J. Phys. Chem. 1988, 92, 3386-3391
3386
with frequency scale expansion were taken throughout for improved band definition and frequency precision. Except for weak or broad bands on sloping backgrounds, frequencies should normally be correct to within 1 cm-'. The intensities obtainable depended on good fortune: recrystallization from D 2 0 gave comparatively large microcrystals, hence the higher intensities in Figure 3(a); economic considerations precluded recrystallization acid from H2l80;the powdery specimen available gave of the 1802
spectra of low intensity. Spectral slit widths were in the range 2-4 cm-I, depending on the spectral range and particle size of specimen.
Acknowledgment. S. Brown and D. J. Light are thanked for technical assistance. Registry No. 4-C1602D-CSH4N,5275 1-34-5; 4-CL6O2H-C5H4N, 55-22-1; 4-C1*02H-C,H4N, 114274-61-2.
Disperslon Curves from Short-Time Molecular Dynamics Simulations. 1. Diatomic Chain Results D. W. Noid,* Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, and Department of Chemistry, The University of Tennessee, Knoxville, Tennessee 37996- 1600
B. T. Broocks, S. K. Gray, Department of Chemistry, Northern Illinois University, DeKalb, Illinois 601 15
and S. L. Marple Martin Marietta Aerospace and Naval Systems, 103 Chesapeake Park Plaza, Baltimore, Maryland 21 220 (Received: October 19, 1987; In Final Form: December 29, 1987)
The multiple signal classification method (MUSIC) for frequency estimation is used to compute the frequency dispersion curves of a diatomic chain from the time-dependent structure factor. In this paper, we demonstrate that MUSIC can accurately determine the frequencies from very short time trajectories. MUSIC is also used to show how the frequencies can vary in time, Le., along a trajectory. The method is thus ideally suited for analyzing molecular dynamics simulations of large systems.
I. Introduction In the past two decades, the study of nonlinear phenomena in few-body systems has received much attention.'$* A widely used technique for studying the classical dynamics in these systems is the power spectra method.*v3 These studies involve solving numerically Hamilton's equations for various time-dependent properties and Fourier transforming a correlation function to generate a spectral representation by using the fast Fourier transform (FFT) method.4 This technique has been commonly used to characterize chaotic and quasiperiodic motion, compute transition frequencies by using the correspondence principle, follow energy transfer, and in a few studies to compute dispersion curves of larger systems. The most significant limitation of this procedure is the frequency resolution of a classical spectral estimation using the fast Fourier transform, the frequency resolution being roughly the inverse of simulation time of the system being studied. A -33-ps simulation is needed to produce (in theory) I-cm-' resolution. In very large many-body systems, the molecular dynamics (MD) calculations become very CPU intensive, and relatively few M D calculated dispersion curve calculations have been r e p ~ r t e d . ~ Also, - ~ as we
0) Lichtenberg, A. J.; Lieberman, M. A. Regular and Stochastic Motion; Springer: New York, 1983. (2) Noid, D. W.; Koszykowski, M. L.; Marcus, R. A. Annu. Reu. Phys. Chem. 1981, 32, 267. (3) Noid, D. W.; Koszykowski, M. L.; Marcus, R. A. J . Chem. Phys. 1977, 67, 404. For a review, see: Koszykowski, M. L.; Pfeffer, G. A,; Noid, D. W. N.Y. Acad. Sci. 1987, 497, 127. (4) e.g., Press, W. H.; Flannery, B. P.; Teukolsky, S. A,; Vetterling, W. T. Numerical Recipes; Cambridge University Press: New York, 1987. (5) Bishop, M.; Berne, B. J. J . Chem. Phys. 1973, 59, 5337. (6) Henry, B. I.; Oitmoa, J. Ausf. J . Phys. 1985, 38, 191. 0022-3654/88/2092-3386$01.50/0
