Relaxation, vibrational models

If the system under consideration possesses non-adiabatic electronic couplings within the excited-state vibronic manifold, the latter approach no longer is applicable. Recently, we have developed a simple model which allows for the explicit calculation of RF s for electronically nonadiabatic systems coupled to a heat bath [2]. The model is based on a phenomenological dissipation ansatz which describes the major bath-induced relaxation processes excited-state population decay, optical dephasing, and vibrational relaxation. The model has been applied for the calculation of the time and frequency gated spontaneous emission spectra for model nonadiabatic electron-transfer systems. The predictions of the model have been tested against more accurate calculations performed within the Redfield formalism [2]. It is natural, therefore, to extend this... 

Here, p = AEq o/ eff is the dimensionless energy gap between the upper state and the closest lower-energy state in units of the effective vibrational energy, Veff (cm ). C is the electronic factor, and S is the Huang-Rhys dimensionless excited-state distortion parameter in units of vibrational quanta v ff. As shown in Eq. (2), /c ,p is strongly dependent onp. Additionally, for a given reduced energy gap p, the introduction of even small excited-state distortions, S, can rapidly enhance the radiationless multiphonon relaxation rate such that this dominates the total 0 K relaxation. This model is easily extended to elevated temperatures, where substantial increases in may be observed [7,8]. 

The careful analysis of the experimental data within the framework of the developed theories of interfacial electron transfer in a nanoscale electrochemical metal/redox molecule/metal configuration led to the conclusion that the experimentally observed enhanced tunnehng current fenh (Fig- 33D) could be represented best by a two-step electron transfer process accompanied with partial vibrational relaxation. This model was developed by Kuznetsov and Ulstrup and is represented by the following theoretical formahsm [255,261,262,300]... 

The basic concepts o hich the relaxation-localization model is based are simple. The molecular character of the solid state leads to weak interactions between the molecular entities and hence to a high degree of disorder. Contributions to the disorder are both static (e.g., local variations in composition and/or structure) and dynamic (e.g., thermally induced vibrations) in nature. The static disorder localizes both injected charges as molecular ions and injected excitations as molecular excitons. Once localized, these entities interact strongly with the (dynamic) charges which they induce in the surrounding dielectric medium. The induced charges in turn can be described... 

In this MCT model, all the dynamic processes are intrinsically included and coupled. The fast vibration, the slow structural relaxation, and the intermediate decay appear as solutions of the equations of motion. The vibrational dynamics is characterized by two frequency parameters. Cl and w, of about 1 THz, in reasonable agreement with the mean firequeney of the vibrational modes found in previous studies. The crossover from the fast to the slow processes correctly describes the intermediate relaxation. This model does not require extra over-damped vibrational modes, as required by the previous interpretations. According to this interpretation, the intermediate relaxation is simply the merging of the local vibrational dynamies into the slow collective diffusive process. This merging is not trivial, as it has already been proved in glass-former liquids, because the two processes are strongly coupled. 

Hill J R ef a/1995 Vibrational relaxation of oarbon monoxide in model heme oompounds 6-ooordinate metalloporphyrins (M = Fe, Ru, Os) Chem. Phys. Lett. 224 218-23... 

The purpose of these comparisons is simply to point out how complete the parallel is between the Rouse molecular model and the mechanical models we discussed earlier. While the summations in the stress relaxation and creep expressions were included to give better agreement with experiment, the summations in the Rouse theory arise naturally from a consideration of different modes of vibration. It should be noted that all of these modes are overtones of the same fundamental and do not arise from considering different relaxation processes. As we have noted before, different types of encumbrance have different effects on the displacement of the molecules. The mechanical models correct for this in a way the simple Rouse model does not. Allowing for more than one value of f, along the lines of Example 3.7, is one of the ways the Rouse theory has been modified to generate two sets of Tp values. The results of this development are comparable to summing multiple effects in the mechanical models. In all cases the more elaborate expressions describe experimental results better. 

By changing from the simplest to larger aliphatic and cyclic ketones, structural factors may be introduced which favor alternative unimolecular primary photoprocesses or provide pathways to products not available to the simple model compound. In addition, both the increase in molecular size and irradiation in solution facilitate rapid vibrational relaxation of the electronically excited reactant as well as the primary products to thermally equilibrated species. In this way the course of primary and secondary reactions will also become increasingly structure-selective. In a,a -unsym-metrically substituted ketones, the more substituted bond undergoes a-cleavage preferentially. 

Valiev-Ivanov model 219, 275 vibrational broadening 123 vibrational dephasing 111, 113-15, 123 vibrational relaxation, and angular momentum relaxation 92 vibrational transition, adiabatic dephasing 92... 

Often the electronic spin states are not stationary with respect to the Mossbauer time scale but fluctuate and show transitions due to coupling to the vibrational states of the chemical environment (the lattice vibrations or phonons). The rate l/Tj of this spin-lattice relaxation depends among other variables on temperature and energy splitting (see also Appendix H). Alternatively, spin transitions can be caused by spin-spin interactions with rates 1/T2 that depend on the distance between the paramagnetic centers. In densely packed solids of inorganic compounds or concentrated solutions, the spin-spin relaxation may dominate the total spin relaxation 1/r = l/Ti + 1/+2 [104]. Whenever the relaxation time is comparable to the nuclear Larmor frequency S)A/h) or the rate of the nuclear decay ( 10 s ), the stationary solutions above do not apply and a dynamic model has to be invoked... 

From the above discussion, we can see that vibrational relaxation plays a very important role in ultrafast phenomena. In this appendix we shall present a model that can describe vibrational relaxation and dephasing (especially pure dephasing). Suppose that... 

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