Subject quantum mechanical tunneling

Although the use of the correct energy levels for calculating the density of states is strictly a quantum correction of the classical RRKM theory, there are two other effects that are much more fundamental to the theory quantum mechanical tunnelling and fluctuations. The first of these is dealt with in Chapter 2, and the second is the main subject of Chapter 3. 

It is worth noting that, as far as they are less than several nanometers thick, the passive films are subject to the quantum mechanical tunneling of electrons. Electron transfer at passive metal electrodes, hence, easily occurs no matter whether the passive film is an insulator or a semiconductor. By contrast, no ionic tunneling is expected to occur across the passive film even if it is extremely thin. The thin passive film is thus a barrier to the ionic transfer but not to the electronic transfer. Redox reactions involving only electron transfer are therefore allowed to occur at passive film-covered metal electrodes just like at metal electrodes with no surface film. It is also noticed, as mentioned earlier, that the interface between the passive film and the solution is equivalent to the interface between the solid metal oxide and the solution, and hence that the interfacial potential is independent of the electrode potential of the passive metal as long as the interface is in the state of band edge pinning. 

It appears that the electron transfer in some biological systems is temperature independent at temperatures close to absolute zero. It is therefore inferred that such reactions proceed through a quantum mechanical tunneling mechanism, sometimes involving distances up to 30 A (3 nm). Such electron transfers seem to be involved in biological redox reactions of chloroplasts and mitochondria. It appears that the electron-transfer reactions will be one of the central subjects of inorganic reaction mechanism studies in the future. 

More subtle than the lack of ZPE in bound modes after the collision is the problem of ZPE during the collision. For instance, as a trajectory passes over a saddle point in a reactive collision, all but one of the vibrational (e.g., normal) modes are bound. Each of these bound modes is subject to quantization and should contain ZPE. In classical mechanics, however, there is no such restriction. This has been most clearly shown in model studies of reactive collisions (28,35), in which it could be seen that the classical threshold for reaction occurred at a lower energy than the quantum threshold, since the classical trajectories could pass under the quantum mechanical vibrationally adiabatic barrier to reaction. However, this problem is conspicuous only near threshold, and may even compensate somewhat for the lack of tunneling exhibited by quantum mechanics. One approach in which ZPE for local modes was added to the potential energy (44) has had some success in improving reaction threshold calculations. 

The two chief experimental criteria for tunneling in chemical reactions are an abnormal isotope effect (the tunnel effect is much more pronounced for hydrogen than for deuterium), which does not concern us here, and a curved Arrhenius plot. The reason for this is that the effect becomes most marked at low temperatures, when the fraction of systems which are able to cross the barrier becomes considerably higher than that calculated from classical considerations. As a result, the rate decreases with decreasing temperature less than expected, and the Arrhenius plot becomes concave upward. We cannot go into the quantum-mechanical details, and refer the reader to the literature on the subject. (See, e.g., Refs. 2b, 23, 77, 99, 105.)... 

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