Saddle regions transition states

The adiabatic potential energy surface below the intersection point possesses three equivalent minimum regions. One of these surface minima is seen in the cross section in Figure 3 and is labeled as point Min. The three minima on the surface are separated by three saddle-point transition states. Point TS in the Figure 3 cross section is the transition state between the two other... 

The transition state concept, once understood in static terms only, as the saddle point separating reactants and products, may be fruitfully expanded to encompass the transition region, a landscape in several significant dimensions, one providing space for a family of trajectories and for a significant transition state lifetime. The line between a traditional transition structure and a reactive intermediate thus is blurred The latter has an experimentally definable lifetime comparable to or longer than some of its vibrational periods. 

R. A. Marcus I used the words saddle-point avoidance, incidentally, to conform with current terminology in the literature. More generally, one could have said, instead, avoidance of the usual (quasi-equilibrium) transition-state region (i.e., the most probable region if viscosity effects were absent). 

Using the same method as for the first excited electronic state, we select a level shift in the region Xi < p < X2. This procedure may indeed lead to a transition state but in this way we always increase the function along the lowest mode. However, if we wish to increase it along a higher mode this can only be accomplished in a somewhat unsatisfactory manner by coordinate scaling. Nevertheless, this method has been used by several authors with considerable success.14 The problem of several first-order saddle points does not arise in electronic structure calculations since there is only one first excited state.15... 

Moving along the intemuclear line we find a point in a saddle-shaped region, analogous to a transition state, where the surface again (caveat) has zero slope (all first derivatives zero), and is negatively curved along the z-axis but positively curved in all other directions (Fig. 5.41), i.e. 

Transition-state theory was developed in the 1930s. The derivation presented in this section closely follows the original derivation given by H. Eyring [1]. From Eq. (6.1), the reaction rate may be given by the rate of disappearance of A or, equivalently, by the rate at which activated complexes (AB) pass over the barrier, i.e., the flow through the saddle-point region in the direction of the product side. 

In the first step, we define the relevant activated complexes as microcanonical transition states having a total energy H = E and a value for the reaction coordinate qi that lies between q and q + dq. The separation of the reaction coordinate from the other degrees of freedom in the saddle-point region implies that the Hamiltonian in this region can be written as... 

The region of the potential energy surface indicating the transition state is illustrated in Figure 7-2, while a modem sculpture reminiscent of a potential energy surface at and around the saddle point is shown in Figure 7-3. 

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