shall discuss later, structural changes can occur that can blur the FFT spectrum since it involves a long-time average. Fortunately, alternative spectral estimation procedures have recently been developed that can produce much higher resolution than the FFT method. In this paper, we have used the highresolution MUltiple SZgnal Classification method (MUSIC) developed by S ~ h m i d tto~ compute ,~ the dispersion curve for a model diatomic chain using data collected over only -0.2 ps. This method assumes that the function considered (in this case the dynamic structure factor) consists of a given number of frequencies and additive white noise. The time dependence of the dispersion curve sampling only local dynamics is also discussed. In the next section, our diatomic chain model is presented along with a discussion of the MUSIC method. Results of the calculation are presented in section 111, followed by our conclusion in section IV. 11. Theoretical Method Below we outline the specifics of our diatomic chain model (1I.A) and outline the MUSIC spectral method (1I.B). A . Diatomic Chain Model. All of the calculations presented use parameters for a linear H F chain with 500 atoms. The model thus does not adequately describe actual HF crystals which occur in zigzag chains.I0 The model does, however, present a serious (7) Riehl, J. P.; Diestler, D. J. J . Chem. Phys. 1976, 64, 2593. (8) Schmidt, R. 0. IEEE Trans. Antennas Propag. 1986, 34, 276. (9) For a good discussion of this method, see: Marple, S . L. Digital Spectral Anolysis with Applications; Prentice-Hall: Englewood Cliffs, NJ, 1987. (10) e.g., Tubine, R.; Zervi, G. J . Chem. Phys. 1969, 51, 4509; Springberg, M. Phys. Rev. Left. 1987, 59, 2287.
0 1988 American Chemical Society
The Journal of Physical Chemistry,
Short-Time Molecular Dynamics Simulations test of the MUSIC method since very low frequency modes occur that normally require, by FFT methods, long-time integrations. Moreover, simple analytical formulas apply to certain limits that permit a very quantitative assessment of the frequency resolution of the method. The Hamiltonian used is
249
Z[VHF(qF! i=l
- qH,)
+
vFH(qHi+,
- qF,)]
+ VFH(qHI - qF2JO + L, (1)
= D[1
+
(4)
and we will denote x[m] = p_(K,[m]At) to be the mth component of this vector. The vector X, is used to generate the pth order Toeplitz autocorrelation matrix E , of dimension (p 1) X (p + 1)
+
E = ~(j;.~*[n]j;.~~[n])
- e-a(rr)]2
(5)
or more explicitly
and the nonbonded interaction used is a Lennard-Jones potential
L represents the fixed length of the line the chain is located on, and periodic boundary conditions are used so that HI and FZs0 are connected as indicated by the last term in (1). The parameters employed were D = 0.21006 au, a = 1.22 aO-l, p = 1.75 ao, e = 0.009 560 5 au, u = 3.1823 ao, hH= 1837 me, m F = 34637 m,, and L = 1330.5 ao, where the covalently bonded HF Morse parameters are from ref l l and the hydrogen bonded H-F parameters were chosen to be approximately consistent with experimental and theoretical data in ref 12. The initial conditions used involved starting each atom at its equilibrium position and then randomly selecting a momentum subject to zero center of mass velocity. Initially all the energy is kinetic energy, but almost immediately it is shared equally among the kinetic and potential terms in the Hamiltonian. The temperature Tis then defined by kBT/2 = (Elun), where (Ekin) is the average kinetic energy per atom and kBis the Boltzmann constant. We found that a 500-atom system and our choice of initial conditions lead to a fairly well defined temperature. From the Hamiltonian in ( l ) , Hamilton’s equations were solved by using the program ODE^^ using the CRAY XMP computer with an estimated absolute error lo-” per step. From each trajectory, we computed the particle density correlation function p for various values of K, the wave vector, using coordinates of the H and F atoms with
As is ~ e l l - k n o w n , ~ - ’the ~ ’ ~Fourier transform of p(K,t) yields the dynamic structure factor S(K,w)
S(K,w) = leiwLp(K,t)dt
92, No. 12, 1988 3387
for t = 0 to (N - 1)At g_enerated from the molecular dynamics, 1 is defined by a complex data vector X of dimension p
where the bonded potential H F used is a Morse potential vHF(r)
Vol.
