Adiabatic process, transmission coefficient

To summarize, in this article we have discussed some aspects of a semiclassical electron-transfer model (13) in which quantum-mechanical effects associated with the inner-sphere are allowed for through a nuclear tunneling factor, and electronic factors are incorporated through an electronic transmission coefficient or adiabaticity factor. We focussed on the various time scales that characterize the electron transfer process and we presented one example to indicate how considerations of the time scales can be used in understanding nonequilibrium phenomena. 

The Marcus classical free energy of activation is AG , the adiabatic preexponential factor A may be taken from Eyring s Transition State Theory as (kg T /h), and Kel is a dimensionless transmission coefficient (0 < k l < 1) which includes the entire efiFect of electronic interactions between the donor and acceptor, and which becomes crucial at long range. With Kel set to unity the rate expression has only nuclear factors and in particular the inner sphere and outer sphere reorganization energies mentioned in the introduction are dominant parameters controlling AG and hence the rate. It is assumed here that the rate constant may be taken as a unimolecular rate constant, and if needed the associated bimolecular rate constant may be constructed by incorporation of diffusional processes as ... 

These relaxation times correspond to rates which are about 106 slower than the thermal vibrational frequency of 6 x 1012 sec 1 (kBT/h) obtained from transition state theory. The question arises how much, if any, of this free energy of activation barrier is due to the spin-forbidden nature of the AS = 2 transition. This question is equivalent to evaluating the transmission coefficient, k, that is, to assess quantitatively whether the process is adiabatic or nonadiabatic. 

Fig. 5 Adiabatic and non-adiabatic ET processes. In the adiabatic process (Fig. 5a), Vel > 200 cm and the large majority of reaction trajectories (depicted as solid arrows) which reach the avoided crossing region remain on the lower energy surface and lead to ET and to the formation of product (i.e., the electronic transmission coefficient is unity). In contrast, non-adiabatic ET is associated with Vel values <200 cm-1, in which case the majority of reaction trajectories which reach the avoided crossing region undergo non-adiabatic transitions (surface hops) to the upper surface. These trajectories rebound off the right-hand wall of the upper surface, enter the avoided crossing region where they are likely to undergo a non-adiabatic quantum transition to the lower surface. However, the conservation of momentum dictates that these trajectories will re-enter the reactant well, rather than the product well. Non-adiabatic ET is therefore associated with an electronic transmission coefficient which is less than unity.

These two possibilities of reaction processes make much difference in calculating the rate of any chemical reaction. According to the transition state theory formalism, these two types of reactions (i.e., adiabatic and nonadiabatic) influence the value of the transmission coefficient, k, which is a preexponential term in the absolute rate expression. The value of k is considered unity for the adiabatic reaction and less than unity for a nonadiabatic reaction. 

If Pad in Eq. (116) is unity, the reaction is adiabatic, which corresponds to a transmission coefficient, k = 1, in the absolute reaction rate expression. If Pad is less than unity, the probability of a nonadiabatic process, Pnonad [cf. Eq. (117)], is not zero, and then the transmission coefficient, k, becomes less than unity. 

The electron transfer is assumed to be an adiabatic process in the Ehrenfest sense so that the transmission coefficient, k, in the transition-state theory expression (Section 1.6.2), Eq. (6.7), is equal to one. 

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