J. Phys. Chem. 1983, 87,3028-3033
3028
FEATURE ARTICLE Vibrational Relaxation in Llqulds: Quantum States in a Classical Bath Davld W. Oxtoby James Franck Instnute and Deparfment of Chemistry, University of Chicago, Chicago, Illinois 60637 (Received: February 28, 1983)
Vibrational motion in liquids is highly quantum mechanical: distinct vibrational quantum states are clearly seen and can usually be identified with the corresponding quantum states for molecules in the gas phase. The relaxation of these states occurs through coupling to a bath of rotational and translational degrees of freedom, which can be described through classical mechanics. In this article we outline some promising new theoretical approaches to vibrational phase and energy relaxation in liquids and show their relationship to experimentally determined vibrational line shapes and population relaxation rates. We focus particularly on the qualitatively new types of behaviors that result when the time scales for vibrational and bath relaxation become comparable.
Introduction Since 1970 there has been a resurgence of interest in the study of vibrational relaxation in liquids, and considerable progress has been made through experiment, theory, and computer simulation toward a deeper understanding of the physical processes involved. There are a number of reasons for this interest. First, there has been a growing awareness of the practical importance of vibrational relaxation in a number of areas, such as the effect of energy relaxation in excited electronic states on the products and yields of liquid-state photochemical reactions, and the possibility of developing a liquid-phase chemical laser taking advantage of the long relaxation times in cryogenic liquids. Second, it has been recognized that vibrations are sensitive probes of local structure and dynamics in molecular liquids, and thus provide microscopic information about a state of matter which is still relatively poorly understood. In this article I will present some of the contributions of my research group to the theory of liquid-state vibrational relaxation. In most cases a clear distinction may be made between vibrational degrees of freedom in liquids, on the one hand, and rotational and translational degrees of freedom on the other. Vibrational frequencies are typically in the range of 500-4000 cm-’ and are thus larger than the thermal energy kBT (at room temperature the energy kBT corresponds to 200 cm-’). As a result, only a few vibrational states are thermally populated. This, combined with the fact that vibrational modes retain their identity in the liquid phase, means that the quantum nature of vibrational states is quite important. On the other hand, rotational and translational modes have lower frequencies so that more states are thermally populated. Rotational states are usually so strongly mixed in liquids that it is not very meaningful to describe them as discrete levels; classical mechanics is thus sufficient to describe rotational and translational dynamics in liquids. Vibrational relaxation then occurs through the coupling of a quantum vibrational system to a classical “heat bath” of rotational and translational degrees of freedom. (There are of course exceptions to this classification of quantum and classical degrees of freedom: low-frequency vibrational modes in large molecules may often be treated classically, while in some small molecules like H2 the quantum nature of the rota0022-3654/83/2087-3028$0 1.5010
tional modes may be important.) The appropriate language for describing a quantum system relaxing through coupling with a classical bath is that of the density matrix. The diagonal density matrix elements pii(t) describe the evolution of the population of vibrational level i, while the off-diagonal matrix elements p i j ( t ) describe the phase coherence between levels i and j. Popul&ion relaxation involves vibrationally inelastic processes, and corresponds to what is called TI relaxation for spin systems (as in NMR), while phase relaxation involving vibrationally elastic processes corresponds to T2 relaxation. This analogy of vibrational to spin relaxation, while a useful one, should not be pushed too far. A crucial distinction is that in spin systems relaxation times are usually in the microsecond or millisecond range and are thus much slower than bath relaxation times in liquids (except possibly in highly viscous liquids near a glass transition). Vibrational phase and energy relaxation times, however, can be as short as a few picoseconds, and may thus be comparable to the bath relaxation times; this has important consequences for the dynamics of the coupled systems. Experimental measurements of vibrational phase and energy relaxation may be carried out in either the time or the frequency domain. Population (or energy) relaxation measurements in the time domain involve pulsed excitation (by stimulated Raman scattering or infrared absorption) followed by detection of level populations after a time delay t (using anti-Stokes Raman scattering or fluorescence). In this way relaxation times have been measured ranging from seconds (for liquid nitrogen’) to picoseconds.2 The major experimental problem in the former case is the rigorous exclusion of impurities which speed relaxation, while in the latter it is the development of picosecond excitation and detection techniques. Picosecond spectroscopy may also be used to measure phase relaxation;2 by monitoring the intensity at the phase-matched angle, one may study the relaxation of phase coherence. This technique is referred to as four-wave mixing or coherent anti-Stokes Raman spectroscopy (CARS). (1)S. R. J. Brueck and R. M. Osgood, Chem. Phys. Lett., 39, 568 (1976);W.F.Calaway and G. E. Ewing,J . Chem. Phys., 63,2842(1975). (2) A. Laubereau and W. Kaiser, Rev. Mod. Phys., 50, 607 (1978); Annu. Reu. Phys. Chem., 26, 83 (1975).