rxxl01
(11) Stine, J. R.; Noid, D. W. Chem. Phys. Lett. 1981, 77, 287. (12) Yarkoni, D. R.; O’Neil, S. V.; Schaefer, H. F.; Baskin, C. P.; Bender, C. F. J . Chem. Phys. 1974, 60, 855. Michael, D. W.; Dykstra, C. E.; Lisy, J. M. J . Chem. Phys. 1984, 81, 5998. (13) Shampine, L. F.; Gordon, M. K. Computer Solution of Ordinary Differential Equations: The Initial Value Problem; Freeman: San Francisco, 1975. (14) Painter, P. C.; Coleman, M. M.; Koenig, J. L. The Theory of Vibrational Spectroscopy and Its Application to Polymeric Materials; Wiley: New York, 1982.
***
rxx[-Pl
.**
‘XX[P+lI
where r,.. is an autocorrelation sequence
= c(x[n+m]x*[n])
r,[m]
(6)
which can be estimated from the data samples as N-m-l
1
(7) Operator e indicates the mean or expected value of the time index and PJm] is the estimate of r,,[m]. 2z This matrix can be factored into a signal autocorrelation matrix S and a noise autocorrelation matrix i?.
z=g+
(8) Furthermore, we can solve for the eigenvectors and eigenvalues of 2 such that
where 8, are the eigenvectors of Eassociated with the NSIG largest eigenvalues Xi. Here, NSIG eigenvectors are associated with the NSIG sinusoids assumed present in the data, Le., the major frequencies expected in the spectrum. The rest correspond to p - NSIG noise eigenvectors and a set of numerically smaller eigenvalues Xi. Finally, the MUSIC frequency estimate PMuSIc(w) is then 1
pMUSIC(w) =
(3)
The major peaks in S(K,w) lead to a dispersion relation w ( K ) . For a one-dimensional diatomic chain, two peaks are observed which are optical and acoustic branches, wOpT and wAc, respectively. B. MUSIC Method. In order to perform the frequency analysis of p(K,t) in (3), we have used a class of spectral techniques based on an eigenanalysis of an autocorrelation m a t r i ~ .The ~ important feature of this technique is the factoring of the data autocorrelation = matrix R into two vector subspaces, one a signal subspace and the other a noise subspace. From a set of N samples of p(K,t)
rxx[-ll
(10)
P
eT(w)(
vkvkT)e(W) k=NSIG+l
where exp(i2a[O]wt) = 1
e(w)
=
[* :
exp(i2.n[p]w At)
1
(11)
The sum in (9) is over the eigenvectors of the noise subspace. These eigenvectors will be orthogonal to signal eigenvectors, Le., V 8 ( w ) = 0. Thus, the PMusIc estimate is effectively infinite when the frequency w is found in the signal. However, in practice, the values just become large. We have found that the relative power levels have qualitative significance but are not quantitative. A more detailed discussion of this method is found in ref 9. This method has been implemented in the subroutine EIGENFREQ’ which we have used to compute the dispersion curves discussed in the next section. It is instructive to contrast the standard FFT method with the MUSIC method in terms of actual user interaction. In both methods one must input a time sequence of data points evenly spaced by At over a range [0, tmJ. Similarly, both methods
3388 The Journal of Physical Chemistry, Vol. 92, No. 12, 1988
Noid et al.
