Classical transition state theory

From classical transition state theory, the phenomenological rate constant of ET can be expressed as... 

P. Pechukas and E. Poliak, Classical transition state theory is exact if the transition state is unique, J. Chem. Phys. 71, 2062 (1979). 

As in classical transition state theory, the PMF, w(z), can be computed from the equilibrium average ... 

So far, only the nuclear reorganization energy attending electron transfer has been discussed, yielding the expressions above of the free energy of activation in the framework of classical transition state theory. A second series of important factors are those that govern the preexponential factor, k, raising in particular the question of the adiabaticity or nonadiabaticity of electron transfer between a molecule and the electronic states in the electrode. 

We start from classical transition state theory which provides the basic factorization of the electron transfer rate constant ... 

Poliak, E., Child, M.S., and Pechukas, P, (1980). Classical transition state theory A lower bound to the reaction probability, J. Chem. Phys. 72, 1669-1678. 

Corrections to transition-state theory due to quantum tunneling along the reaction coordinate give a thermal rate constant that is larger than the prediction obtained from classical transition-state theory. 

If we suppose now, as is the case in classical transition state theory, that the molecules in the ground (adsorbed state) and transition states (delocalized movement state) are in equilibrium, it can be easily shown that the equilibrium constant for this equilibrium process is [12,109]... 

The difference between the QM/MM-calculated energy barriers for the rate-determining steps of the two enzymatic reaction systems is consistent with the experimental observation that the fccat value (1.6 x 10" s ) [124] for AChE-catalyzed hydrolysis of ACh was about 150-fold larger than that (fccat = 1.07 X 10 s ) [97] for BChE-catalyzed hydrolysis of (+)-cocaine. Based on the widely used classical transition-state theory (CTST) [125], the experimental fccat difference of 150-fold suggests an energy barrier difference of 3.0 kcal/mol when T = 298.15 K, which is in good agreement with the calculated barrier difference of 3.7 kcal/mol [113]. 

A number of the R -f O2 reactions show a negative temperature dependence. It is not possible to explain this by classical transition state theory (where the transition state is of a defined structure and located at a constant position on the reaction coordinate) and the reaction needs to be modelled using the variational techniques discussed earlier for methyl -I- methyl and methyl -f H (Section 2.5.4). 

The profile of the potential energy surface obtained by Brudnik et al. 25 at the G2 level is shown in Fig. 17. When the loosely bound intermediates are not stabilized by collisions, they can be omitted in the reaction mechanism. The kinetics of the reaction can, in a first approximation, be described by the rate constant obtained from classical transition state theory. The rate constant calculations of Brudnik et al 25 show that this approach is realistic at temperatures below 1000 K. The temperature dependence of the rate constants calculated for CF3O + H20 can be expressed as... 

The stability of proteins can be viewed from kinetic as well as from thermodynamic considerations. Here we give the thermodynamic description and note that the kinetic description would be equivalent in view of the thermodynamic basis of the classical transition state theory. An example of the treatment of kinetic data of the stability of enzymes is given by Weemaes et al. [78]. 

An example of pressure effects on the dynamic properties of liquids is given in the study by Hasha et al. of conformational isomerization in liquid cyclohexane. In contrast to classical transition-state theory, stochastic models predict that for such reactions the transmission coefficient, k, should depend on the collision frequency between the solvent and the solute molecules, which is a measure of the coupling of the reaction coordinate with the... 

In classical transition-state theory, the expression for the rate constant of a bi-molecular reaction in solution is... 

The calculated energies and barriers can be readily related to the reaction rates using classical transition state theory (TST). The relationship between the rate constant /f of a reaction and the free energy of activation (AG ) can be expressed as ... 

An important feature of classical transition state theory is that it is an upper hound to the correct result for any choice of the dividing surface. That is, since all reactive trajectories must cross the dividing surface, but all trajectories that cross it are not necessarily reactive (because they might recross it at a later time and be nonreactive), any error in the TST approximation, Eq. (12), is to count some nonreactive trajectories as reactive. Thus, while the exact rate expression does not depend on the choice of the dividing surface, the TST rate does, and by virtue of this bounding property the best choice of the dividing surface is the one which makes kTS, a minimum. This is the variational aspect of TST any parameters which specify the shape or location of the dividing surface are best chosen to minimize the TST rate (7). 

Figure 2 Reaction probability for the collinear H + H2 reaction on the Porter-Karplus potential surface from a microcanonical classical trajectory calculation (CLDYN) and microcanonical classical transition state theory (CLTST) as a function of total energy above the barrier height (1 eV = 23.06 kcal/mole).

There are several ways to derive the RRKM equation (Forst, 1973). The one adopted here is based on classical transition state theory and was first proposed by Wigner (Wigner, 1937 Hirschfelder and Wigner, 1939). Although there are several other statistical formulations of the unimolecular rate [phase space theory (Pechukas and Light, 1965), statistical adiabatic channel model (SACM) (Quack and Troe, 1974),... 

This formula may be readily applied to a variety of problems because, in the spirit of classical transition-state theory [79-84], only equilibrium information is required in order to estimate the rate constant. Additionally, the excess free energy in Eq. (4.5) can be calculated directly from PIMC techniques with umbrella sampling (this has now been done many times see, e.g.. Sections IV.B.l and IV.B.2). Equation (4.7) has also been rederived by Stuchebrukhov [95] using a different analysis. 

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