3330
J. Phys. Chem. 1996, 100, 3330-3343
Intermolecular Dynamics of Substituted Benzene and Cyclohexane Liquids, Studied by Femtosecond Nonlinear-Optical Polarization Spectroscopy Yong Joon Chang† and Edward W. Castner, Jr.* Chemistry Department, BrookhaVen National Laboratory, Upton, New York 11973-5000 ReceiVed: July 19, 1995; In Final Form: NoVember 5, 1995X
Femtosecond time-resolved optical-heterodyne detected Raman-induced Kerr effect spectroscopy (OHD-RIKES) is shown to be a powerful and comprehensive tool for studying the intermolecular dynamics occurring in liquids. The observed dynamics include both the underdamped, or coherent inertial motions, and the longer time scale diffusive relaxation. The inertial dynamics include phonon-like intermolecular vibrations, intermolecular collisions, and librational caging motions. Data are presented and analyzed for a series of five liquids: cyclohexane, methylcyclohexane, toluene, benzyl alcohol, and benzonitrile, listed in order of increasing polarity. We explore the effects of aromaticity (e.g., methylcyclohexane vs. toluene), symmetry reduction (cyclohexane vs methylcyclohexane), and substitution effects (e.g., substituted benzene series) on the ultrafast intermolecular dynamics, for a group of molecular liquids of similar size and volume. We analyze the intermolecular dynamics in both the time and frequency domains by means of Fourier transformations. When Fourier-transformed into the frequency domain, the OHD-RIKES ultrafast transients of the intermolecular dynamics can be directly compared with the frequency domain spectra obtained from the far-infrared absorption and depolarized Raman techniques. This is done using the Gaussian librational caging model of LyndenBell and Steele, which results in a power-law scaling relation between dipole and polarizability time correlation functions. Last, we use a theoretical treatment of Maroncelli and co-workers to model for some of these liquids the solvation time-correlation function for the solvation of a charge-transfer excited-state chromophore based on the measured neat solvent dynamics.
Introduction The intermolecular dynamics in molecular liquids have been widely studied in the frequency domain by several different spectroscopic techniques, including depolarized Rayleigh/Raman scattering, far-infrared absorption spectroscopy, and dielectric absorption/dispersion measurements. Several excellent monographs1-3 and reviews4-8 have been published on these subjects. While all of the above experiments can provide accurate data, one of the past limitations of these techniques has been that it was very difficult to measure a broad frequency range covering the entire bandwidth of the intermolecular dynamics in a single experiment. For example, for a viscous liquid such as triacetin, the slower diffusive rotational reorientation will have a multiexponential character, with the longest time constant of 92 ps at room temperature, corresponding to a narrow frequency domain bandwidth of 0.12 cm-1.9 At the other extreme of very rapid time scales, the highest frequency intermolecular motions occur in liquid water, for which the peak of the first underdamped libration occurs at 20 fs, corresponding to a frequency domain band centered at about 600 cm-1, extending from ∼300 to ∼1000 cm-1.10 Femtosecond nonlinear-optical polarization spectroscopy allows the direct time-domain measurement of the intermolecular dynamics in liquids, over a range of time scales that include the slowest diffusive dynamics, the fastest underdamped intermolecular vibrations, and all motions in between. In particular, femtosecond optical-heterodyne detected, Raman-induced Kerr effect spectroscopy11-15 (fs-OHD-RIKES) allows the complete characterization of the intermolecular motions using only one † Present address: Department of Chemistry, Mail Code-0341, University of California, San Diego, La Jolla, CA 92093-0341. * To whom correspondence should be addressed. E-mail: castner1@ bnl.gov. X Abstract published in AdVance ACS Abstracts, February 1, 1996.
0022-3654/96/20100-3330$12.00/0
experiment, on one sample. The Fourier transform nature of the physics of the experiment allows presentation of the data in either the time or frequency domains. Presentation in the time domain allows the calculation of quantities that are directly comparable to the angular-velocity time-correlation functions obtainable from molecular dynamics simulations. Frequencydomain presentation of the spectra allows direct comparison of the femtosecond laser data with far-infrared and depolarized Rayleigh/Raman spectra. A number of other ultrafast laser techniques also probe the molecular nonlinear optical response, such as the transient-grating impulsive stimulated Raman spectroscopy experiment (ISRS)16-21 and other interferometric techniques.22 A unique advantage of fs-OHD-RIKES is that the heterodyne detection allows a direct linear measure of the third-order anisotropic response, as opposed to the more common quadratic measure of the same response. There has also been extensive progress in measuring the intermolecular dynamics in the low-frequency domain, as illustrated by the recent work of Friedman and She.23,24 They used a depolarized stimulated Raman gain experiment to measure the low-frequency Raman spectrum for several molecular liquids in the frequency range -200 to 200 cm-1, without interference from Rayleigh scattered light. The spectra obtained from this technique are directly comparable to the fs-OHD-RIKES data Fouriertransformed into the frequency domain (vide infra). We have measured the room temperature fs-OHD-RIKES transients of a series of five liquids: cyclohexane (C6H12), methylcyclohexane (MeCH), toluene, benzyl alcohol (BzOH), and benzonitrile (BzCN). The analysis of our transients is carried out in both the time and frequency domains, using Fourier transform and nonlinear least-squares numerical techniques. Our results enable us to construct a detailed analysis of all of the intermolecular motions occurring in the five liquids. In addition to the femtosecond nonlinear optical experiments, © 1996 American Chemical Society
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SCHEME 1
we have performed ab initio geometry optimization computations, with the goal of better understanding the collective dynamics in liquids on the basis of detailed information on the ground-state electronic structure of the isolated molecule. One reason to choose this particular set of five solvents is that they are all six-membered carbon ring molecules, which means that they have roughly similar molecular volumes and moments of inertia. At the same time, the electrostatic properties of these molecules are very different. The aliphatic cyclohexane and methylcyclohexane liquids have negligible electrostatic interactions. Being aromatic, the three liquids toluene, benzyl alcohol, and benzonitrile have strong electrostatic interactions because of the π-orbital character of their electronic structure. The aromatic liquids all have substantial electric quadrupole moments. Within the group of aromatics, the dipole strengths vary greatly, as seen in Table 1. Last, liquid benzyl alcohol will have strong hydrogen-bonding interactions between hydroxyl groups on adjacent molecules (Scheme 1). The ultrafast optical Kerr effect has been used to study liquid dynamics for more than 25 years.25 Optical heterodyne detection (OHD) was incorporated for Raman spectroscopy first in the frequency domain.26,27 OHD was first used in ultrafast Kerr effect spectroscopy to linearize the measured molecular response,28 as well as to increase the signal-to-noise ratio of the experiment29,30 and to allow direct Fourier transform mapping between the measured birefringence transient and the frequency spectrum.12,13,15,31 Using femtosecond laser pulses, the technique has been refined by McMorrow and Lotshaw to study CS2,11,14,32-34 halomethanes,11,35 pyridine,15,31,36 benzene11,36 and substituted analogues,13,37 and acetonitrile.12,31 Our own work has emphasized the study of complex hydrogen-bonding liquids,38 including a series of amides,39,40 solutions of amides with water and acetonitrile as cosolvents,41 water,9,10 acetic acid,42 and ethylene glycol.9 Several other groups have studied a number of other solvents, including acetonitrile,43 the n-
TABLE 1: Macroscopic Solvent Parameters96,a solvent
η (cP)
C6H12
0.9751 (293 K) 0.898 (298 K) MeCH 0.734 (293 K) 0.685 (298 K) toluene 0.586 (293 K) 0.5525 (298 K) BzOH 5.69 (298 K) BzCN
1.237 (298 K)
µ (D) 0.0
0
2.02431 (293 K) 0.0 2.02 (293 K) 0.31 2.3807 (298 K) 1.66 13.1 (293 K) 4.01 25.2 (298 K)
Veff ∆Hvap V (Å3) (kJ/mol) Rmkp (Å3) 101
32.89 180.6 (298 K) 118 35.359 213.1 (298 K) 111 37.990 1.07 177.5 (298 K) 107.0 50.484 1.79 172.5 (478.6 K) 94.2 55.48 9.72 171.1 (298 K)
a η is viscosity, µ is the permanent electric dipole moment, V is the volume estimated from the calculated van der Waals surface, ∆Hvap is the enthalpy of vaporization, Rmkp the Maroncelli normalized dipole density,72,79 and Veff the effective volume calculated from the molecular weight and density.