TABLE I: Comparison of Dispersion Curve Frequencies“
MUSIC^ nK
1 20 40 60 80 100 120 140 160 180 200 220 240
K , a,,-’ 0.0047 0.0944 0.1889 0.2833 0.3778 0.4722 0.5667 0.6611 0.7556 0.8500 0.9445 1.0389 1.i338
harmonic perturbation theory
FFT‘
WAC’2
WOPT+2
5 125 245 357 438 495 512 509 468 402 301 193 69
4329 4326 4319 4319 4314 4304 4302 4292 4309 4319 4324 4326 4331
WAC+8
125 259 359 435 493 518 518 476 393 309 192 75
Won**
WAC
WOE
4329 4321 4321 4312 4312 4304 4312 4312 4312 4329 4329 4329
6 127 247 352 435 49 1 515 507 466 396 302 189 64
433 1 4329 4324 4317 4309 4303 4300 4301 4306 4313 4320 4327 4330
“All frequencies are in units of cm-’. *Computed with -0.2-ps trajectory. ‘Computed with -4-ps trajectory.
compute the spectral density or intensity over a frequency range of [-/At, 7r/At]. However, the similarities end here. In the FFT method, the resolution is 27r/t,,,. The frequency resolution of MUSIC theoretically is infinitesimal. (One must, of course, represent the frequency estimate on a discrete grid.) To compute the MUSIC estimate of the spectrum of a given signal, one must specify the number of expected major frequencies (NSIG) and the dimension of the total vector space (p). Some experimentation is necessary to find optimal parameters as discussed by Marple.’ In practice, if NSIG is greater than the actual number of frequencies, and if p > 3(NSIG), then we have found unambiguous and stable results. Therefore MUSIC requires somewhat more understanding of the function than the FFT method. Finally, we note that in some ways the method has features of a normal coordinate analysis for obtaining frequencies. The computational complexity scales at N 3 whereas for the FFT method it is N log N , where N is the number of data samples. Fortunately, the CPU time for MUSIC is trivial compared with the computational effort of the molecular dynamics calculations for our system with 500 atoms. Moreover, the fact that very short time sequences lead to accurate frequency estimates makes this method ideally suited for a number of nontrivial applications. 111. Calculation and Results As discussed in the preceding section, a -4-ps trajectory was computed with 1200 time steps with a Af equal to 0.0033 ps. The 500-atom H F chain was chosen to have a temperature of approximately 5 K to ensure that quasiperiodic-type motion would occur. From the 500 values of (qF,,qH,]at 1200 values of time, the complex correlation function p(K,t) was generated for values of K = 2nnK/Lwhere the range of nK is 0-250. For Figure 1 , we have plotted the MUSIC spectral estimate for S(K,w) with nK = 100 or K = 0.4722 ao-I. This spectrum was generated with only a 64-point data sample with a time range of 0.2 ps. Arbitrarily we computed 4096 frequencies in the range -5014.5 to +5014.5 cm-’ with a spacing of about 2 cm-l. (Any number could be used to get any desired resolution.) The theoretical resolution of the FFT method with this data would be approximately 167 cm-’! In Figure 1, we have plotted both the positive and negative branches of the frequency in one plot. A small displacement of the peaks is observed. The sample frequency (27rlAf) corresponded to 10029 cm-’. To further test the method, the function was added to the data sample with w = 1254 cm-l. MUSIC was able to compute the frequency to four digits, even for values of A = 0.05. We have also assumed that the signal space (NSIG) contained 2,4, or 6 frequencies as discussed in the previous Section and obtained similar results. The power spectral density P(w) was rescaled in units of decibels from the maximum signal strength PSDMAX by
P(w)= 10 lOg,o (PSD(w)/PSDMAX)
(12)
to compute the intensity in decibels. In Figure 2, the corresponding FFT spectral estimate is plotted using all 1200 data points. The
TABLE I 1 Temperature Dependence of w ~ ~ / w ~ ~ ”
5K
K 0.0047
500 K
10000 K 5
284 4069 484 4300 534 4108
414 3315 698 3795 815 3450
5
4326 245 4319 438 4314 512 4302
0.1889 0.3778 0.5667
