Transition state theory linear case, linearization

The case of m = Q corresponds to classical Arrhenius theory m = 1/2 is derived from the collision theory of bimolecular gas-phase reactions and m = corresponds to activated complex or transition state theory. None of these theories is sufficiently well developed to predict reaction rates from first principles, and it is practically impossible to choose between them based on experimental measurements. The relatively small variation in rate constant due to the pre-exponential temperature dependence T is overwhelmed by the exponential dependence exp(—Tarf/T). For many reactions, a plot of In(fe) versus will be approximately linear, and the slope of this line can be used to calculate E. Plots of rt(k/T" ) versus 7 for the same reactions will also be approximately linear as well, which shows the futility of determining m by this approach. 

Consider the case of a single collective solvent coordinate y. This coordinate is linearly coupled to the solute at the transition state by generalized Langevin theory [71-75], (It is not necessary to couple the solvent to the solute for the calculation of reactant properties because we retain the equilibrium-reactant approximation.) The form of the coupling is [60,61]... 

The orbital requirements for radical attack on any polyene are given in Table 6. If H3, HC2 and Cl8 (see Walsh diagram, Fig. 2) can be taken as models, then three-center transition states will be linear. If, however, cyclic transition states can be formed, HMO theory indicates a preference for them (Fig. 1). Unfortunately, attempted radical displacements have not been observed, simply because the radicals take other reaction paths (Pryor, 1966). The transition states may have been linear, but for abstraction from rather than displacement on carbon (Bujake et al., 1961). If the radical and molecule generated in these cases remain in... 

It is very fortunate that in this very important case, it was possible to devise an exact theoretical form of a classical transition state. This theory was discovered in the late 1970s by Pechukas, Child, and Poliak [39] and may be intuitively understood. We wish to replace the linearized motion by an analysis of the full Hamiltonian (21). Let us examine Fig. 8. 

Transition states for analogous reactions with rt-nucleophiles, such as alkenes, were calculated by Radom and co-workers to have a co-linear structure <1998JA7063>. Radom and co-workers <2000CEJ590> disagree with Modena et al. <1999JA3944, 2000GEJ589> about the structures of transition states for attack by monocentric and bicentric nucleophiles, arguing, on the basis of theory, that both are probably collinear or nearly collinear. Apparently the matter is not settled, possibly because the theoretical calculations are for the gas phase whereas the experiments were done in solution. Moreover, the substrates were different in the two cases. 

In the previous section, we met three situations where the localized bond model fails. In each case, we had to represent bonding in terms of MOs covering three or more atoms, formed by overlapping of AOs on them. Now the general rules for formation of such many-center MOs are basically the same as for normal two-center bonds. The AOs involved must be of comparable energies (i.e., all from valence shell AOs of the participating atoms) and they must overlap in space. How they overlap is not important. There is no basic distinction between the 7r-type overlap of p AOs in benzene (Fig. 1.37), the linear a-type overlap in the transition state of Fig. 1.39, and the nonlinear a-type overlap of AOs in diborane (Fig. 1.36). All three situations are equivalent in terms of MO theory. The distinction between them arises solely from quantitative considerations of the efficiency of overlap and the energies of the orbitals involved. 

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