alkanes,44 the n-alcohols,44,45 DMSO,44,45 pyrrole,46 water9,10,47 and aqueous solutions,48,49 and alkyl nitriles, halomethanes,50 benzene and substituted benzenes,51-53 and carbon tetrachloride,51,52,54 and cyclodiene compounds.53 A related ultrafast nonlinear optical technique for studying liquid dynamics has been developed and used by the Nelson group, namely, the transient-grating impulsive stimulated Raman scattering experiment.16,20,55-57 They have studied the temperature21 and pressure17 dependence of the intermolecular response of CS2, in addition to room-temperature studies on benzene, chlorobenzene, and CH2Br2.20,57 Fayer and co-workers have applied a similar technique, transient grating optical Kerr effect spectroscopy (TG-OKE),58 to glasses, plastic phases, and liquid crystals such as methoxybenzylidenebutylidenebutylaniline (MBBA), poly(2-vinylnaphthalene), biphenyl, pentacyano biphenyl, and 2-ethylnaphthalene.58-64 Higher-order nonlinear optical techniques provide further insight into the collective nature of the intermolecular interac-
3332 J. Phys. Chem., Vol. 100, No. 9, 1996 tions. Using a Raman echo experiment, the Berg65 and Yoshihara66 groups observed purely homogeneous vibrational dephasing for the C-H stretches of acetonitrile and for the CtN stretch of benzonitrile, respectively. Recent theoretical work by Khidekel and Mukamel details the connections between microscopic molecular dynamics and multidimensional IR and Raman spectroscopy measurements via photon echos.67 One of the key advantages of ISRS over fs-OHD-RIKES is that polarization selection may be used to separate out different parts of the molecular response tensor, e.g., to eliminate the electronic contribution to the response from the measured signal.18 An advantage of fs-OHD-RIKES (aka OHD-OKE) experiments relative to the ISRS or TG-OKE experiments is that the measured signal is linear in the molecular response for OHD-RIKES, whereas the ISRS signal is quadratic in the molecular response. This means that Fourier transform deconvolution of the instrument response and representation of the molecular response for the OHD-RIKES data are straightforward, which is not the case for ISRS or TG-OKE. A method for combining the best features of both OHD-RIKES and ISRS/ TG-OKE experiments has been presented recently by Vo¨hringer and Scherer, which they demonstrated by measuring the responses of CCl4 and benzene.51 These authors applied OHD to linearize the transient-grating diffracted signal beam in a standard ISRS experiment. Different combinations of polarization between the pumping, probing, and scattered beams may be used to select different components of the third-order molecular response. Recently, Chang et al. have also demonstrated the utility of the OHD-ISRS experiment on a CS2 sample.68 In our view, the most exciting question that could be addressed with the new technique is this: Can the symmetric and asymmetric intermolecular vibrations between molecules of different symmetries be differentiated by measuring the spectral densities of the polarized and depolarized third-order tensor elements using OHD-ISRS? We recall here that OHDRIKES experiments are sensitive only to the anisotropic part of the Raman response, analogous to depolarized Raman experiments. A major motivation to carry out the experiments presented here was a desire to further understand the effects of dielectric and collisional friction on barrier-crossing chemical reactions in solution. In particular, we seek to understand the microscopic solute (or reactant) response to solvent interactions. We carried out the experiments on the five solvents presented here to better understand the neat solvent dynamics that are relevant for timedependent dielectric friction measurements (via rotational diffusion) on a series of anthracenes.69 If one considers the diffusive rotational motion of a solute, then the relative rotational self-diffusion of the neat solvent is often on a similar time scale. More importantly, we seek the answer to the following question: What is the effect of the underdamped intermolecular vibrational spectrum of the neat solvent on the solute-solvent dynamical coupling? A recent study by Ladanyi and Stratt directly explores the solute-solvent coupling using instantaneous normal-mode (INM) techniques and molecular dynamics (MD) simulations.70 Their work demonstrates clearly that the differences between the molecular shapes and charge distributions between solute and solvent can substantially change the dynamical coupling. A recent review by Stratt provides a fine introduction to the use of INM techniques for predicting ultrafast liquid dynamics and solvation dynamics.71 Experimental knowledge of the total intermolecular fluctuation spectrum of the pure liquid is a first step toward a complete picture of solute-solvent dynamical coupling interactions, solvation dynamics, and reactions in solution. Using a Brownian
Chang and Castner oscillator model for the solvent dynamics, Cho et al. have previously made a connection between the neat solvent dynamics measured for acetonitrile using OHD-RIKES, and solvation dynamics of a styryl dye in acetonitrile measured by the timedependent fluorescence Stokes shift.43 We have previously used a theory of Maroncelli et al.72 to predict the solvation dynamics for several dipolar and hydrogen-bonding liquids, using the OHD-RIKES neat-solvent dynamics.9,38,39 An even more detailed model of solute-solvent dynamical interactions involves the attempts to extend Brownian oscillator models to predict the electronic dephasing of a chromophore in solution (measured by photon echo experiments) from the neat-liquid spectral density (measured by OHD-RIKES).44,73 Ever more sophisticated photon echo experiments are now being employed to explore the solute-solvent dynamical coupling.74-76 Experimental Section The liquid solvents were of the highest obtainable spectroscopic purity, purchased from Aldrich and Baker. They were used without further purification. The details of the femtosecond optical-heterodyne detected, Raman-induced Kerr effect spectroscopy experiment have been described in detail previously by McMorrow and Lotshaw.11,12 The specifics of our experimental apparatus have also been described in detail.9,38 The experiments were done at room temperature, which fluctuated from day to day. The actual temperatures for the data sets analyzed here were as follows: cyclohexane, 24.1 ( 0.5 °C; methylcyclohexane, 18.0 ( 1.0 °C; toluene, 21.0 ( 0.5 °C; benzyl alcohol, 20.8 ( 0.5 °C; benzonitrile, 20.5 ( 0.5 °C. Laser pulse cross correlations were measured using a 200 µm thick, type I phase-matched KDP crystal. Several scans are averaged into one data set, using an R928 photomultiplier and Stanford Model 850 digital lockin amplifier detection system. Cross-correlation scans were typically measured for 256 steps of 1.0 µm delay, corresponding to a 6.67 fs/step delay. The fs-OHD-RIKES data were measured using two different scan lengths, both with equally spaced data points. To characterize the slower, diffusive reorientational motions, longer distance scans were measured, typically of 400 points, with a time delay per step of either 0.33 or 0.5 ps/step. The total length of scan is determined by the requirement that the experimentally measured scan must begin and end at zero intensity level, to ensure that we have captured all of the diffusive relaxation. To characterize the more rapid inertial and intermolecular vibrational dynamics, scans of 1024 steps with 1.0 µm/step spacing were taken, for a time window of 6.8 ps. In both the short- and long-scan cases, three to five scans were recorded in immediate succession and averaged to form one data set. A total of three or four data sets are then summed and normalized prior to further data analysis. Theory and Data Analysis In the fs-OHD-RIKES experiment, a macroscopic nuclear and electronic polarization is induced in the liquid sample by passage of a tightly focused femtosecond laser pulse. The experiment is electronically nonresonant, meaning that there is no measurable absorption of the laser pulse in the sample. The macroscopic sample polarization results from the interaction of the third power of the laser pulse electric field with the third-order molecular nonlinear optical response. The induced polarization is linearly proportional to this nonlinear optical response. As discussed previously, the Raman-induced Kerr-effect signal measures both the electronic and nuclear contributions to the
