Crossover temperature tunneling reactions

We note that it is the lower crossover temperature Tcl that is usually measured in experiments. The above simple analysis shows that this temperature is determined by the intermolecular vibrational frequencies rather than by the properties of the reaction complex or by the static barrier. It is not surprising then, that in most solid-state reactions the observed value of Tcl is of order of the Debye temperature of the crystal. Although the result (2.81) was obtained using the approximation w, exponential term turns out to be exact for arbitrary values of the two frequencies. It is instructive to compare (2.81) with (2.27) and see that friction slows tunneling down, while the q mode promotes it. 

The potential (6.37) corresponds with the previously discussed projection of the three-dimensional PES V(p,p2,p3) onto the proton coordinate plane (pi,p3), shown in Figure 6.20b. As pointed out by Miller [1983], the bifurcation of reaction path and resulting existence of more than one transition state is a rather common event. This implies that at least one transverse vibration, q in the case at hand, turns into a double-well potential. The instanton analysis of the PES (6.37) was carried out by Benderskii et al. [1991b], The existence of the onedimensional optimum trajectory with q = 0, corresponding to the concerted transfer, is evident. On the other hand, it is clear that in the classical regime, T > Tcl (Tc] is the crossover temperature for stepwise transfer), the transition should be stepwise and occur through one of the saddle points. Therefore, there may exist another characteristic temperature, Tc2, above which there exists two other two-dimensional tunneling paths with smaller action than that of the one-dimensional instanton. It is these trajectories that collapse to the saddle points at T = Tcl. The existence of the second crossover temperature Tc2 for two-proton transfer was noted by Dakhnovskii and Semenov [1989]. 

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