Partition function transition state theory

Finally, the generalization of the partition function q m transition state theory (equation (A3.4.96)) is given by... 

These equations lead to fomis for the thermal rate constants that are perfectly similar to transition state theory, although the computations of the partition functions are different in detail. As described in figrne A3.4.7 various levels of the theory can be derived by successive approximations in this general state-selected fomr of the transition state theory in the framework of the statistical adiabatic chaimel model. We refer to the literature cited in the diagram for details. 

This is connnonly known as the transition state theory approximation to the rate constant. Note that all one needs to do to evaluate (A3.11.187) is to detennine the partition function of the reagents and transition state, which is a problem in statistical mechanics rather than dynamics. This makes transition state theory a very usefiil approach for many applications. However, what is left out are two potentially important effects, tiiimelling and barrier recrossing, bodi of which lead to CRTs that differ from the sum of step frmctions assumed in (A3.11.1831. 

Figure 10. Arrhenius plot of the thermal rate constants for the 2D model system. Circles-full quantum results. Thick solid (dashed) curve present nonadiabatic transition state theory by using the seam surface [the minimum energy crossing point (MECP)] approximation. Thin solid and dashed curves are the same as the thick ones except that the classical partition functions are used. Taken from Ref. [27].

To understand how collision theory has been derived, we need to know the velocity distribution of molecules at a given temperature, as it is given by the Maxwell-Boltzmann distribution. To use transition state theory we need the partition functions that follow from the Boltzmann distribution. Hence, we must devote a section of this chapter to statistical thermodynamics. 

Because the frequency of a weakly bonded vibrating system is relatively small, i.e. kBT hu we may approximate its partition function by the classical limit k T/hv, and arrive at the rate expression in transition state theory ... 

If hu0 is small compared with kT, the partition function becomes kT/hv0. The function kT/h which pre-multiplies the collision number in the transition state theory of the bimolecular collision reaction can therefore be described as resulting from vibration of frequency vq along the transition bond between the A and B atoms, and measures the time between each potential transition from reactants to product which will only occur provided that the activation energy, AE°0 is available. 

The rate of reaction from transition state theory is given by equation (4.31) as Rate of reaction = A v Therefore, in terms of partition function... 

Transition state theory in terms of partition function)... 

In earlier sections of this chapter we learned that the calculation of isotope effects on equilibrium constants of isotope exchange reactions as well as isotope effects on rate constants using transition state theory, TST, requires the evaluation of reduced isotopic partition function ratios, RPFR s, for ordinary molecular species, and for transition states. Since the procedure for transition states is basically the same as that for normal molecular species, it is the former which will be discussed first. 

Of course, one is not really interested in classical mechanical calculations. Thus in normal practice the partition functions used in TST, as discussed in Chapter 4, are evaluated using quantum partition functions for harmonic frequencies (extension to anharmonicity is straightforward). On the other hand rotations and translations are handled classically both in TST and in VTST, which is a standard approximation except at very low temperatures. Later, by introducing canonical partition functions one can direct the discussion towards canonical variational transition state theory (CVTST) where the statistical mechanics involves ensembles defined in terms of temperature and volume. There is also a form of variational transition state theory based on microcanonical ensembles referred to by the symbol p,. Discussion of VTST based on microcanonical ensembles pVTST is beyond the scope of the discussion here. It is only mentioned that in pVTST the dividing surface is... 

The use of transition state theory as a convenient expression of rate data is obviously complex owing to the presence of the temperature-dependent partition functions. Most researchers working in the area of chemical kinetic modeling have found it necessary to adopt a uniform means of expressing the temperature variation of rate data and consequently have adopted a modified Arrhenius form... 

The conversion of (kgT/h) to (co/it) may be derived within the Eyring Transition State Theory as due to the inclusion in the prefactor of the reactant vibrational (harmonic) partition function. 

The rate constants, k+ and k of the forward and backward reactions are finally derived from (12) and (13) according to the transition-state theory, i.e. assuming that the transition and the initial states, on the one hand, and the transition and final states, on the other, are in equilibrium (Glasstone et al., 1941). Thus, estimating the partition function of these three states in the classical way gives (18) and (19), where p is the reduced mass of the two reactants in the homogeneous case and m the mass of the reactant in the electrochemical case. 

Occasionally, the rates of bimolecular reactions are observed to exhibit negative temperature dependencies, i.e., their rates decrease with increasing temperature. This counterintuitive situation can be explained via the transition state theory for reactions with no activation energy harriers that is, preexponential terms can exhibit negative temperature dependencies for polyatomic reactions as a consequence of partition function considerations (see, for example, Table 5.2 in Moore and Pearson, 1981). However, another plausible explanation involves the formation of a bound intermediate complex (Fontijn and Zellner, 1983 Mozurkewich and Benson, 1984). To... 

Following Fey nman s original work, several authors pmsued extensions of the effective potential idea to construct variational approximations for the quantum partition function (see, e g., Refs. 7,8). The importance of the path centroid variable in quantum activated rate processes was also explored and revealed, which gave rise to path integral quantum transition state theory and even more general approaches. The Centroid Molecular Dynamics (CMD) method for quantum dynamics simulation was also formulated. In the CMD method, the position centroid evolves classically on the efiective centroid potential. Various analysis and numerical tests for realistic systems have shown that CMD captures the main quantum effects for several processes in condensed matter such as transport phenomena. 

Some important systems, which certainly do not fulfill the assumptions of harmonic transition state theory are gas phase reactions. In the gas phase, there are zero-modes such as translation and rotation, and these lead to totally different configuration integrals than those obtained from a normal mode analysis. For these species one can in a simple manner modify the terms going into the HTST rate by incorporating the molecular partition functions [3,119]. 

By considering the symmetry of the normal modes of transition states Murrell and Laidler showed that problems encountered when calculating the statistical factors of transition states (which are needed to calculate the partition function in transition state theory) were associated with configurations of too high a symmetry to be transition states (61, 62). 

It is known that the isotope effect may occur not only because of tunneling. Within the framework of the transition-state theory, which does not take tunneling into account, the isotope effect is explained by the variations of the energies of the ground vibrational levels and by the variations of the partition functions of the reagents and of the activated complex upon changing one isotope for another [53]. To make it clear to what extent the isotope effect in reactions (a) (d) is connected with tunneling, it is useful to mention... 

The statistical mechanical contribution to transition state theory uses partition functions. These are statistical mechanical quantities made up from translational, rotational, vibrational and electronic terms, though the electronic terms can normally be ignored if the reaction occurs in the ground state throughout. 

Comparison of collision theory, the partition function form and the thermodynamic form of transition state theory... 

Theoretical calculations are less fundamental and rigorous for solution reactions. This is a consequence of the difficulty of calculating partition functions in solution. The main focus for solution reactions has been on the thermodynamic formulation of transition state theory. 

In connection with transition-state theory, one will also occasionally meet the concept of a statistical factor [13]. This factor is defined as the number of different activated complexes that can be formed if all identical atoms in the reactants are labeled. The statistical factor is used instead of the symmetry numbers that are associated with each rotational partition function (see Appendix A.l) and, properly applied, the... 

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