Non-adiabatic limit

In the purely non-adiabatic limit the phase (5.52) coincides with that calculated in [203] and for very long flights (rt b,v" v) or high energies (.E e) it reduces to what can be obtained from the approximation of rectilinear trajectories. However, there is no need for these simplifications. The SCS method enables us to account for the adiabaticity of collisions and consider the curvature of the particle trajectories. The only demerit is that this curvature is not subjected to anisotropic interaction and is not affected by transitions in the rotational spectrum of the molecule. 

The difference between the two results is in the pre-exponential term. In the quantum mechanical treatments either vet= vaKe or vet = 0.5(tu + reyl depending on the model adopted. In the Marcus equation the pre-exponential term is ZKee p( w lRT) and the time dependence is introduced through the collision frequency, Z. In any case, in the non-adiabatic limit, ve < v , and kobs is given by equation (41). In the adiabatic limit, ve > vn and kobs is given by equation (42), with vn in the range 10n —1013 s-1. 

The integral on the right-hand side of Eq. (11.98) diverges unless (w) falls off faster than 1 /u at infinity. Thus, the inequality established above might not be valid close to the adiabatic limit. In the adiabatic limit where (w) = , we use instead Eqs (11.90) and (11.92) to determine the difference between the transmission coefficients in the adiabatic (Kramers) limit and in the non-adiabatic limit. We find... 

In the non-adiabatic limit, the rate constant for ET is determined in the lowest (i.e., the second) order in /. Therefore, the rate constant is given by Fermi s Golden Rule,... 

The non-adiabatic limit is justified when the transfer integral J for electron transfer is sufficiently small. On the other hand, when J is large enough, we approach the... 

In the adiabatic limit, the reaction can be described by the adiabatic potentials which are composed of the upper and the lower branch and whose energy separation is given by 27 at Q = Qc, as shown in Figure 1. The relative positions of the branches are quite different for the normal and the inverted cases the positions of the diabatic potentials appropriate for the non-adiabatic limit, shown in Figure 5, change to those shown in Figure 6 in the adiabatic limit for both the normal and the inverted case. In the normal case, the reactant-state potential corresponds to the left-hand well in the lower branch of the adiabatic potential. In the inverted case, on the other hand, it corresponds to the upper branch, which consists of a single well. 

On the basis of the Landau-Zener formula (Eq. 59), the parameter for discriminating between the adiabatic and the non-adiabatic limits of ET should be given by... 

This dimensionless parameter has been called the adiabaticity parameter. When y 1, the adiabatic limit is realized, while the non-adiabatic limit is realized in the limit of y 1. As the average energy quantum hw of phonons approaches zero, y of Eq. 61 diverges, and only the adiabatic limit becomes justifiable. Bom and Oppen-... 

Eq. 73 which gives the rate constant in the non-adiabatic limit can be expressed as... 

The rate-constant formula (Eq. 20) in the non-adiabatic limit is applicable irrespective of whether the phonon Hamiltonians Hr and Hn for the reactant and the... 

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