Phase-space transition states reaction paths

Our question now becomes, why does the cylinder follow the specific path that it does We already have a preliminary answer to this question very large mode-mode couplings are required for activation of the reaction coordinate, so trajectories approaching the transition state must pass through a relatively restricted window in phase space to achieve these couplings. 

In chemical terms, normally hyperbolic invariant manifolds play the role of an extension of the concept of transition states. The reason why it is an extension is as follows. As already explained, transition states in the traditional sense are regarded as normally hyperbolic invariant manifolds in phase space. In addition to them, those saddle points with more than two unstable directions can be considered as normally hyperbolic invariant manifolds. Such saddle points are shown to play an important role in the dynamical phase transition of clusters [14]. Furthermore, as is already mentioned, a normally hyperbolic invariant manifold with unstable degrees of freedom along its tangential directions can be constructed as far as instability of its normal directions is stronger than its tangential ones. For either of the above cases, the reaction paths in the phase space correspond to the normal directions of these manifolds and constitute their stable or unstable manifolds. 

The first method comes from the idea that the connections among normally hyperbolic invariant manifolds would form a network, which means that one manifold would be connected with multiple manifolds through homoclinic or heteroclinic intersections. Then, a tangency would signify a location in the phase space where their connections change. This idea offers a clue to understand, based on dynamics, those reactions where one transition state is connected with multiple transition states. In these reaction processes, the branching points of the reaction paths and the reaction rates to each of them are important We expect that analysis of the network is the first step toward this direction. 

The "unified" statistical model was introduced originally in order to reconcile, or unify, two different kinds of statistical theories, transition state theory which is appropriate for reactions proceeding via a direct" reaction mechanism, and the phase space theory of Lightand Nikitin which is designed to describe reactions proceeding via a long-lived collision complex. It is particularly straightforward to apply this theory within the framework of the reaction path Hamiltonian. 

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