“Computed by MUSIC method with 0.2-ps trajectory and in units of cm-’. The relative stability of these results is indicated in Figure 4. theoretical frequency resolution of the FFT calculation in this case is 8 cm-I. In Table I, a detailed comparison is presented for values of nK = 1, 20,40, ..., 240. The high-resolution frequencies labeled as acoustic or optic in the third or fourth column were chosen from the most intense optic or acoustic peaks found from the list of 4096 spectral points generated by MUSIC. Correspondingly, the values listed in the fifth and sixth columns were found from the list of 1200 points generated with the standard FFT routine. In the last two columns, we have computed the optic and acoustic frequencies from the well-known result for the diatomic chain,14 assuming harmonic oxcillator interactions w2
=
f, + f 2 ~
2P
(fl
+f2)2
4 f h sin2 (K*d/2)
1”:
[7 MHMF -
(13)
where wopT is found for the positive sign and wACC with the negative sign. The parameters for (13) consistent with our model in section 1I.A are f l = 0.625 31 au, f2 = 0.054 au, d = 5.3220 a,, and p = 1745 me. The MUSIC frequencies are in better agreement with the harmonic oscillator results. At 5 K, we believe that the perturbative frequencies from (13) are very accurate. (In some calculations, a constant phase 7r/2 was added to p(K,f) and the computed frequencies remained constant to within the theoretical limits.) Parts a, b, and c of Figure 3 contain the MUSIC computed dispersion curve at 5, 500, and 10000 K, respectively, for several values of K. An interesting feature here is the seemingly stable acoustic branch of the dispersion curve. However, the 500 and 10000 K optical branch appears to break up into subbranches. By use of 64 time points, a time-dependent spectrum can be computed for various values of K . In Figure 4 MUSIC estimated frequencies are plotted as a function of time corresponding to the three temperatures in Figure 3. The 5 K time-dependent spectrum shows a very stable time-dependent behavior. The 500 K spectrum shows more structure but on a short time scale appears regular. However, the 10 000 K time-dependent spectrum shows a large amount of spectral fluctuations which probably are an indication of short-time chaotic behavior. A more thorough study of the
Short-Time Molecular Dynamics Simulations
The Journal of Physical Chemistry, Vol. 92, No. 12, 1988 3389 5
I
4
3 A
3
x 2
1 -77.06 -5000
I'
-2500
0
2500
5000 4
-5000
-2500
b-,
2500
5000
0
2500
5000
0
2500
5000
0
w (cm-') Olb
4J
3n
-2500
0
2500
5000
w (cm-l)
4-
-33.914
-5000
-2500
0
2500
5000
w(Cm-1)
Figure 1. (a) MUSIC spectral estimate of S(K,w) for 500-atom chain for nK = 100 ( K = 0.4722 ac') and T = 5 K with a 0.2-ps trajectory. (b) As in (a) except T = 500 K. (c) As in (a) except T = 10000 K.
nature of chaos in this system, including estimates of the K entropy, is in progress.I5 Finally, in Table 11, we have used MUSIC on the first 64 data points (Le., t,,, = 0.2 ps) to estimate the optic and acoustic mode frequencies for several values of K and the three temperatures mentioned above. As is apparent from the table, the value of uACC is much more stable than uOPT.The temperature change from 5 to 500 K shifted u A C approximately 20-50 cm-I, while uOpT changes -200-300 cm-'. Going from 500 to 10000 K, the change in wAc was 100-300 cm-', while won changed 600-800 cm-'.
-
(15) Broocks, B.; Gray, S . K.; Noid, D. W., manuscript in
preparation.
-go00
-2500
dcm-')
Figure 2. (a) Standard FFT estimate of S(K,w) as in Figure l a with a 4-ps trajectory. (b) As in (a) except T = 500 K. (c) As in (a) except T = 10000 K.
In all cases, the acoustic mode frequency increased and uOPT decreased. The tendency of uOpTto decrease with increasing T is related to the fact that the optical frequency branch is strongly connected with intramolecular HF frequency which naturally decreases with increasing internal energy due to anharmonicity. For the acoustic mode, the form of the potential is not as significant, as the increased frequency probably represents a higher
Noid et al.
