Microcanonical variational theory

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. 

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. 

Quack and Troe suggested that the microcanonical variational theory is conceptually interesting but hardly worthwhile computationally (when viewed as an approximation to the SACM, although such computations may find use in other methods). 

The transmission coefficient described above is appropriate" for correcting the adiabatic theory or equivalently the microcanonical variation theory,... 

Equation (ASA. 110) represents the canonical fonn T= constant) of the variational theory. Minimization at constant energy yields the analogous microcanonical version. It is clear that, in general, this is only an approximation to the general theory, although this point has sometimes been overlooked. One may also define a free energy... 

Because T -> V energy transfer does not lead to complex formation and complexes are only formed by unoriented collisions, the Cl" + CH3C1 -4 Cl"—CH3C1 association rate constant calculated from the trajectories is less than that given by an ion-molecule capture model. This is shown in Table 8, where the trajectory association rate constant is compared with the predictions of various capture models.9 The microcanonical variational transition state theory (pCVTST) rate constants calculated for PES1, with the transitional modes treated as harmonic oscillators (ho) are nearly the same as the statistical adiabatic channel model (SACM),13 pCVTST,40 and trajectory capture14 rate constants based on the ion-di-pole/ion-induced dipole potential,... 

Calculational estimates of the lifetimes of the trimethylene diradical " based on microcanonical variational unimolecular rate theory and direct dynamics simulations have been reported. The lifetimes derived from theory, 91 and 118 fs, are comparable to the experimental estimate, 120 20 fs. Similar lifetime estimates from theory for tetramethylene are comparable, or slightly below, the experimental value. ... 

The SN2 identity exchange reaction of chloride with chloroacetonitrile (equation 23) was studied in an FT-ICR spectrometer and also theoretically by means of statistical theories (RRKM with the microcanonical variational transition state) and high-level ab initio calculations238. 

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). 

The transition state is taken to be the (3V - 7)-dimensional plane orthogonal to the reaction path at that value of s which minimizes the TST rate. For reactions at given energy [microcanonical variational transition-state theory (/uVT)], the bimolecular rate constant is given by [125]... 

S. C. Tucker and E. Poliak, J. Stat. Phys., 66, 975 (1992). Microcanonical Variational Transition State Theory for Reaction Rates in Dissipative Systems. 

Microcanonical variational transition state theory (VTST)... 

W (E, J) has an obvious interpretation as the number of open channels at the location q of the transition state. As one varies the position q of the transition state one finds the optimum choice at the position of a minimum value, W(E.J.q) . Therefore, such approximations are also called microcanonical variational transition state theory. Sometimes the variational procedure has been used with the minimum density of states criterion in equation (117) ... 

Similar to a microcanonical variational optimization, one can also carry out a canonical variational transition state theory defining a -dependent partition function by equation (121) ... 

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

Classical Methods in Microcanonical Variational Transition State Theory... 

Fig. 2. Cumulative reaction probability for collinear H + H2 H2+H and collinear D 4- H2 DH + H as a function of energy above the barrier (E-V ). For each reaction the results obtained by conventional transition state theory and microcanonical variational transition state theory are both shown.

It may be iisefiil to mention here one currently widely applied approximation for barrierless reactions, which is now frequently called microcanonical and canonical variational transition state theory (equivalent to the minimum density of states and maximum free energy transition state theory in figure A3,4,7. This type of theory can be understood by considering the partition fiinctions Q r ) as fiinctions of r similar to equation (A3,4.108) but with F (r ) instead of V Obviously 2(r J > Q so that the best possible choice for a... 

As a result of possible recrossings of the transition state, the classical RRKM lc(E) is an upper bound to the correct classical microcanonical rate constant. The transition state should serve as a bottleneck between reactants and products, and in variational RRKM theory [22] the position of the transition state along q is varied to minimize k E). This minimum k E) is expected to be the closest to the truth. The quantity actually minimized is N (E - E ) in equation (A3.12.15). so the operational equation in variational RRKM theory is... 

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