0 1983 American Chemical Society
Feature Article
Frequency-domain studies of vibrational energy relaxation involve either ultrasonic absorption3 or Brillouin line width4 measurements. They provide information about energy relaxation of thermally populated (and therefore usually low frequency) vibrational modes. There are two disadvantages with these techniques: first, they are useful only for relaxation times in the nanosecond range, and second, they often measure the combined relaxation of several levels rather than that of a single level. Frequency-domain measurements of phase relaxation are quite extensive. The isotropic Raman line shape is given by the Fourier transform of the vibrational coordinate correlation function: which is proportional to the off-diagonal density matrix element pio(t) (0 is the vibrational ground state). In this way direct determinations of phase relaxation are possible. The line shapes of anisotropic W a n scattering and infrared absorption are also affected by vibrational phase relaxation; however, here the situation is more complicated because rotational relaxation plays a role as well. This article will focus on theoretical approaches first to phase and then to energy relaxation. It is not intended as a comprehensive survey of the field, since several such surveys exist already.2p68 Instead, I will focus on areas in which I and my research collaborators have been particularly active.
Phase Relaxation As mentioned earlier, vibrational phase relaxation leads to broadening of the isotropic Raman line shape. For a well-separated vibrational transition (no overlapping with hot bands or other transitions) there are three primary sources of line broadening and thus of phase relaxation. The first is lifetime broadening: because of the finite lifetime of a quantum state, there is a broadening due to the uncertainty principle. In liquids where population (TI) relaxation times have been measured, this uncertainty broadening may be estimated; it generally makes a rather small contribution to the line width and will not be considered further here. The second and most important contribution to vibrational phase relaxation may be referred to as “environmental broadening”; it is sometimes also called “pure dephasing”. I t arises from the fact that the vibrational frequency of a molecule i is perturbed by its interaction with other molecules and therefore has a component Awi(t)which fluctuates with time. If it were possible to freeze the environment at a particular time, one would observe a distribution of frequency shifts and therefore a broadened vibrational line shape. This is referred to as the static limit. In practice, however, the time dependence of the environment is important and therefore the line is at least partially “motionally narrowed” from its static limit. In more quantitative terms, the isotropic Raman line shape is given by the Fourier tranform of the vibrational coordinate autocorrelation function ( Qi(t)Qi(0)) where the angular brackets define an ensemble average. The vibrational coordinate Qi at time t differs from Qi(0)by a phase factor exp[i($Jt) - 4(0)] where the phase difference is given by (3) K. F. Herzfeld and T. A. Litovitz, “Absorption and Dispersion of Ultrasonic Waves”, Academic Press, New York, 1959. (4)See, for example, G. Huf-Desoyer, A. Asenbaum, and M. Sedlacek, J. Chem. Phys., 70,4924(1979). (5) S.Bratos and E. Marechal, Phys. Reu. A, 4,1078 (1971). (6) D. W. Oxtoby, Adu. Chem. Phys., 40,1 (1979). (7)D.W. Oxtoby in ”Photoselective Chemistry”, Part 2, J. Jortner, Ed., Wiley, New York, 1981,p 487. (8)D. W. Oxtoby, Annu. Rev. Phys. Chem., 32, 77 (1981).
The Journal of Physlcal Chemistry, Vol. 87, No. 16, 1983 3020
&(t)- $i(0) =
1’
dt’wi(t? = at
0
+ Jtdt’
Aoi(t?
