Invariant structures phase-space transition states

The location of the saddle point in phase space is specified by and Pi = 0, where qi is the reaction coordinate. On top of the saddle point, the reaction coordinate is completely separated from the rest of the degrees of freedom. Therefore, a set of orbits where Pi) is fixed on the saddle point while the rest are arbitrary is invariant under dynamical evolution. Its dimension in phase space is 2n — 2. Such invariant manifolds are considered as the phase-space structure corresponding to transition states, and will play a crucial role in the following discussion. 

Oped a so-called reactive island theory the reactive islands are the phase-space areas surrounded by the periodic orbits in the transition state, and reactions are interpreted as occurring along cylindrical invariant manifolds through the islands. Fair et al. [29] also found in their two- and three-dof models of the dissociation reaction of hydrazoic acid that a similar cylinderlike structure emerges in the phase space as it leaves the transition state. However, these are crucially based on the findings and the existence of (pure) periodic orbits for all the dof, at least in the transition states. Hence, some questions remain unresolved, for example, How can one extract these periodic orbits from many-body dof phase space and How can the periodic orbits persist at high energies above the saddle point, where chaos may wipe out any of them ... 

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