RRKM theory saddle point

The Transition State Hypothesis. The general idea that a transition state is located at a saddle point on the PES, as detailed in Section 1.3, is familiar to most organic chemists. However, the original concept of a transition state started out as something rather different. In the development of both transition state and RRKM theory, the transition state was defined as the location of a plane (actually a hyperplane) in phase space, perpendicular to the reaction coordinate. ... 

One example of non-IRC trajectory was reported for the photoisomerization of cA-stilbene.36,37 In this study trajectory calculations were started at stilbene in its first excited state. The initial stilbene structure was obtained at CIS/6-31G, and 2744 argon atoms were used as a model solvent with periodic boundary conditions. In order to save computational time, finite element interpolation method was used, in which all degrees of freedom were frozen except the central ethylenic torsional angle and the two adjacent phenyl torsional angles. The solvent was equilibrated around a fully rigid m-stilbene for 20 ps, and initial configurations were taken every 1 ps intervals from subsequent equilibration. The results of 800 trajectories revealed that, because of the excessive internal potential energy, the trajectories did not cross the barrier at the saddle point. Thus, the prerequisites for common concepts of reaction dynamics such TST or RRKM theory were not satisfied. 

The MP2/6-31G direct dynamics simulation study was later extended to cover the dynamics from the central barrier for the SN2 reaction of Cl I C2H5CI.104 The majority of the trajectories starting from the saddle point moved off the central barrier to form the Cl- C2H5CI complex. The results were different from those obtained previously for the CH3C1 reaction, in which extensive recrossing was observed. The reaction of C2H5CI was, in this sense, consistent with the prediction by the RRKM theory. However, some of the... 

First, we want to derive an expression for the microcanonical rate constant k(E) when the total internal energy of the reactant is in the range E to E + dE. From Eq. (7.43), the rate of reaction is given by the rate of disappearance of A or, equivalently, by the rate at which activated complexes A pass over the barrier, i.e., the flow through the saddle-point region. The essential assumptions of RRKM theory are equivalent to the assumptions underlying transition-state theory. 

A third assumption in RRKM theory is that the special coordinates, dq and dp, are perpendicular to all other coordinates, and that they can therefore be separated from the other coordinates. This separability is most valid at the saddle point and at very low energies where the vibrational modes of the transition state can be treated as normal modes. However, as the energy is increased, there will be coupling between the reaction coordinate and the rest of the modes, especially for the case of a complicated reaction, for example, HCl loss om C2H5CI+. The separability assumption is one that... 

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