3390 The Journal of Physical Chemistry, Vol. 92, No. 12. 1988
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Figure 3. (a) Dispersion curve for 500-atom HF chain with T = 5 K. (b) As in (a) except T = SO0 K. (c) As in (a) except T = 10000 K.
Figure 4. (a) Time-dependent frequencies of S(K,w) as in Figure l a versus initial time Ti (in units of -0.003 ps) at 5 K. (b) As in (a) except T = 500 K. (c) As in (a) except T = 10000 K.
collision rate of the effectively gaseous H F molecules. It should be noted that the FFT results would not have been able to give any useful frequencies as this amount of chaos (at 10000 K) completely blurs the entire spectrum (see Figure 2c).
IV. Conclusion In this paper, we have used a new method to compute the dispersion curves in a model diatomic chain. This method successfully computes acoustic and optical mode frequencies with
J. Phys. Chem. 1988, 92, 3391-3394 very short molecular dynamics simulation, especially in the quasiperiodic regime. It is also now possible to generate a time-dependent spectrum which only samples short time intervals and can give new insight into energy flow in these large systems. In the realm of small molecules, McDonald and MarcusI6 first demonstrated the use of time-dependent spectra. It is also possible that the MUSIC method would be of some use in discerning intramolecular tran~iti0ns.l~The technique is especially useful for generating spectra of polymer systems since structural changes would blur a normal FFT spectra and the M D simulations are more time consuming. We have made use of this advantage in a calculation of spectral shifts in stressed polyethylene.'* Also, in subsequent work, we have found this method to be very useful in computing semiclassical spectral transitions in the chaotic (16) McDonald, J. D.; Marcus, R. A. J . Chem. Phys. 1976, 65, 2180. (17) Martens, C. C.; Davis, M. J.; Ezra, G. S. Chem. Phys. Lett. 1987, 142, 519. (18) Noid, D. W.; Pfeffer, G. A., submitted for publication in J . Polym. Phys.
3391
regime,19computing local frequencies to locate resonance zones,20 and to compute frequencies from trajectories generated from quantum chemistry methods.21 The computational requirements for MUSIC are somewhat more than the FFT method but are trivial compared with CPU times needed for the molecular dynamics calculation.
Acknowledgment. This research was sponsored by the Division of Materials Sciences, Office of Basic Energy Sciences, U.S. Department of Energy, under Contract No. DE-AC05-840R21400 with Martin Marietta Energy Systems, Inc. S.K.G. acknowledges partial support from the donors of the Petroleum Research Fund, administered by the American Chemical Society. We also acknowledge helpful discussions with Professor G. A. Pfeffer in the early stages of this work. (19) Noid, D. W.; Gray, S. K. Chem. Phys. Lett. 1988, 145, 9. (20) Wozny, C. E.; Gray, S. K.; Noid, D. W., manuscript in preparation. (21) Noid, D. W.; Bloor, J. E.; Spotswood, M.; Koszykowski, M. L., to be submitted for publication in Chem. Phys. Lett.
Molecular Aspects of Nonequillbrium Solvation: A Simulation of Dipole Relaxation Omar A. Karim, A. D. J. Haymet,*.+ Department of Chemistry, University of California, Berkeley, California 94720
Matthew J. Banet, and John D. Simon*?+ Department of Chemistry B-014 and the Institute for Nonlinear Science R-002, University of California, San Diego. La Jolla. California 92093 (Received: November 30, 1987)
The molecular dynamics method has been used to simulate solvent relaxation around a dissolved solute molecule after a sudden change in its permanent dipole moment. The solvation process was followed for two cases: (1) a 4-D (debye) moment that is suddenly rotated 180° without a change in magnitude and (2) a 4-D moment that is instantaneously changed to 12 D without a change in direction. The evolution to equilibrium is found to occur on a time scale different than that predicted by dielectric continuum models. In addition, we have examined the relaxation dynamics of various solvent shells. Our simulations show that molecular aspects of the solvent, which are not included in dielectric continuum models, are important in understanding the mechanism of solvations. These simulation data provide insight into recent experimental studies of solvation measured by monitoring the time-resolved Stokes shift of probe molecules in polar solvents.
