J. Phys. Chem. 1995,99, 2502-25 11
2502
Short-Time Dynamics of Solvation: Linear Solvation Theory for Polar Solvents Branka M. Ladanyi" Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523 Richard M. Stratt" Department of Chemistry, Brown University, Providence, Rhode Island 02912 Received: August 9, 1994; In Final Form: October 11, 1994@
Almost all of the current effort at understanding the dynamics of solvation in polar liquids has been focused on the dielectric properties of the solvent. For times under 300 fs, however, such a perspective tends to hide a feature which is common to both polar and nonpolar solvents, namely, that short-time solvation is govemed by the instantaneous normal modes (INMs) of the solvent. We illustrate this point here by applying the simplest level of INM theory-a linear theory-to a dipolar solute dissolved in acetonitrile. What we find is that the solvation process can be represented in the frequency domain as a solvation spectrum, which in turn can be dissected into portions resulting from a variety of different molecular processes. This analysis leads inexorably to the conclusions that short-time solvation, at least in this polar solvent, is primarily the result of solvent librations and overwhelmingly the result of the first solvation shell. At somewhat longer times, the solvent motions responsible for the solvation are far more intertwined, but most of the basic events in polar solvation, ranging from the earliest purely inertial dynamics through the onset of the more highly coupled behavior, can apparently be thought of in instantaneous-normal-mode terms. Within this framework one finds that polar liquids have a surfeit of high-frequency modes available and do have a somewhat specialized weighting of the various modes, but, aside from such issues, it is probably fair to say that short-time solvation is largely independent of dielectric considerations.
I. Introduction For a variety of reasons, much of the experimental effort at understanding the process of solvation has been directed at the time-dependent fluorescence of dye molecules in polar solvents.'-6 Naturally since what was usually being monitored was the progress of the Stokes shift of the emission, it made sense to focus on situations in which the solvent significantly altered the energetics of either the ground or excited electronic state of some solute-and polar solvents were known to be quite capable of causing large shifts in electronic spectra.' However perhaps an equal motivation was the feeling that equilibrium solvation was dominated by the energy lowering afforded by placing dipoles in dielectric media, so one would reasonably expect dynamics to be similarly constrained. Indeed, the majority of the theoretical work in the area has taken just this perspective that one should derive solvation dynamics from the detailed dielectric properties of the ~ o l v e n t . ~ -In ' ~ support of this idea, it might be pointed out that as theories have started to include more and more information about the dielectric constant as a function of frequency and wavevector, the results have consistently improved.8 It should be emphasized, nonetheless, that while such a viewpoint is useful, it is not as far reaching as might be hoped. For one thing, different kinds of experimental probes of solvation have now started to appear, most notably via transient hole burning,I4that do permit the study of solvation by nonpolar solvent^.'^^'^ Moreover the ultimate goal of the solvation dynamics theories should not be simply to reproduce the observed fluorescence decays. Eventually, we would like to be able to ascribe the various contributions to the dynamics to well-defined classes of specific molecular motions, something that dielectric-based theories, even microscopically based ones, @
Abstract published in Advance ACS Abstracts, February 1, 1995.
are not quite as well suited for. The purpose of this paper is to illustrate how recent theoretical developments allow even solvation by a polar solvent to be profitably factored into distinguishable molecular contributions. Thorough simulation studies, of course, do make all of this molecular-level information available to US.'^-*^ It is largely through the conjunction of such studies with the corresponding experiment^^^-,^ that the universal features of solvation correlation functions C(t) have been revealed. One typically finds, in both ~ o l a r ' and ~ - ~nonpolar16 ~ solvents, that some 70-80% of the decay of correlation occurs in the first 100 fs or so, followed by a much slower, but frequently subpicosecond, relaxation of the remainder. The initial ultrafast decay, which strongly resembles a Gaussian govemed by some solvation f r e q u e n ~ y ,wsolv ~~.~~
has been shown by compelling simulation (as well as by more analytical route^)*^.^^ to reflect the inertial dynamics of the solvent-the ballistic motion of individual solvent molecules. At the other end of the scale, continuum dielectric t h e o r i e ~ ' - ~ .seem * ~ to explain nicely why the asymptotically long-time decay should be governed by the longitudinal dielectric relaxation time 2 ~ : ~ ~
C(t)= exp(-t/t,)
(1.2)
Hence the realm where solvent molecules behave collectively enough to be interesting, but not so collectively as to obscure individual molecular events, appears to be the middle ground of the subpicosecond but still non-Gaussian relaxation. It is here where we suggest that our methodology may be useful in helping to interpret the computer simulations. The very
0022-3654/95/2099-2502$09.00/0 0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 9, 1995 2503
Short-Time Dynamics of Solvation
studies. However, intermolecular potentials are often of the simulations that demonstrated that the initial Gaussian decay interaction-site form, meaning that the natural variables for each could be reproduced without allowing either intersolvent molecule are the locations of a set of intramolecular sites.36 i n t e r a ~ t i o nor~ coordinated ~ solvent dynamics22simultaneously Unfortunately, with rigid molecules, these site variables are made the telling point that there are coherent oscillations in the linearly dependent and therefore cannot be used as a basis for intermediate region of the correlation functions that really are INM analysis. In the three-collinear-sites model for acetonitrile a direct result of coupled solvent motions-and it is this behavior to be used in this paper, for example, the three sites correspond one wants to be able to dissect into its component molecular to 3(3) = 9 Cartesian coordinates, but there are only five events. independent degree of freedom-the center-of-mass translation The approach we shall be taking is based on the instantaneous and two angles reflecting the orientation of the dipolar axis. normal mode (INM)analysis of liquid dynamics.30,32-42The Nonetheless, as we describe in this section, it is a simple idea is that the displacement of the solution's configuration R, matter to accomodate site-site potentials within INM theory. at any time t from what it was in its initial configuration & (at One just needs to take care of some bookkeeping. Suppose time 0) can be decomposed into a set of independent, collective the solution we are studying has a single solute, labeled 0, N modes qa(t),where a ranges from 1 to the number of degrees solvent molecules, j = 1, ...,N, and the intermolecular potential of freedom of the system. Each different starting point generates can be written in a pairwise interaction-site form.43 That is, let a different set of modes, but for time intervals t short enough, us assume that the total (ground-state) potential energy V and not only can these instantaneous modes be shown to be a the solvation energy A (the difference in the solute-solvent rigorous representation of the time evolution, they can be shown potential energies between the excited- and ground-solute states) to have simple harmonic dynamics, meaning that they can be can be expressed in terms of pair potentials u, v, and w : written analytically in terms of sines and cosines o f t times the mode frequencies, ~ a As. a ~result, ~ any time correlation V = Vsolute-solvent + Uso~vent-solvent function pertinent to the liquid can in principle be expressed in terms of one or more Fourier transforms of some appropriately weighted probability distributions of INM f r e q u e n ~ i e s . ~ ~ ~ ~ ~ One fairly simple way of applying this framework to solvation time correlation functions is to expand any changes in A, the N difference in solvation energies of the excited and ground solute states, in powers of the instantaneous normal modes.30 Stopping at linear order N
