Potential energy surface transition rates

The reaction systems on which the quantum chemical calculations were done and their results that are reported in this chapter are compared with existing experimental data that were obtained mainly in our laboratory. An important issue that should be mentioned at this point is the necessity to compare calculated rate parameters to those reduced in the experiment. One can calculate a very convincing potential energy surface, transition states, and intermediates on a surface and evaluate from them rate parameters. If, however, the calculated values cannot be anchored into experimental results, no one can be sure whether this particular surface is really the relevant one. 

Rather than using transition state theory or trajectory calculations, it is possible to use a statistical description of reactions to compute the rate constant. There are a number of techniques that can be considered variants of the statistical adiabatic channel model (SACM). This is, in essence, the examination of many possible reaction paths, none of which would necessarily be seen in a trajectory calculation. By examining paths that are easier to determine than the trajectory path and giving them statistical weights, the whole potential energy surface is accounted for and the rate constant can be computed. 

Fig. 28. Schematic of potential energy surfaces of the vinoxy radical system. All energies are in eV, include zero-point energy, and are relative to CH2CHO (X2A//). Calculated energies are compared with experimentally-determined values in parentheses. Transition states 1—5 are labelled, along with the rate constant definitions from RRKM calculations. The solid potential curves to the left of vinoxy retain Cs symmetry. The avoided crossing (dotted lines) which forms TS5 arises when Cs symmetry is broken by out-of-plane motion. (From Osborn et al.67)...

We note at this point that the nonadiabatic-transition state method used here (6,19,77) is not expected to be able to give quantitative agreement with experimental rate constants. There are too many factors that are treated approximately (or not at all) in this theory for such performance to be possible. One of the key difficulties is that calculated rate constants are very sensitive to the accuracy of the potential energy surface at room temperature, an error of lkcalmol-1 on the relative energy of the MECP relative to reactants will equate, roughly speaking, to an error by a factor of five on the calculated rate constant. Even though we... 

A simplified approach is statistical rate theory (transition state theory) with the help of which the overall rate constant k(T) may be obtained from potential energy surface (PES) in a single jump averaging out all of the intermediate details. It is generally not possible to extract microscopic details such as energy-dependent cross sections from conventional kinetics experiments. The preferable approach is to calculate microscopic quantities from some model and then perform the downward averaging for comparison with measured quantities. 

A similar relationship is also derived by the absolute reaction rate theory, which is used almost exclusively in considering, and understanding, the kinetics of reactions in solution. The activated complex in the transition state is reached by reactants in the initial state as the highest point of the most favorable reaction path on the potential energy surface. The activated complex Xms in equilibrium with the reactants A and B, and the rate of the reaction V is the product of the equilibrium concentration of X and the specific rate at which it decomposes. The latter can be shown to be equal to kT/h, where k is Boltzmannn s constant and h is Planck s constant ... 

In siunmary, although the application of detailed chemical kinetic modeling to heterogeneous reactions is possible, the effort needed is considerably more involved than in the gas-phase reactions. The thermochemistry of surfaces, clusters, and adsorbed species can be determined in a manner analogous to those associated with the gas-phase species. Similarly, rate parameters of heterogeneous elementary reactions can be estimated, via the application of the transition state theory, by determining the thermochemistry of saddle points on potential energy surfaces. 

Fig. 7. Energetics of a bimolecular rate process. Top Representation of the potential energy surface along coordinate axes corresponding to the interatomic distance of B-to-C and A-to-B, where incremental displacements along the potential energy axis are shown as a series of isoenergetic lines (each marked by arbitrarily chosen numbers to indicate increased energy of the transition-state (TS) intermediate relative to the reactants). Bottom Typical reaction coordinate diagram for a bimolecular group transfer reaction.

In transition-state theory, the absolute rate of a reaction is directly proportional to the concentration of the activated complex at a given temperature and pressure. The rate of the reaction is equal to the concentration of the activated complex times the average frequency with which a complex moves across the potential energy surface to the product side. If one assumes that the activated complex is in equilibrium with the unactivated reactants, the calculation of the concentration of this complex is greatly simplified. Except in the cases of extremely fast reactions, this equilibrium can be treated with standard thermodynamics or statistical mechanics . The case of... 

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