Transmission activated complex theory

Unfortunately, there are usually large discrepances in the different treatments of activated complex theory, concerning the definitions and the results of calculations of the "transmission coefficient" or the"tunneling correction"/19/ ... 

An equation in the form (78.Ill), with an unspecified transmission coefficient, was first derived by EYEING et al./9/, but is incorrectly considered as a rate expression of the activated complex theory. Actually, as pointed out by CHRISTOV /20 b /> if represents a semiclassical equation of the simple collision theory. 

A is very large are there gross discrepancies. However, there are other difficulties. According to the theory as developed in Section 9.3, the mean transmission coefficient is less than 1. Thus activated complex theory should always overestimate and therefore the 4-factor as well. Even recog-... 

Where k is the transmission coefficient, A(f is the difference in standard Gibb s free energy between the reactants and the transition state, Rg is the universal gas constant. Based on activated complex theory, the standard volume of activation (AE ) of a reaction is related to the pressure dependence of the reaction rate constant as expressed by. 

In transition state theory, dynamic effects are included approximately by including a transmission coefficient in the rate expression [9]. This lowers the rate from its ideal maximum TS theory value, and should account for barrier recrossing by trajectories that reach the TS (activated complex) region but do not successfully cross to products (as all trajectories reaching this point are assumed to do in TS theory). The transmission coefficient can be calculated by activated molecular dynamics techniques, in which molecular dynamics trajectories are started from close to the TS and their progress monitored to find the velocity at which the barrier is crossed and the proportion that go on to react successfully [9,26,180]. It is not possible to study activated processes by standard molecular dynamics because barrier crossing events occur so rarely. One reason for the... 

At the same time transition state theory requires information about the activated complexes, assumes equilibrium only for reactants, but not products and requires introduction of a special partition function (minus one degree of freedom). Another question which remains is the applicability of statistical thermodynamics, if the life time of activated complexes is ca. 10 13 s. For instance the application of transmission coefficient contradicts the basic principles of TST, namely statistical equilibrium between reactants and activated complexes. 

In the Eyring s theory of the absolute reactions rates there are three essential lacks a) the concentration of the activated complexes can be found from the consideration of condition of their equilibrium with the initial (or final) substances b) the activated complex devoided of one freedom degree along the coordinate of the reaction c) the transmission coefficient is the empirical co-multiplier. 

Thus, the approach based on taking into account the hierarchy of the time scales on the level of an elementary reaction usually leads to equations of the absolute rates of reaction theory, but, at this, allows to escape the contrasting postulate about an equilibrium of activated complexes with the initial substances, does not request the artificial introduction into the expression for the constant rate of the transmission coefficient and gives its analytical determination of the type (1.54) and (1.56). 

The theory proposes that the activated complex or transition state will proceed to products when the A— B bond has a thermal energy k T, so that the rate constant, k, will be proportional to the bond s vibrational frequency, v = k T/h s, with a proportionality constant, K, known as the transmission coefficient (k = Boltzmann s constant, 1.381xl0 erg K h s Planck s constant, 6.626xl0 erg s). It also is assumed that the activated complex is always in equilibrium with the reactant with a normal equilibrium constant, it = [A—B] /[A— B], so that... 

The expression (11.1) shows that the rate constant k depends on the choice of the activated complex model which determines the values of F" and of the transmission coefficient y. A precise evaluation of y is equivalent to the solution of the dynamical many-body problem analogous to problems of the collision theory. This problem is very difficult to solve and therefore y is usually assumed to be unity. 

We saw in Chapter 7 that the transmission coefficient k takes into account the fact that the activated complex does not always pass through to the transition state and the term kT/h arises from consideration of motions that lead to the decay of the activated complex into products. It follows that, in the case of an electron transfer process, K kT/h) can be thought of as a measure of the probability that an electron will move from D to A in the transition state. The theory due to R.A. Marcus supposes that this probability decreases with increasing distance between D and A in the DA complex. More specifically, for given values of the temperature and A G, the rate constant varies with the edge-to-edge distance... 

Sometimes the expression shown in Eq. (26.4-15) is multiplied by a transmission coefficient k, which represents the fraction of activated complexes that react. The transmission coefficient is commonly used as a correction factor to make the theory agree with experiment. It is probably better to omit this factor and to admit that the theory is approximate and cannot be expected to give exact agreement with experiment. 

The rate constant is defined by equation (2), according to the Theory of Absolute Reaction Rates (67,83). In equation (2), k refers to the specific reaction rate, the equilibrium between the normal and activated states of the reactants, AF the free energy, AH the heat, AJB the eneigy, AT the volume change, and AS the entropy, all of activation, p the hydrostatic pressure, T the absolute temperature, and R the gas constant. The expression K kT/h) is the universal frequency for the decomposition of the activated complex in all chemical reactions. In this, k is the transmission coefficient, usually equal to 1, IT the absolute temperature, Jb the Boltzmann constant, and h Planck s constant. 

In deriving the TST rate-coefficient formula it was assumed that the rate of reaction is identical with the rate of passage of activated complexes in one direction [Eq. (3.48)]. If TST is evaluated in terms of reactive trajectories through a dividing surfaces in phase space (equivalent to a transition state) it can be shown that the theory is exact (Pechukas and Poliak, 1979), provided that all trajectories move into the product region and none of them are reflected. To allow for reflective failures in the free passage assumption, the conventional TST expression is multiplied by a transmission coefficient k... 

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