dPJ(t) ) -(kJfJ+1 + kJfJ-1) PJ(t) + kJ-1fJPJ-1(t) + dt kJ+1fJ PJ+1(t) (3.6) Taking into account the possible MJ () -J, ..., J) states available for each J level and using the detailed balance requirement quickly leads us to an explicit expression for these rate constants
kJfJ-1 ) kJ-1fJ )
k0 J 3 2J + 1
k0 J e-2βBJ, (J ) 0, 1, ...) 3 2J - 1
(3.7)
which we can recognize as those of the standard spectroscopic master equation governing the optical transitions of a rigid rotor in the presence of a heat bath.25 The overall rate constant k0, which includes all of the effects of the solvent coupling strength and the solvent density of states,26 is simply taken to define an unspecified time scale τ ) (k0/3)-1. A summary of the explicit derivation is provided in the Appendix. The results of solving this master equation for the situation in which the initial J value is 35 (the initial rotational energy is 2043 cm-1) are shown in Figure 5. Despite the delta-functionlike initial conditions, these results show no evidence of any bimodal behavior. Moreover when the nonequilibrium average energy predicted by this approach ∞
E h ROT(t) )
∑
J)0
∞
E qm J PJ(t) )
∑ BJ(J + 1) PJ(t)
(3.8)
J)0
is used to construct the nonequilibrium rotational energy response function, eq 3.1, the results (Figure 6) show the insensitivity to initial conditions that one would have expected from linear response theory. Both Figures 5 and 6, in fact, look very much like what one would have obtained from the (analytically solvable) master equation appropriate to a relaxing harmonic oscillator.27,28 So, is a master equation description simply an inadequate way to understand the nonlinear response and the bimodality seen in the actual dynamics? What are missing from the treatment to this point, of course, are any consequences of the most interesting feature of the problem: the coupling of the solvent geometry to the solute dynamics. We expect the solute-solvent coupling to undergo a rapid decrease once the solvent cavity is created, something that our master equation would seem incapable of including. However, a simple qualitative way to see this effect would be to postulate that there are two possible values for our overall rate constant, one value for the equilibrium liquid structure and an attenuated value for the cavity structure, and to assume that one only sees the latter when J is greater than some threshold value Jc. In other words, for each J f J′ transition, we can replace k0 in eq 3.7 with
{
Figure 5. Elementary master equation predictions for PJ(t), now the probability distribution of quantum mechanical states J. As in Figure 3, the initial rotational energy level J(0) ) 35 (corresponding to EROT(0) ) 2043 cm-1).
k0, Jmax < Jc , Jmax ) max(J, J′) k0(J, J′) ) Rk 0, Jmax g Jc
Figure 6. Nonequilibrium energy relaxation predicted by our elementary master equation. We show, for comparison, the rotational energy response functions SE(t) for the system starting in rotational energy level J(0) ) 35, (EROT(0) ) 2043 cm-1), in rotational energy level J(0) ) 14, (EROT(0) ) 340 cm-1), and in a Boltzmann distribution of initial states with a “temperature” similar to that of the first case, namely one for which 〈EROT(0)〉 ) 1998 cm-1.
