From Thermal Activation to Tunneling

Although the inadequacies of the one-dimensional model [Goldanskii, 1959, 1979] are well understood by now, we begin our discussion with the simplest one-dimensional version of theory, because it will permit us to elucidate some important features of tunneling reactions inherent in more realistic approaches. The one-dimensional model relies on the following assumptions  

The reaction coordinate is selected from the total set of PES coordinates. 

Tunneling and over-barrier transitions proceed along the same coordinate. 

The rate constant is a statistical average of the reactive flux from the initial to the final state  

According to (2.6) both the apparent activation energy a and the apparent prefactor ka = k0exp(-2S(Ea)/h) decrease with decreasing temperature. As T— 0, the activation energy a— 0 and r(Ea)—but the rate constant k given by (2.6) approaches a finite value kc. As temperature increases, the vibrational period t decreases. It is evident, though, that it cannot be smaller than 2tt (o, where 

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Figure 7. Diagrams of nuclear configuration for H abstractions and cycloadditions to olefins, showing the possible intersections,

Fig. 16.10 The electron transfer time (inverse rate) in a chemically modified photoreaction center of bacteriochlorophyll, showing a crossover from thermally activated sequential hopping behavior at high temperature to a superexchange tunneling behavior at low temperature. (Open circles are experimental data from M. E. Michel-Beyerle et al., unpublished fall and dashed lines are theoretical fits from the articles by M. Bixon and J. Jortner cited at the end of this chapter.)...

The diffusion coefficient corresponding to the measured values of /ch (D = kn/4nRn, is the reaction diameter, supposed to be equal to 2 A) equals 2.7 x 10 cm s at 4.2K and 1.9K. The self-diffusion in H2 crystals at 11-14 K is thermally activated with = 0.4 kcal/mol [Weinhaus and Meyer 1972]. At T < 11 K self-diffusion in the H2 crystal involves tunneling of a molecule from the lattice node to the vacancy, formation of the latter requiring 0.22 kcal/mol [Silvera 1980], so that the Arrhenius behavior is preserved. Were the mechanism of diffusion of the H atom the same, the diffusion coefficient at 1.9 K would be ten orders smaller than that at 4.2 K, while the measured values coincide. The diffusion coefficient of the D atoms in the D2 crystal is also the same for 1.9 and 4.2 K. It is 4 orders of magnitude smaller (3 x 10 cm /s) than the diffusion coefficient for H in H2 [Lee et al. 1987]. 

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