Periodic orbit dividing surface

Periodic boundary conditions, Monte Carlo heat flow simulation, nonequilibrium molecular dynamics, 79—81 Periodic-orbit dividing surface (PODS) geometric transition state theory, 196-201 transition state trajectory, 202-213 Perturbation theory, transition state trajectory, deterministically moving manifolds, 224-228... 

Figure 8. Linear transition state (TS) and nonlinear periodic orbit dividing surface (PODs) F.

We shall make more use of the notion of normally hyperbolic invariant manifold (NHIM). This invariant surface is the n-DOF generalization of the periodic orbit dividing surface, even if originally defined in a much more general framework (a bibliography may be found in Ref. 24). Its correct definition is put forward in Section IV.A and is used in all examples coming thereafter. 

The idea that the vibrational enhancement of the rate is due to the attraetive potential for excited vibrational states of the reactant is closely related to the observation made long ago based on transition state theoiy [25,26]. Poliak [25] found that for vibrationally highly excited reactants the repulsive pods (periodic orbit dividing surface) is way out in die reactant valley, and the corresponding adiabatic barrier is shallow. Based on this theory one can explain why dynamical thresholds are observed in reactions with vibrationally excited reactants. The simplicity of the theory and its success for mostly collinear reactions has a real appeal. However, to reconcile the existence of a vibrationally adiabatic barrier with the capture-type behavior - which seems to be supported by the agreement of the calculated and experimental rate coefficients [23] -needs further study. 

Consider then an adiabatic well in the hyperspherical coordinate system. Classically, the motion of the periodic orbit at the well would be an oscillation from a point on the inner equipotential curve in the reactant channel to a point on the same equipotential curve in the product channel. This is qualitatively the motion of what are termed "resonant periodic orbits" (RPO s). For example the RPO s of the IHI system are given in Fig. 5. Thus, finding adiabatic wells in the radial coordinate system corresponds to finding RPO s and quantizing their action. Note that in Fig. 5 we have also plotted all the periodic orbit dividing surfaces (PODS) of the system, except for the symmetric stretch. By definition, a PODS is a periodic orbit that starts and ends on different equi-potentials. Thus the symmetric stretch PODS would be an adiabatic well for an adiabatic surface in reaction path coordinates. However, the PODS in the entrance and exit channels shown in Fig. 5 may be considered as adiabatic barrieres in either the radial or reaction path coordinate systems. Here, the barrier in radial coordinates, has quantally a tunneling path between the entrance and exit channels. 

We also found [41] that besides total energy, the velocity across the transition state plays a major role in many-dof systems to migrate the reaction bottleneck outward from the naive dividing surface S(qi = 0). A similar picture has been observed by Pechukas and co-workers [37] in 2D Hamiltonian systems, that is, as energy increases, pairs of the periodic orbit dividing surfaces (PODSs) appearing on each reactant and product side migrate outwards, toward reactant and product state, and the outermost... 

II. PERIODIC ORBIT DIVIDINGS SURFACES - PODS a. Classical Variational Transition State Theory... 

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