Eckart type potential

We take a periodically perturbed Eckart type-potential [28] as a model system of the fringed tunneling. The Hamiltonian of the model system is given as follows ... 

The tunnelling correction P is the transmission probability through the potential barrier averaged over all possible crossing points and potential energies . An asymmetrical banier of the Eckart type l is assumed in the present model. 

The tunneling correction may distinctly improve the values of calculated rate constants for reactions with high energy barriers, especially at low temperatures.1 9 A reasonable correction of the rate constant may be obtained, even in a one-dimensional approximation, using the Eckart type of potential.10 In this approach, the asymmetric potential is characterized by the forward and backward barrier heights and the imaginary frequency of the transition state.1 A number of different kinds of tunneling corrections have been evaluated by... 

The strength of the coupling is controlled by the parameter e in Eq. (66). The vector field generated by the corresponding classical Hamilton function has an equilibrium point at (q p2, q, pi, p2, p ) = 0. For e sufficiently small (for given values of parameters of the Eckart and Morse potentials), the equilibrium point is of saddle-center-center stability type. 

There are some reservations about the strict applicability of the treatment outlined above to actual reactions. In the first place a parabolic energy barrier becomes unrealistic for configurations far removed from the transition state, and a more appropriate type of barrier is shown by the full curve in Figure 22(a), in which the broken curve represents a parabola. There is one potential energy function of this kind for which an explicit expression for the permeability can be obtained, commonly known as the Eckart barrier. For a symmetrical barrier (AH = 0) the equations are... 

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