Transition state approximate correction

We counted the contribution of only those trajectories that have a positive momentum at the transition state. Trajectories with negative momentum at the transition state are moving from product to reactant. If any of those trajectories were deactivated as products, their contribution would need to be subtracted from the total. Why Because those trajectories are ones that originated from the product state, crossed the transition state twice, and were deactivated in the product state. In the TST approximation, only those trajectories that originate in the reactant well are deactivated as product and contribute to the reactive flux. We return to this point later in discussing dynamic corrections to TST. 

The main approximation of such one-dimensional corrections is that the tunnelling is assumed to occur along the MER This may be a reasonable assumption for reactions having either early or late (close to either reactant or product) transition states. For reactions where both bond breaking and formation are significant at the TS (as is usually the case), the dominant tunnel effect is comer cutting (Figure 16.5), i.e. the favoured... 

Quantum mechanical effects—tunneling and interference, resonances, and electronic nonadiabaticity— play important roles in many chemical reactions. Rigorous quantum dynamics studies, that is, numerically accurate solutions of either the time-independent or time-dependent Schrodinger equations, provide the most correct and detailed description of a chemical reaction. While hmited to relatively small numbers of atoms by the standards of ordinary chemistry, numerically accurate quantum dynamics provides not only detailed insight into the nature of specific reactions, but benchmark results on which to base more approximate approaches, such as transition state theory and quasiclassical trajectories, which can be applied to larger systems. 

Arrhenius proposed his equation in 1889 on empirical grounds, justifying it with the hydrolysis of sucrose to fructose and glucose. Note that the temperature dependence is in the exponential term and that the preexponential factor is a constant. Reaction rate theories (see Chapter 3) show that the Arrhenius equation is to a very good approximation correct however, the assumption of a prefactor that does not depend on temperature cannot strictly be maintained as transition state theory shows that it may be proportional to 7. Nevertheless, this dependence is usually much weaker than the exponential term and is therefore often neglected. 

Expression (2-16) is approximately correct for first-order desorption and for values of vt[ between 108 and 1013 K l. It is very often applied to determine from a single TDS spectrum. The critical point however is that one must choose a value for v, the general choice being 1013 s, independent of coverage. As we explain below, this choice is only valid when there is little difference between the entropy of the molecule in the ground state and that in its transition state 125, 27], The Redhead formula should only be used if a reliable value for the prefactor is available ... 

From this it appears that A Eft Ea, the Arrhenius activation energy. A more correct treatment gives A Eft = Ea - RT for reactions in solution. However, since RT at 25°C is only 2.5 kj mol-1, the approximation that A Eft = Ea is often used. The preexponential term, in parentheses in Eq. 9-82, depends principally on A St, the entropy change accompanying formation of the transition state. The quantities AGt, AFft, and A St are sometimes measured for enzymatic reactions but useful interpretations are difficult. 

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