Tunneling quantum mechanical considerations

The first quantum-mechanical consideration of ET is due to Levich and Dogonadze [7]. According to their theory, the ET system consists of two electronic states, that is, electron donor and acceptor, and the two states are coupled by the electron exchange matrix element, V, determined in the simplest case by the overlap between the electronic wave functions localized on different redox sites. Electron transfer occurs by quantum mechanical tunneling but this tunneling requires suitable bath fluctuations that bring reactant and product energy levels into resonance. In other words, ET has... 

It should be noted here that quantum mechanical considerations are not required for calculation of the total proton transport rate. Inter-molecular proton transfer which might have contributions from proton tunnelling, is triggered by a longitudinal acoustical phonon which can be described by classical mechanics. This is in accordance with a small isotope effect for the equivalent conductance of H and The... 

If the characteristic electron tunneling distance, is assumed to remain unchanged with the presence of the PEDOTiPSS layer, 2 = fi, and then Equation 6.3 can be solved. From quantum mechanical considerations fp is estimated to range from 0.1 to 2 nm. The PEDOTiPSS thickness is taken as 1.57 nm [original calculation given a 1 to 4 ratio of SWCNTs to PEDOTiPSS]. The CNT diameter is taken as 1 nm. Avalue of 0.012 to 0.156 wt% is then obtained for the given... 

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 accurate prediction of enzyme kinetics from first principles is one of the central goals of theoretical biochemistry. Currently, there is considerable debate about the applicability of TST to compute rate constants of enzyme-catalyzed reactions. Classical TST is known to be insufficient in some cases, but corrections for dynamical recrossing and quantum mechanical tunneling can be included. Many effects go beyond the framework of TST, as those previously discussed, and the overall importance of these effects for the effective reaction rate is difficult (if not impossible) to determine experimentally. Efforts are presently oriented to compute the quasi-thermodynamic free energy of activation with chemical accuracy (i.e., 1 kcal mol-1), as a way to discern the importance of other effects from the comparison with the effective measured free energy of activation. 

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.)... 

It can then be calculated by using the Landau-Zener theory as discussed in Sec.6.1.II. In general, however, nuclear motion should be treated quantum-mechanically hence, the tunneling of solvent molecules must be taken into account, as done in the considerations in Sec.6.2 II. The reaction probability, i.e., the probability of an electron... 

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