Double-well systems

Quantum mechanical tunneling. Tunneling is the phenomenon by which a particle transfers through a reaction barrier due to its wave-like property.Figure 1 graphically illustrates this for a carbon-hydrogen-carbon double-well system Hydrogen... 

Figure 6. Population decay of the initial state in a barrierless double-well system calculated using multilevel Redfield theory [25]. The vibrational frequency is 60 cm1. (----------)...

Gray-Rice Theory Versus Reactive Island Theory Isomerization in Double-Well Systems Isomerization in a Triple-Well System Isomerization of 3-Phospholene Isomerization of HCN — CNH Isomerization of Cyclobutanone (C4H6O)... 

Consider the rate constants for isomerization in symmetric and asymmetric double-well systems predicted by the several theories discussed above. The two-well potential for coupled x and y DOFs is given by... 

The energy of the charge transfer transition in an unsymmetrical double-well system is given by ... 

Finally, although /zvmax for a symmetrical double-well system is independent of the degree of electronic interaction, the free energy of activation does depend on Hah- Thus when AG° may be neglected, the ratio /zvmax/AG for a double-well system is given by ... 

Evidently /iv ax/AG is <4 for a weakly coupled, endergonic charge-transfer reaction and >4 for a weakly coupled, exergonic charge-transfer reaction or for charge transfer in a moderately coupled symmetrical double-well system. The value of /zVmax/AG Can thus provide information about the degree of electronic interaction. However, in practice the latter is more readily obtained from the intensity of the charge transfer transition. 

Another way to apply the same idea can be derived from formal considerations. We now consider a double well system that correspond to the reaction 7 P and suppose that we can generate trajectories as before. Define the population function... 

In the preceding subsections we have presented the turmelling effect for a double well system. In chemistry, any given reaction can usually be represented as a double well system with reactants and products as the two minima and the transition state located in the barrier separating both wells. Sure, chemical reactions are not one-dimensional systems as it was assumed in the preceding Figures but the formulas presented up to now remain valid irrespective of the dimensionality of the double well. 

From an analysis of a double-well system, Gillan has observed that the equilibrium density of the centroid of the imaginary-time path for the reaction coordinate (t). 

These requirements are usually only fulfilled by model Hamiltonians. Therefore, the number of systems which can be described by accurate path integral treatments is rather small. A relevant example has been the investigation of a double well system coupled linearly to a harmonic bath [23]. 

---