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nonlinear optical response R(t):
R(t) ) σ(t) + ∑ri(t)
(1)
i
where σ(t) represents the hyperpolarizability contribution and the ri(t) are the components of the time-dependent nuclearcoordinate response.11,12 This separability between the electronic and nuclear dynamics is an immediate consequence of the Born-Oppenheimer approximation. The polarizationsensitive detection of the molecular response selects only the anisotropic part of the third-order response tensor R(t), analogous to depolarized Raman spectroscopy. Because our ultrashort laser pulses are not instantaneous, the observed signal T(t) in the fs-OHD-RIKES experiment is a convolution of the molecular nonlinear optical response R(t) with the second-order autocorrelation function of the laser pulse, G2(t):12,29,30,77
T(t) ∝ ∫dt R(t - τ) G2(t)
(2)
We assume that our measured intensity cross correlation between pump and probe pulses is equal to G2(t). The time-domain part of the signal arising from rotational diffusion of the solvent molecules is fit to a (multi-)exponential decay from about 1 ps until the end of the data set. Standard nonlinear least-squares fitting is used in either the Igor (Wavemetrics, Inc.) or MatLab (MathWorks, Inc.) program packages. The rotational diffusion correlation function has an inertial risetime, which we model32-34,41 using the following equation:
r1(t) ) Arot∑[ci exp(-t/τi)][1 - exp(-2ω0t)]
(3)
i
Here ω0 is chosen to be the first moment of the low-frequency spectral density. A value of 0.5, 1.0, or 2.0 ps for the starting point of the fit was arbitrarily chosen as a point where the oscillatory inertial dynamics have already been damped out. In practice, by choosing to fit from times longer than about 1.0 ps, we may ignore the second inertial rise time term altogether. By arbitrarily choosing a starting point for the fit at 0.5, 1.0, or 2.0 ps, we have included some of the nondiffusive interactioninduced subpicosecond response in our fit to the longer timescale rotational diffusion. Thus, the response fitted to the fastest exponential, τ1, is not a diffusive relaxation but is contaminated by the inertial dynamics. The longer time constants τ2 and τ3 are indicative of the true reorientational diffusional relaxation. Because the value of τ1 depends strongly on the choice of fit starting delay time, it must be regarded solely as an artifactual device which aids us in separating the diffusive from the inertial relaxation. Following the analysis of McMorrow and Lotshaw,12,13,15 we can obtain the pure-nuclear-coordinate relaxation in the frequency domain, assuming Fourier transform limited laser pulses. The frequency-domain representation of the fs-OHD-RIKES experiments is given by the imaginary part of the Fourier transform deconvoluted transient, i.e.
Im[RRIKES(ω)] ) Im{F[T(t)]/F [G2(t)]}
(4)
where T(t) represents the measured time-domain transient, and G2(t) the measured second-order intensity autocorrelation of femtosecond laser pulse. The spectral resolution in the frequencydomain fs-OHD-RIKES data is increased by extending the time domain data from 1024 to 8192 points. The additional data points are calculated from the multiple-exponential decay law obtained from the fit to the diffusive reorientational dynamics.
The resulting Im[RRIKES(ω)] spectrum has 4096 data points spaced by 0.62 cm-1. The relation between the depolarized Raman spectrum, SDRS(ω), and the frequency domain representation of the fs-OHD-RIKES data is given by45
Im[RRIKES(ω)] ) SDRS(ω)[1 - exp(-pω/kT)]/p2
(5)
Because there are several distinct contributions to the intermolecular dynamics, we often find it convenient to perform a line-shape analysis on the OHD-RIKES frequency spectrum. Our experience with these and other liquids has shown us that we need to include the following features in our model: a (multi-)exponential long-time decay to account for the diffusive reorientation, a skewed low-frequency band or/and an intermediate subpicosecond time constant to account for the collisional and interaction-induced dynamics, and one or more Gaussian librational bands centered at frequencies between 50 and 200 cm-1. Prior to the line-shape analysis, the long-time rotational diffusion response is first subtracted from the raw time domain data. This then enables us to characterize the spectral features that correspond to the inertial dynamics in the frequency-domain following Fourier transformation. The analysis of the band profiles in the Im[RRIKES(ω)] spectrum is carried out by nonlinear least-squares fitting of the spectra to a number of several different line shapes. Normal, skewed, and antisymmetrized versions of Lorentzian and Gaussian functions were fit to the observed librational bands, and for the lowest frequency inertial bands, a smooth asymmetric function given below by eq 6 was used. Equation 6 was
IBL(ω) ≈ ωR exp(-ω/ω0)
(6)
introduced by Bucaro and Litovitz,78 who used an isolated binary collision model to account for collisional interactions in liquids, and has been used by Cho et al.45 and ourselves9,39 to fit the lowest frequency collisional and interaction-induced bands in several molecular liquids. Though the collisional designation for this band shape was certainly correct for CCl4, we recognize that in some liquids, this band shape that we fit will have contributions from both translational and rotational inertial dynamics. The best overall fit was always obtained by using a sum of an antisymmetrized Gaussian line shape for the librational motions, plus eq 6 for the lowest frequency bands, which include the collision-induced dynamics. The form of the Gaussian function is given by
IG(ω) ) exp[-(ω - ω1)2/∆ω] - exp[-(ω + ω1)2/∆ω]
(7)
where ω1 is the band center and ∆ω is the bandwidth. The antisymmetrized version of the Gaussian line shape is used to satisfy the requirement that Im[RRIKES(ω)] goes to zero at zero frequency. The fact that Lorentzian line shapes were never adequate and (antisymmetrized) Gaussian bands always fit well is at least a crude indication that the intermolecular vibrational dynamics have an inhomogeneous line shape. An inverse complex Fourier transform of the imaginary, frequency domain OHD-RIKES spectrum yields the purenuclear-coordinate response in the time domain. Thus the pure nuclear-coordinate response r(t) is given by12,13
r(t) ) 2F -1{Im[RRIKES(ω)]}H(t - t0)
(8)
3334 J. Phys. Chem., Vol. 100, No. 9, 1996
Chang and Castner
