Statistical adiabatic channel model SACM

Rather than using transition state theory or trajectory calculations, it is possible to use a statistical description of reactions to compute the rate constant. There are a number of techniques that can be considered variants of the statistical adiabatic channel model (SACM). This is, in essence, the examination of many possible reaction paths, none of which would necessarily be seen in a trajectory calculation. By examining paths that are easier to determine than the trajectory path and giving them statistical weights, the whole potential energy surface is accounted for and the rate constant can be computed. 

For highly exothermic SN2 reactions, which have a central barrier significantly lower in energy than that of the reactants, association of the reactants may be the rate controlling step in TST.1 The SN2 rate constant can then be modeled by a capture theory9 such as VTST,10 average dipole orientation (ADO) theory,11 the statistical adiabatic channel model (SACM),12 or the trajectory capture model.13... 

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,... 

Another advantage of the quantum calculations is that they provide a rigorous test of approximate methods for calculating dissociation rates, namely classical trajectories and statistical models. Two commonly used statistical theories are the Rice-Ramsperger-Kassel-Marcus (RRKM) theory and the statistical adiabatic channel model (SACM). The first one is thoroughly discussed in Chapter 2, while the second one is briefly reviewed in the Introduction. Moreover, the quantum mechanical approach is indispensable in analyzing the reaction mechanisms. A resonance state is characterized not only by its position, width and the distribution of product states, but also by an individual wave function. Analysis of the nodal structure of resonance wave functions gives direct access to the mechanisms of state- and mode-selectivity. 

Figure 20 Overview of the calculated dissociation rates of HOCl as function of the excess energy. The solid line is the prediction of the statistical adiabatic channel model (SACM). Adapted from Ref. 67.

The model of Quack and Troe,18 the statistical adiabatic channel model (SACM), is an approximate prescription for calculating the number of open adiabatic channels. A universal function g(R) is assumed for interpolating between all eigenvalues of reactants and products,19... 

Figure 5. Comparison of the observed product rotational state distributions (solid bars) and the results of two statistical models the statistical adiabatic channel model (SACM, open bars) and phase space theory (PST, hatched bars). The distributions are an average of those observed, or those calculated, at several excitation wavelengths in the region of (a) the 5tbu main band, (b) the 5voh combination band, (c) the 6vOH main band, and (d) the 6v0H combination band of HOOH. (Reproduced with permission from Ref. 41.)...

There are several ways to derive the RRKM equation (Forst, 1973). The one adopted here is based on classical transition state theory and was first proposed by Wigner (Wigner, 1937 Hirschfelder and Wigner, 1939). Although there are several other statistical formulations of the unimolecular rate [phase space theory (Pechukas and Light, 1965), statistical adiabatic channel model (SACM) (Quack and Troe, 1974),... 

Figure 7.24 Adiabatic vibronic reaction channels in the statistical adiabatic channel model (SACM) as a function of the reaction coordinate, s. e(, and e(, are, respectively, the continuum and bound state energy levels of the molecule. are the final product energies when s = < . These degenerate final product states are adiabatically related to the various barriers at s. This figure is taken with permission in slightly modified form from Mies (1969).

In chapter 7 the statistical adiabatic channel model (SACM) (Quack and Troe, 1974, 1975) was described for calculating unimolecular reaction rates. This theory assumes the reaction system remains on the same diabatic potential energy curve while moving from reactant to products. Two parameters, a and (3 are used to construct model diabatic potential curves. The unimolecular rate constant, at fixed E and 7, for forming products with specific energy , (e.g., a specific vibrational energy in one of the fragments) is... 

In the limit of high pressure, collisions maintain the thermal distribution of reactant molecules over their internal energy states and consequently TST can be used to determine the thermal rate constants for dissociation and association. However, in the case where there is no maximum in the reaction path leading from reactants to products, it is necessary to take account of angular momentum (/) constraints as well as internal energy. The transition state is not found at a single separation but rather it depends on Eint and J. Then, in the language of the statistical adiabatic channel model (SACM), the partition function for the transition state can be expressed as ... 

A large number of approximate theories have been proposed for ion-dipole reactions. Some of these include the average dipole orientation (ADO) approximation and its extension to include conservation of angular momentum (the AADO method ), various transition-state theories involving variational and statistical modifications, the semiclassi-cal perturbed rotational state (PRS) approximation, classical trajectory studies, the adiabatic invariance method, and the statistical adiabatic channel model (SACM). 

In Table II we compare rate coefficients calculated [20] for the He + HCt reaction using three different theories - the ACCSA, the statistical adiabatic channel model (SACM) of Troe [14] and classical trajectory calculations [16]. The trajectory calculations have been parameterized to give the simple formula... 

Important further work by Troe and Nikitin and co-workers " considered the calculation of the capture rate from the perspective of the statistical adiabatic channel model (SACM). Ramillon and McCarroll demonstrated that the adiabatic capture method of Clary and the SACM method of Troe are identical in concept. However, there are still some minor differences in the approaches used to evaluate the rotational energies. Direct comparison for a number of ion-dipole capture rates found good agreement down to about 50 K, but there were increasing discrepancies at lower temperatures. These discrepancies are apparently a... 

When the interaction depends on the orientation of the neutral molecule, as is the case, for example, for ion-dipole reactions, the simple treatment outlined above is no longer appropriate. The adiabatic channel method is often used in this context [32]. The rotational energy levels in the separated reactants are coupled with the orbital energy levels to define a set of channels for the collision complex. The number of open states, N(EJ), is the number of channels with an energy maximum below the energy E. Examples of this approach include the adiabatic channel centrifugal sudden approximation (ACCSA) of Clary [33] and the statistical adiabatic channel model (SACM) of Troe and co-workers [34]. 

The statistical adiabatic channel model (SACM) " is one realization of the laiger class of statistical theories of chemical reactions. Its goal is to describe, with feasible computational implementation, average reaction rate constants, cross sections, and transition probabilities and lifetimes at a detailed level, to a substantial extent with state selection , for bimolecular reactive or inelastic collisions with intermediate complex formation (symbolic sets of quantum numbers v, j, E,J. ..)... 

Another important statistical approach to this same problem is the statistical adiabatic channel model (SACM) of Quack and Troe, - which adiabatically correlates the eigenstates of the orthogonal modes along the reaction coordinate, thereby generating rovibrational adiabatic channels. The adiabatic approximation reduces the multidimensional dynamical problem to essentially a one-dimensional barrier-crossing problem. The catch, of course, is that it is extremely difficult to compute the requisite adiabatic channels, though no more difficult than a rigorous quantum mechanical implementation of VTST would be. An authoritative account of adiabatic channel methods is to be found in Statistical Adiabatic Channel Models. 

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