Potential energy tunneling

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 2.3. Tunnelling of a wave with kinetic energy E through a rectangular potential energy barrier, height V. The narrower the barrier, the smaller the mass of the particle and the smaller the difference between V and E, the greater the tunnelling probability. If the amplitude of the wave has not reached zero at the far side of the barrier, it will stop decaying and resume the oscillation it had on entering the barrier (but with smaller amplitude).

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.

Factors that enhance tunnelling are a small particle mass and a narrow potential energy barrier. In biology, electron transfer is known to occur over large distances (up to about 25 X 10 m). Given the mass of protium is 1840 times that of the electron, the same probability for protium... 

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... 

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