A =zw(0j) j= 1
gives what we shall call a linear solvation theory, enabling us to identify a single solvation spectrum, the relevant probability density of modes, as simply the distribution of INMs in the solution weighted by the coefficients (A'J2. These coefficients, in turn, act as a set of susceptibilities, classifying the modes by their varying abilities to solvate the solute. Thus by projecting out specific portions of this solvation spectrum, the thought is that one should be able to quantify the extent to which librational, translational, or even more narrowly defined kinds of motions are responsible for the solvation. In what follows we shall show how this program-which was outlined in a previous paper30 (hereinafter refered to as paper 1)-can be implemented for the canonical polar aprotic solvent, acetonitrile. We begin, in section 11, by reviewing the instantaneous normal modes and linear solvation theory in more detail, paying particular attention to how the theory can be formulated when site-site potentials are being employed. We also spell out just how one can carry out projections of the solvation spectrum into different physical solvation channels. Section III presents the specifics of our model and our calculational procedures, and section IV sets out our results. We conclude, in section V, with some discussion of our findings. 11. Linear Instantaneous-Normal-Mode Solvation Theory
A. Instantaneous-Normal-Mode Analysis with Interaction-Site-Model Potentials. Typically one sets up an instantaneous normal mode treatment of the dynamics of a molecular liquid in terms of the center-of-mass coordinates of each molecule and the Euler angles describing the molecule's o r i e n t a t i ~ n . ~This ~.~~ procedure makes it straightforward to divide any given motion into its rotational and translational components, a rather desirable feature in the context of solvation
which themselves are sums of pairwise potentials between all the sites on the interacting molecules. Thus for two solvent molecules j and k with intramolecular sites labeled by a and b, respectively:
with rja,bthe distance between site a on molecule j and site b on molecule k. We shall assume similar formulas can be written for the solute-solvent potentials. Much as we have done before,32 the instantaneous normal modes can be constructed for our (N f 1) linear molecules by diagonalizing, at each liquid configuration &, the 5(N 1) x 5(N 1) dynamical matrix D corresponding to the total potential:
+
+
Djp,kv= (mjpmkV)-'l2 a2vlarjpark,,
01,~) = (x,Y,z,e,@) (2.3)
where the rigid-molecule coordinates and effective masses for each molecule j are defined to be xj, Y j , zj
rjp={:
Mj 01 = x, y , z ) mju={4 01 = 6 ) 4 sin2 ejo 01 = 4)
(2.4)
Here x, y , and z are the Cartesian coordinates of the center of mass, 0 and 4 are the angles specifying the orientation with respect to the laboratory frame, M is the molecular mass, I the moment of inertia, and 8 0 is the value of 8 at configuration &. The frequencies of the INMs, ua,are then prescribed by the eigenvalues, and the modes themselves, qa(t), are related to the
Ladanyi and Stratt
2504 J. Phys. Chem., Vol. 99, No. 9, 1995
actual coordinate displacement through the matrices of eigenvectors U :
Once we know G(t),the experimental correlation function can be computed, if so desired, just by integration: C(t) = 1 - ((SA)2)-1J‘(t- t)G(r)dt
But the locations of the sites on the jth molecule can be derived fromthe center-of-mass location rj, and the orientational unit vector Qj:
rja= rj
+ djaQj
(2.6)
where da is the (signed) distance between site a and the center of maw4 Given this fact, the dynamical matrix can eventually be reexpressed in terms of derivatives with respect to site coordinates. After some algebra we find from eqs 2.1-2.3 and 2.6 that
(2.13)
In any case, within the linear approximation set forth in eq 1.3, it is a simple matter to substitute the instantaneous-normalmode dynamics into eq 2.10 to arrive at the desired correlation functions. Inasmuch as this process has already been exhibited in paper I,30we content ourselves here with presenting the final results:
G(f) = k,TS dw @solv(w) cos ut
(2.14)
C(t) = 1 - (kBT/((SA)’))Sdw @solv(w) (1 - COS
(2.15) where the solvation spectrum esolv(w)is given by the weighted probability density of instantaneous normal modes: esolv(w)
= ( C ( A ’ a ) 2 S ( w- ma)>
(2.16)
a
k t j
[U’(rja,kb>/rja,kblrja,kb c a , p v u ) }
where the 3 x 3 tensor fa&) site-site pair potentials:
(2.7)
contains the derivatives of the
(u(r) in the case of solvent-solvent terms and v(r) for solventsolute terms) with i. the unit vector in the r direction and 1 the 3 x 3 unit matrix, and the three-dimensional transformation vectors C are defined to be C,,Jj) = 8rja/arjp 2
Ca,,uvti)= a rjdarjp arjv
(2.9)
Practical formulas for these coefficients in terms of site coordinates are provided in the Appendix. B. Linear Solvation Theory. The quantity that describes the time evolution of solvation within linear response theory is the normalized solvation correlation function:
By construction, the susceptibilities that perform the weighting, A’a, are just linear combinations of derivatives of the solvation energy with respect to the rigid-molecule coordinates listed in eq 2.4, each evaluated at the time 0 configurations &. Physically, though, the susceptibilities serve to quantify the extent to which each INh4 a is capable of influencing the solute-solvent interaction energy. The explicit sum-over-modes form of the solvation spectrum suggests that it should be possible to partition the solvation response according to the different kinds of liquid dynamics responsible. For example, one can define rotationally and translationally projected versions of our susceptibilities:
A’ = g o t a
a
+ stram
(2.18)
a
sFt= DfPT: P
17319
C(t) = (SA(t)SA(O))/ ((SA)’)
(2.10)
where 6A is the fluctuation in the (differential) solvation energy A defined by eq 2.1, the angular brackets denote an equilibrium average over the time t = 0 state of the solution, and the denominator is equal to the mean-square fluctuation in solvation energy:
SA(t) = A(t) - (A) ((SA)’) = (A2)- (A)’
(2.1 1)
However, while C(t)is the object most closely allied with timedependent fluorescence experiments, as we noted in paper I,30 it turns out to be simpler for us to compute a mathematically equivalent function, the solvation velocity correlation function
J
WC=X.j,Z
in which we have selected out those motions involving solely reorientation or solely center-of-mass translation with the aid of the projection matrices:
J
,U=XJ,Z
G(t):
G(t)= (A(t)A(O))= -((dA)’) d’C(t)/dtZ
(2.12)
The solvation spectrum can thus be divided into rotational, translational, and coupled portions by substituting eq 2.18 into
J. Phys. Chem., Vol. 99, No. 9, 1995 2505
Short-Time Dynamics of Solvation
TABLE 1: Intermolecular Potentials and Molecular Properties site Q.le (EalkdlK 0,lA A. Solvent Acetonitrile"
eq 2.16:
(2.22)
Me C N
a
(2.23) U
b
(2.24)
0.269 0.129 -0.398
50 50
3.6 3.4 3.3
B. Diatomic Soluteb 0.5 146 -0.5 146
3.8 3.8
mec = 1.46 A, me^
= 2.63
191
A.
