Relaxation, vibrational stochastic approach

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... 

Following the solid-state approach, equations have been derived [8,9] also for the electron spin relaxation of 5 = V2 ions in solution determined by the aforementioned processes. Instead of phonons, collisions with solvent should be taken into consideration, whose correlation time is usually in the range 10"11 to 10 12 s. However, there is no satisfactory theory that unifies relaxation in the solid state and in solution. The reason for this is that the solid state theory was developed for low temperatures, while solution theories were developed for room temperature. The phonon description is a powerful one when phonons are few. By increasing temperature, the treatment becomes cumbersome, and it is more convenient to use stochastic theory (see Section 3.2) instead of analyzing the countless vibrational transitions that become active. 

It is important to point out that this does not imply that Markovian stochastic equations cannot be used in descriptions of condensed phase molecular processes. On the contrary, such equations are often applied successfully. The recipe for a successful application is to be aware of what can and what cannot be described with such approach. Recall that stochastic dynamics emerge when seeking coarsegrained or reduced descriptions of physical processes. The message from the timescales comparison made above is that Markovian descriptions are valid for molecular processes that are slow relative to environmental relaxation rates. Thus, with Markovian equations of motion we cannot describe molecular nuclear motions in detail, because vibrational periods (10 " s) are short relative to environmental relaxation rates, but we should be able to describe vibrational relaxation processes that are often much slower, as is shown in Section 8.3.3. 

Since the direct simulation of vibrational relaxation in condensed phases is clearly a difficult and lengthy procedure for molecules with realistic vibrational frequencies, Shugard et al. ° proposed an alternative approach based on work of Adelman and Doll and applied it to relaxation of diatomic impurities in solid matrices. The motion of atoms near the impurity was simulated directly and the effect of more distant atoms was taken into account through a stochastic force, which was constructed from the phonon spectrum of the solid. This method still requires that the relaxation time not be too long compared to the vibrational period (i.e., that the vibrational frequency not be too high) but the calculation is much faster than a full molecular dynamics simulation since only a few degrees... 

Vibrational Relaxation. Stochastic processes, including vibrational relaxation in condensed media, have been considered from a theoretical standpoint in an extensive review,502 and a further review has considered measurement of such processes also.503 Models have been presented for vibrational relaxation in diatomic liquids 504 and in condensed media,505 using a master-equation approach. An extensive development of quantum ergodic theory for relaxation processes has been published,506 and quantum resonance effects in electronic to vibrational energy transfer have been considered.507 A paper has also considered the coupling between vibrational relaxation and molecular electronic transitions.508 A theory has also been outlined for the time-resolved electronic absorption spectrum of a molecule undergoing collisional vibrational relaxation.509... 

Nitzan, A., Shugard, M., Tully, J.G. Stochastic classical trajectory approach to relaxation phenomena. 11. Vibrational relaxation of impurity molecules in Debye solids, J. Chem. Phys. 1978,69, 2525. 

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