Configurational dividing surface, phase space

B. See Trajectories Recrossing the Configurational Dividing Surface in Phase Space... 

Let us divide the phase space Ri, Pi into two subspaces by some critical surface The first subspace corresponds to the reactants, and the second subspace corresponds to the reaction products. Determine the reaction coordination = r in such a way that it is perpendicular to the critical surface S. The qr value on the 5 surface that characterizes the configuration of the activated complex will be designated as qf It is assumed that the reaction has occurred if the representative point had crossed the critical surface in the direction of the reaction products. Then the reaction rate is determined as a flow of representative points along q through S" in the direction of the products. 

As mentioned earlier, Komatsuzaki and Berry [26] have recently developed a promising approach to analyzing many-dimensional reacting systems. By seeking appropriate canonical transformations that yield local approximate constants of motion associated with the reactive mode, they were able to transform the conventional dividing surface in configuration space to a manydimensional separatrix in phase-space. Specifically, suppose the original phase-space variables are denoted by = q, q2,..., qn ,P, Pi, , Pn) the... 

In order to deal properly with reactions that have no saddle point, it is necessary to go back to the notion that a unimolecular reaction is represented by a flux in phase space. Recall that the TS is defined as the surface in phase space which divides reactants from products, and at which the phase space is a minimum. For reactions with a substantial energy barrier, the dividing surface will be located at the saddle point because energy is such a dominating factor in determining the transition-state sum of states. However, for loose transition states it is necessary to search directly for the minimum flux configuration. The existence of such a minimum flux configuration is due to the interplay be-... 

For example, the projection onto the configurational reaction coordinate qi is an important device to reveal how iS(gi(p,q) = 0) differs from the conventional dividing surface S( i = 0). Remember that in an energy range close to the threshold energy in which the normal mode picture is approximately valid, the phase-space S qi = 0) cohapses onto the traditional configuration-space surface where q = 0. Similarly, the projection onto Pj... 

RRKM theory is also at the basis of localization of loose transition states in the PES. Another assumption of the theory is that a critical configuration exists (commonly called transition state or activated complex) which separates internal states of the reactant from those of the products. In classical dynamics this is what is represented by a dividing surface separating reactant and product phase spaces. Furthermore, RRKM theory makes use of the transition state theory assumption once the system has passed this barrier it never comes back. Here we do not want to discuss the limits of this assumption (this was done extensively for the liquid phase [155] but less in the gas phase for large molecules we can have a situation similar to systems in a dynamical solvent, where the non-reacting sub-system plays the role... 

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