r,b
= 2.28
A.
a
Hence the solvation response itself, eq 2.14, can be broken into these same categories. That these are, in fact the correct projectors is demonstrated by confinning that they have all the usual properties of projection operators p o t +pmns pot ptrans
=1
=potpot
= ptransptrans
Indeed, these same projectors are what we have used elsewhere32,34to separate the (unweighted) instantaneous-normalmode density of states into its component parts:
a
= D"'(w)
+ Dtrans(u)
Drot(0)= ( ( 5 N ) - ' x P 3 ( 0
- w,))
(2.25) (2.26)
a
which, in tum, are just the ingredients necessary to compute the INM versions of the single-molecule center-of-mass and angular-velocity autocorrelation functions. For the jth molecule, instantaneous normal mode theory predicts
(2.28) V,,t(t>
= b j ( O > * OjCO) kBT = 5-Jd~ I
D"'(w) COS ut
state and a completely nonpolar excited state. Since we were envisioning that the nonelectrostatic portions of the solutesolvent potentials could be regarded as remaining largely the same in both electronic states, the differential solute-solvent energy A was taken to be simply the electrostatic portion of the ground-state potential. A complementary study in which A was set equal to the nonelectrostatic component alone (representing a somewhat extreme nonpolar solvation situation in which the well depth of the interaction with the solvent doubles on electronic excitation) was also ~ n d e r t a k e n . ~ ~ Our specific calculations were performed on samples of 256 rigid molecules comprising either pure acetonitrile or 255 acetonitrile solvent molecules and one solute molecule. For acetonitrile we used the three-site Edwards et al?6 model, in which the methyl group (CH3) is a single interaction site (Me), making the molecules effectively linear. This representation takes the solvent-solvent pair potentials to be sums of sitesite Lennard-Jones (LJ) and Coulomb potentials:
(2.29)
re~pectively.~~~3~ In much the same way, different operators focusing on other subsets of the dynamics (and, in particular, on motion within the first solvation shell) can be constructed as well.
111. Models and Computational Details The major choice confronting us in carrying out this study concerned the models to be used for the ground and excited states of the solute. We chose not to attempt to mimic any of the rather complicated dye molecules that have been employed in real time-dependent fluorescence studies, opting instead for a sizeable, hypothetical, diatomic solute with a polar ground
where and a, are the LJ potential well depths and the collision diameters, respectively. The ground-state solute-solvent pair potentials Vu&) were taken to have the same form, with the solute bond length, site masses, and Lennard-Jones (LJ) parameters similar to those of B r ~ , 4but, ~ unlike bromine, were given site partial charges of el2 and -e/2. In accord with our previous remarks, the Wnb(T) pair potentials, the differences between the solvent-excited-state solute and the solvent-ground-state solute potentials, were set equal to the Coulombic portion of the vab(r) potentials. All of the potential parameters involved in these functions and all of the bond lengths are summarized in Table 1. The net result of these choices is that the solute mass and moment of inertia are larger than those of the solvent by about a factor of 4, but the solvent and ground-state solute are of roughly comparable polarity (with dipole moments of 4.12 and 5.47 D, respectively; 1 D = 3.336 x C M). Both the pure liquid and the solution were simulated at conditions corresponding to 293 K and atmospheric pressure. A microcanonical ensemble MD simulation and time steps of 8 fs were used to generate the molecular trajectories with the aid of a leapfrog algorithm for linear molecules, which applies a length constraint to the unit vector along the molecular axis.48 Ewald sums49were employed for Coulombic interactions. The results, shown in the next section, represent averages over trajectories of 320 ps duration in each case. In the calculation of the solute-solvent electrostatic energy A, its derivatives with respect to molecular coordinates, AjP, and the force constant matrix elements DjP,b,we used real-space lattice sums50so as to include the long-ranged portions of Coulomb interactions as accurately as possible. The INM eigenvalues and eigenvectors for each system were calculated at a set of 100 configurations separated by 3.2 ps
Ladanyi and Stratt
2506 J. Phys. Chem., Vol. 99, No. 9, 1995
0.0081
I
8
8
I
-
*
8
I
8
-
8
I
I
r
9
I
I
1.0,
I
1
’.On
I
1
1
total
-
INM
I
-100
0
100 ;/cm
-
’
200
300
Figure 1. Instantaneous-normal-mode spectrum of neat liquid acetonitrile. The total spectrum and its projections into translational and rotational parts are as indicated. Following the usual convention, the imaginary modes are shown as negative frequencies in this and the succeeding figures.
(400 time steps). They were computed by standard methods5’ for real symmetric matrices, using the Numerical Algorithms Group (NAG) Fortran library5*routine folujf and fo2umf.