dependence one would expect from nonlinear response. Another way to regard this model, though, is to note that the overall rate constant is what embodies the magnitude of the rotational friction. The angular momentum dependence in eq 3.9 is therefore equivalent to postulating an angular-momentumdependent friction, precisely what Gelin and Kosov suggested might be pertinent to this system.16 The similarity of the two perspectives is actually quite quantitative. Gelin and Kosov point out that a rotational friction of the form16
ζ(J) ) (3.9)
(with R a constant, 0 < R 0) is always less than 1). However, as the relaxation proceeds and SE(t) f 0, this equation also predicts that the asymptotic decay rate should eventually become independent of the initial conditions, which is not what one sees. Indeed, a glance at curves produced by eq 4.6 (not shown) confirms that trying to fit this equation to the data presented in Figure 1 results in poor agreement at long times, for this very reason.21 In principle, one should be careful not to begin fitting these long-time formulas until after the subpicosecond delay required for the initial solvent expulsion (Figure 10).11 In addition, we could augment eq 4.3b so as to include a direct (local-solventpopulation-independent) dependence on ∆E(t). Marginally better agreement can be obtained (also not shown) by incorporating both of these effects.21 The real physical issue, though, is that we have not allowed for the possibility of dynamical heterogeneity. If we actually want to allow for the possibility of distinct subpopulations of hot and cold rotors, we should not even be trying to describe our system with a single energy variable ∆E(t) (or, for that matter, with a single population variable ∆n(t)). It is possible to make a rough treatment of the effects of dynamical
(4.8)
rather than the instantaneous rotational energy EROT(t).32 Suppose then that we do adopt this approach and that we also make the other minor modifications we have alluded to. We still describe our coupled dynamics by eq 4.4, but now we use eq 4.7 and we allow for a direct dependence of the solvent rearrangement dynamics on solute energy by permitting kc to be an adjustable parameter (removing the restriction imposed by the second half of eq 4.4c). The end result is that eq 4.5 continues to be the steady-state expression for the transient local solvent population, but the final formula for the relaxation is a slight generalization of eq 4.6
∆E(t) R 1 + γ∆E(t) ) -k0t - ln , (t g t0) (4.9) ln γ 1 + γ∆E(t0) ∆E(t0) R)
(
)
k1kc kb kc kb ) , γ) -R k0ka ka kbnjin(∞) ka
with k0 ) k0(∆E(t0)) now defined by eq 4.7 and the first half of eq 4.4c.33 The results obtained from eq 4.9 by fitting R, γ, k0(0) ) k1(0)njin(∞), and b to the ∆E(t) curves produced by the molecular dynamics simulations (with t0 fixed, somewhat arbitrarily, at 0.497 ps) are shown in Figure 12. The agreement now is actually quite impressive.34 The long-time decay rate constants, which eq 4.9 predicts should be given by k0(∆E(t0)), seem to be determined reasonably accurately by eq 4.7. Interestingly, we can follow our reasoning in section III and extract a characteristic angular momentum quantum number Jc marking the dividing line between the hot and cold populations by setting b ) Jc-4. Our fit gives Jc ) 27, a value remarkably close to the cutoff value Jc ) 25 adopted in our revised master equation treatment in section III. The upshot seems to be that not only can the basic slowness of the coupled solute-solvent dynamics can indeed be understood with a simple kinetic model, but the way in which the dynamical heterogeneity modifies the asymptotic relaxation can as well. V. Concluding Remarks One of the principal surprises of the example we have been studying is that two intrinsically rapid subsystems, a rapidly rotating small molecule and a simple atomic liquid, can somehow conspire to make the entire system, both solute and solvent, exhibit slow dynamics. Fundamentally, the phenomenon has its origin in the nature of the coupling between the solute and the solvent in our example: the rotating solute strongly
376 J. Phys. Chem. B, Vol. 112, No. 2, 2008
Figure 12. Nonequilibrium energy relaxation predicted by our coupled solute-solvent kinetic model compared with that observed in our molecular dynamics simulations. The curves correspond to the same set of 7 initial rotational energies (with the same color coding) depicted in Figure 10. The values of the fitting parameters are shown in the upper right-hand corner of the figure with the numbers for R and γ reported in units of (kBT)-1.
perturbs the local solvent structure in just the right way to remove much of the rotational friction the solvent would have generated. The real surprise is that this kind of effect is robust enough to be self-perpetuating for more than 30 rotational periods.35 In this paper, we explored what was behind this long-time behavior. We showed that the coupling to the solvent leads to an initial transfer of solvent from the innermost half to the outermost half of the first solvation shell, with anywhere from 0.1 to 0.3 solvent atoms (on the average) undergoing this initial expulsion. That a coupling of this form is capable of leading to solute and solvent relaxations that both slow with increasing initial rotational energy is something we verified by constructing a simple kinetic model. This model, in turn, pointed out that this same coupling to the solvent automatically generates an angular-momentum-dependent rotational friction, the quintessentially nonlinear response behavior of the solute that we discussed in our previous paper. We found that this friction makes a fairly sharp transition between normal and abnormally low values at a characteristic value of the angular momentum. The sharpness of this transition itself has an interesting consequence: it gives the entire system the possibility of behaving in a fashion that is notably dynamically heterogeneous. Instead of finding the normal fluctuations in initial liquid configurations leading to a chaotic array of outcomes, we found that, at least for high initial rotational energies, the subsequent dynamics remains segregated. The relatively small number of configurations with low rotational friction at the outset are identifiably the ones that lead to a long-lived population of hot rotors. Indeed, our results suggest that it is this dynamical heterogeneity that is ultimately responsible for the specific initial-energy-dependent variation seen in the long-time energy and solvent-geometry relaxation rates.
