Evaluation of vibrational relaxation rates

In this Section we apply the general formalism developed in Section 13.3 together with the interaction models discussed in Section 13.2 in order to derive explicit expressions for the vibrational energy relaxation rate. Our aim is to identify the molecular and solvent factors that determine the rate. We will start by analyzing the implications of a linear coupling model, than move on to study more realistic nonlinear interactions. 

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The validity of the conclusions was checked by evaluation of the dissociation rate from the A state in the gas phase in which the low-friction limit of Eq. (4.3) applies. The agreement with exjjeriment was extremely good, indicating that the use of a constant friction is reasonable for these shallow potentials. Vibrational relaxation of the 12 molecule in the X state will be much slower... 

In a stochastic approach the frequency-depiendent friction appears in the definition of the energy dependence of the relaxation rate P(E), defined by Eq. (4.12), and is evaluated for a Morse potential by Eq. (4.14). In this section the applicability of these relationships and the friction kernel B(d)( )) of Eq. (3.27) is tested in a variety of approaches for the case of u = 1, a diatomic. The use of frequency-dependent friction in the evaluation of D(E) for a system with many degrees of freedom is an area of ongoing activity. While many of the features of a stochastic approach to vibrational relaxation are found in inelastic scattering theories or master equation kernels, it is the characteristic of... 

We have seen that vibrational relaxation rates can be evaluated analytically for the simple model of a hannonic oscillator coupled linearly to a harmonic bath. Such model may represent a reasonable approximation to physical reality if the frequency of the oscillator under study, that is the mode that can be excited and monitored, is well embedded within the spectrum of bath modes. However, many processes ofinterest involve molecular vibrations whose frequencies are higherthan the solvent Debye frequency. In this case the linear coupling rate (13.35) vanishes, reflecting the fact that in a linear coupling model relaxation cannot take place in the absence of modes that can absorb the dissipated energy. The harmonic Hamiltonian... 

Coming back to the timescale issue, it is clear that direct observation of signals such as shown in Fig. 13.2 cannot be achieved with numerical simulations. Fortunately an alternative approach is suggested by Eq. (13.26), which provides a way to compute the vibrational relaxation rate directly. This calculation involves the autocorrelation function of the force exerted by the solvent atoms on the frozen oscillator coordinate. Because such correlation functions decay to zero relatively fast (on timescales in the range of pico to nano seconds depending on temperature), its numerical evaluation requires much shorter simulations. Several points should be noted ... 

Vibrational relaxation time x, evaluated from the linewidth of the isotropic spectrum is plotted in Fig. 4 together with that of the liquid. The relaxation time for the supercritical fluid is larger than that in the liquid. The x, much depends upon density in the supercritical fluid, whereas it does not in the liquid, showing a good contrast between them. The difference in the behavior can be understood in terms of the collision rate of the molecules and inhomogeneity with respect to the environment of the oscillators. 

Keeping the above considerations in mind it is relatively easy, starting from an a priori correlation function approach, to derive successive approximate expressions for binary and nonbinary terms in vibrational relaxation in liquids, to define the limits of validity of binary dynamics, and to obtain easily evaluated analytical expressions for energy relaxation rates that can be directly compared to experiment. 

Thus the activation volume AV for the rate constant kp of an individual ES reaction pathway can be evaluated if the pressure dependencies of the photoreaction quantum yield, of intersystem crossing and of the ES lifetime can be separately determined. However, such parameterization becomes considerably more complex if several different excited states are involved or if a fraction of the photosubstitution products are formed from states that are not vibrationally relaxed with respect to the medium. Currently, parameterization of pressure effects on photosubstitutions has been attempted for a limited number of metal complexes. These include certain rhodium(III) and chromium(III) amine complexes and some Group VI metal carbonyls, which will be summarized here. 

Note added in proof In view of the failure of the harmonic oscillator model to account for the observed rate of activation in unimolecular dissociation reactions (the dissociation lag problem) these calculations have been repeated for a Morse anharmonic oscillator with transition between nearest and next-nearest neighbor levels [S. K. Kim, /. Chem. Phys. (to be published)]. The numerical evaluation of the analytical results obtained by Kim has not yet been carried out. From the results obtained by us and our co-workers [Barley, Montroll, Rubin, and Shuler, /. Chem. Phys. in press)] on the relaxation of vibrational nonequilibrium distributions of a system of Morse anharmonic oscillators it seems clear, however, that the anharmonic oscillator model with weak interactions (i.e., adiabatic perturbation type matrix elements) does not constitute much of an improvement on the harmonic oscillator model in giving the observed rates of activation. The answer to tliis problem would seem to lie in a recalculation of the collisional matrix elements for translational-vibrational energy exchange which takes account of the strong interactions in highly energetic collisions which can lead to direct dissociation. 

The main applications covered in this study are the accurate determination of rotation and rotation-vibration molecular energies the determination of the molecular geometry of simple molecules the evaluation of force field and of the vibration- and rotation-vibration interactions the measurement of pressure broadening and pressure shift of the spectral lines the determination of electric dipole moments via laser-Stark spectroscopy the studies of intramolecular dynamics the calculation of rate constants, equilibrium constants and other thermodynamic data the evaluation of relaxation times. 

The electron-spin time-correlation functions of Eq. (56) were evaluated numerically by constructing an ensemble of trajectories containing the time dependence of the spin operators and spatial functions, in a manner independent of the validity of the Redfield limit for the rotational modulation of the static ZFS. Before inserting thus obtained electron-spin time-correlation functions into an equation closely related to Eq. (38), Abernathy and Sharp also discussed the effect of distortional/vibrational processes on the electron spin relaxation. They suggested that the electron spin relaxation could be described in terms of simple exponential decay rate constant Ts, expressed as a sum of a rotational and a distortional contribution ... 

These relaxation times correspond to rates which are about 106 slower than the thermal vibrational frequency of 6 x 1012 sec 1 (kBT/h) obtained from transition state theory. The question arises how much, if any, of this free energy of activation barrier is due to the spin-forbidden nature of the AS = 2 transition. This question is equivalent to evaluating the transmission coefficient, k, that is, to assess quantitatively whether the process is adiabatic or nonadiabatic. 

It is not possible to evaluate k directly, for it appears with the entropy of activation in the temperature-independent part of the rate constant. An estimate of k requires an extrathermodynamic assumption. In two cases of iron(II) spin equilibria examined by ultrasonic relaxation the temperature dependence of the rates was precisely determined. If the assumption is made that all of the entropy of activation is due to a small value of k, minimum values of 10-3 and 10-4 are obtained. Because there is an increase in entropy in the transition from the low-spin to the high-spin states, this assumption is equivalent to assuming that the transition state resembles the high-spin state. There is now evidence that this is not the case. Volumes of activation indicate that the transition state lies about midway between the two spin states. This is a more chemically reasonable and likely situation than the limiting assumption used to evaluate k. In this case the observed entropy of activation includes some chemical contributions which arise from increased solvation and decreased vibrational partition functions as the high-spin state is compressed to the transition state. Consequently, the minimum value of k is increased and is unlikely to be less than about 10 2. 

The procedure described here is an example for combining theory (that relates rates and currents to time correlation functions) with numerical simulations to provide a practical tool for rate evaluation. Note that this calculation assumes that the process under study is indeed a simple rate process characterized by a single rate. For example, this level of the theory cannot account for the nonexponential relaxation of the v = 10 vibrational level of O2 in Argon matrix as observed in Fig. 13.2. 

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