Tunnelling, potential energy surfaces

Reality suggests that a quantum dynamics rather than classical dynamics computation on the surface would be desirable, but much of chemistry is expected to be explainable with classical mechanics only, having derived a potential energy surface with quantum mechanics. This is because we are now only interested in the motion of atoms rather than electrons. Since atoms are much heavier than electrons it is possible to treat their motion classically. Quantum scattering approaches for small systems are available now, but most chemical phenomena is still treated by a classical approach. A chemical reaction or interaction is a classical trajectory on a potential surface. Such treatments leave out phenomena such as tunneling but are still the state of the art in much of computational chemistry. 

Figure 2.4. Reaction coordinate diagram for a simple chemical reaction. The reactant A is converted to product B. The R curve represents the potential energy surface of the reactant and the P curve the potential energy surface of the product. Thermal activation leads to an over-the-barrier process at transition state X. The vibrational states have been shown for the reactant A. As temperature increases, the higher energy vibrational states are occupied leading to increased penetration of the P curve below the classical transition state, and therefore increased tunnelling probability.

The method is composed of the following algorithms (1) transition position is detected along each classical trajectory, (2) direction of transition is determined there and the ID cut of the potential energy surfaces is made along that direction, (3) judgment is made whether the transition is LZ type or nonadiabatic tunneling type, and (4) the transition probability is calculated by the appropriate ZN formula. The transition position can be simply found by... 

The effect of the solvent upon the breaking of the symmetry of the potential energy surface for proton transfer has a profound consequence for the reaction dynamics for proton transfer. The tunneling of the proton out of the reactant state... 

Figure 3. Equipotential sections through the potential energy surface for an exchange reaction, as in Figure 2. The heavy horizontal line indicates the solvent configuration appropriate to the activated complex and is the solvent configuration at which inner-sphere tunneling takes place.

Before discussing tunneling in VTST where the discussion will focus on multidimensional tunneling, it is appropriate to consider the potential energy surface for a simple three center reaction with a linear transition state in more detail. The reaction considered is that of Equation 6.3. The collinear geometry considered here is shown in Fig. 6.1a it is in fact true that for many three center reactions the transition state can be shown to be linear. The considerations which follow apply to a onedimensional world where the three atoms (or rather the three nuclei) are fixed to a line. We now consider this one-dimensional world in more detail. The Born-Oppenheimer approximation applies as in Chapter 2 so that the electronic energy of... 

Here it was found that the tunneling factor k is very close to unity. However this result is uncertain because the magnitude of k is sensitive to the choice of the potential energy surface which is not as well established for reaction 6.20 as it is for 6.6. For that matter, learning whether a given reaction rate is significantly influenced by tunneling either on the basis of theory or experiment is not a trivial problem as will be pointed out in further discussion. 

This model has obvious shortcomings. For example, the interaction with the solvent in the initial state is straightforward since the proton is in the ionic form, whereas in the final state, the proton is the nonionic adsorbed H atom and its interaction with the solvent should be negligible. No consideration of this fact was made in the potential of the final state Uf m Eq. (43). However, this treatment incorporates the basic feature of the proton transfer reaction interaction with the solvent, tunneling as well as classical transition of the proton, and the effect of the electric field on the potential energy surfaces of the system. 

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