Tunneling thermally activated

Figure 6.12 Increase in the reaction quantum yield owing to nuclear tunnelling. Thermal activation takes the system over the barrier and across a funnel from the product surface to the ground-state reactant surface, which may take some system back to the reactants. In contrast, nuclear tunnelling places the reactive systems in a region of the product surface beyond the funnel with the reactants, and the systems cannot return to their original sate.

This relation may be interpreted as the mean-square amplitude of a quantum harmonic oscillator 3 o ) = 2mco) h coth( /iLorentzian distribution of the system s normal modes. In the absence of friction (2.27) describes thermally activated as well as tunneling processes when < 1, or fhcoo > 1, respectively. At first glance it may seem surprising... 

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]. 

Figure 23. This caricature demonstrates the predicted phenomena of energy level crossing in domains whose energy bias is comparable or larger than the vibronic frequency of the domain wall distortions. The vertical axis is the energy measured from the bottom state the horizontal axis denotes temperature. The diagonal da ed line denotes roughly the thermal energies. A tunneling center that would become thermally active at some temperature Tq will not possess ripplons whose frequency is less than To.

Quantum tunnelling in chemical reactions can be visualised in terms of a reaction coordinate diagram (Figure 2.4). As we have seen, classical transitions are achieved by thermal activation - nuclear (i.e. atomic position) displacement along the R curve distorts the geometry so that the... 

Figure 2.4. Reaction coordinate diagram for a simple chemical reaction. The reactant A is converted to product B. The R curve represents the potential energy surface of the reactant and the P curve the potential energy surface of the product. Thermal activation leads to an over-the-barrier process at transition state X. The vibrational states have been shown for the reactant A. As temperature increases, the higher energy vibrational states are occupied leading to increased penetration of the P curve below the classical transition state, and therefore increased tunnelling probability.

In contrast to classical overbarrier reactions, QMT can occur from the lowest vibrational quantum levels without thermal activation. Under these circumstances at the lowest temperatures, the degree of tunneling, and hence the reaction rate, is independent of temperature. At some point as the temperature is raised, higher vibrational levels become populated. As illustrated in Figure 10.1, the effective barrier is narrower for excited vibrational levels, and hence tunneling becomes more facile, leading to an increase in rate. Finally, as temperatures are raised further, classical reaction begins to compete, and usually dominates at room temperature (but, not always). 

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