Kinetic energy quantum-mechanical tunneling

Figure 3.5 Graphical representation of the quantum mechanical tunnelling effect between tip and sample. The probability P of a particle with kinetic energy E tunnelling through a potential barrier cf> is shown as a function of sample-tip separation z.

In Chapter 7 general kinetics of electrode reactions is presented with kinetic parameters such as stoichiometric number, reaction order, and activation energy. In most cases the affinity of reactions is distributed in multiple steps rather than in a single particular rate step. Chapter 8 discusses the kinetics of electron transfer reactions across the electrode interfaces. Electron transfer proceeds through a quantum mechanical tunneling from an occupied electron level to a vacant electron level. Complexation and adsorption of redox particles influence the rate of electron transfer by shifting the electron level of redox particles. Chapter 9 discusses the kinetics of ion transfer reactions which are based upon activation processes of Boltzmann particles. 

Returning to the one-dimensional box of constant width, if the potential does not increase suddenly to infinity at one of the walls then the wave-function does not vanish there. Due to the continuity requirement of the wavefunction, it decreases exponentially to zero inside the wall of finite height. Therefore, there is a non-zero probability that the particle will penetrate the wall, although its kinetic energy is lower than the potential barrier (Fig. 2.6). This effect is called quantum-mechanical tunnelling. 

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. 

From the above discussion, it is clear that various kinetic criteria (reaction orders, Tafel slopes, and log/-AGads plots on different electrode materials, where appropriate) can be used to determine the reaction mechanism and the coverage conditions. For reactions involving protons, the separation factors (H-T or H-D) can also be used as another criterion, since the barrier heights vary characteristically with mechanism because of the relatively large zero point energy difference between the isotopes and their different quantum mechanical tunneling properties. This is evaluated in Refs. 50 and 51, and may be used to confirm other evidence. 

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