IV. Results We begin our presentation by considering what an INM analysis of neat liquid acetonitrile reveals. The probability density of the instantaneous-normal-mode frequencies and its decomposition into rotational and translational parts, eqs 2.25 2.27, are shown in Figure 1. What is probably most striking about these spectra is how smoothly they fit into the pattem established by other l i q ~ i d s . ~Regardless ~ - ~ ~ of the molecular details, the translational portion apparently retains its universal, low-frequency-dominated,“right-triangle” shape; neither wate9‘ nor A$3,35differ from CH3CN all that much in their translational density of states. The rotational segment, though, shows a clear progression from nonpolar to polar aprotic liquids (such as CH3CN) to waters3‘ As attractive interactions become stronger and more directional, the right-triangle shape begins to disappear and the whole rotational peak moves to higher frequency, suggesting that the rotational motions are becoming stiffer. The rotational peak in acetonitrile is still not a prominent feature of the spectrum as a whole, the way it is in water, but it is noticeably skewed from the translational peak. In a similar vein, there is a notably continuous progression in the fraction of imaginary modes in the various liquids. While there can be as many as 33% imaginary modes in liquid Ar (depending on thermodynamic condition^),^^ our CH3CN shows only 21% imaginary modes, and liquid water only 6%.34 Evidently propagation along the unstable, negative curvature directions of the potential surface becomes less and less feasible as the attractive forces become more important. The most direct dynamical implications of acetonitrile’s INM spectra are translational and angular velocity autocorrelation functions, eqs 2.28 and 2.29, which we exhibit in Figure 2. Just as we have observed in previous application^,^^^^^ provided we omit the imaginary modes, the instantaneous-normal-mode expressions will accurately predict the subpicosecond features of these correlations functions, including the rotational back-
INM,s
-0.51 0.0
I
I
0.2
0.4
0.6
t/PS Figure 2. Translational (upper panel) and rotational (lower panel) velocity autocorrelation functions of acetonitrile in the neat liquid. In both cases, the solid line is the exact molecular dynamics result, the dot-dashed line is the full instantaneous-normal-mode prediction, and the dashed line is the instantaneous-normal-mode prediction when only real (stable) modes are included.
scattering that leads to the minimum around 170 fs. Indeed, rotational motion always seems to be particularly well described in terms of an INM analysis, perhaps because of its somewhat shorter time scale than center-of-mass translation. Of course, one could imagine that success in understanding single-molecule dynamics in the neat liquid need not carry over to the putatively different problem of solvation dynamics. However, as pointed out by Maroncelli et al.?4 there is a strong link between the two-a link which we shall find as well. The INM approach to solvation dynamics begins with the solvation spectrum, eq 2.16, shown in Figure 3. To the extent that we can rely on a linear solvation theory, the solvation spectrum is how the solvation process should be regarded in the frequency domain and is therefore our gateway to decomposing the process into its component events. Using the projection operators given in Section II, we find, for example, that the vast majority of this spectrum corresponds to rotational motion. Even without this projection, one might have surmised this feature just by comparing the solvation spectrum with the neat liquid’s density of states (Figure 4). Except for differences in normalization, the solvation spectrum resembles the rotational portion of the total spectrum far more than it does the translational portion. In much the same fashion, we can project out the fraction of the solvation spectrum arising from solvent molecules which are in the first solvation shell of the solute at f = 0. The definition of such a first shell is somewhat arbitrary-we classified any solvent whose center-of-mass was within 6.38 8, of one of the two solute sites as being within the first shell (giving us an average of 14.8 solvent near neighbors)-but reasonable alternative definitions seemed to lead to little difference in the resulting time correlation functions.55 By using first-solvation-shell projectors and dividing the INM susceptibilities between first shell and the remaining (outer) shell contributions, in a manner analogous to eq 2.18, we observe
J. Phys. Chem., Vol. 99, No. 9, 1995 2507
Short-Time Dynamics of Solvation
First & outer solvation shell contributions
io
1 " " I " " I " " I " " l
total
I 0 7
5-
0
\
4
\
3
5
Y
...
8 9
'\.I
I I
-100
I .
0
I
,
I I I I , , I , I , , I
100
200
300 -io0
Figure 3. Solvation spectrum of acetonitrile with a dipolar solute. The total spectrum and its projections into rotational, translational, and
rotation-translation-coupled parts are as indicated. Note that these spectra are defined so that the area under the curves reflects the magnitude of the solute coupling to the solvent dynamics. These areas are therefore not dimensionless quantities. Note also that the area under the cross-rotation-translation curve is zero by construction.60
Figure 4. Comparison of the solvation spectrum and the instantaneousnormal-mode spectra of liquid acetonitrile. The total, translational, and rotational INM spectra are the same neat-liquid results shown in Figure 1. The solvation spectrum is the total spectrum given in Figure 3, @solv, but made dimensionless here (Dsolv)by normalizing to an area of * / 5 , the same area as the rotational INM spectrum.
that the overwhelming bulk of the spectrum is, in fact, directly the result of motion within the first shell (Figure 5). Since a subsequent rotationaYtranslational breakdown of this first-shell spectrum (not shown) reveals that rotations are just as prominent in the first shell as they are for the solvent as a whole, one is led to the inescapable conclusion that the most essential ingredient in the short-time solvation performed by our polar solvent is the reorientation and libration of near-neighbor solvent molecules. This conclusion is rather striking, though it does agree with the findings in a number of simulation s t ~ d i e s , ~ ~
0
100
200
300
Figure 5. Decomposition of the solvation spectrum of acetonitrile into different solvation-shell components. The total spectrum and its projections into first-solvation-shell contributions, remaining (outershell) solvent contributions, and inner-shelllouter-shell-coupled parts are as indicated. As with Figure 3, these curves have an absolute normalization and the cross curve is defined so as to have zero area.
because one could argue, as Bagchi and co-workers have,I2 that the long range of the electrostatic potentials makes for a rather collective response from the solvent-and that it is the collective response that generates the ultrafast solvation. The extent to which our results disagree with this collective picture is an issue we shall return to in the next section. We should emphasize that these conclusions are inescapable only for times short enough for our linearized version of INM theory to work. We therefore need to establish the domain of validity of the theory by comparing our predictions for time correlation functions with actual molecular dynamics results; these comparisons are shown in Figure 6 for the full-solvent response and in Figure 7 for the first-shell response. The linearized theory is clearly not fully quantitative, but the results within the first 300 fs (and in some cases much longer) are quite respectable. The existence and even the time of the first minimum in the solvation velocity correlation function G(t)are predicted successfully. While the small subsequent minimum at 600 fs is not anticipated, at least the first few hundred femtoseconds seem amply suited to an instantaneous normalmode description. Actually, a comparison of the "experimental" correlation functions C(r) in Figures 6 and 7 is revealing in yet another way. It is plain that the "long" time relaxation of the first solvation shell is significantly slower than that of the complete solvent, a feature the linear-INM theory, perhaps surprisingly, correctly predicts.57 However, the two versions of the solvation spectra are not all that different. We therefore need to come to grips with where else within the INh4 picture, besides the solvation spectra, such solvent relaxation information can reside. This issue as well will be the subject of the next section. V. Discussion
As computer simulations begin to reproduce the details of solvation dynamics experiments in finer and finer detail, it seems plausible that simulated time correlation functions will end up , mirroring ~ ~ , ~ ~ the laboratory findings ever more closely. With the
Ladanyi and Stratt
2508 J. Phys. Chem., Vol. 99, No. 9, 1995
r
First solvation shell I
I
I
I
I
I
I
I
0
5
I 0
n
5
\
r=
u
W
u
u
v
u
-2
, *o1*
u
I
1.c
0.5-
W
n
t
u
W
t
0.0 -
-0.5 0.0
I
0.2
I
0.4
I
0.6
0.8
-0.5
3
0.0
Figure 6. Solvation time correlation functions for a dipolar solute in liquid acetonitrile. (a) Upper panel: The solvation-velocity time correlation function defined by eq 2.12. (b) Lower panel: The full solvation time correlation function defined by eq 2.10. In both cases, the solid line is the exact molecular dynamics result, the dot-dashed line is the full instantaneous-normal-mode prediction, and the dashed line is the instantaneous-normal-mode prediction when only real (stable) modes are included. Note that c(t)is normalized to unity at t = 0 but that G(0) has a nontrivial value proportional to the so-called solvation frequency (eq 5.1).