Tao and Stratt Although our main evidence for the existence of dynamical heterogeneity comes from molecular dynamics, our evidence for its importance comes from kinetic modeling. It is therefore worth emphasizing that the kinetic and master equation models we used are not only reasonably simplistic they are by no means unique. It is possible to get agreement similar to that seen in Figure 12, for example, by using an expression for the initial energy dependence different from that of eq 4.7.21 However, we should also note that we were unable to find any similar kinetic models that reproduced the asymptotic molecular dynamics unless we included heterogeneity in some fashion. A more elaborate approach to the modeling, including the dynamics of our sub-populations explicitly, would probably help to establish a more convincing case. This entire study has, of course, dealt with dynamical processes that we are calling slow only in comparison to our subpicosecond expectations. Nonetheless, it is intriguing that so much of the phenomenology of the slow dynamics of glass forming systems has counterparts in this rather different kind of system. The presence of dynamical heterogeneity36 and a well-defined propensity for correlations between instantaneous structure and subsequent dynamics,37 along with fluctuationdissipation theorem violations,38 and non-mixing (“nonergodic”) behaviors39 are all mainstays of supercooled liquids. Systems of the sort that undergo nominally ultrafast relaxation, it seems, may eventually prove to be a useful place to learn about the molecular basis of ultraslow dynamics. Acknowledgment. We thank Daniel Kosov and our experimental collaborator, Stephen Bradforth, for continued thoughtful discussions. We are also grateful to Casey Hynes and William H. Miller for comments and suggestions offered at the early, formative, stages of this project. This work was supported by NSF Grant Nos. CHE-0518169 and CHE-0131114. Appendix: State-to-State Rotational Transition Rates The transition rates used in our master equation, eqs 3.6 and 3.7, are simply the Fermi’s-golden-rule rates for transitions between isolated-molecule rigid rotor states YJM mediated by the solvent perturbation given in eq 3.5
V ) V0
x4π3 Y
10
The rate of the J′ f J transition for energetically downhill (J′ g J) transitions is an average over all of the (equally probable) initial states (-J′ < M′ < J′) consistent with J′, and a sum over all of the final states (-J < M < J) states consistent with J
kJ′fJ ) k0
() 4π
1
J′
J
∑ ∑
3 2J′ + 1 M′)-J′ M)-J
|〈J,M|1,0|J′,M′〉|2 (A.1)
provided we absorb V0 and solvent density of states into the overall rate constant k0. The properties of 3-j symbols40 tell us that only final states with J′ ) J ( 1 and M′ ) M contribute, allowing us to evaluate the nonzero rate constants
kJ+1fJ ) k0(2J + 1)(2J + 3) ) k0(J + 1)SJ+1fJ
(
J 1 J+1 0 0 0
)
2
SJ+1fJ
Slow Solvent Structural Relaxation
SJ+1fJ ) )
1
M ) (J + 1)
∑
2J + 3 M ) -(J + 1)
J. Phys. Chem. B, Vol. 112, No. 2, 2008 377
(
J 1 J+1 -M 0 M
)
2
1 1 3 2J + 3
and leading to the simple final expresssion
kJ+1fJ )
k0 J + 1 3 2J + 3
(A.2)
The corresponding upward transition rate is specified by the detailed balance requirement in terms of the equilibrium probability distributions, eq 3.4b.
