Hamiltonian systems phase-space transition states

The point q = p = 0 (or P = Q = 0) is a fixed point of the dynamics in the reactive mode. In the full-dimensional dynamics, it corresponds to all trajectories in which only the motion in the bath modes is excited. These trajectories are characterized by the property that they remain confined to the neighborhood of the saddle point for all time. They correspond to a bound state in the continuum, and thus to the transition state in the sense of Ref. 20. Because it is described by the two independent conditions q = 0 and p = 0, the set of all initial conditions that give rise to trajectories in the transition state forms a manifold of dimension 2/V — 2 in the full 2/V-dimensional phase space. It is called the central manifold of the saddle point. The central manifold is subdivided into level sets of the Hamiltonian in Eq. (5), each of which has dimension 2N — 1. These energy shells are normally hyperbolic invariant manifolds (NHIM) of the dynamical system [88]. Following Ref. 34, we use the term NHIM to refer to these objects. In the special case of the two-dimensional system, every NHIM has dimension one. It reduces to a periodic orbit and reproduces the well-known PODS [20-22]. 

A unimolecular reaction can be viewed as a reaction flux in phase space. It is best to have in mind a potential energy surface with a real barrier in the product channel, that is, a saddle point. Figure 6.4 shows both the reaction coordinate and a picture of the phase space associated with the molecule and the transition state. Recall, that a molecule of several atoms having a total of m internal degrees of freedom can be fully described by the motion of m positions (q) and m momenta (p). At any instant in time, the system is thus fully described by 2m coordinates. A constant energy molecule (a microcanonical system) has its phase space limited to a surface in which the Hamiltonian H = E. Thus, the dimensionality of this hypersurface is reduced to 2m — 1. 

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