where F -1 denotes inverse Fourier transformation, and H(t) is the Heaviside step function. The r(t) function is obtained by inverse Fourier transforming 2048 data points of the fit that was used to analyze the inertial part of the Im[RRIKES(ω)] spectrum. After calculating the inverse FFT, the long rotational diffusion response is added back to the inertial impulse response. We use a correctly scaled, deconvoluted version of the multipleexponential decay function that was previously obtained by fitting the long time part of the OHD-RIKES transient. A recent theory of Maroncelli and co-workers72,79 makes a connection between single-molecule rotation frequencies and inertial solvation frequencies. They proposed the time correlation function Cv(t) as a model for the solvation dynamics that occur in response to a change in solute charge distribution. The solvation time correlation function is given by
Cv(t) ) {C1(t)}RMKP
correlation functions of different ranks are related by the eigenvalues of the spherical harmonics of the form L(L + 1) describing the correlation functions. Thus, applying eq 10, we obtain the relation between the first and second rank reorientational time correlation functions:
C1(t) ) [C2(t)]1/3
(11)
Next, we must clarify the connection between our OHDRIKES response and C2(t). For room-temperature liquids, we assume that we may invoke the classical or high temperature approximation (for which hν/kT , 1).56 We then have the relation between the OHD-RIKES nuclear-coordinate response r(t) and C2(t) as
r(t) R - ∂C2(t)/∂t
(9A)
(12)
Thus, the second rank correlation function C2(t) is obtained as
where
RMKP )
[ ] / [( 4πFµ2 3kBT
9∞
(0 + 2)2
)( )] 1-
∞ 0
(9B)
and C1(t) is the reorientational dipole autocorrelation function of rank L ) 1; C1(t) ) 〈µ(t)‚µ(0)〉. RMKP is a translational factor, equal to the dipole density for a dielectric continuum model.79 It is clear from the dependence of the exponent RMKP on the dipole moment that this theory applies only to strongly dipolar liquids. However, for the case of nondipolar liquids, such as toluene and the cyclohexanes studied here, it would be interesting to extend this theory to higher orders in the generalized electric multipole expansion. Though the Maroncelli et al. model has limitations, we use it here because it is an elegant yet simple molecular-level picture describing solvation dynamics. Because our previous use of this model9 was misunderstood by Deuel et al.,50 we describe in detail the approximations and assumptions we use to connect our OHD-RIKES data with the Maroncelli et al. model. Further theoretical justification of this model has recently been provided by Raineri and Friedman80 and by Roy and Bagchi.81 The rotational time correlation function used in the Maroncelli et al. theory is of rank L ) 1 with respect to the spherical harmonics. The correct rotational correlation function describing depolarized Raman scattering and OHD-RIKES transients is of rank L ) 2, i.e., the polarizability autocorrelation function C2(t) ) 〈R(t)‚R(0)〉. One of the approximations that we must make is to find a direct connection between the rank L ) 1 and rank L ) 2 rotational correlation functions. Such a connection has been shown for the intermolecular dynamics of liquids with strong intermolecular torques (arising from dipole-dipole electrostatic interactions) by Lynden-Bell and Steele.82,83 Their theory assumes that the strong dipolar interactions lead to a caging effect in complex liquids, where a harmonic libration results from the cage. They showed using a cumulant expansion that for an orientational correlation function CL(t), based on the librational motion of frequency Ω0 and inhomogeneously broadened Gaussian bandwidth ∆, that
kT t ∫ dτ (t - τ) cos(Ω0τ) exp(-∆2τ2/2) I 0 (10)
lnCL(t) ) -L(L + 1)
The term I in the above denominator is the molecular moment of inertia. Equation 10 is given by Lynden-Bell and Steele as eq 14 of ref 83. The idea is that for a given intermolecular motion, such as the libration described above, the rotational
∫0tr(t′) dt′ C2(t) ≈ 1 - ∞ ∫0 r(t′) dt′
(13)
where in this part of the analysis we are careful to include the rotational diffusion term, given by r1(t), eq 3, to the overall impulse response, eq 8. Then the dipolar solvation dynamics correlation function CV(t) may be calculated from
Cv(t) ≈
{
}
∫0tr(t′) dt′ 1- ∞ ∫0 r(t′) dt′
RMKP/3
(14)
by combining eqs 9 and 11. The dipole densities, RMKP, are listed in Table 1 for toluene, benzyl alcohol, and benzonitrile. This solvation time-correlation function, constructed from our third-order pure-nuclear-coordinate response, predicts the effect that ultrafast solvent dynamics will have on an electron-transfer reaction coordinate. Specifically, Cv(t) is the predicted energy time-correlation function that applies for the nonequilibrium relaxation of a polar solvent about a molecule or ion with an instantaneous change of charge distribution. In calculating the time correlation functions C2(t) and Cv(t), the number of data points used for r(t) function depends on the long-time diffusive response of a given liquid. We have used up to 10 000 data points with a step size of 0.053 ps/point to account for the entire range of the long-time rotational diffusion contribution to these time-correlation functions. Molecular Orbital Computational Results The equilibrium (isolated molecule) geometries, dipole and higher-order multipole moments, rotational constants, and atomcentered charge models, (obtained by fitting the electrostatic potential), for the five solvents discussed here were calculated using the ab initio electronic structure program packages Gaussian9284 and Mulliken.85 In all of the ab initio calculations, as well as in all of our analysis and discussions, we have used the chair (D3d point group) geometry for C6H12. Two-electron correlation and exchange effects were included in post-HartreeFock calculations using the Mulliken code, exclusively with the 6-31G** polarization basis set using the Becke/Lee-YangParr implementation of the density functional theory (Mulliken RHF/BLYPDFT algorithm). Table 2 contains the summary of the rotational constants for each of the three orthogonal axes,
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TABLE 2: Solvent Rotational Constants from Gaussian92 Geometry Optimizations and Effective “Free-Rotor” Times for Rank L ) 2 Time-Correlation Functiona solvent/ point group
rotational constant A (GHz)
rotational constant B (GHz)
rotational constant C (GHz)
effective free-rotor time τfr(41°) (fs)
C6H12 (D3d) MeCH (Cs) toluene (Cs) BzOH (Cs) BzCN (C2V)
4. 297 4.200 5.624 4.624 5.731
4.297 2.179 2.538 1.443 1.560
2.460 1.586 1.768 1.194 1.226
537 633 567 663 612
a T ) 293 K assumed. Effective free-rotor time τ (41°) (fs) ) (41/ fr 360)2πx(Ieff/kT); Ieff ) h/(8π2Beff), and Beff ) (A + B + C)/3.
obtained from the Gaussian92 geometry-optimization output or by using the geometry optimized Mulliken output and computing the diagonalized inertia tensor. The rotational constants A, B, and C are computed as, e.g., A ) h/8π2Iaa.86 To obtain an average value for the time constant that would be predicted for the thermally averaged rotational in the liquid, we consider the work of Bartoli and Litovitz.87 They observed that, for a second rank orientational correlation function such as our C2(t), an angular displacement of 41° is required to achieve a 1/e decay lifetime. They have used the parameter τfr(41°) as a measure of the thermally averaged time it should take for a molecule to reorient itself by 41° of angular motion. Since the Boltzmann thermal-average angular velocity is given by (kT/Ieff)0.5, they obtain the equation for this parameter as:
τfr(41°) ) (41/360)2π(Ieff/kT)0.5
Figure 1. OHD-RIKES transients, normalized to unity, are shown in a stacked plot. From the bottom, the data are for cyclohexane, methylcyclohexane, toluene, benzonitrile, and benzyl alcohol.