The molecular structure and dynamics of solvation of ions and depending on the nature of the perturbation: for the case of a dipoles have been the subject of numerous recent theoretical and point charge, T~ is related to T D by T L = (c,/to) T D . In a polar experimental studies.I-l7 These studies have been encouraged, in part, by a desire to understand the dynamic role of the solvent (1) Marcus, Y. Ion Soluation; Wiley: New York, 1985. in chemical reactions. In the past decade, there have been sig(2) Impey, R. W.; Madden, P. A.; McDonald, I. R. J . Phys. Chem. 1983, 87, 5071. nificant advances in the theoretical modeling of the equilibrium (3) Robinson, G. W.; Thistlewaite, P. J.; Lee, J. J . Phys. Chem. 1986, 90, properties of polar solvents."J8-21 On the other hand, none4244. quilibrium systems, and relaxation of perturbed systems back to (4) Friedrich, V.; Kivelson, D. J . Chem. Phys. 1987, 86, 6425. equilibrium, have proven difficult to study. (5) van der Zwan, G.; Hynes, J. T. J . Phys. Chem. 1985, 89, 4181. Most nonequilibrium models employ a dielectric c o n t i n ~ u m ~ ~ , ~ ~(6) Bagchi, B.; Oxtoby, D. W.; Fleming, G. R. Chem. Phys. 1984, 257. (7) Karim, 0. A.; McCammon, J. A. J. Am. Chem. SOC.1986,108, 1762. representation of the solvent. In these models, the time-dependent (8) Calef, D. F.; Wolynes, P. G. J . Chem. Phys. 1983, 78, 4145. properties of the medium are treated by the relaxation behavior (9) Loring, R. F.; Mukamel, S. M. J . Chem. Phys. 1987, 87, 1272. of the frequency-dependent dielectric constant, t(w), for which, (10) Wolynes, P. G. J . Chem. Phys. 1987, 86, 5133. (11) Rossky, P. J. Annu. Reu. Phys. Chem. 1985, 36, 321. in general, a Debye form is used. t(w)
=
t,
€0 +1+ ~ W T D
In this equation, E,,, e,, and T D are the static dielectric constant, the high-frequency dielectric constant, and the Debye relaxation time, respectively. In addition to T D , a second relaxation time, the longitudinal relaxation time 71, is used commonly to gauge solvation dynamics. The functional form for T L varies slightly National Science Foundation Presidential Young Investigators 1985-1990, Alfred P. Sloan Fellows.
0022-3654/88/2092-3391$01.50/0
(12) J., Ed.; (13) (14)
Hubbard, J. B.; Wolynes, P. G. In Physics oflonic Soluation;Ulstrup, Elsevier: Amsterdam, 1986. Maroncelli, M.; Fleming, G. R. J . Chem. Phys. 1987, 86, 6221. Castner, E. W., Jr.; Maroncelli, M.; Fleming, G. R. J . Chem. Phys. 1987, 86, 1090. (15) Nagaragan, V.; Brearley, A. M.; Kang, T.-J.; Barbara, P. F. J . Chem. Phys. 1987, 86, 3183. (16) Su,S.-G.; Simon, J. D. J . Phys. Chem. 1987, 91, 2693. (17) Maroncelli, M.; Castner, E. W., Jr.; Webb, S . P.; Fleming, G. R. In Ultrafast Phenomena V; Siegman, A. E., Fleming, G. R., Eds.; SpringerVerlag: New York, 1986; p 303. (18) Mezei, M.; Beveridge, L. J . Chem. Phys. 1981, 74, 622. (19) Chandler, D.; Andersen, H. C. J . Chem. Phys. 1973, 57, 1930. (20) Pettitt, B. M.; Rossky, P. J. J . Chem. Phys. 1982, 77, 1451.
0 1988 American Chemical Society
