Kinetic isotope effect general theory

This chapter mainly focuses on the reactivity of 02 and its partially reduced forms. Over the past 5 years, oxygen isotope fractionation has been applied to a number of mechanistic problems. The experimental and computational methods developed to examine the relevant oxidation/reduction reactions are initially discussed. The use of oxygen equilibrium isotope effects as structural probes of transition metal 02 adducts will then be presented followed by a discussion of density function theory (DFT) calculations, which have been vital to their interpretation. Following this, studies of kinetic isotope effects upon defined outer-sphere and inner-sphere reactions will be described in the context of an electron transfer theory framework. The final sections will concentrate on implications for the reaction mechanisms of metalloenzymes that react with 02, 02 -, and H202 in order to illustrate the generality of the competitive isotope fractionation method. 

Various quantum-mechanical theories have been proposed which allow one to calculate isotopic Arrhenius curves from first principles, where tunneling is included. These theories generally start with an ab initio calculation of the reaction surface and use either quantum or statistical rate theories in order to calculate rate constants and kinetic isotope effects. Among these are the variational transition state theory of Truhlar [15], the instanton approach of Smedarchina et al. 

Studies of kinetic-isotope effects have also provided valuable information about the mechanisms of reactions, and have been very helpful in elucidating biological mechanisms. Unfortunately, thje theory is somewhat complicated, and there are a few pitfalls. Only a very general and brief outline can be given here. For further details, with special reference to biological mechanisms, the reader is referred to the book by Laidler and Bunting listed in Selected Reading at the end of this chapter. 

B. C. Garrett and D. G. Truhlar, Generalized transition state theory calculations for the reactions D + H2 and H + D2 using an accurate potential energy surface Explanation of the kinetic isotope effect, /. Chem. Phys. 72 3460 (1980). 

Absolute rate coefficients and Arrhenius parameters have been obtained for the cycloaddition reaction of S( F2,1,0) atoms with a representative series of olefins and acetylenes. The activation energies are small, and they exhibit a trend with molecular structure which is expected for an electro-philic reagent The A-factors show a definite trend which can be attributed to steric repulsions and a generalized secondary a-isotope effect explained by activated complex theory. Secondary a-H/D kinetic isotope effects have been measured and their origin discussed. Hartree-Fock type MO calculations indicate that the primary product of the S( F) + olefin reaction is a ring-distorted, triplet state thi-irane, with a considerable energy barrier with respect to rotation around the C-C bond. 

All above conclusions are involved as special cases in the general consequences of the collision theory rate equation (51j III) derived in Sec.7.III. The corresponding consequences from the statistical formulation (67.Ill) of the reaction rate theory were also discussed there. The current interpretations of kinetic isotope effects are based on transition state theory. The correction for proton tunneling is first taken into consideration by BELL et al./155/. More extensive work in this direction has been carried out by CALDIN et al. /I53/. In this treatment estimations of the tunneling correction are made using one-dimensional (parabolic) barrier by neglecting the coupling of the proton motion with other motions of reactants or solvent. 

Absolute reaction rate theory requires a quasi-equilibrium to be set up between reactants and transition state. For a generalized example of a secondary kinetic isotope effect, we can express this ... 

Thus, the above analysis shows that the regularities of the kinetic isotope effect in enzymatic hydrolysis reactions confirm the basic results of the quantum-mechanical theory of an elementary act and contradict the results of the bond-stretching model. The concepts of the quantum-mechanical theory are found to be useful for discussing some specific aspects of the action of enzymes. Hence it is important to discuss the general corollaries of the theory as applied to enzymatic reactions and other biological processes. Some aspects of this problem will be discussed in the following section. 

There exists now an enormous literature on the applications of the classical or semiclassical collision and transition state theories to different types of chemical reactions in gas phase and in solution (see,for instance, /1,3,19a,35f49/). For our purposes it is sufficient to show the applicability of the general formulations presented in Chapter III to some simple gas phase and dense phase reactions. In this way we would like to demonstrate, first, the computational possibility of these formulations, and, second, their utility for an understanding of the influence of various factors, such as nonseparability effects, quantum effects, isotope effects a.o. on the kinetic parameters. 

In the particular reaction used as an illustration, there are one primary and two secondary isotope effects and a good deal of ancillary kinetic evidence to fall back on. Generally, however, the burden of uncertainty is much too great for a minute and not always a very precisely determined secondary isotope effect to bear. Small wonder then that many workers in the field have preferred to lash themselves firmly to the formal framework of isotope effect theory After a model for the transition state is assumed, bond lengths and angles can be altered, force constants increased and decreased ad libitum, until the experimental result is more or less comfortably accommodated in the Procrustean bed of theory. How much useful mechanistic information is derived by this procedure is open to question. 

The most serious limitatioa to the application of reaction rate theory to secondary isotope effects is our complete inability to ascertain the geometry and vibrational frequencies of the transition state by extra-kinetic methods. In the discussion of specific examples, reference will be made to a number of attempts to calculate kinetic secondaiy isotope effects in terms of eq. (III-14) or its equivalent, on the basis of different transition state models. As Miller (60), who carried out numerous calculations of this kind puts it, they may be regarded as an exercise in the adjustment of theoretical models of activated complexes to experimental data. Of course, this adjustment generally is possible because the disposable parameters are many. Still, the adjustment is hardly arbitrary, and if numerical agreement between a calculated isotope effect and experiment cannot be taken as proof of the correctness of the model, a model that yields an isotope effect in sharp disagreement with experiment can be excluded as being in all probability unrealistic. 

Kinetic effects have been of great interest in cosmochemistry since the mid-1990s and we now review the generally accepted theory of evaporation and condensation in some detail. This discussion focuses on isotope fractionations, but is equally applicable to elemental fractionations. 

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