The thermally averaged rate

We now proceed with the thermally averaged 2 1 rate, Eq. (12.34), rewritten in 

We will evaluate this rate for the two models considered above. In the model of (12.28a) 

We have thus found that the Ari. ( 2 is given by a Fourier transform of a quantum time correlation fimction computed at the energy spacing that characterizes the 

Using the same procedure as above, show that this leads to 

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In this case, the thermal average rate constant given by Eq. (3.20) becomes... 

To simplify the expression of the thermal average rate W, f given by Eq. (3.32), we shall assume that both potential energy surfaces of the two excited vibronic manifolds of the DA system consist of a collection of harmonic oscillators, that is,... 

The thermally averaged rate constant is the velocity averaged and can be determined by averaging the velocity... 

There are three types of rate constants appearing in the previous subsection, i.e., WbUiav, the state-to-state rate constant, Wav (or Wbu), the single-vibronic level rate constant, and Wa b (or Wb a), the thermal average rate constant. In the BOA approximation,... 

Eq. (60) indicates that Wa >b(E) can be obtained from the thermal average rate constant with temperature [1 being determined for Eq. (61). 

Neglecting the super-exchange contribution the thermal average rate constant for the singlet-singlet transfer can be expressed as... 

It is also interesting to consider the thermally averaged rate constant k T), which can be obtained from... 

The quantum flux-flux autocorrelation formalism, developed by Miller, Schwartz, and Tromp [78] and by Yamamoto [79], represents an exact quantum mechanical expression for a chemical reaction rate constant. According to the flux-flux autocorrelation formalism, the thermally averaged rate constant k T) is given by... 

Finally, the thermally averaged rate coefficients of the QCT and GCE models are compared with rates computed using recommended Arrhenius parameters and experimental data " " " in Fig. 16. At all temperatures, the agreement between the QCT and GCE results is excellent. Note that the QCT analysis predicts a significantly higher rate than that of Park at temperatures above 5000 K. 

The assumption that thermal relaxation in the bath is fast relative to the timescale determined by this rate (see statement (2) of Problem 12.1) makes it possible to define also the thermally averaged rate to go from state 2 to state 1... 

Problem 12.1. Refer to the general discussions of Sections 9.1 and 10.3.2 in order to explain the following two statements (1) Eq. (12.33) for the partial rates and hence Eq. (12.34) for the thermally averaged rate are valid rate expressions only if the partial rates (12.33) are larger than the inverse of /ipi( 2,v) = /iZ2v ( 2,v — i,v ). (2) The thermally averaged rate, of Eq. (12.34), is meaningful only if it is much smaller than the rate of thermal relaxation between the levels v of the initial 2 manifold. 

To describe these approximations, we provide the treatment of dynamics within a correlation function formalism (Yamamoto 1960 Miller et al. 1983). Let the reaction-path coordinate be denoted by u and the other coordinates by U. Define the saddle point on the path as u = 0. Then the thermally averaged rate constant for transition from m < 0 to u > 0 is given by... 

Here m is the reduced mass of the collision system. Clearly this formula is independent of energy e and so the thermal-averaged rate constant is also given by (9). For the Langevin case the rate constant of an ion-molecule reaction just depends on the ratio a//w. 

The dynamics of molecular motion must be treated quantum mechanically if one is to have a quantitative description of chemical reactions. Since transition state theory is such a good approximation in classical mechanics—particularly at the lower energies that are most important for determining the thermally averaged rate k(T)—one would like to quantize it. Unfortunately there does not seem to be a way to quantize the basic transition state idea without also introducing other approximations. The heuristic argument goes as follows. 

Golden-rule mANsirioN rates 12.4.2 The thermally averaged rate... 

With these expressions we have given a complete set of matrix elements to evaluate the thermally averaged rate constants for the radiationless decay between states s and I [138-143] in successive order of approximation with respect to X. Following the concept of Section 3.3, we have in second order... 

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