Saddle regions phase space

This independency is related to the fact that the semiclassical calculation of the scattering amplitudes involves classical orbits belonging to an invariant set that is complementary to the set of trapped orbits in phase space [56]. The trapped orbits form the so-called repeller in systems where all the orbits are unstable of saddle type. The scattering orbits, by contrast, stay for a finite time in the scattering region. Even though the scattering orbits are controlled... 

In the Smale horseshoe and its variants, the repeller is composed of an infinite set of periodic and nonperiodic orbits indefinitely trapped in the region defining the transition complex. All the orbits are unstable of saddle type. The repeller occupies a vanishing volume in phase space and is typically a fractal object. Its construction is based on strict topological rules. All the periodic and nonperiodic orbits turn out to be topological combinations of a finite number of periodic orbits called the fundamental periodic orbits. Symbols are assigned to these fundamental periodic orbits that form an alphabet... 

In this section we will develop the phase-space structure for a broad class of n-DOF Hamiltonian systems that are appropriate for the study of reaction dynamics through a rank-one saddle. For this class of systems we will show that on the energy surface there is always a higher-dimensional version of a saddle (an NHIM [22]) with codimension one (i.e., with dimensionality one less than the energy surface) stable and unstable manifolds. Within a region bounded by the stable and unstable manifolds of the NHIM, we can construct the TS, which is a dynamical surface of no return for the trajectories. Our approach is algorithmic in nature in the sense that we provide a series of steps that can be carried out to locate the NHIM, its stable and unstable manifolds, and the TS, as well as describe all possible trajectories near it. 

It is well known (see, e.g.. Ref. 13) that the normal form transformations do not converge in the sense that normalization to all orders generally does not yield a meaningful result. However, this is of no consequence for our purposes. We view the technique more as the input to a numerical method for realizing the NHIM, its stable and unstable manifolds, and the TS. In this sense the limitations of machine precision make normalization beyond a certain finite order meaningless. This is a local result valid in the neighborhood of the equilibrium point of center center saddle type. However, once the phase-space structure is established locally, it can be numerically continued outside of the local region. 

In contrast, conventional reaction rate theory replaces the dynamics within the potential well by fluctuations at equilibrium. This replacement is made possible by the assumption of local equilibrium, in which the characteristic time scale of vibrational relaxation is supposed to be much shorter than that of reaction. Furthermore, it is supposed that the phase space within the potential well is uniformly covered by chaotic motions. Thus, only information concerning the saddle regions of the potential is taken into account in considering the reaction dynamics. This approach is called the transition state theory. 

The validity of the transition state concept that separates the phase space into the two distinct regions has not been clarified in cases in which the system passes through higher-rank saddles at least as frequently as through the lowest, first-rank, saddle. 

The reaction bottleneck or the transition state is located somewhere in the vicinity of the first-rank saddle, and is not delocalized through the whole region of configurational (or phase) space. [Note, however, that a delocalized, rigorous transition state certainly exists for some situations (e.g., Refs. 10-12).]... 

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