The vibrational relaxation rate

The model Hamiltonian (13,6)—(13.8) and (13.13) and (13.14) can be used as a starting point within classical or quantum mechanics. For most diatomic molecules of interest hu ksT, which implies that our treatment must be quantum mechanical. In this case all dynamical variables in Eqs (13,6)—(13.8) and (13,13)—(13.14) become operators. 

Q is the bath partition function, Q =. Following the procedure that leads 

Here the matrix elements in the molecular subspace, for example (. sb)// are operators in the bath space, (.Wsb) .t(Z) = exp(z BZ/ )( SB)z7exp(—iHBt/K and (... )t = Tr[e B. .. ]/Tr[e - B]j where Tr denotes a trace over the eigenstates of Hb At this point expression (13.16) is general. In our case /7s b = Eq where F is an operator in the bath sub-space and q is the coordinate operator of our oscillator. This implies that (77sb)z./ = QifF and consequently 

Note that, using Eq. (12.45) (or applying Eq. (6.75)) we find that this result satisfies detailed balance, that is, kf i = Furthermore, this res- 

For a harmonic oscillator, a transition between levels n) and n — 1) involves the matrix element p = h/2ma))n, so that Eq. (13.17) becomes 

Q is the bath partition function, Q = J2a Following the procedure that leads from Eq. (12.34) or (12.37) to (12.44) we can recast this rate expression in the time correlation form 

Here the matrix elements in the molecular subspace, for example (. sb)i/ are operators in the bath space, (HsB)if(t) = exp(iHBt/h)(HsB)if exp(-iHBt/h) and 

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From Eqs (7.5) and (7.6) we can deduce that the pure dephasing rate is Yio( ) = 0.2ps 1 and the vibrational relaxation takes place in the timescale of 0.5 ps for the 100-cm 1 mode. More results related to vibrational relaxation have been reported by Martin group [1-5], In this chapter we choose 0.3 ps for the 100-cm-1 mode, and the vibrational relaxation rates for other modes are scaled with their vibrational frequencies. 

In suggesting an increased effort on the experimental study of reaction rates, it is to be hoped that the systems studied will be those whose properties are rather better defined than many have been. By far and away more information is known about the rate of reactions of the solvated electron in various solvents from hydrocarbons to water. Yet of all reactants, few can be so poorly understood. The radius and solvent structure are certainly not well known, and even its energetics are imprecisely known. The mobility and importance of long-range electron transfer are not always well characterised, either. Iodine atom recombination is probably the next most frequently studied reaction. Not only are the excited states and electronic relaxation processes of iodine molecules complex [266, 293], but also the vibrational relaxation rate of vibrationally excited recombined iodine molecules may be at least as slow as the recombination rate [57], Again, the iodine atom recombination process is hardly ideal. 

Medium effects on the linewidths of electronic transitions are quite pronounced in certain instances. For example, it is quite general that the 0-0 band of an electronic transition will be the sharpest line in the vibronic spectrum of that state. The additional line broadening over that of the 0-0 band usually amounts to 1-5 cm-1, and increases in certain progressions with increasing vibrational separation from the origin. These line broadenings must gauge the vibrational relaxation rates, which are clearly much smaller than the usual rates of electronic relaxation. The assistance... 

Here, y 0 represents the vibrational relaxation rate constant for the transition v — 1 -> v — 0 and is given by... 

With the same interaction model used in Eq. (130), the vibrational relaxation rate constants associated with the vibrational population transitions bv + 1 -+bv and bv bv— 1 are given by... 

Since the vibrational relaxation rate is evidently some 2 orders of magnitude smaller than the OH vibrational frequency, it was reasonable... 

Equation (6.70) results from Eq. (6.60) together with the definition of the vibrational relaxation rate kyg E) = dE/dt. 

As in the weak coupling limit of the rate (12.55), analyzed in Section 12.5.3, the weak couphng limit (13.66) of the vibrational relaxation rate also has the characteristic form of an energy-gap law It decreases exponentially, like... 

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 ... 

This result is akin to Eq. (13.22), which relates the vibrational relaxation rate of a single harmonic oscillator of frequency m to the Fourier transform of the force autocorrelation function at that frequency. In the Markovian limit, where (cf. Eq. (8.60)) Z(Z) = 2y<5(Z), we recover Eq. (14.78). 

A similar behavior has been observed under the optical excitation of the = 1 level of the CO state perturbed by the isoenergetic level of the triplet state. Again, a large delay of the emission from the u = 0 level with respect to the u = 1 level decay must be considered as due to the transit through long-lived triplet states. Since this delay may be fitted by the vibrational relaxation rate within the triplet manifold (directly measured in independent experiments), a three-step relaxation path ... 

The isolated binary collision (IBC) model for vibrational relaxation in liquids was developed by Herzfeld, Litovitz, and Madigosky. Its fundamental assumption is that the vibrational relaxation rate from level / to level y( //) given by the product of two factors... 

The weak coupling theory prediction for the vibrational relaxation rate between two levels i and j was shown in Section II.B to have the form... 

This is clearly the dramatic temperature dependence of the vibrational relaxation rates predicted by theories of multiphonon relaxation. Apparently, in this case one deals with a true multiphonon relaxation process. It is reasonable to expect that also in many other heavier diatomics and in polyatomic molecules with small rotational constants or high barriers to free rotation the multiphonon relaxation mechanism will dominate. 

A more interesting point was made by Bader and Berne, who noted that the vibrational relaxation rate of a classical oscillator in a classical harmonic bath is identical to that of a quantum oscillator in a quantum harmonic bath [71]. On the other hand, when the relaxation of the quantum system is calculated using the corrected correlation function of the classical bath [Eq. (31)], the predicted rate is slower by a factor of j/3h(i) coth(/3h(o/2), which can be quite substantial for high-frequency solutes. The conclusions of a number of recent studies were shown to be strongly affected by this inconsistency [42,43,72]. Quantizing the solvent by mapping the classical correlation functions onto a quantum harmonic bath corrects the discrepancy. 

Similar to IC, the vibrational relaxation rate formula can be expressed as... 

Another system aniline C6H5NH2 has also been studied. We found that the vibrational relaxation rates of symmetric and antisymmetric stretching modes of NH2 take in the ps range in good agreement with experiment. 

Measurements of relaxation times fall broadly into two classes, those which monitor the populations of some chosen states, and those which measure in some way the impedance of the system to the propagation of a thermal disturbance many laser experiments fall into the first class, whereas ultrasonic dispersion or shock-tube measurements fall into the second. Although artefacts can occur if unsuitable population v. time profiles are used [76.P3], there is, in general, no real difficulty in using equation (2.14) to obtain the vibrational relaxation rate we need not discuss this point further at the moment. Problems may well arise, though, in the determination of rotational relaxation rates in this way, as I will show. 

Another conclusion we can draw from equation (3.9) is that the steady reactant distribution retains some memory of the initial distribution, but we should not be misled into reading too much significance into it. In a typical static reaction experiment, where the vibrational relaxation rate constant might be 10 s and the rate constant the starting... 

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