Tunneling incoherent

Usually, if VR is fast enough, the H tunneling process can be described in terms of rate constants for the forward and the backward H-transfer. 

An exception to this rule is, however, the case of exchange of the nuclei of dihydrogen pairs bound to a transition metal center, where the exchange exhibits a barrier. This phenomenon will be sketched only briefly for further information the reader is referred to Chapter 21 and to a recent minireview [36]. 

The situation is illustrated for the gas phase in Fig. 6.1(e). The nuclear spins -characterized by arrows - play a decisive role. The lower turmel state is symmetric with respect to a permutation o and the upper state is antisym- 

As this chapter focuses on hydrogen transfers in liquids and solids, it will be assumed that the transfer constitutes a rate process which can be described in terms of rate constants, for which the usual rate theories can be applied, in particular those derived from transition state theory. 

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As argued in section 2.3, when the asymmetry e far exceeds A, phonons should easily destroy coherence, and relaxation should persist even in the tunneling regime. Such an incoherent tunneling, characterized by a rate constant, requires a change in the quantum numbers of the vibrations coupled to the reaction coordinate. In section 2.3 we derived the expression for the intradoublet relaxation rate with the assumption that only the one-phonon processes are relevant. 

In the genuine low-temperature chemical conversion, which implies the incoherent tunneling regime, the time dependence of the reactant and product concentrations is detected in one way or another. From these kinetic data the rate constant is inferred. An example of such a case is the important in biology tautomerization of free-base porphyrines (H2P) and phtalocyanins (H2PC), involving transfer of two hydrogen atoms between equivalent positions in the square formed by four N atoms inside a planar 16-member heterocycle (fig. 42). 

In the theory of deuteron spin-lattice relaxation we apply a simple model to describe the relaxation of the magnetizations T and (A+E), for symmetry species of four coupled deuterons in CD4 free rotators. Expressions are derived for their direct relaxation rate via the intra and external quadrupole couplings. The jump motion between the equilibrium positions averages the relaxation rate within the same symmetry species. Spin conversion transitions couple the relaxation of T and (A+E). This mixing is included in the calculations by reapplying the simple model under somewhat different conditions. The results compare favorably with the experimental data for the zeolites HY, NaA and NaMordenite [6] and NaY presented here. Incoherent tunnelling is believed to dominate the relaxation process at lowest temperatures as soon as CD4 molecules become localized. 

The presented approach loses its applicability when exchange becomes infrequent and CD4 localized. As in solids we have the dominating contribution from incoherent tunnelling to the relaxation at low temperatures. 

A discussion of the indirect dissipation mechanism is more pertinent to the present topic of incoherent tunneling reactions. In this mechanism, the reaction coordinate is coupled to one or several active modes that characterize the reaction complex. These modes are damped because of coupling to a continuous bath. The overall effect of active oscillators and bath may be represented by an effective spectral density 7eff(o>). For instance, in the case of one harmonic active oscillator with frequency friction coefficient 17, 7eff(w) is proportional to the imaginary part of its susceptibility and equals [Garg et al., 1985]... 

This increase in A with increasing n is simply due to shortening of the tunneling distance with increasing vibration amplitude 8ln, and it is equivalent to the effect of increasing temperature for the incoherent tunneling rate [Benderskii et al., 1992b],... 

In other crystals the splitting is too small for spectroscopic measurements, but the rate constants for incoherent tunneling have been found from relaxation measurements. The results obtained by Kapphan [1974] are listed in Table 9.1. The temperature dependences of the characteristic time scales for orientational relaxation of OH- dipoles in several different crystals are depicted in Figure 9.2. Below 5 K, the relaxation times are inversely proportional to T, but scale as T 4 at higher temperatures. 

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