kJfJ+1 ) )
eq-qm PJ+1
Peq-qm J
kJ+1fJ )
2J + 3 -2βB(J + 1) e kJ+1fJ 2J + 1
k0 J + 1 -2βB(J+1) e 3 2J + 1
(A.3)
The rate constants given in eq 3.7 can be obtained from eqs A.2 and A.3 just by replacing J in the latter equations by J - 1. References and Notes (1) Chandler, D. Introduction to Modern Statistical Mechanics; Oxford University Press: Oxford, 1987; Chapter 8. (2) Zwanzig, R. Nonequilibrium Statistical Mechanics; Oxford University Press: Oxford, 2001; Chapters 7 and 8. (3) Stratt, R. M.; Maroncelli, M. J. Phys. Chem. 1996, 100, 12981. Stratt, R. M. In Ultrafast Infrared and Raman Spectroscopy; Fayer, M. D., Ed.; Marcel Dekker: New York, 2001. (4) Maroncelli, M.; Fleming, G. R. J. Chem. Phys. 1988, 89, 5044. Maroncelli, M. J. Chem. Phys. 1991, 94, 2084. (5) Carter, E. A.; Hynes, J. T. J. Chem. Phys. 1991, 94, 5961. (6) Whitnell, R. M.; Wilson, K. R.; Hynes, J. T. J. Phys. Chem. 1990, 94, 8625. Whitnell, R. M.; Wilson, K. R.; Hynes, J. T. J. Chem. Phys. 1992, 96, 5354. (7) Tuckerman, M.; Berne, B. J. J. Chem. Phys. 1993, 98, 7301. (8) Ladanyi, B. M.; Stratt, R. M. J. Phys. Chem. 1998, 102, 1068. (9) Jang, J.; Stratt, R. M. J. Chem. Phys. 2000, 112, 7524. Jang, J.; Stratt, R. M. J. Chem. Phys. 2000, 112, 7538. (10) Moskun, A. C.; Jailaubekov, A. E.; Bradforth, S. E.; Tao, G.; Stratt, R. M. Science 2006, 311, 1907. (11) Tao, G.; Stratt, R. M. J. Chem. Phys. 2006, 125, 114501. (12) The term “linear response” has been given a wide range of meanings in the solute-energy-relaxation literature. See ref 11 and Geissler, P. L.; Chander, D. J. Chem. Phys. 2000, 113, 9759. Laird, B. B.; Thompson, W. H. J. Chem. Phys. 2007, 126, 211104. As we have emphasized in ref 11, the linear response failures relevant here are not the fairly routine cases that automatically occur when the equilibrium dynamics around a solute excited state differs from that around a ground-state solute. The cases interesting in our context are those that have the additional attribute of slow solvent dynamics induced by the solute excitation. (13) Fonseca, T.; Ladanyi, B. M. J. Phys. Chem. 1991, 95, 2116. Fonseca, T.; Ladanyi, B. M. J. Mol. Liq. 1994, 60, 1. Skaf, M. S.; Ladanyi, B. M. J. Phys. Chem. 1996, 100, 18258. (14) Turi, L.; Mina´ry, P.; Rossky, P. J. Chem. Phys. Lett. 2000, 316, 465. (15) Aherne, D.; Tran, V.; Schwartz, B. J. J. Phys. Chem. B 2000, 104, 5382. Bedard-Hearn, M. J.; Larsen, R. E.; Schwartz, B. J. Phys. ReV. Lett. 2006, 97, 130403. (16) Gelin, M. F.; Kosov, D. S. J. Chem. Phys. 2006, 125, 224502. (17) Gordon, R. G. J. Chem. Phys. 1966, 44, 1830. (18) Moskun, A. C.; Bradforth, S. E. J. Chem. Phys. 2003, 119, 4500. (19) Alexander, M.; Yang, X.; Dagdigian, P. J.; Berning, A.; Werner, H. J. J. Chem. Phys. 2000, 112, 781. (20) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1987; Chapter 3. (21) The choices for the initial solute rotational energies and solvent conditions used in this study were motivated by those employed in the
ultrafast experiments described in ref 10 but were not attempts to mimic the experiments. In particular, our observation that liquid-state Ar seemed to cause the same kinds of relaxation behavior as that seen in experiment suggested that we concentrate our efforts in the simpler solvent. The actual experiments looked at CN rotors generated photochemically in ambienttemperature liquid water and alcohol solutions. Our simulated energetics, however, are more faithful to the experimental conditions. The rotationally hot channel for the actual photoreaction generated CNs with a broad range of rotational states centered near J ) 40 (corresponding to an average rotational energy of about 3000 cm-1, close to the largest initial rotational energies we study here). For a more detailed description of the connections with the experiments, as well as for other calculational details, see Tao, G. Ph.D. Thesis; Brown University: Providence, RI, 2007 and the on-line supplementary material for ref 10: (www.sciencemag.org/cgi/content/full/ 311/5769/1907/DC1). (22) Handbook of Mathematical Functions; Abramowitz, M.; Stegun, I. A., Eds.; Dover: New York, 1972; pp 896, 897. (23) Although