(1)
where a is the average vibrational transition frequency in the liquid (which may differ from its value wo in the gas phase) and Awi(t)gives the fluctuations in frequency due to the environment. The vibrational phase relaxation is then given b 9 6
and depends on the statistical properties of Awi(t). If we consider the autocorrelation function of Awi(t), ( Awi(t). Awi(0)),then we may define a characteristic time T, (the “bath relaxation time”) through T, E
Jmdt
(Awi(t)Awi(O))
(Ad)
(3)
then the nature of the line shape depends on the relative magnitude of the two characteristic frequencies ( Aw;)1/2 and T ; ~ . When ( A O J ? ) ~ />> ~ T ,1 we have the “static” limit considered above, while when ( A o ? ) ~ / ~ T> F2, but it can only contribute to Awi(t) if the oscillator is anharmonic so that Qll # Qoo. Line shapes thus cannot be described by models of harmonic oscillators in liquids. The correlation times 7, calculated appear to be rather short (0.15 ps for Nz, 0.12 ps for HC1) and not very different from each other. The very large differences between the two spectra (the experimental line widths are 0.067 cm-’ for N2 and 60 cm-’ for HCl) are thus due almost entirely to differences in the magnitude of Awi. In our calculations ( A u : ) ~ / ~ was 1.3 cm-’ for N2,but 26 cm-’ for HCl. Dispersion interactions dominate in both cases, although for N2there is substantial cancellation between the dispersion and short-range parts of the potential. For HCl, the electrostatic contribution is also quite important, while in neither case does the vibration-rotation coupling contribute significantly. Nitrogen falls in the weak coupling, or rapid modulation, limit since ( A w i z ) l / z ~=c0.04 > kBT) then the IBC model should give reasonable results, although the definition of collision rate may require further study. On the other hand, for hwij 5 kBT (i.e., in the range of 0-200 cm-' typically) collective effects may be important and should be included. In the latter case there is a reasonable likelihood of success through computer simulation or correlation function modeling. We have proposed31a general formulation for correlation function calculations of vibrational relaxation in liquids. We showed how group theoretical techniques may be employed to construct symmetry-adapted generalized forces to describe the intramolecular relaxation of a molecule in a solvent. This use of symmetry greatly reduces the number of calculations that need to be done. The relaxation rate is then expressed as a projection of generalized forces (stresses) onto the molecular strains, obtained from matrix elements of vibrational coordinates by using gasphase wave functions. This projection may lead to propensity rules favoring certain relaxation pathways in the liquid. The stress correlation functions may be simulated or modeled; we considered32in particular both a binary collision and a h y d r ~ d y n a m i cmodel ~ ~ and obtained reasonable results for nanosecond time scale intramolecular relaxation of triatomic molecules in inert solvents. The frequency differences in this case were 100 cm-' or smaller; for larger frequency differences the correlation function approach is less successful. Until now, we have considered only the evaluation of rate constants, and have implicitly assumed that such a description suffices. However, there is a fundamental assumption made in using rate equations, namely, that the relaxation of a particular molecule depends only on the state it is in at that particular instant, and is independent of its past history or the path it followed in reaching that state. This "lack of memory" is referred to as the Markovian limit; it holds if there is a separation of time scales such that the bath relaxation is fast relative to the vibrational population relaxation. Such is surely the case for diatomic and triatomic molecules which show relaxation times in the nanosecond range or slower, since bath (rotational and translational) relaxation times are on the order of picoseconds. However, for slightly larger molecules, vibrational relaxation will lie in the picosecond time scale, a domain now accessible by picosecond spectroscopy. The question then arises of whether the rate equation description still holds for these very fast relaxations, or whether qualitatively new non-Markovian effects may arise. This question has been addressed in three of our recent papers.34-36 Our first paper34presented a simulation of a two-level system strongly coupled to a classical bath. Rather than simulate a realistic system-bath coupling we considered two very simple stochastic models; in one the coupling to the bath was described by a two-state jump model, with a Poisson distribution of times between jumps, while the second was a Gaussian random process. The first process, because of its abrupt changes in interaction, might be considered to roughly simulate the effect of random hard collisions, while the second corresponds to long-range interactions with a number of neighbors. The system was (30) D. W. Oxtoby, Mol. Phys., 34, 987 (1977). (31) S. Velsko and D. W. Oxtoby, J. Chem. Phys., 72, 4853 (1980). (32) S. Velsko and D. W. Oxtoby, J. Chem. Phys., 72, 2260 (1980). (33) H. Metiu, D. W. Oxtoby, and K. Freed, Phys. Rev. A , 15, 2005 (1970). (34) R. J. Abbott and D. W. Oxtoby, J . Chem. Phys., 72,3972 (1980). (35) B. Bagchi and D. W. Oxtoby, J. Chem. Phys., 86, 2197 (1982). (36) B. Bagchi and D. W. Oxtoby, J. Chem. Phys., 77, 1391 (1982).
