Bottleneck separatrix

A Zeroth-Order Calculation of the Rate Constant for Crossing Intramolecular Bottlenecks The Rate Constant for Crossing the Separatrix... 

The most important element of the Davis-Gray theory of unimolecular reaction rate is the identification of bottlenecks to intramolecular energy flow and the intermolecular separatrix to molecular fragmentation. Davis and Gray s work was motivated by the discovery of bottlenecks in chaotic transport by MacKay, Meiss, and Percival [8,9] and by Bensimon and Kadanoff [10]. 

Figure 14. An Hel2 surface of section for an unstable trajectory which forms a collision complex. The total energy is —2661.6 cm . Also shown are the reaction separatrix and the intramolecular bottleneck, (a) Graph showing the full dynamics of the trajectory. (b)-(f) Graphs illustrating the trajectory over five consecutive time ranges. These graphs are arranged to demonstrate the manner in which the trajectory moves with respect to the bottleneck and the separatrix. [From M. J. Davis and S. K. Gray, J. Chem. Phys. 84, 5389 (1986).]...

The Davis-Gray theory teaches us that by retaining the most important elements of the nonhnear reaction dynamics it is possible to accurately locate the intramolecular bottlenecks and to have an exact phase space separatrix as the transition state. Unfortunately, even for systems with only two DOFs, there may be considerable technical difficulties associated with locating the exact bottlenecks and the separatrix. Exact calculations of the fluxes across these phase space structures present more problems. For these reasons, further development of unimolecular reaction rate theory requires useful approximations. 

Clearly, the A and B isomer states should be inside the separatrix, and the state C should be in the phase-space region outside of the separatrix but inside the energy boundary. A schematic diagram of this three-state isomerization model is presented in Fig. 20. From the results of previous analyses of predissociation we expect that within the A and B domains there are, in general, intramolecular bottlenecks to energy transfer. However, these bottlenecks are... 

Implicit in the application of reactive islands theory, or in the application of other models of reaction dynamics that make use of the properties of a reaction separatrix, is the idea that the chaotic phase space may be considered to be completely structureless (within the restrictions imposed by the homoclinic/heteroclinic tangling of the separatrix manifolds). However, the presence of internal bottlenecks within one or more isomers may cause this approximation to break down.2 io3-i05,iu Accurate procedures for developing corrections to account for internal bottlenecks in two-dimensional systems have been developed. - > > 2 Unfortunately, much progress remains to be made toward developing quantitative procedures for dealing with internal bottlenecks in many-dimensional systems. 

The existence of Arnold diffusion is irrelevant to the properties of separatrix manifolds, which still mediate the transport of chaotic trajectories within the regions of phase space they control. However, if Arnold diffusion is present in a given multidimensional system, the possibility exists for chaotic motion initially trapped between two nonreactive (trapped) KAM layers to eventually become reactive. This would presumably manifest itself as an apparent bottleneck to the rate of population decay, as chaotic trajectories slowly leak out from the region occupied by regular KAM surfaces into the portion of phase space more directly accessible to the hypercylinders. However, transport via the Arnold diffusion mechanism typically manifests itself on time scales much larger than those that we observe in numerical simulations (Arnold diffusion usually occurs on the order of thousands of mappings, or vibrational periods), and so it seems improbable that this effect would be observed in a typical reaction dynamics simulation. It would be interesting to characterize the effect of Arnold diffusion in realistic molecular models. 

The theoretical model includes the partial separatrix as intermolecular bottleneck and one cantorus as intramolecular bottleneck. From Ref. 34. 

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