possible exception of a few nagging questions about the accuracy of linear response theory,21the evidence seems to be that as long as one includes a sufficiently realistic portrait of the size and charge distribution of the straightforward molecular dynamics should suffice to mimic timedependent-fluorescence experiments, at the very least. The question that remains, though, is how best to assign the features of these time correlation functions to specific molecular occurrences in the solvent. The earliest portion of the solvation relaxation is, as has been can be noted, entirely single molecule in understood simply by summing up the contributions that would be obtained from all of the solvent molecules were they allowed completely free rotation and translation. The absence of any cooperative dynamics in this “inertial” stage suggests that the (intrinsically collective) instantaneous normal modes may not be all that useful a language for discussing this phase of the r e l a x a t i ~ n .However, ~~ by working out the relevant power series in time, it is easy to show that the solvation frequency defined in eq 1.I can be written rigorously in terms of the lNh4 solvation spectrum:59
meaning that it can be partitioned into its rotational and translational components using eq 2.21:60 (5.2)
with
0.2
0.4
0.6
0.8
t/PS Figure 7. Contributions to the solvation time correlation functions arising from the first solvation shell alone. (a) Upper panel: The solvation-velocity time Correlation function defined by eq 2.12. (b) Lower panel: The full solvation time correlation function defined by eq 2.10. The individual curves are as described in the caption to Figure
6.
for example. Indeed, if we compare the integral of our solvation spectrum with that of its projections, we find that 76.2% of the ultrafast relaxation is attributable to rotation and only 23.8%to translational motion, despite the fact that 60% of the solvent degrees of freedom are translational. The predominance of rotational dynamics in polar solvation is even more pronounced at somewhat longer times. The fact that the entire shape of the solvation spectrum is almost precisely that of its pure rotational component-and the translational portion mainly serves to cancel the ro-translational coupling-has consequences for times longer than the (order r2) inertial domain. While center-of-mass translation may be playing some quantitative role, librational motion apparently continues to be the primary mechanism involved in the relaxation, even beyond the first 50 fs. A direct comparison of the solvation velocity correlation function, G(r), with the single-molecule angular velocity autocorrelation function (Figure 8) confirms this s u p p ~ s i t i o n .The ~ ~ real dynamics in this regime, however, is no longer single-molecule based. If the success of the INM formation in reproducing the first oscillation in G(r) between 150 and 300 fs is any indication, the contributing rotational motion is now better regarded as being fundamentally collect i ~ e Indeed, . ~ ~ by this point, probably all of the liquid’s time evolution is best thought of in these terms, though it is apparently the rotational kinematics that one normally sees the most of in polar solvents. Of course, with nonpolar solvents,61 or even solvated electrons in ~ a t e r ,it~is~the . ~effective ~ size of the solute “cavity” that tends to define the energetics, so one would expect to find translational motion to reassert its preeminence-and the INM solvation spectrum to reflect this shift. In support of this prediction, we note that when the nonelectrostatic form of the solute-solvent energy referred to
J. Phys. Chem., Vol. 99, No. 9, 1995 2509
Short-Time Dynamics of Solvation
-0.5
V 1
1
1
1
~
1
1
1
1
1
0.0 t/PS
Figure 8. Comparison of the solvation velocity (G) and angular
velocity (W") time correlation functions for acetonitrile. Both curves are exact molecular dynamics results normalized to unity at time zero. in section I11 was employed, only 44.9% of relaxation was rotational, a figure close to the 40% one would have expected if all degrees of freedom participated equally. Much the same analysis and therefore many of the same kinds of remarks can be made concerning the dominant role of the first solvation shell in controlling solvation dynamics. By the analogs of eqs 5.2 and 5.3, we find that 88% of the ultrafast relaxation arises from the first solvent shell, and by noting the remarkable parallelism in the shapes of total and fiist-shell solvation spectra we see that this behavior must extend through the first 300 fs or so. Beyond this point, though, Figures 6b and 7b show that relaxation would be noticeably slower were it not for the presence of the remainder of the solvent. The ability of INM theory to anticipate this phenomenon is somewhat of a puzzle but may have its origins in the fact that the INM correlation function, eq 2.15, depends not only on the solvation spectrum but also on the equilibrium mean-square fluctuations in the solvation energy, ((dA)2),as well. Since these fluctuations are larger for the first shell alone than they are for the entire solvent (in part because of the fluctuations in the number of near neighbors), the first shell correlations would be expected to decay more slowly. The INM perspective on this long time behavior is interesting in view of the rather different analysis in the literat~re?~ which regards the asymptotic relaxation as a symptom of the highly collective character of dynamics in dielectric liquids. That this tail really is the result of strongly coupled dynamics is quickly demonstrated by performing an exact decomposition of C(t). From the molecular dynamics trajectories themselves it is straightforward to write
C(t) = (dA(0)dA(t))/((dA)2) = shell(0)dA1"she"(t))/((dA)2)
+
(dAouter shells (0)dAouter shells
( t ) ) / ( ( W 2+ ) ( o ) w A ) 2 )+ shells(0)dAIStshe"(t))l((dA)2)
(dA1st shell(0)dAouter (dAouter
-
shells
where dAshell(t)= Ashell(t) - (A)she11, and by A(t) for a given shell we refer to the value of the (differential) solute-solvent energy arising solely from molecules in that shell at time t.
.