placed in its instantaneous excited state at t = 0, and the subsequent evolution of the wave function (and therefore of the populations) was calculated through an exact solution of the Schrodinger equation. As expected, when the bath relaxation was taken to be fast (the Markovian limit) the population relaxed exponentially as in the rate equation; in the opposite limit some new effects arose, although they were rather small. In particular, the relaxation in the Poisson bath case showed small oscillations rather than strict exponential decay. While suggestive, these simulations are inadequate in two ways. The first is that the initial conditions were somewhat unrealistic, in that the molecule was assumed to be in its instantaneous excited state at t = 0 without any consideration of how it got there. In realistic situations, the excitation would take place through coupling to a radiation field and would not be instantaneous. The other problem with simulation is the sometimes rather long calculations involved in order to minimize statistical error. For these reasons, we have considered in our subsequent work the stochastic Liouville equation (SLE) approach of K ~ b o . ~This ' is a powerful technique for describing the relaxation of a quantum system coupled (with arbitrary strength) to a classical stochastic bath. Until now it has been applied only in a limited number of cases: by Freed% to ESR line shapes and by G r i g ~ l i nto i ~nonradiative ~ relaxation problems. The stochastic Liouville equation begins by describing the bath by a set of stochastic variables {A)whose probability distribution relaxes according to the equation
a
- P ( x , ~ )= r,P(X,t) at
where rxis the relaxation operator. For the Poisson bath considered earlier, rh is a 2 X 2 matrix, while for the Gaussian bath it is a Fokker-Planck operator. The density matrix a(X,t) then satisfies the stochastic Liouville equation3'
which shows clearly the two contributions to the relaxation from the quantum part (with the commutator) and the classical part. If a(X,t) is then expanded in eigenfunctions of rx
dht) =
c a,(t)lb,) n
then the set of density matrices satisfy coupled linear differential equations which may be solved numerically to obtain the population (or phase) relaxation. We applied this method first35to the two-level system simulated earlier. We treated a more realistic excitation situation, where a radiative coupling (through a short, intense pulse) was introduced between a ground state and one of the two levels. We verified that the simulation and SLE results coincided, as indeed they must since the SLE approach is exact. We observed much more pronounced oscillations in this case, indicating that the excitation process is very important and cannot be separated from the subsequent relaxation. The Poisson bath showed stronger oscillations than the Gaussian, suggesting that (37) R. Kubo, Adu. Chem. Phys., 15, 101 (1969); J . Phys. SOC.Jpn., 26, (Suppl.), 1 (1969). (38) J. H. Freed, G. V. Bruno, and C. F. Polnaszek, J . Phys. Chem., 75, 3385 (1971). (39) P. Grigolini, J. Chem. Phys., 74, 1517 (1981).
The Journal of Physical Chemi.Wy, Vol. 87, No. 76, 1983 3033
Feature Article
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SAME BATH INDEP.
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sitive to slow bath degrees of freedom like rotations. Since our calculations seem to show that non-Markovian effects are smaller for more realistic baths and excitation pulses, dramatic effects will be hard to see. However, such experiments are well worth pursuing.
Concluding Remarks A great deal of progress has been made in recent years in the field of vibrational relaxation. Experiments have advanced to the point where they are giving much more specific quantitative information (e.g., line shapes are being measured rather than simply widths, and state-to-state rate constants are being determined rather than simply overall energy relaxation rates). On the theoretical side there have been significant advances in the development of new formalisms. In another sense, however, the field is just beginning to open up. What is necessary in the future is to go beyond formalisms to a more detailed microscopic interpretation of the information provided by vibrational relaxation experiments. On the experimental side, it is necessary to carry out more systematic studies of series of related molecules or of a single molecule under a wide range of conditions; measurements of a single relaxation rate are virtually useless for microscopic interpretation. On the theoretical side, simulations with more accurate potential surfaces would be desirable, and more realistic modeling of corrleation functions must be attempted. We are reaching the point where much more detailed information about the interactions responsible for relaxation is becoming available. Vibrational phase and energy relaxation are properties that are very sensitive both to potential surfaces and to liquid-state dynamics. On the one hand, this might be discouraging to a theoretician since it suggests that quantitatively accurate calculations will only be possible for the simplest systems; on the other hand, such a sensitive probe has the capability of providing extensive new information about liquid-state structure and dynamics. Acknowledgment. I acknowledge all of my co-workers in this field for their many contributions to our joint research, especially R. Abbott, B. Bagchi, D. Levesque, S. Velsko, and J. -J. Weis. This research was supported by the Petroleum Research Fund, administered by the American Chemical Society, and by the National Science Foundation (Grant No. CHE 81-06068).