(15)
The values of τfr(41°) for the five liquids are given in Table 2. In doing this calculation, we assume T ) 293 K, and we have computed Ieff ) h/(8π2Beff), where Beff is the average of the rotational constants:
Beff ) (A + B + C)/3
(16)
Table 6 contains the computed gas-phase dipole moments, along with the diagonalized traceless quadrupole tensor. These electric multipole parameters were obtained from the geometry optimizations using the Mulliken program. The molecular volumes were calculated using the piecewise quartic molecular surface program by Connolly,88 where the molecular van der Waals surface of the molecule is integrated to obtain the volume. The molecular Cartesian input coordinates to this program were obtained from the geometry optimizations done using the Gaussian92 programs. The van der Waals radii input data were obtained from the comprehensive article by Bondi.89 The molecular volumes obtained by this method are summarized in Table 1 for later comparison with diffusive reorientation time constants. We have also included in Table 1 an estimate of the effective molecular volume, Veff, obtained from the ratio of the molecular weight over the density. Experimental Results
Figure 2. Bottom: The longer time delay part of the OHD-RIKES transient for benzonitrile is shown. The data are shown as the large dots, and the solid line is a triple-exponential fit to the data in the range from 0.50 to 120 ps. Top: The residuals for the triple exponential fit are shown in the upper curve. The oscillations in the first few picoseconds are the remaining oscillations from the 175 cm-1 bending mode, not an improper fit.
The fs-OHD-RIKES transients for cyclohexane, methylcyclohexane, toluene, benzonitrile, and benzyl alcohol are shown in Figure 1 for a time range of about 5 ps. Common to all of these birefringence transients is a very rapid rise, following the leading edge of the laser pulse at prezero time delays, and a (multi-)exponential decay to zero amplitude at the longest times. The underdamped, or inertial, dynamics dominate the OHDRIKES nuclear-coordinate part of the molecular response. These inertial motions include intermolecular collisions, librational caging effects (arising from strong electrostatic interac-
tions, such as dipole-dipole effects), and intramolecular bending modes and occur in the range from less than 100 fs to beyond 2 ps for benzonitrile. The diffusive reorientation characteristics of the liquids are obtained from OHD-RIKES scans on a longer time scale. Figure 2 shows the data for the diffusive relaxation of benzonitrile, over a 130 ps time window. The residuals to the nonlinear least-squares three-exponential fit demonstrate our ability to clearly discriminate between one, two, three, and stretched exponential forms for the decay profile. The stretched exponential fits were never adequate. The parameters obtained
3336 J. Phys. Chem., Vol. 100, No. 9, 1996
Chang and Castner
Figure 3. Frequency-domain OHD-RIKES spectra, after Fourier transform deconvolution of the instrument response, via eq 4. From bottom to top, the data sets are for cyclohexane, methylcyclohexane, toluene, benzonitrile, and benzyl alcohol.
TABLE 3: Summary of Longer-Time Scale Fits of the OHD-RIKES Transients to Eq 3 solvent
A1
τ1 (ps)
C6H12 MeCH toluene BzOH BzCN
0.05421 1.897 × 10-4 0.6185 1.416 × 10-3 0.5106
0.44 0.64 0.57 1.22 0.59
A2
τ2 (ps)
A3
τ3 (ps)
4.193 × 10-3 1.70 3.517 × 10-5 2.64 0.3384 1.88 0.3447 5.90 2.96 × 10-4 11.68 1.15 × 10-4 62.6 0.09627 3.38 0.1730 21.2
from the best fits are given in Table 3. Recall here that the shortest time constant, τ1, does not characterize the diffusive but rather the interaction-induced dynamics. The frequency domain representations of the OHD-RIKES data for the five liquids are shown in Figure 3. These data are obtained by the Fourier transform deconvolution procedure outlined by McMorrow and Lotshaw,12,13,15 where the imaginary part of the ratio between the Fourier transforms of OHD-RIKES transient and laser-pulse autocorrelation function is obtained using eq 4. For the three aromatic liquids, a low-frequency intramolecular bending or torsional mode is observed, at 220 cm-1 for toluene, 175 cm-1 for benzonitrile, and as a shoulder at 160 cm-1 for benzyl alcohol. The slowest rotational diffusion features in the time-domain transients appear as a sum of one or two Lorentzian features near 0 cm-1. The cyclohexane and methylcyclohexane frequency domain spectra appear rather different. Because of the faster diffusive reorientation occurring in cyclohexane relative to methylcyclohexane, the Lorentzian peak at the lowest frequencies in cyclohexane is substantially broadened. While not apparent to the unaided eye, the fitting procedure reveals that a Lorentzian component is also present in the methylcyclohexane response, though of a smaller amplitude. Figure 4 presents the best-fitted results from our line-shape analysis of the nondiffusive part of the spectra shown in Figure 3. The intramolecular bending vibrations occurring in the three aromatic liquids are not included in the fit. Prior to the nonlinear
Figure 4. Results of the line-shape analysis of the frequency-domain OHD-RIKES spectra are shown. The solid lines are the OHD-RIKES frequency-domain data, and the dashed and dot-dashed lines are the fits to the librational antisymmetrized Gaussian, eq 7, and collisional, eq 6, line shapes, respectively. From bottom to top, the data sets are for cyclohexane, methylcyclohexane, toluene, benzonitrile, and benzyl alcohol.
least-squares lineshape fit to eqs 6 and 7, the rotational diffusion decay law is fit in the time domain to eq 3. This multiexponential decay law is subtracted from the transient before carrying out the Fourier transform/deconvolution procedure. The lowest frequency feature in each of the three aromatic liquids and in methylcyclohexane is a skew-shaped function best fit by eq 6, shown in the figure as the dot-dashed line. Notably, a very good fit for cyclohexane is obtained without use of eq 6, the first of 20 liquids we have studied to not require such a skewed function. Instead, the cyclohexane spectrum fits best to an antisymmetrized Gaussian, with only a negligible improvement obtained when two functions are used as for the other liquids. Further consideration of the cyclohexane spectra in Figure 3 shows that the rapid diffusive reorientation time constant (1.7 ps) overlaps substantially with the 0.44 ps time constant, as well as the low-frequency collisional band peaked at 40 cm-1 (and fit by eq 7 with ω1 ) 22.3 cm-1). The same antisymmetrized Gaussian function, used to fulfill the causal requirement that the spectrum go to zero amplitude at zero energy, fits the higherenergy part of the intermolecular spectrum quite well in all of the other four liquids. While the spectral decomposition for methylcyclohexane presented in Figure 3 is not very convincing, the knee at about 20 cm-1 in the aromatic liquid spectra demonstrates the clear existence of more than one component in the broad band spectra peaked at 50, 60, and 70 cm-1 for toluene, benzonitrile, and benzyl alcohol, respectively. A similar multicomponent OHD-RIKES spectrum was previously observed for liquid benzene by McMorrow and Lotshaw.36 We emphasize that the above values are the effective maxima for the broad bands in the spectra. The librational line shapes for the three aromatic liquids are centered at about 70 cm-1, as shown by the data in Table 4. We obtain a time-domain impulse response function via
Substituted Benzene and Cyclohexane Liquids
J. Phys. Chem., Vol. 100, No. 9, 1996 3337 Discussion Intermolecular Interactions. To better understand the ultrafast dynamics in molecular liquids, we first consider the origin of some of the intermolecular forces between molecules. We neglect ternary and higher-order molecular interactions, considering only pairs of molecules. Several excellent monographs and texts provide much detail on the subject of intermolecular interactions.90-93 Following the discourse of Maitland, et al.91 we recall that the intermolecular interaction energy, described by an empirical potential such as the LennardJones or Buckingham potential, has a short-range repulsive part and a long-range attractive part. The long-range interactions or van der Waals attractions are categorized as electrostatic, induction, and dispersion energies, i.e.
