Tunneling trajectory

Keeping this in mind one may proceed to classify tunneling trajectories in the following way choose a certain known trajectory at a given T then decrease T by 5T, linearize the problem near that trajectory, and continuously transform to a new trajectory with a longer period. This way a family of trajectories arises which is parametrized by the period, i.e., by the temperature. In the... 

Two trajectories that join the minima and have special significance are the minimum energy path q = cs/mC02 and the sudden tunneling trajectory q =... 

Figure 3.2. Time contour (a) and real part (b) of the tunneling trajectory for a separable system of a parabolic barrier-harmonic oscillator (schematic). Curves E, E and E" are equipotentials. Vertical and horizontal dashed lines show the loci of vibrational and translational turning points. Points A, B, and C indicate the corresponding times and positions along trajectory. (From Altcorn and Schatz [1980]).

Figure 6.20. (a) Projection of a three-dimensional PES K(p,p2,p3) for two-proton transfer in formic acid dimer onto the (p, p,) and (p, p3) planes. In contrast with points A and B, in points C and D the potential along the p3 coordinate is a double well resulting in bifurcation of the reaction path [from Shida et al., 1991b]. (b) The contour lines correspond to equilibrium value of p3 and potential (6.37) when V(Q) = V0(Q4 - 2Q2), V0 = 21 kcal/ mol, C = 5.()9V0, A = 5.351/, Qn = 0.5. When Q > Qc, two-dimensional tunneling trajectories exist in the shaded region between curves 1 and 2. Curve 3 corresponds to synchronous transfer. 

Figure 5. Energy/coordinate scheme of the trajectories of Fig. 3. Trajectory (i) is a tunneling trajectory.

Figure 3. Effects of the periodic perturbation, (a) Integration path on the complex time plane. (b) Deformation of the potential by the periodical perturbation. In the case where Im t = Im ti 0— that is, the part of integration path indicated by the same broken line in (a)— the oscillation of complexified potential is amplified exponentially as shown by the broken lines, (c) Change of the tunneling trajectory with increase of the perturbation strength. In the bottom figure, a trajectory stating at ti in the close neighborhood of t c is drawn.

As mentioned in the previous subsection, the adiabatic solution (34) together with the Melnikov method enables us to prove items 1, 2a, 2b, and 3. Then the significant properties of tunneling trajectories and of the branches consisting of them, which are numerically observed, can be explained in terms of the adiabatic approximation associated with the Melnikov method. 

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