Statistical transition state method

The rest of this paper will be devoted to the consideration of the second kind of reactions. I shall endeavour rather to emphasise the basic assumptions of the theory than to derive ready formulas. Especially on account of some discrepancies with experiment, I think that it may be useful to see that the transition state method is based, in addition to well-established principles of statistical mechanics, on only three assumptions, two of which arc generally accepted. 

In this way the transition state method was first applied to actual energy surfaces. Cf. H. Pelzer and E. Wigner, 2. physik. Chem., B, 1932, 15, 445. The quantum corrections were included ibid., 1932, 19, 203. Formulae similar to those resulting from the transition state method were obtained not much later by W. H. Rodebush, J. Chem. Physics, 1933, 44° by O. K. Rice and H. Gershinowitz, ibid., 1934, 3, 853. However, these are very ingenious guesses rather than real derivations based on statistical mechanics. Consequently, the results are not quite identical with those of the transition state method. 

E. Wigner (1938) The transition state method. Trans. Faraday Soc. 34, p. 29 T. Yamamoto (1960) Quantum statistical mechanical theory of the rate of exchange chemical reactions in the gas phase. J. Chem Phys. 33, p. 281... 

It must be admitted that the mass action law has been successfully applied to homogeneous elementary reactions, while the statistical independence of the system underlying it is premised in the absolute reaction rate theory of Eyring et al. (1) and in the transition state method of Evans and Polanyi 2). The statistical independence is not however insured in the treatment, especially of heterogeneous elementary reactions constituting heterogeneous catalyses as exemplified below this constitutes the second limitation of the kinetics. 

A quantitative theory of rate processes has been developed on the assumption that the activated state has a characteristic enthalpy, entropy and free energy the concentration of activated molecules may thus be calculated using statistical mechanical methods. Whilst the theory gives a very plausible treatment of very many rate processes, it suffers from the difficulty of calculating the thermodynamic properties of the transition state. 

Pratt L R 1986 A statistical method for identifying transition states in high dimensional problems J. Chem. Phys. 85 5045-8... 

The hrst step in theoretical predictions of pathway branching are electronic structure ab initio) calculations to define at least the lowest Born-Oppenheimer electronic potential energy surface for a system. For a system of N atoms, the PES has (iN — 6) dimensions, and is denoted V Ri,R2, - , RiN-6)- At a minimum, the energy, geometry, and vibrational frequencies of stationary points (i.e., asymptotes, wells, and saddle points where dV/dRi = 0) of the potential surface must be calculated. For the statistical methods described in Section IV.B, information on other areas of the potential are generally not needed. However, it must be stressed that failure to locate relevant stationary points may lead to omission of valid pathways. For this reason, as wide a search as practicable must be made through configuration space to ensure that the PES is sufficiently complete. Furthermore, a search only of stationary points will not treat pathways that avoid transition states. 

One disadvantage of statistical approaches is that they rely on two of the assumptions stated in the introduction, namely, that reactions follow the minimum energy path to each product channel, and that the reactive flux passes through a transition state. Several examples in Section V violate one or both of these assumptions, and hence statistical methods generally cannot treat these instances of competing pathways [33]. 

Various statistical treatments of reaction kinetics provide a physical picture for the underlying molecular basis for Arrhenius temperature dependence. One of the most common approaches is Eyring transition state theory, which postulates a thermal equilibrium between reactants and the transition state. Applying statistical mechanical methods to this equilibrium and to the inherent rate of activated molecules transiting the barrier leads to the Eyring equation (Eq. 10.3), where k is the Boltzmann constant, h is the Planck s constant, and AG is the relative free energy of the transition state [note Eq. (10.3) ignores a transmission factor, which is normally 1, in the preexponential term]. 

Although the collision and transition state theories represent two important methods of attacking the theoretical calculation of reaction rates, they are not the only approaches available. Alternative methods include theories based on nonequilibrium statistical mechanics, stochastic theories, and Monte Carlo simulations of chemical dynamics. Consult the texts by Johnson (62), Laidler (60), and Benson (59) and the review by Wayne (63) for a further introduction to the theoretical aspects of reaction kinetics. 

A method for the estimation of thermodynamic properties of the transition state and other unstable species involves analyzing parts of the molecule and assigning separate properties to functional groups (Benson, 1976). Another approach stemming from statistical mechanics is outlined in the next section. 

Beyond the clusters, to microscopically model a reaction in solution, we need to include a very big number of solvent molecules in the system to represent the bulk. The problem stems from the fact that it is computationally impossible, with our current capabilities, to locate the transition state structure of the reaction on the complete quantum mechanical potential energy hypersurface, if all the degrees of freedom are explicitly included. Moreover, the effect of thermal statistical averaging should be incorporated. Then, classical mechanical computer simulation techniques (Monte Carlo or Molecular Dynamics) appear to be the most suitable procedures to attack the above problems. In short, and applied to the computer simulation of chemical reactions in solution, the Monte Carlo [18-21] technique is a numerical method in the frame of the classical Statistical Mechanics, which allows to generate a set of system configurations... 

Abstract The statistical thermodynamic theory of isotope effects on chemical equilibrium constants is developed in detail. The extension of the method to treat kinetic isotope effects using the transition state model is briefly described. 

Transition state theory yields rate coefficients at the high-pressure limit (i.e., statistical equilibrium). For reactions that are pressure-dependent, more sophisticated methods such as RRKM rate calculations coupled with master equation calculations (to estimate collisional energy transfer) allow for estimation of low-pressure rates. Rate coefficients obtained over a range of temperatures can be used to obtain two- and three-parameter Arrhenius expressions ... 

Using statistical-dynamical methods and transition state theory, Zhang and co workers demonstrated that excited carbonyls dissociate promptly to prodnce OH radicals (11%) or isomerize to form dioxirane (32%) or are collisionally stabilized (57%) . 

R. Schinke Actually, Dr. Klippenstein is currently applying his methods of defining the best transition state to HNO and it will be interesting to see how his results agree or disagree with our estimations of the statistical rate using the usual scattering (Jacobi) coordinates. 

We distinguish two limiting cases dissociation through a narrow transition state and dissociation through a wide transition state where we define the region that separates the reactants and the products (i.e., the point of no return ) as the transition state. The first case may be qualitatively considered as a direct process with the ultimate dissociation starting at the transition state. The second case may be treated by statistical methods without including dynamical constraints. We will discuss both limits separately and illustrate them with typical examples. 

In some other cases, more elaborate statistical mechanics methods are needed to calculate the free energies of the reactants and the transition state. This occurs whenever the range of geometries sampled by the system goes well beyond the vicinity of the relevant stationary point, that is, the reactant minimum or the saddle point. Some examples of this type of behavior will be described below. Also, in some cases, atomic motion is not well described by classical mechanics, and although TST incorporates some quantum mechanical aspects, it does not typically include others, and more advanced methods are needed to describe reactions in such cases. Again, some examples will be given below. 

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