Harmonic oscillator vibrational energy relaxation

Equation (13.39) implies that in the bilinear coupling, the vibrational energy relaxation rate for a quantum hannonic oscillator in a quantum harmonic bath is the same as that obtained from a fully classical calculation ( a classical harmonic oscillator in a classical harmonic bath ). In contrast, the semiclassical approximation (13.27) gives an error that diverges in the limit T 0. Again, this result is specific to the bilinear coupling model and fails in models where the rate is dominated by the nonlinear part of the impurity-host interaction. 

Z is proportional to the gas pressure, and, since Z1>0, the collision number for energy transfer, is constant for a particular transition, the actual value of fi is inversely proportional to the pressure. For convenience relaxation times are usually referred to a pressure of 1 atm. Equation (1) is an approximation, and requires modification to take into account the reversibility between quantum states 0 and 1. For example, the correct equation for vibrational relaxation of a simple harmonic oscillator of fundamental frequency, v, is... 

In this example the master equation formalism is appliedto the process of vibrational relaxation of a diatomic molecule represented by a quantum harmonic oscillator In a reduced approach we focus on the dynamics of just this oscillator, and in fact only on its energy. The relaxation described on this level is therefore a particular kind of random walk in the space of the energy levels of this oscillator. It should again be emphasized that this description is constructed in a phenomenological way, and should be regarded as a model. In the construction of such models one tries to build in all available information. In the present case the model relies on quantum mechanics in the weak interaction limit that yields the relevant transition matrix elements between harmonic oscillator levels, and on input from statistical mechanics that imposes a certain condition (detailed balance) on the transition rates. 

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. 

Fig. 8.2. The vibrational energy levels and the associated (umiormalized) linear harmonic oscillator wavefunctions. The Huang-Rhys parameter, S = Edihui, and thns S = u +. Ed is the reorganization (or relaxation) energy.

This example is one of the many relaxation processes attributed to the col-lisional vibrational-translational energy exchange. Without discussing other processes, we shall only mention those most important for the kinetics of nonequilibrium reactions vibrational relaxation of anharmonic oscillators in a heat bath, relaxation of one- and two-component mixture of harmonic oscillators with resonant and quasi-resonant energy exchange and relaxation of anharmonic oscillators [87, 164, 343, 460] (see Section IV. 15). 

Detailed studies show that the relaxation time T ib can be calculated from the frequency v found from the inflection point in the dispersion curve, or from the maximum of sound absorption ([192]). For molecules simulated by harmonic oscillators, Tyib is expressed via (Pi,o) by Eq. (8.45). Then, one can ultimately find the transition probability (Pio) or a related quantity Z ib = l/Pi,o which is the number of collisions needed for the deactivation of the first vibrational energy level. 

The validity of Eq. (15.11) even at the limit Na Nm means in fact that for harmonic oscillators relaxing in a heat bath there exists a closed equation for the mean energy. This is intimately connected with the linear dependence of transition probabilities on the vibrational quantum number (see Eq. (14.1)) and implies that the energy relaxation rate is independent of the initial distribution of harmonic oscillators over vibrational states. Still another peculiarity of this system is known the initial Boltzmann distribution corresponding to the vibrational temperature Tq =t= T relaxes to the equilibrium distribution via the set of the Boltzmann distributions with time-dependent temperatures. If (E ) is explicitly expressed by a time-dependent temperature, this process is again described by Eq. (15.11). 

We have seen that vibrational relaxation rates can be evaluated analytically for the simple model of a harmonic 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 of interest 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... 

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