Determination of Caustics and Propagation in Tunneling Region

As stated above, further solution of the differential equation is not possible beyond this point and we have to reformulate the problem. To achieve this we use a certain transformation. To clarify the idea, let us first consider a one-dimensional problem in which the caustics just becomes the turning point qo. As the trajectory reaches qo, the momentum p q) - qo becomes zero and A = dp/dq 1/V o diverges, 

For numerical implementation, the sequential steps to carry out these procedures are summarized as follows  

Equations (9.5)-(9.8) are then used to compute the matrix elements of A in terms of the elements of A by evaluating partial derivatives of the new Jacobi matrix D(p)/D(q). The second-derivative coefficients in the new representation are derived by rotating the old coefficients as in Equation (9.3) for consistency, and then substituting the new momenta and coordinates into the old coefficients. For instance, replacing pn and qN with -qN and Pn, respectively, gives 

Numerical examples are provided below for the two cases (1) two-dimensional Henon-Heiles system and (2) three-dimensional adiabatic chemical reaction. 

The two-dimensional Henon-Heiles Hamiltonian (in atomic units) is given by 

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