6
0.5
1 1.o
Figure 9. Solvation time correlation functions resulting from the various solvation shells. The total correlation function is indicated by a solid line, the portion of the correlation function arising from the first-shell solvent molecules alone by a dashed line, the fraction coming from non-first-shellsolvent by the upper dot-dash curve, and that arising from the cross correlations between the inner and outer shells by lower dot-dash curve. All of these correlation functions are exact molecular dynamics results.
Similarly (A) for a given shell is the equilibrium contribution from that shell. Since we still take ((dA)*) to be the total fluctuation, the first term then gives us an unnormalized version of Clstshell(t),the second term Cuter shells(t),and the third and fourth terms CroSs(t), all of which are shown in Figure 9. Clearly both the inner and outer shell responses are relatively slow, but the cross term serves to cancel these slow decays quite neatly in the total response. Hence the longer time decay is manifestly the result of collective These observations differ in varying measures from the ideas put forth by Bagchi and co-workers.I2 For the very earliest (order t2) portion of the relaxation, their suggestion that the collective behavior and long-ranged forces are necessary to understand the extreme rapidity of the decay turns out to be somewhat of an overstatement. It is true that the equilibrium distributions of molecular positions and orientations are controlled by the specifics of the intermolecular forces, so the situation at t = 0 is indeed set by those forces. However, the onset of the subsequent dynamics is, as we have e m p h a ~ i z e d , ~ ~ explicitly and rigorously noncollective. Even the instantaneous normal mode picture of the dynamics is more collective than one really needs in this regime. At these very early times, the INM formulation is best regarded as an exact-but in this time domain, rather unnecessary-mathematical reformulation of what is essentially one-molecule-at-a-time motion.30 The INM language, nonetheless, is quite helpful in rendering quantitative answers to questions such as the range and character of the solvent dynamics critical to solvation. The dominance of the solvation spectrum by the first-shell solvents and by rotational motion is unambiguous-which is not to say that there are no other contributions, but it does make somewhat less relevant any discussion of the comparative successes of approximate theories that do and do not include, say, translational motion.11,'2 Indeed, the sensitivity of these approximate theories to the inclusion of translation is apparently somewhat deceptive. On the other hand, theories that are not based on explicitly molecular modes may be at somewhat of an advantage for time
2510 J. Phys. Chem., Vol. 99, No. 9, 1995
scales appropriate to genuinely diffusive motion, the main question being just where such a regime begins in the context of solvation. In closing we might also emphasize that none of this analysis makes any reference whatsoever to any damping of the solvent's modes. The instantaneous normal modes seem to do a reasonable job of explaining not only the velocity autocorrelation functions of polar solvents but also the early time course of solvation as well, all without using the language of under- and overdamped modes popularized by the useful, but purely phenomenological, Brownian-oscillator description^.^^ By the same token, though, the kinds of INM analysis illustrated here are evidently not going to be useful beyond a half a picosecond. It is possible that transcending our linear solvation theory may help matters somewhat, but eventually the molecular origin of damping, what we would call the time development of the instantaneous normal modes themselves, will have to play a role. It will be interesting to see if solvation dynamics experiments can be devised that are capable of watching this transition take place.
Acknowledgment. It is a pleasure to dedicate this paper to Stuart Rice, one of the pioneers in the study of excitations in liquids. We thank Graham Fleming, Minhaeng Cho, Mark Maroncelli, Paul Barbara, Mark Berg, and Matthew Zimmt for sharing their insights. We also thank Michael Buchner for letting us know about his preliminary INM work on acetonitrile. This work was supported by NSF grants to B.M.L. (CHE9019707) and to R.M.S. (CHE-9119926). Appendix We present here the coefficients use to accomplish the rigidbody to site-coordinates transformations defined by eqs 2.6 and 2.9. In addition to the laboratory-frame Cartesian unit vectors 9 ( y = x,y,z), let us define the three mutually perpendicular unit vectors for each molecule j :
A,G)
Au)
= abo)/aej= Amo)x
Here 60')defines the molecule's orientation in the laboratory frame, so that if site YZ is the "head" of the molecule (say the N site in CH3CN), making rjn its position, and rj is the location of the molecule's center of mass, then
XG) = [(rjn- rj)/djn] 2 and similarly for Yo') and ZU). In terms of these vectors, the five first-derivative coefficients Ca,po') are
C,,O') = P,
y = x, y , z
If we further define the vectors
Ladanyi and Stratt
= -[1 - z0')2]-1{Ao') - Z0')2> we can express the 25 second derivative coefficients, Ca,pv(j):
