Reactant valley

These methods try to bracket the transition state from both the reactant and the product side [72, 73]. For example, in the method of Dewar etal [73], two stmctiires, one in the reactant valley and one hi the product valley, are optimized simultaneously. The lower-energy stmcture is moved to reduce the distance separating the two stmctures by a small amount, e.g. by 10%, and its stmcture is reoptimized under the constraint that the distance is fixed. This process is repeated until the distance between the two stmctures is sufficiently small. 

This question is closely related to the coherent-incoherent transition problem absent from the standard situation in the gas phase namely, a true rate constant can be defined only when the tunneling dynamics is incoherent, i.e., once prepared in the initial state (reactant valley), the system... 

Let us now consider a chemical reaction whose initial and final states are different. Then the potential energy surface will not be symmetrical. This geological analogy will be helpful Suppose the valleys are formed by erosion. Then the valley that has eroded faster (or for a longer time) will be both deeper and longer than the less eroded valley, with the necessary consequence that the saddle between the two valleys is shifted toward the shallower valley. Figure 5-4 shows such a surface on which the reactant valley is longer and deeper than the product valley clearly the transition state is located closer to the final state than to the initial state as a result of this disparity in stabilities. 

The potential energy is illustrated in Fig. 6.3a. While one can in principal calculate the exact quantum mechanical Born-Oppenheimer surface, the figure presents a semi-empirical surface constructed to yield the exact spectral properties of reactants and products and the correct activation energy (taken as the difference in energy between the energy (potential) in the reactant valley (x = oo) and the maximum of the MEP (minimum energy pathway).)... 

Reactants or products in a bimolecular system are represented as asymptotic regions of the surface (one or more internal coordinates Rj becoming infinite) at which the potential is independent of these coordinates/ . For example, the bimolecular reaction AB + CD, has a reactant valley which is flat in four dimensions Vis independent o R, R jj,RgQ,Rgjjwhen these coordinates become infinite. 

We have already emphasized that it is more difficult to establish that a point is a transition state than to establish that one is a position of equilibrium. It is therefore not surprising that techniques for finding such a point directly are not well established. The most common approach has been to start from a position of equilibrium or from a product or reactant valley, and to advance up the reaction coordinate, but we have already seen in the example of figure 4 that this will fail in some cases. 

Fig. 2.6. Schematic representation of a reaction treated in the Kramers approximation. The shape of the probability density distribution is assumed to have reached equilibrium (i.e., time independence) at the bottom of the reactant valley. Only the weight of P x. t) (total number of reactants) diminishes by activated diffusion across the barrier.

PES is characterized by an early transition state that is, the saddle point is strongly shifted toward the reactant valley so that the reaction barrier is overcome without appreciable lengthening of the Cl-Cl bond. Since the angle between the valleys is less than 90° and the intramolecular vibration frequencies are much greater than a>0, criterion (2.86) indicates that this reaction takes place in the vibrationally adiabatic regime. 

The rate of reaction is the total number of units (X-—Y-—Z) per unit volume which pass through the critical configuration per unit time from reactant valley to product... 

Since the M—N distance in the activated complex is only slightly greater than the intemuclear distance in the reactant molecule, then the reaction entity has only moved very slightly along the reactant valley. This means that the activated complex lies in the entrance valley, giving an early barrier. This conclusion is verified by the large P—M distance in the activated complex, which indicates that P is still far from MN in the activated complex. 

But remember that H+H is the simplest of all reactions. Moving more in the direction of true chemistry, consider next a reaction for which only two nuclei are hydrogens (instead of three) — the F+H reaction. This reaction is over 1 eV exothermic in going from the reactant valley, over a small (1 kcal) barrier, to the product valley. The exothermicity of reaction means that there are several energetically accessible (open) vibrational channels for this system even at the threshold for reaction. If we include all the rotational levels with each vibration, and the proper (2j+l) rotational degeneracies, we have an unthinkably large number of coupled equations to solve — over 1200 channels. (See Fig. 10.) To solve this problem, we must... 

Figure 14. The solid lines demonstrate H-atom motion trajectories in reaction of H abstraction by CH3 radical from CH3OH molecule with energies of relative motion equal 4.6 and 2.4 kcal/mol respectively. Vibrational motion in reactant valley stipulated by intermolecular vibrations (from ref. 49). The dashed lines refer to fixed distances between CHj and CHjOH.

The transition state concept is useful for this reason we can analyze the structure of the transition state very much as though it were a molecule, and attempt to estimate its stability. Any factor that stabilizes the transition state relative to the reactants tends to lower the energy of activation that is to say, any factor that lowers the top of the energy hill more than it lowers the reactant valley reduces the net height we must climb during reaction. Transition state stability will be the basis—whether explicit or implicit—of almost every discussion of reactivity in this book. 

The idea that the vibrational enhancement of the rate is due to the attraetive potential for excited vibrational states of the reactant is closely related to the observation made long ago based on transition state theoiy [25,26]. Poliak [25] found that for vibrationally highly excited reactants the repulsive pods (periodic orbit dividing surface) is way out in die reactant valley, and the corresponding adiabatic barrier is shallow. Based on this theory one can explain why dynamical thresholds are observed in reactions with vibrationally excited reactants. The simplicity of the theory and its success for mostly collinear reactions has a real appeal. However, to reconcile the existence of a vibrationally adiabatic barrier with the capture-type behavior - which seems to be supported by the agreement of the calculated and experimental rate coefficients [23] -needs further study. 

When the forces between reactants are derived from a precalculated PES it is possible to produce informative pictures with reactant valleys and product valleys perhaps connected by saddles indicating transition states. Time-laps photography or movies of dynamical events may show probabilities in terms of nuclear wave functions evolving on one surface and then transfer to another surface if nonadiabatic coupling terms are present. 

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