Dimensionality 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]. 

Figure 7.4 Three-dimensional phase space plot of the dividing surfaces and the example trajectories. The two non-reacting trajectories shown in Figure 7.3 are now plotted in the phase space with one axis representing the momenta. While they cross the naively taken dividing surface (qi = 0) more than once, they do not cross the true transition state taken as a surface in the phase space. Note the momentum dependence of the true TS.

In order to understand the problem of finding TS with three or more DOFs, it is useful to address the question of dimensionalities, in configuration and phase space. In classical, Hamiltonian dynamics, transition states are grounded on the idea that certain surfaces (more precisely, certain manifolds) act as barriers in phase space. It is possible to devise barriers in phase space, since in phase space, in contrast to configuration space, two trajectories never cross [uniqueness of solutions of ODEs, see Eq. (4)]. In order to construct a barrier in phase space, the first step is to construct a manifold if that is made of a set of trajectories [8]. 

Because hindered rotors involve densely spaced energy levels, it is possible to treat the problem classically. This has been done for the case of a two-dimensional hindered rotor (Jordan et al., 1991) which is particularly important in the dissociation of loose transition states, a topic to be discussed in the following chapter. The classical phase space integral is solved using the Hamiltonian ... 

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. 

The transitional mode Hamiltonian is given by the last four terms in equation (7.36). In the work of Wardlaw and Marcus, the phase space volume for the transitional modes was calculated versus the center of mass separation R, so that R is assumed to be the reaction coordinate. In recent work, Klippenstein (1990, 1991) has considered a more complex reaction coordinate. The multidimensional phase space volume for the transitional modes can not be determined analytically, but must be evaluated numerically, for example, by a Monte Carlo method of integration (Wardlaw and Marcus, 1984). The density of states is then obtained by dividing the phase space volume by h", where n is the dimensionality of the integral, and differentiating with respect to the energy. The total sum of states of the transition state is obtained by convoluting the density of the transitional modes with the sum of the conserved modes, N(E,J) so that... 

So far, there has been no means to address why, as the system passes through the transition state, there is such a distinct dependence of probability on the direction of climbing. The visualization scheme of the phase-space dividing surface lets us probe deeper into such questions. Figures 2.10 and 2.11 show projections of the ith order dividing surface 5 (gi (p, q) = 0) onto the two-dimensional (q, <5(2) subspace at = 0.1 and 0.5e for saddle I and saddle II, respectively. As the total energy increases, the projections of the phase-space dividing surfaces, q) = 0) (i = 1,2), broaden and extend... 

This enables us to extract and visualize the stable and unstable invariant manifolds along the reaction coordinate in the phase space, to and from the hyperbolic point of the transition state of a many-body nonlinear system. PJ AJI", Pj, Qi, t) and PJ AJ , Pi, q, t) shown in Figure 2.13 can tell us how the system distributes in the two-dimensional (Pi(p,q), qi(p,q)) and PuQi) spaces while it retains its local, approximate invariant of action Jj (p, q) for a certain locality, AJ = 0.05 and z > 0.5, in the vicinity of... 

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