References and Notes (1) Maroncelli, M. J. Mol. Liq. 1993, 57, 1. (2) Barbara, P. F.; Jarzeba, W. Adv. Photochem. 1990, 15, 1. (3) Weaver, M. J.; McMannis, G. E.; Jarzeba, W.; Barbara, P. F. J . Phys. Chem. 1990, 94, 1715. (4) Maroncelli, M.; MacInnis, J.; Fleming, G. R. Science 1989, 243, 1674. (5) Simon, J. D. Acc. Chem. Res. 1988, 21, 128. (6) Hynes, J. T.; Carter, E. A.; Ciccotti, G.; Kim, H. J.; Zichi, D.; Ferrario, M.; Kapral, R. In Perspectives in Photosynthesis; Jortner, J., Pullman, B., Eds.; Kluwer: Dordrecht, 1990. (7) Reichardt, C. Solvents and Solvent E'ects in Organic Chemistry, 2nd ed.; VCH: Weinheim, 1988; Chapter 6. (8) Raineri, F. 0.;Zhou, Y.; Friedman, H. L.; Stell, G. Chem. Phys. 1991, 152,201. Raineri, F. 0.;Resat, H.; Pemg, B.-C.; Hirata, F.; Friedman, H. L. J . Chem. Phys. 1994, 100, 1477. Raineri, F. 0.; Pemg, B.-C.; Friedman, H. L. Chem. Phys. (submitted). (9) Wei, D. E.; Patey, G. N. J. Chem. Phys. 1990,93, 1399. Chandra, A.; Wei, D. E.; Patey, G. N. J. Chem. Phys. 1993, 99, 4926. Chandra, A,; Patey, G. N. J. Chem. Phys. 1994, 100, 1552. (10) Bagchi, B. Annu. Rev. Phys. Chem. 1989, 40, 115. Bagchi, B.; Chandra, A. Adv. Chem. Phys. 1991, 80, 1. (11) Chandra, A.; Bagchi, B. Chem. Phys. Lett. 1988, 151,47. Bagchi, B.; Chandra, A. Chem. Phys. Lett. 1989,155,533. Vijayadamodar, G. V.; Chandra, A.; Bagchi, B. Chem. Phys. Lett. 1989, 161, 413. Chandra, A,; Bagchi, B. Chem. Phys. Len. 1990, 165, 93. Bagchi, B.; Chandra, A. J . Chem. Phys. 1989,90,7338. Chandra, A.; Bagchi, B. J. Chem. Phys. 1989, 91,2594. Chandra, A,; Bagchi, B. J. Chem. Phys. 1989,91,7181. Chandra, A.; Bagchi, B. J. Phys. Chem. 1989,93,6996. Chandra, A.; Bagchi, B. J. Phys. Chem. 1990,94, 1874. Chandra, A,; Bagchi, B. J. Chem. Phys. 1991, 94, 8367. Bagchi, B.; Vijayadamodar, G. V. J. Chem. Phys. 1993, 98, 335 1. (12) Chandra, A,; Bagchi, B. J . Chem. Phys. 1991, 94, 3177. Bagchi, B.; Chandra, A. J . Chem. Phys. 1992, 97, 5126. Roy, S.; Bagchi, B. J. Chem. Phys. 1993, 99, 1310. Roy, S . ; Komath, S.; Bagchi, B. J . Chem. Phys. 1993,99, 3139. Roy, S.; Bagchi, B. J . Chem. Phys. 1993,99,9938. (13) Fried, L. E.; Mukamel, S. J . Chem. Phys. 1990, 93, 932. (14) Yu, J.; Kang, T. J.; Berg, M. J . Chem. Phys. 1991, 94, 5787. Fourkas, J. T.; Berg, M. J . Chem. Phys. 1993, 98, 7773. Fourkas, J. T.; Benigno, A.; Berg, M. J. Chem. Phys. 1993, 99, 8552. (15) Bagchi, B. J . Chem. Phys. 1994, 100, 6658. (16) Saven, J. G.; Skinner, J. K. J . Chem. Phys. 1993, 99, 4391. (17) Maroncelli, M.; Fleming, G. R. J . Chem. Phys. 1988, 89, 5044. (18) Karim, 0. A,; Haymet, A. D. J.; Banet, M. J.; Simon, J. D. J . Phys. Chem. 1988, 92, 3391. (19) Bader, J. S . ; Chandler, D. Chem. Phys. Lett. 1989, 157, 501. (20) Carter, E. A,; Hynes, J. T. Chem. Phys. 1991, 94, 5961. (21) Fonseca, T.; Ladanyi, B. M. J. Phys. Chem. 1991, 95, 2116. Fonseca, T.; Ladanyi, B. M. J . Mol. Liq. 1994, 60, 1. (22) Maroncelli, M. J . Chem. Phys. 1991, 94, 2084. (23) Perera, L.; Berkowitz, M. L. J. Chem. Phys. 1992, 96,3092. Perera, L.; Berkowitz, M. L. J . Chem. Phys. 1992, 97, 5253. (24) Neria, E.; Nitzan, A. J . Chem. Phys. 1992, 96, 5433. (25) Rosenthal, S. J.; Xie, X.; Du, M.; Fleming, G. R. J. Chem. Phys. 1991, 95, 4715. (26) Cho, M.; Rosenthal, S . J.; Scherer, N. F.; Ziegler, L. D.; Fleming, G. R. J. Chem. Phvs. 1992. 96. 5033. (27) Jimenez, R.; Fleming, G. R.; Kumar, P. V.; Maroncelli, M. Nature 1994, 369, 47 1. (28) van der Zwan, G.; Hynes, J. T. J. Phys. Chem. 1985, 89, 4181.
Short-Time Dynamics of Solvation (29) Bruehl, M.; Hynes, J. T. J. Phys. Chem. 1992, 96, 4068. Smith, B. B.; Staib, A,; Hynes, J. T. Chem. Phys. 1993, 176, 521. (30) Stratt, R. M.; Cho, M. J . Chem. Phys. 1994, 100, 6700. (31) Simulations of polar liquids actually raise the question of whether a longitudinal relaxation time t~ is all that well defined. See: Kivelson, D.; Friedman, H. L. J. Phys. Chem. 1989, 93, 7026. The time correlation function of the longitudinal component of the collective dipole density ML(k,t) always seems to be highly nonexponential at small wave vectors, even when the decay of the transverse part is largely exponential (as in the case of acetonitrile). However, there does seem to be at least a rough correlation between the experimental values of TL as predicted by continuum theory and asymptotic experimental solvation t i m e ~ . l - ~ (32) Buchner, M.; Ladanyi, B. M.; Stratt, R. M. J. Chem. Phys. 1992, 97, 8522. (33) Xu, B.-C.; Stratt, R. M. J . Chem. Phys. 1990, 92, 1923. Adams, J. E.; Stratt, R. M. J . Chem. Phys. 1990, 93, 1332. Adams, J. E.; Stratt, R. M. J . Chem. Phys. 1990,93, 1632. Wan, Y.; Stratt, R. M. J. Chem. Phys. 1994, 100, 5123. (34) Cho, M.; Fleming, G. R.; Saito, S.; Ohmine, I.; Stratt, R. M. J . Chem. Phys. 1994, 100, 6672. (35) Seeley, G.; Keyes, T. In Spectral Line Shapes; Szudy, J., Ed.; Ossolineum: Wroclaw, Poland, 1989; Vol. 5 , p. 649. Madan, B.; Keyes, T.; Seeley, G. J . Chem. Phys. 1990,92,7565. Madan, B.; Keyes, T.; Seeley, G. J . Chem. Phys. 1991, 94, 6762. Seeley, G.; Keyes, T.; Madan, B. J. Chem. Phys. 1991, 95, 3847. Seeley, G.; Keyes, T.; Madan, B. J. Phys. Chem. 1992, 96, 4074. Madan, B.; Keyes, T. J . Chem. Phys. 1993, 98, 3342. Keyes, T. J. Chem. Phys. 1994, 101, 5081. (36) Moore, P.; Keyes, T. J. Chem. Phys. 1994, 100,6709. Keyes, T.; Moore, P.; Seeley, G. Proc. XXIII Meeting on Statistical Physics; Cuernavaca, Mexico, Jan 1994. (37) Wu, T.-M.; Loring, R. F. J. Chem. Phys. 1992, 97, 8568. Wu, T.