Variational RRKM theory

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... 

Variational RRKM theory is particularly important for imimolecular dissociation reactions, in which vibrational modes of the reactant molecule become translations and rotations in the products [22]. For CH —> CHg+H dissociation there are tlnee vibrational modes of this type, i.e. the C—H stretch which is the reaction coordinate and the two degenerate H—CH bends, which first transfomi from high-frequency to low-frequency vibrations and then hindered rotors as the H—C bond ruptures. These latter two degrees of freedom are called transitional modes [24,25]. C2Hg 2CH3 dissociation has five transitional modes, i.e. two pairs of degenerate CH rocking/rotational motions and the CH torsion. 

In fact, the lifetime of excited CHT itself was too short to measure, but k( ) values of 2.5 x 107, 1.6 x 106 and 2x 106 s 1 were recorded, respectively, for Me-CHT, Et-CHT, and Me2-CHT. These variations of lifetime with structure are as predicted by RRKM theory. The individual values could be matched to within a factor of 2. 

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. 

It is important to note that this assumption yields an RRKM rate coefficient, RRKM, that is an upper bound to the ergodic rate coefficient, ergodic, since every reactive trajectory (with xr J) necessarily has a positive velocity through the dividing surface. Thus, RRKM theory may be implemented in a variational manner, with the best approximation to ergodic obtained from the dividing surface S that provides the smallest rrkm-... 

The quantized results of Eqs. (2.8) and (2.10) negate, at least in principle, the variational property of classical RRKM theory, since both the one-dimensional tuimeling probabilities and the step function may exceed the true reactivities. In practice, however, variational minimizations still prove useful. 

The minimization of the canonical transition state partition function as in Eq. (2.13) is generally termed canonical variational RRKM theory. This approach provides an upper bound to the more proper E/J resolved minimization, but is still commonly employed since it simplifies both the numerical evaluation and the overall physical description. It typically provides a rate coefficient that is only 10 to 20% greater than the E/J resolved result of Eq. (2.11). 

The proper evaluation of the quantized energy levels within the SACM requires a separable reaction coordinate and thus numerical implementations have implicitly assumed a center-of-mass separation distance for the reaction coordinate, as in flexible RRKM theory. Under certain reasonable limits the underlying adiabatic channel approximation can be shown to be equivalent to the variational RRKM approximations. Thus, the key difference between flexible RRKM theory and the SACM is in the focus on the underlying potential energy surface in flexible RRKM theory as opposed to empirical interpolation schemes in the SACM. Forst s recent implementation of micro-variational RRKM theory [210], which is based on interpolations of product and reactant canonical partition functions, provides what might be considered as an intermediate between these two theories. 

The variation of the first-order rate coefficient with pressure can be interpreted in terms of rrk theory using s = 6, and by rrkm theory on the assumption that all of the vibrations are active . ... 

The rate constants were calculated with the transition state theory (TST) for direct abstraction reactions and the Rice-Ramsperger-Kassel-Marcus (RRKM) theory for reactions occuring via long-lived intermediates. For reactions taking place without well-defined TS s, the Variflex [35] code and the ChemRate [36] code were used for one-well and multi-well systems, respectively, based on the variational transition-state theory approach... 

J. D. Kress and S. J. Klippenstein, Comparison of variational RRKM theory with quantum scattering theory for the Ne + H2 - NeH+ + H reaction, Chem. Phys. Lett. 195 513... 

The variational version of RRKM theory (VTST) can be used to locate the transition state on the basis of the minimum sum of states. However, if this level of effort does not appear appropriate for the particular reaction, it is perfectly possible to fit a given data set with the vibrator model of the RRKM theory simply by adjusting the transition-state vibrational frequencies until a fit is obtained (as was done in the calculations of figures 7.3 and 7.4). In fact, such a fitting procedure is one means for determining whether the reaction is characterized by a loose or a tight transition state. 

The modification to the RRKM theory that makes possible accurate modeling of loose transition states is variational transition state theory (Pechukas, 1981 Miller, 1983 Forst, 1991 Wardlaw and Marcus, 1984, 1985, 1988 Hase, 1983, 1987). In this approach the rate constant k E, J) is calculated as a function of the reaction coordinate, R. The location of the minimum flux is found by setting the derivative of the sum of states equal to zero and solving for / . Thus, we evaluate... 

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