Canonical VTST

Variational transition-state theory has been formulated on various levels [5, 23-27]. At first, there is the group of canonical VTST (CVTST) treatments, which correspond to the search for a maximum of the free energy AG(r) along the reaction path r [23, 24]. It was noticed early that for barri-erless potentials this approach leads to an overestimate of the rate constant because, in the language of SACM, channels are included that are closed. Therefore, an improved version (ICVTST) was proposed [25] that truncates Q at the position r of the minimum of (t(r) by including only states... 

Cf. R. A. Marcus, J. Chem. Phys. 45,2630 (1966). This paper contains this criterion (p. 2635), but mistakenly ascribes it to Bunker, who actually uses, instead, a minimized density of states criterion [D. L. Bunker and M. Pattengill, J. Chem. Phys. 48, 772 (1968)]. This minimum number of states criterion has been used by various authors, for example, W. L. Hase, J. Chem. Phys. 57, 730 (1972) 64, 2442 (1976) M. Quack and J. Troe (Ref. 21) B. C. Garrett and D. G. Truhlar, J. Chem. Phys. 70, 1593 (1979). The transition state theory utilizing it is now frequently termed microcanonical variational transition state theory (/iVTST). A recent review of /tVTST and of canonical VTST is given in D. G. Truhlar and B. C. Garrett, Ann. Rev. Phys. Chem. 35,159 (1984). 

This defines a different transition state for each energy, and is an improvement over canonical VTST, which does not incorporate any energy dependence of recrossing. CTST gives very good estimates of rate coefficients for many reactions. For some types of reactions, most notably reactions without energy barriers, it is not very accurate. In these cases VTST should be used. Implementation of VTST requires a computer, as... 

Several VTST techniques exist. Canonical variational theory (CVT), improved canonical variational theory (ICVT), and microcanonical variational theory (pVT) are the most frequently used. The microcanonical theory tends to be the most accurate, and canonical theory the least accurate. All these techniques tend to lose accuracy at higher temperatures. At higher temperatures, excited states, which are more difficult to compute accurately, play an increasingly important role, as do trajectories far from the transition structure. For very small molecules, errors at room temperature are often less than 10%. At high temperatures, computed reaction rates could be in error by an order of magnitude. 

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 formation of the transition state from the excited molecule is referred to as a microcanonical process, while the formation of the transition state in conventional TST in Chapter4 and in VTST in Chapter 6 is referred to as canonical process. The terms microcanonical and canonical in statistical mechanics refer respectively to processes at constant energy and processes at constant temperature. 

The application of the VTST to reactions in solution has to face several computational problems, of the type we have discussed for the canonical static description of the reaction mechanism. Moreover, there is another important problem to face, not present when this approach is applied to reaction in vacuo, which literally adds new dimensions to the model. We shall consider now this point, even if shortly. 

Instead of this type of model, we suggest utilization of TST in the recently developed variational versions (Lauderdale and Truhlar 1985,1986 Truhlar et al. 1986 Truong et al. 1989b). These can be either microcanonical (E-,) or canonical (T) forms. These also include multidimensional tunneling via least-action techniques as well as surface atom motion via embedded cluster models. They do require a PES, but in our opinion, it is preferable to at least indicate a PES rather than make a multitude of assumptions about the dynamics. Because of the speed of the VTST methods, large computing facilities would not be necessary, and a well-documented program exists for these calculations, which should be available by the time of publication of this review (from QCPE at the University of Indiana). 

A viable alternative for small systems is variational transition state theory or VTST (see Truhlar et al. 1985). Recall that TST makes use of the non-recrossing rule assumption. When recrossing does occur, the assumption results in the over-counting of transitions from reactants to products that is, the TST rate constant is an upper bound. In VTST, a divide is sought that minimizes these transitions resulting in a minimum rate constant and this divide becomes the basis for the VTST rate constant. We consider, as the simplest example, canonical variational ensemble transition state theory (CVT). 

CVT = canonical variational transition state theory /xVT = microcanonical variational transition state theory TST=transition state theory VTST = variational TST. 

It has been shown that the pOMT transmission coefficients are comparable in accuracy with the LAT transmission coefficients for atom-diatom reac-tions. Often we just say OMT without including the microcanonical specification in the algorithm (OMT can also mean canonical OMT in which we first thermally average the SCT and LCT probabilities and then choose the larger transmission coefficient). The resulting VTST/OMT rate constants... 

At this point one can include optimized multidimensional tunneling in each (i = 1,2,..., 7) of the VTST calculations. The tunneling transmission coefficient of stage 2 for ensemble member i is called and is evaluated by treating the primary zone in the ground-state approximation (see the section titled Quantum Effects on Reaction Coordinate Motion ) and the secondary zone in the zero-order canonical mean shape approximation explained in the section titled Reactions in Liquids , to give an improved transmission coefficient that includes tunneling ... 

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