-M.; Loring, R. F. J. Chem. Phys. 1993, 99, 8936. (38) Rosenberg, R. 0.; Thirumalai, D.; Mountain, R. D. J . Phys. Condens. Matter 1989, 1, 2109. Straub, J. E.; Thimmalai, D. Proc. Natl. Acad. Sci. USA. 1993, 90, 809. (39) Buch, V. J. Chem. Phys. 1990, 93, 2631. (40) Bauer, J. D.; Calef, D. Chem. Phys. Lett. 1991, 187, 391. (41) Space, D.; Rabitz, H.; Askar, A. J . Chem. Phys. 1993, 99, 9070. (42) Stassen, H.; Gburski, Z. E. Chem. Phys. Lett. 1994, 217, 325. (43) Note that painvise potentials can be used to include many-body polarizability effects if Drude-like representations of polarizability are employed. See, for example: Sprik, M.; Klein, M. L. J . Chem. Phys. 1988, 89, 7556. Watanabe, K.; Klein, M. Chem. Phys. 1989, 131, 157. (44) That is, if we adopt a molecule-fixed frame in which molecule j lies along the z axis, the origin is its center of mass, and one end of the molecule is designated as positive (e.g., the N in MeCN), then is d,, is the z coordinate of site a for molecule j . (45) Actually our choices for A , the differential solute-solvent interaction, apply equally well to a variety of choices of excited-state interactions beside the ones we mention. In the polar solvation study, for example, any excited-state interaction differing from the ground-state interaction by no more than a rescaling of the electrostatic part of the potential would give precisely the same normalized solvation correlation function. Thus, we would obtain the same results if the excited-state solute dipole were zero, twice the size of the ground-state dipole, or the same magnitude as the ground-state dipole but reversed in direction. (46) Edwards, D. M. F.; Madden, P. A.; McDonald, I. R. Mol. Phys. 1984, 51, 1141. This three-site model omits any dynamics arising from the torsion of the methyl group. See: Lynden-Bell, R. M.; Madden, P. A,; Stott, D. T.; Tough, R. J. Mol. Phys. 1986, 58, 193 for a discussion. (47) Rappk, A. K.; Casewit, C. J.; Colwell, K. S.; Goddard, W. A. 111; Skiff, W. M. J . Am. Chem. SOC.1992, 114, 10024.
J. Phys. Chem., Vol. 99, No. 9, 1995 2511 (48) Fincham, D. Mol. Simulation 1993, 11, 79. (49) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford: New York, 1989; Chapter 5. (50) Ladd, A. J. C. Mol. Phys. 1978, 36, 463. Neumann, M.; Steinhauser, 0.;Pawley, G. S. Mol. Phys. 1984, 52, 97. (51) Wilkinson, J. H.; Reinsch, C. Handbookfor Automatic Computation; Springer-Verlag: New York, 1971; Vol. 11: Linear Alnebra, pp 212-240. (52) The NAG Fortran Library Manual, Mark I6; NAG: Downers Grove, IL,1993; Vol. 5. (53) See, in particular, the liquid-phase (293 K)studies of CS2 by Moore and K e y e ~ . ~ ~ (54) Maroncelli, M.; Kumar, V. P.; Papazyan, A. J . Phys. Chem. 1993, 97, 13. (55) Our distance criterion was based on the existence of a clear minimum in the solute-sitelsolvent-center-of-mass radial distribution function-which was located at 6.38 A for both positive and negative solute sites. An alternative definition based on the analogous minimum in the solute-center-of-mass/solvent-center-of-mass radial distribution function leads to an estimate of only 11.6 solvent near neighbors, but scarcely changes our first-solvation-shell projections. (56) Muiiio, P. L.; Callis, P. R. J . Chem. Phys. 1994, 100, 4093. (57) The suggestion that dielectric relaxation around a charge proceeds from the outside in was first advanced by Onsager: Onsager, L. Can. J. Chem. 1977.55, 1819. Since then the subject has been addressed repeatedly. See, for example: Calef, D. F.; Wolynes, P. G. J. Chem. Phys. 1983, 78, 4145. Chandra, A.; Bagchi, B. J. Chem. Phys. 1989, 91, 2594. Tachiya, M. Chem. Phys. Lett. 1993, 203, 164. Papazyan, A,; Maroncelli, M. J. Chem. Phys. 1993, 98, 6431. (58) Rosenthal, S. J.; Jiminez, R.; Fleming, G. R.; Kumar, P. V.; Maroncelli, M. J . Mol. Liq. 1994, 60, 25. (59) Within the Marcus picture of solvation, in which one identifies the (single) solvation coordinate as the solute-solvent interaction energy itself, the ratio of kBT to ((6A)’) is the force constant for this solvent “mode”, making the area under the solvation spectrum the reciprocal of the effective mass of the mode. See ref 20 and Fonseca, T.; Ladanyi, B. M.; Hynes, J. T. J. Phys. Chem. 1992, 96, 4085. (60) The solvation frequency has no contribution from the cross rotational-translational portion of the solvation spectrum because the integral over this portion is identically zero. This observation derives from the fact that the rotational and translational projection operators project onto distinct spaces, a feature one can confirm by integrating eq 2.24 and then making use of eqs 2.19 and 2.20. Similar reasoning establishes that there is no contribution from the cross term between the inner and outer solvation shells. (61) Berg, M. Chem. Phys. Lett. 1994, 228, 317. (62) Alfano, J. C.; Walhout, P. W.; Kimura, Y.; Barbara, P. F. J. Chem. Phys. 1993, 98, 5996. Kimura, Y.; Alfano, J. C.; Walhout, P. K.; Barbara, P. F. J. Phys. Chem. 1994, 98, 3450. (63) Rossky, P. J.; Schnitker, J. J . Phys. Chem. 1988, 92, 4277. Motakabbir, K. A.; Schnitker, J.; Rossky, P. J. J. Chem. Phys. 1992, 97, 2055. Schwartz, B. J.; Rossky, P. J. Phys. Rev. Lett. 1994, 72, 3282. Schwartz, B. J.; Rossky, P. J. J. Chem. Phys. 1994, 101, 6903, 6917. (64) The relevance of cross correlations to Onsager’s view of dielectric relaxation has been emphasized by Papazyan and M a r ~ n c e l l i .There ~ ~ is an interesting contrast, in this regard, with Bader and Chandler’s results for ion solvation in water.I9 In that system, the first shell is so strongly bound to the solute (and so weakly connected with the remainder of the solvent) that inner-shelUouter-shell cross correlations are small. (65) Mukamel, S. Annu. Rev. Phys. Chem. 1990,41, 647. Fried, L. E.; Mukamel, S. Adv. Chem. Phys. 1993, 84, 435.
JP9421085
