Microcanonical VTST

The most sophisticated and computationally demanding of the variational models is microcanonical VTST. In this approach one allows the optimum location of the transition state to be energy dependent. So for each k(E) one finds the position of the transition state that makes dk(E)/dq = 0. Then one Boltzmann weights each of these microcanonical rate constants and sums the result to find fc ni- There is general agreement that this is the most reliable of the statistical kinetic models, but it is also the one that is most computationally intensive. It is most frequently necessary for calculations on reactions with small barriers occurring at very high temperatures, for example, in combustion reactions. 

Analytic potential energy functions for unimolecular reactions without reverse activation energies can be obtained by semi-empirical methods or by ab initio calculations, and enhanced by experimental information such as vibrational frequencies, bond energies, etc. To determine microcanonical VTST rate constants from such a potential function, the minimum in the sum of states along the reaction path must be determined. Two approaches have been used to calculate this sum of states. 

Microcanonical VTST minimizes the microcanonical rate coefficients, k E), and takes into account that the dividing surface location is most likely energy dependent... 

In Section 2.2, the significance of VTST in relation to the important class of reactions involving no barrier to recombination was outlined. The classical tracing methods described above find particularly fruitful application within the context of E- and 7-resolved microcanonical VTST [ VTST( , 7)] calculations of the microcanonical dissociation or association rate coefficients in such reaction systems. In this context, the crucial degrees of freedom which control the kinetics of the reaction are the fragment rotations which turn into hindered rotations and ultimately vibrations as the fragments recombine to form a metastable molecule. These degrees of freedom are referred to as the transitional modes, and a precise count of the sum of states J) is essential for accurate predic-... 

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. 

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). 

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). 

Like Eq. (27.2), Eqs. (27.11) and (27.12) are also hybrid quantized expressions in which the bound modes are treated quantum mechanically but the reaction coordinate motion is treated classically. Whereas it is difficult to see how quantum mechanical effects on reaction coordinate motion can be included in VTST, the path forward is straightforward in the adiabatic theory, since the one-dimensional scattering problem can be treated quantum mechanically. Since Eq. (27.12) is equivalent to the expression for the rate constant obtained from microcanonical variational theory [7, 15], the quantum correction factor obtained for the adiabatic theory of reactions can also be used in VTST. 

Microcanonical variational transition state theory (VTST)... 

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... 

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