Ulong(r) ) Ues(r) + Uind(r) + Udisp(r)
Figure 5. Time-domain nuclear-coordinate impulse responses for the intermolecular part of the molecular response. These data are obtained via inverse Fourier transformation of the intermolecular spectra from Figure 4, using eq 8. From bottom to top, the data sets are for cyclohexane, methylcyclohexane, toluene, benzonitrile, and benzyl alcohol.
TABLE 4: Parameters for Im[RRIKES(ω)] Fitting Using Eq 6 and Antisymmetrized Gaussian (Eq 7) Line Shapes solvent
AGauss (%)
Gaussian ω1 (cm-1)
∆ω (cm-1)
ALit (%)
Litovitz ω0 (cm-1)
R
BzOH BzCN toluene MeCH C6H12
97.2 91.0 92.0 94.6 100
67.0 70.8 70.5 41.7 22.3
100.8 65.5 71.3 78.3 83.1
2.8 9.0 8.0 5.4
11.1 27.7 26.9 31.1
1.4 1.1 1.1 1.1
inverse Fourier transformation of the intermolecular components of the spectra using the spectral decompositions from Figure 4, obtained from the nonlinear least-squares fitting of line shapes to the intermolecular dynamics. By explicitly omitting the intramolecular torsional or bending modes of the functional group substituents bonded to the six-membered ring structures, we can concentrate on the intermolecular dynamics. These inverse Fourier transforms of the spectral components, multiplied by the Heaviside step function to account for causality, as in eq 4, are shown in Figure 5. Note that the rapidly rising and falling decay is not connected with the response time limitations imposed by the laser pulse duration, because this has been accounted for in the prior Fourier transform deconvolution, eq 4. This initial (and inertial) response has a width of approximately 250 fs for the aliphatic ring compounds and of about 200 fs for the aromatic ring molecules. Clear evidence for the presence of librational caging effects is present in Figure 5. In the time-domain molecular response, there is a oscillation peaked at 400-500 fs for each of the aromatic liquids. Such an oscillation is absent in the aliphatic liquids cyclohexane and methylcyclohexane, because of the very weak intermolecular interactions in these liquids. The oscillation in the aromatic liquids is evidence for the underdamped librations, because the intramolecular bending modes have been eliminated via the selective line-shape analysis.
(17)
The first, second, and third terms represent the contributions from electrostatic interactions such as dipole-dipole interactions, induction effects from dipole/induced dipole interactions, and the dispersion, or induced dipole/induced dipole interactions, respectively.91 For hydrogen-bonding liquids such as benzyl alcohol, an additional term can be added to account for the hydrogen-bonded structure of the liquid. We will make estimates of the attractive interaction potential energies for each of the five liquids. Considering the cyclohexane liquids, we see that the minuscule dipole and quadrupole strengths imply that there will be only negligible contributions to the van der Waals attractive potential from either electrostatic or induction effects. The ionization potentials of course are different between the aliphatic and aromatic ring liquids but not dramatically so. When a simple Drude model is used to account for dispersion interactions, the leading term in the power series expansion for the pair dispersion interaction energy is91
Udisp )
-3R2IP 4(4π0)2r6
(18)
where 0 is the vacuum permittivity, R is the molecular polarizability (R/4π0 has units of Å3), r the intermolecular separation, and IP the first ionization potential (in joules). Using the measured liquid value for cyclohexane photoionization energy of 8.43 eV94 and an intermolecular radius of 6.1 Å obtained from neutron-scattering experiments,95 we calculate a dispersion interaction energy of Udisp ) -2.3 × 10-21 J for C6H12. For the other four liquids, we estimate of the intermolecular center-to-center separation as twice the radius obtained from Veff, (Table 1). We obtain similar pair dispersion energies for methylcyclohexane, toluene, benzyl alcohol, and benzonitrile. These dispersion energies are listed in Table 7 along with the intermolecular radii obtained from the volume and the gas-phase ionization potentials. To estimate the contribution to the intermolecular potential energy from the induction forces, we use the high-temperature, orientationally averaged formula91
Uind )
-2Rµ2 (4π0)2r6
(19)
where µ is the molecular dipole moment. Again using the values given above for intermolecular separation and molecular polarizability in liquid benzonitrile, with a gas-phase permanent dipole moment of 4.01 D,96 we find the dipole/induced dipole
3338 J. Phys. Chem., Vol. 100, No. 9, 1996
Chang and Castner
TABLE 5: Summary of Parameters from Fits of Calculated Solvation Time Correlation Function Cv(t) to Eq 24a solvent
τs (ps)
Agauss
ωgauss (ps-1)
AA
τA (ps)
AB
τB (ps)
AC
τC (ps)
toluene BzOH BzCN
12.7 87.0 3.6
0.06114 0.2122 0.0712
9.68 9.9734 10.2
0.08966 0.0725 0.2367
1.012 1.003 0.9041
0.8492 0.2223 0.5515
14.83 15.045 6.059
0.6247
134.6
a
The average solvation time τs is defined as the zeroth moment of Cv(t), i.e., τs ≡ ∫∞0 Cv(t) dt.
TABLE 6: Summary of (Gas-Phase) Electric Multipole Moments for the Six-Membered Ring Molecules, from the Literature and Mulliken RHF/BLYPDFT/6-31G** Geometry Optimizations solvent
R/(4π0) (Å3)
µ (D) (expt)96
C6H12 MeCH toluene BzOH BzCN
0.0 0.0 0.31 1.66 4.01
µ (D) (ab initio)
10.87 13.1 12.26 12.90 12.5
10-4
6.1 x 6.7 x 10-2 0.262 1.38 4.45
Qxx (D Å)
Qyy (D Å)
Qzz (D Å)
〈Q〉 (D Å)
0.4634 -0.9031 4.0726 6.2069 13.4689
-0.2317 -0.0840 3.9239 19.6345 1.1274
-0.2317 0.9871 -7.9965 -25.8414 -14.5963
0.463 1.095 7.9970 26.98 16.24
TABLE 7: Summary of Calculated Pair Interaction Energiesa solvent
r (Å)
IP122 (eV)
Udisp (J)
C6H12 MeCH toluene BzOH BzCN
6.1 7.41 6.97 6.91 6.89
8.43 8.95 8.82 9.14 9.70
-2.32 × 10-21 -1.11 × 10-21 -1.39 × 10-21 -1.68 × 10-21 -1.70 × 10-21
a
Uind (J)
Uµ,µ (J)
-2.06 × 10-24 -6.53 × 10-23 -3.76 × 10-22
-1.31 × 10-25 -1.13 × 10-22 -3.92 × 10-21
Uµ,Q (J)
UQ,Q (J)
-5.35 × 10-24 -1.88 × 10-21 -4.06 × 10-21
-4.40 × 10-27 -1.96 × 10-26 -1.03 × 10-22 -1.45 × 10-20 -1.96 × 10-21
r is the effective intermolecular separation, obtained from Veff. IP is the first ionization potential, and energies U defined in eqs 18-22.
