Thermal average rate constant

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

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

Single-Vibronic-Level and Thermal Average Rate Constant. 142... 

In the above discussion the single-vibronic-level and thermal average rate constants have only been calculated using the displaced oscillator surfaces. 

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... 

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. 

Again in the hi -pressure limit one has a thermally averaged rate constant analogous to equation (37). 

The reaction path Hamiltonian also provides a very useful framework for the rigorous calculation of the Boltzmann (i.e., thermally averaged) rate constant for a chemical reaction using the path integral methods described by Miller, Schwartz, and Tromp. In that paper it is shown that the rate constant can be expressed as the time integral of a flux-flux autocorrelation function... 

Given the cumulative reaction probabilities, we can compute thermally averaged rate constants. 

The integration variable E in equation (26) is effectively E, ,. The condition for the validity of these equations for a thermally averaged rate constant kba(T) is the existence of a well defined Maxwell-Boltzmann distribution of velocities of collision partners or relative collision energies (E — a) at temperature T, which remains unperturbed by the reaction process. If, furthermore the internal state distributions of the reactants also remain at an unperturbed Boltzmann distribution at temperature T, one finds a thermal rate constant for complex formation (or capture ) given by equation (28) ... 

Fig. 5. Thermally averaged rate constant for F + H2 with products summed over.all final states. Experimental curves are C, Igoshin et aZ, reference 44 D, Bulatov et aZ., reference 46 E, Wurzberg and Houston, reference 43 F, Heidner et aZ., reference 45. Theoretical curves are M5, Muckerman, reference 37 JA, Jaffe and Anderson, reference 47 G and H, Garrett et aZ., reference 48. The results of Jaffe and Anderson are QCT results using a surface developed by them. Curve G is a generalized transition state result using M5, while curve H is a generalized transition state result which also takes approximate account of tunneling.

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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