Microcanonical variational transition state theory

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... 

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,... 

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... 

TST = conventional Transition State Theory, ICVT = Improved Canonical Variational Transition state theory, ICVT/SCT = ICVT/Small Curvature Tunneling, ICVT/p,OMT = ICVT/Microcanonical Optimized Multidimensional Tunneling. 

An efficient implementation of microcanonical classical variational transition state theory was applied to Si—H bond fission in SiFF and compared with trajectory calculations on the same potential surface235. 

Variational transition state theory was suggested by Keck [36] and developed by Truhlar and others [37,38]. Although this method was originally applied to canonical transition state theory, for which there is a unique optimal transition state, it can be applied in a much more detailed way to RRKM theory, in which the transition state can be separately optimized for each energy and angular momentum [37,39,40]. This form of variational microcanonical transition state theory is discussed at length in Chapter 2, where there is also a discussion of the variational optimization of the reaction coordinate. 

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). 

Another system where accurate microcanonical rate constants have been calculated is Li + HF - LiF + H with 7 = 0 (172). This reaction has variational transition states in the exit valley. Variational transition state theory agrees very well with accurate quantum dynamical calculations up to about 0.15 eV above threshold. After that, deviations are observed, increasing to about a factor of 2 about 0.3 eV above threshold. These deviations were attributed to effective barriers in the entrance valley these are supernumerary transition states. After Gaussian convolution of the accurate results, only a hint of step structure due to the variational transition states remains. Densities of reactive states, which would make the transition state spectrum more visible, were not published (172). 

The microcanonical and canonical variational transition-state theories are based on the assumption that trajectories cross the transition state (TS) only once in forming products(s) or reactants(s) [70,71]. The correction to the transition-state theory rate constant is determined by initializing trajectories at the TS and sampling their coordinates and momenta from the appropriate statistical distribution [72-76]. The value for is the number of trajectories that form product(s) divided by the number of crossings of the TS in the reactant(s) -> produces), direction. Transition state theories assume this ratio is unity. 

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. 

For a reaction with a defined transition state and without recrossing, reaction rate can be well approximated by many methods. For such reaction, we can assume that there is a dynamics bottleneck located at the transition state (conventional transition state theory, TST) or at a generalized transition state obtained by a canonical (CTV) or microcanonical (/zVT) criterion. In the later cases, the dividing surface is optimized variationally to minimize the recrossing. Evans first proposed to place the transition state at the location that maximizes the free energy of activation which provides a key conceptual framework for modern variational transition state theory [33]. However, recrossing always possibly exists and only a full-dimensional reactive scattering dynamics calculations are able to provide us the exact rate constant on a defined PES. Eor a detailed discussion, one may refer to the reviews by Truhlar et al. [38,136]. 

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... 

Notice that the minimum-number-of-states criterion corresponds correctly to variational transition state theory, whereas an earlier minimum-density-of-states criterion does not. The microcanonical rate constant can be written as... 

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.

R. A. Marcus My interests in variational microcanonical transition state theory with J conservation goes back to a J. Chem. Phys. 1965 paper [1], and perhaps I could make a few comments. First, using a variational treatment we showed with Steve Klippenstein a few years ago that the transition-state switching mentioned by Prof. Lorquet poses no major problem The calculations sometimes reveal two, instead of one, bottlenecks (transition states, position of minimum entropy along the reaction coordinate) [2], and then one can use a method described by Miller and partly anticipated by Wigner and Hirschfelder to calculate the net dux. 

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. 

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