interaction energy to be -3.7 × 10-22 J. The induction interactions for benzyl alcohol and toluene are an order of magnitude smaller, because of the smaller dipole moments. The induction energies are also tabulated in Table 7. The gas phase dipole moments and polarizabilities from Table 6 are used in the calculation. The dipole-dipole interaction energy is given in the hightemperature, orientation-averaged limit as91
Uµ,µ(r) )
-2µ4 3kT(4π0)2r6
(20)
Benzonitrile has the strongest dipole-dipole interactions among the group of five liquids studied here. At T ) 298 K, using the benzonitrile parameters above of µ ) 4.01 D and r ) 6.89 Å, we find that dipole-dipole electrostatic energy is Uµ,µ ) -3.92 × 10-21 J, which is more than twice the dispersion interaction energy. For toluene, benzyl alcohol, and benzonitrile, the electrostatic interactions are complex, because both dipole and quadrupole terms are important. The high-temperature formulas for the dipole-quadrupole and quadrupole-quadrupole interactions are given in eq 21.91
Uµ,Q(r) )
-2µ2〈Q〉2 kT(4π0)2r8
UQ,Q(r) )
-14〈Q〉4 5kT(4π0)2r10
(21)
We digress here for a moment to explain how we obtained an effective axial quadrupole moment 〈Q〉 from the quadrupole tensor resulting from the Mulliken ab initio electronic structure calculation. Following the discussion of Gubbins et al.,97 we learn that for axial (such as linear, symmetric, or spherical top) molecules, only a single element of the quadrupole tensor, Qzz, is required to uniquely describe the quadrupole moment. For nonaxial, or asymmetric rotor molecules, only two unique elements exist, commonly denoted Qxx and Qzz. This occurs because the quadrupole tensor is both diagonalizable and traceless. Gubbins et al. defined an effective axial quadrupole moment as97
〈Q〉 ≡
x23(Q
2 xx
+ Qyy2 + Qzz2)
(22)
The diagonalized quadrupole tensor elements and the effective axial quadrupole moment are given in Table 6. Considering the magnitude of the quadrupole tensor elements for the aromatic liquids, toluene, benzyl alcohol, and benzonitrile, it is clear that dipole-quadrupole and quadrupolequadrupole electrostatic interactions will be substantial for nearest-neighbor molecules in the liquid. The quadrupole moments for the aromatic liquids range from a low of about 8 D Å for toluene, to 16 D Å for benzonitrile, to a large value of 27 D Å for benzyl alcohol. From Table 7, we immediately see that the dispersion interactions dominate for the aliphatic liquids, while the electrostatic forces dominate for the aromatic liquids. For benzyl alcohol and for benzonitrile, the electrostatic interactions are stronger than the dispersion interactions. For benzonitrile, the relative contributions to the electrostatic interactions are roughly 40% dipole-dipole, 40% dipolequadrupole, and 20% quadrupole-quadrupole. For benzyl alcohol, the substantially smaller dipole and large quadrupole moments change the relative weighting to 0.6% dipole-dipole, 11.4% dipole-quadrupole, and 88% quadrupole-quadrupole. We have learned that for the first shell around a single solvent molecule, quadrupolar effects can dominate the electrostatic interactions for aromatic liquids, even for strongly dipolar liquids such as benzonitrile. Summing the three contributions to the attractive intermolecular potential, dispersion, induction, and electrostatic (dipole-dipole) interactions, we obtain a value of -1.202 × 10-20 J for benzonitrile. The dispersion interaction energy for cyclohexane was estimated to be only -2.3 × 10-21 J, implying that the intermolecular interaction strengths are a factor of 5.2 smaller for cyclohexane than for benzonitrile. Benzyl alcohol has the strongest intermolecular interactions of the group, of 1.82 × 10-20 J, which is 7.9 times stronger than for cyclohexane. The principal difference is in the estimated electrostatic interaction energy of -9.94 × 10-21 J in benzonitrile and -1.65 × 10-20 J for benzyl alcohol, not present for cyclohexane, which we believe accounts for the differences between the low-frequency spectra of the aliphatic and aromatic liquids. If we consider that the enthalpy of a modest hydrogen
Substituted Benzene and Cyclohexane Liquids bond may be about 10 kJ/mol per H-bond, this implies an effective pair H bond energy of about 1.7 × 10-20 J. The total electrostatic interaction energies for benzyl alcohol and for benzonitrile are therefore of a similar strength to the energy of an H bond. There is a good correlation between our estimated van der Waals interaction energies, summarized in Table 7, and the macroscopic viscosities given in Table 1. This correlation reinforces the established viewpoint that viscosity may be one of the best macroscopic physical parameters with which to predict the microscopic solvent friction that affects barriercrossing reactions in solution. We must be cautious in our use of the above estimates of pairwise van der Waals interactions and their relation to liquidstate dynamics. While the above analysis is certainly valid for the gas phase and for low-density liquid simulations, ternary and higher intermolecular interactions may dominate the interaction-induced intermolecular spectra in liquids. Recent computer simulations by Steele and co-workers98-100 have shown that for simpler liquids (such as Ar, N2, CO2, or C2H6) at very low densities, the two-body, or pairwise interactions dominate the contributions to the interaction-induced Raman spectrum. However, at higher densities, there is a nearly exact cancellation between two- and four-body interactions by threebody interactions. It is not yet known whether such a cancellation effect will appear in the simulated Raman spectra when accurate simulations of highly anisotropic and strongly interacting liquids are carried out for the cyclohexane and benzene liquids discussed here. Diffusive Reorientation. It has long been recognized that even for weakly interacting molecular liquids such as carbon tetrachloride and cyclohexane87,95,101 that a brief period of inertial (free) rotation is interrupted by intermolecular collisions, which leads to the onset of diffusive orientational behavior. Shimizu102 and Gordon103-107 proposed theoretical models to explain this behavior as inferred from far-infrared, Rayleigh, and Raman spectra. Other theories extended the Gordon M- or J-diffusion models108,109 or used alternative techniques.110 Temperature dependent fs-OHD-RIKES experiments on cyclohexane, as well as acetic acid42 and a series of amides,40 indicate that properly scaled Stokes-Einstein-Debye hydrodynamic theories for diffusive reorientation give a good description of the temperature dependence of the longest time response of the OHD-RIKES signal. This is why we confidently assign these longest exponential relaxation processes, characterized by the time constants τ2 and τ3, to diffusive reorientation. Table 1 summarizes the molecular dipole moments, viscosities, and static dielectric constants from the literature96 as well as the calculated volumes. Since the molecular volumes are all quite similar, the differences in the diffusive reorientation time among the five liquids are related to the different viscosities and electrostatic interactions and not the molecular size. The rotational constants for the five solvents and the effective free-rotor times τfr(41°) calculated for a rank L ) 2 Raman process are found in Table 2. Because the molecules are quite similar in size and structure, so also are the values of τfr(41°), ranging between 537 and 663 fs. Though there appears to be a slight correlation between the τ1 lifetimes obtained from fitting the OHD-RIKES data, and the calculated τfr(41°) values, these numbers must be regarded solely as an artificial aid to obtaining accurate values for the diffusive lifetimes τ2 and τ3. While our estimated accuracy on the longer time constants τ2 (and τ3) is better than (0.15 ps, a greater uncertainty arises in the shortest lifetime τ1. This occurs for two reasons: First, starting the fit at 0.5, 1.0, or 2.0 ps time delay means that fitted lifetimes shorter