the phenomena we are trying to describe are entirely classical, it is convenient (and probably numerical equivalent in our applications) to use a quantum mechanical master equation that looks at transitions between discrete quantized rotational energy levels. Among its other virtues, this approach makes it straightforward to capture the microscopically correct kinematics of rotational energy transfer. (24) Had we assumed that the leading symmetry was that of a homonuclear solute, which is nearly the case for our system (see ref 11), it is unlikely that there would have been any qualitative effect on the overall dynamics predicted by the master-equation. (25) McHale, J. L. Molecular Spectroscopy; Prentice Hall: Upper Saddle River, NJ, 1999; pp 228, 229. (26) A more microscopic application of Fermi’s golden rule to individual quantized rotational states in liquids can be used to derive a Landau-Tellerlike expression for the rate constants in terms of the molecular rotational friction. That molecular friction explicitly incorporates a torque-weighted density of solvent states. See: Jang, J; Stratt, R. M. J. Chem. Phys. 2000, 113, 5901. However, this level of treament is probably not justified here. Except for solutes such as H2 or for solvents such as superfluid helium, rotational behavior in liquids is usually too dissimilar to its gas-phase counterpart for this kind of approach to provide an accurate portrait of the solvent-density-of-states effects. (27) Rubin, R. J.; Shuler, K. E. J. Chem. Phys. 1956, 25, 59. Montroll, E. W.; Shuler, K. E. J. Chem. Phys. 1957, 26, 454. (28) A numerical master equation solution for the relaxation of a highly excited anharmonic oscillator appears to show much the same kind of singlecomponent relaxation profile. Nesbitt, D. J.; Hynes, J. T. J. Chem. Phys. 1982, 77, 2130. (29) Larsen, R. E.; Stratt, R. M. J. Chem. Phys. 1999, 110, 1036. Deng, Y.; Stratt, R. M. J. Chem. Phys. 2002, 117, 1735. Deng, Y.; Ladanyi, B. M.; Stratt, R. M. J. Chem. Phys. 2002, 117, 10752. Graham, P. B.; Matus, K. J. M.; Stratt, R. M. J. Chem. Phys. 2004, 121, 5348. (30) In vibrational energy relaxation in polar solvents, the electrostriction caused by electrostatic forces tends to be extremely important in determining the equilibrium solvent geometry, and therefore overall T1 times, but the relaxation process itself is still dominated by repulsive forces. See: Gnanakaran, S.; Lim, M.; Pugliano, N.; Hochstrasser, R. M. J. Phys.: Condens. Matter 1996, 8, 9201. Ladanyi, B. M.; Stratt, R. M. J. Chem. Phys. 1999, 111, 2008. Hynes, J. T.; Rey, R. In Ultrafast Infrared and Raman Spectroscopy; Fayer, M. D., Ed.; Marcel Dekker: New York, 2001. (31) If the total first solvation shell population is conserved, the kinetics is identical to standard A T B chemical kinetics result, with A the innermost and B the outermost populations. The rate constant ka is then the sum of the forward and backward rate constants. See ref 1. (32) This change turns our differential equation into a differentialdifference (or delay-differential) equation. Saaty, T. L. Modern Nonlinear Equations; Dover: New York, 1981; Chapter 5. (33) In the limit that t0 f 0 and the second half of eq 4.4c is satisfied, R f kb/ka, γ f 0, and eq 4.9 reverts back to eq 4.6. (34) The same equations can also be used to predict the time evolution of the solvent geometry (not shown). At intermediate times, the results are not as accurate as those for the energy relaxation, but at long times they match the initial-rotational-energy dependence of the molecular dynamics reasonably well. (35) The 10 ps survival time that one sees for a 1998 cm-1 initial rotation excitation in Figure 4 corresponds to 34 rotational periods of a classical free rotor. (36) Glotzer, S. C. J. Non-Cryst. Solids 2000, 274, 342. Ediger, M. D. Annu. ReV. Phys. Chem. 2000, 51, 99. (37) Widmer-Cooper, A.; Harrowell, P. J. Phys.: Condens. Matter 2005, 17, S4025. (38) Crisanti, A.; Ritort, F. J. Phys. A 2003, 36, R181. (39) Thirumalai, D.; Mountain, R. D. Phys. ReV. E 1993, 47, 479. (40) Messiah, A. Quantum Mechanics; John Wiley: New York, 1958; Appendix C.