J. Phys. Chem., Vol. 100, No. 9, 1996 3339 than this value will be questionable. While τ2 (and τ3) lifetimes are relatively independent of the starting position for the fit, the shortest lifetimes τ1 show a strong correlation with the choice of starting point for the fit range. Second, the oscillations in the OHD-RIKES data arising from the intramolecular vibrations and librations persist to 0.5 ps for cyclohexane to beyond 2.0 ps for benzonitrile, providing additional problems with using a simple exponential fitting procedure. Because of these uncertainties, we estimate the error bounds on the value of τ1 in Table 3 to be (0.25 ps. In the case of the aromatic liquids, each has a nonexponential rotational diffusion decay. There is some general correlation between the ratios of the fitted lifetimes τ2 and τ3 given in Table 3 with the ratios of the largest to smallest rotational constants, A and C, from Table 2. We believe that this loose correlation is likely to be fortuitous. A detailed study of the molecular dynamics and interactions in benzonitrile was presented by Guillaume et al.111 Their research included an analysis of the infrared and Raman line widths of the CtN stretch in benzonitrile, as well as far-infrared and dielectric relaxation experiments. A number of interesting and important results were obtained. From the line-width analysis, they obtained time constants of 10.6 ( 0.3 ps (Raman, L ) 2; 291 K) and 20 ( 10 ps (infrared, L ) 1; 290 K) for the diffusive rotational relaxation of benzonitrile. A careful analysis of their dielectric relaxation data (L ) 1) provided a corrected Debye relaxation time τD of 29 ( 1 ps. Using a GlarumPowles theory, Guillaume et al.111 concluded that the collective reorientational diffusion time constant was a factor of 1.9 times longer than that of the single-particle time. The assumption made in this analysis is that the rotation time obtained from vibrational line widths contains information only about singleparticle reorientation not the collective response of the overall fluid measured by dielectric relaxation. We clearly obtain a nonexponential diffusive reorientation behavior for benzonitrile from our time-domain OHD-RIKES transients, although this is certainly not obvious when considering frequency domain line widths. If we take the properly weighted average of the time constants τ2 and τ3 for benzonitrile, we obtain an effective time constant of 14.8 ps. Within experimental error, this 14.8 ps effective time constant is not in agreement with the time constant of 10.6 ps obtained from the -CtN stretch line widths. The question of whether there is an anisotropy of reorientation about the different molecular inertial axes of (chair) C6H12 was addressed by Bartoli and Litovitz.101 By correcting for the intramolecular (constant) contribution to depolarized Raman line widths in a series of temperature-dependent experiments, they extracted the orientational part of the line width, ωor. This orientational line width is related to the diffusive reorientation time constant by τraman (ps) ) 1/(2πcωor) (cm-1). For cyclohexane, the 802 and 1157 cm-1 vibrational modes are of A1g symmetry, and they obtained values of τRaman ) 1.5 ( 0.25 ps for reorientation about the major symmetry axis. Studying the 1027 cm-1 symmetric C-C stretch of Eg symmetry and the -CH2 twisting motion at 1266 cm-1, the value of τRaman ) 1.7 ( 0.35 ps was obtained. Our value of 1.70 ( 0.15 ps, obtained directly from the exponential fit to the raw OHD-RIKES data, is in exact agreement with their result to within experimental error. Also, the E symmetry modes sampled a mixture of reorientation about the major and minor inertial axes. The differences in these τRaman lifetimes are within the reported experimental error. This indicates that the diffusive reorientation occurring in this symmetric top molecule results in an observed single-exponential relaxation, despite the oblate character of the symmetric top shape. Given that the rotational constants for the five molecules are all in a similar range, as shown in Table
3340 J. Phys. Chem., Vol. 100, No. 9, 1996 2, we expect that the rotational diffusion might be observed to be single exponential for all five of the molecules studied here. This is not the case, as shown in Table 3. Instead, we see that the multiexponential nature of the reorientational decay is manifested by the aromatic liquids because of their strong intermolecular interactions. Inertial Dynamics. For third-order (Raman) processes such as we measure with fs-OHD-RIKES experiments, we measure a polarizability correlation function. There are two principal contributions to the dynamic polarizability of a liquid: a singlemolecule polarizability (such as could be observed in the gas phase) and an interaction-induced polarizability that arises from the dynamic interactions between two or more liquid molecules simultaneously. The ongoing challenge for spectroscopists is to determine which parts of the spectrum for a given liquid belong to the single-molecule dynamics and which spectral components should be assigned to the interaction-induced dynamics and whether or not cross terms are important. For certain liquids, either the symmetry of the molecule or the type and strength of electrostatic interactions will determine whether there can be a clear separation of time scales between single-molecule, interaction-induced, and librational dynamics. By using projection operator techniques, the total liquid polarizability can be rewritten in terms of reorientational and collisional contributions, which can in turn be projected out from molecular dynamics trajectories. Illustrations of this technique have been given for CS2 by Madden112 and by Geiger and Ladanyi.113,114 More recently, Ladanyi and Liang have applied this technique to CH3CN and CH3OH.115 In general, interactioninduced effects are found to be present on all time scales. However, by choosing molecules of varying symmetry or molecules with greatly different electrostatic interactions, we can make some progress in determining what motions are arising from the “librational caging” and which molecules cannot display this behavior. Specifically, C6H12 and MeCH cannot have librational caging dynamics, because there are no strong electrostatic interactions which could exert the strong intermolecular torques required for this phenomena. At present, we must rely on careful computer simulations done using accurate intermolecular potentials if we are to be able to assign rotational or translational character to a given spectral feature in our intermolecular dynamics. Though the precise values for the fitted values of τ1 are artifactual, there is no doubt that there is a rapid response on the time scale of about 500 fs in all liquids. A number of articles in the recent literature assign these lifetimes to collision-induced and interaction-induced effects.11,12,23,24,33,36,116 We are inclined to agree with this general assignment. Molecular dynamics simulations72,79 and instantaneous normal mode (INM) calculations95,117-119 demonstrate that the so-called “free-streaming” motions are a good description of orientational motions at short times. However, these motions are rapidly interrupted by collisions and so occur only for small angular jumps. This makes it difficult to discern the difference between a smallangle rotational jump and a half-period of a librational motion. Small-angle rotations are nonetheless occurring but of a less correlated nature than those contributing to the librational caging effect. We emphasize that it is not the case that these molecular liquids are free from intermolecular torques, but rather that for extremely short times of
