Statistical theories adiabatic channel model

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,... 

Figure 3. Thermal rate constants for capture of HC1 by H3 (PST locked-dipole capture corresponding to phase-space theory, Eq. (16) SACM statistical adiabatic channel model, Eqs. (26)-(34) [15] SACMci classical SACM, Eqs. (28H31) [15] CT classical trajectories, Eqs. (26) and (27) [1]).

At low temperature the classical approximation fails, but a quantum generalization of the long-range-force-law collision theories has been provided by Clary (1984,1985,1990). His capture-rate approximation (called adiabatic capture centrifugal sudden approximation or ACCSA) is closely related to the statistical adiabatic channel model of Quack and Troe (1975). Both theories calculate the capture rate from vibrationally and rotationally adiabatic potentials, but these are obtained by interpolation in the earlier work (Quack and Troe 1975) and by quantum mechanical sudden approximations in the later work (Clary 1984, 1985). 

Statistical methods represent a background for, e.g., the theory of the activated complex (239), the RRKM theory of unimolecular decay (240), the quasi-equilibrium theory of mass spectra (241), and the phase space theory of reaction kinetics (242). These theories yield results in terms of the total reaction cross-sections or detailed macroscopic rate constants. The RRKM and the phase space theory can be obtained as special cases of the single adiabatic channel model (SACM) developed by Quack and Troe (243). The SACM of unimolecular decay provides information on the distribution of the relative kinetic energy of the products released as well as on their angular distributions. 

This concludes the discussion of early theories of the reaction step. These have largely been superseded by RRKM theory and the Statistical Adiabatic Channel Model, which are discussed in the next two sections. 

Phase space theory, flexible RRKM theory, and the statistical adiabatic channel model... 

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. 

The determination of the microcanonical rate coefficient k E) is the subject of active research. A number of techniques have been proposed, and include RRKM theory (discussed in more detail in Section 2.4.4) and the derivatives of this such as Flexible Transition State theory. Phase Space Theory and the Statistical Adiabatic Channel Model. All of these techniques require a detailed knowledge of the potential energy surface (PES) on which the reaction takes place, which for most reactions is not known. As a consequence much effort has been devoted to more approximate techniques which depend only on specific PES features such as reaction threshold energies. These techniques often have a number of parameters whose values are determined by calibration with experimental data. Thus the analysis of the experimental data then becomes an exercise in the optimization of these parameters so as to reproduce the experimental data as closely as possible. One such technique is based on Inverse Laplace Transforms (ILT). 

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.)...

These equations lead to forms for the thermal rate constants that are perfectly similar to transition state theory, although the computations of the partition functions are different in detail. As described in figure A3.4.7 various levels of the theory can be derived by successive approximations in this general state-selected form of the transition state theory in the fiamework of the statistical adiabatic channel model. We refer to the literature cited in the diagram for details. 

Figure B2.5.7 shows the absorption traces of the methyl radical absorption as a function of time. At the time resolution considered, the appearance of CH is practically instantaneous. Subsequently, CH disappears by recombination (equation B2 5.28T At temperatures below 1500 K, the equilibrium concentration of CH is negligible compared with (left-hand trace) the recombination is complete. At temperatures above 1500 K (right-hand trace) the equilibrium concentration of CH3 is appreciable, and thus the technique allows the determination of both the equilibrium constant and the recombination rate [54, 55]. This experiment resolved a famous controversy on the temperature dependence of the recombination rate of methyl radicals. While standard RRKM theories [57, 58] predicted an increase of the high-pressure recombination rate coefficient k, (T) by a factor of 10-30 between 300 K and 1400 K, the statistical-adiabatic-channel model predicts a...

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),... 

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... 

Q The statistical adiabatic channel model proposed by Quack and Troe. (D) A recent adaptation of RRKM theory due to Marcus. ... 

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... 

Relative rate coefficients define relative product state (product channel and product energy) distributions. These can often be described by statistical theories of unimolecular reactions, such as the statistical adiabatic channel model, described in Statistical Adiabatic Channel Models. 

Systems with deep potential wells and consequently a high density of states are a real challenge for exact quantum mechanical theories. Advances in numerical approaches and computer technology have made possible exact calculations for realistic molecular systems only recently. In the following we briefly describe one particular system, HCO, for which the results of exact dynamics calculations using an accurate PES can be compared with state-of-the-art experimental data at an unprecedented level of sophistication. Because of lack of space the discussion must be very short but is intended to stimulate interest in other examples. Traditional descriptions of fragmentation on ground-state potentials use mainly statistical approximations (see Statistical Adiabatic Channel Models). 

Classical Dynamics of Nonequilibrium Processes in Fluids Integrating the Classical Equations of Motion Control of Microworld Chemical and Physical Processes Mixed Quantum-Classical Methods Multiphoton Excitation Non-adiabatic Derivative Couplings Photochemistry Rates of Chemical Reactions Reactive Scattering of Polyatomic Molecules Spectroscopy Computational Methods State to State Reactive Scattering Statistical Adiabatic Channel Models Time-dependent Multiconfigurational Hartree Method Trajectory Simulations of Molecular Collisions Classical Treatment Transition State Theory Unimolecular Reaction Dynamics Valence Bond Curve Crossing Models Vibrational Energy Level Calculations Vibronic Dynamics in Polyatomic Molecules Wave Packets. 

Path Integral Methods Reaction Path Hamiltonian and its Use for Investigating Reaction Mechanisms Reactive Scattering of Polyatomic Molecules State to State Reactive Scattering Statistical Adiabatic Channel Models Time Correlation Functions Transition State Theory Unimolecular Reaction Dynamics. 

The Statistical Adiabatic Channel Model as an Ab Initio Theory and as an Empirical Few Parameter Model Conclusions and Outlook Related Articles References... 

PST = phase space theory SACM = statistical adiabatic channel model TST = transition state theory. 

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. ..)... 

Figure 1 places the statistical adiabatic channel model in the general landscape of dynamical and statistical theories of chemical reactions. While this diagram stems originally from a review written in 1977 shortly after the development of the statistical adiabatic channel model (see also Refs. 5-28, cited in Figure I), it is largely still valid today, when it is noted... 

We shall derive in this section the fundamental equations for the kinetic quantities in the adiabatic channel model from the point of view of the statistical S-matrix in scattering theory, which may seem to be the most logical approach following Refs. 2 and 17. In theoretical quantum dynamics we start from the time-dependent Schrodinger equation (3) ... 

A particular property of the adiabatic channel model is the ability to provide complete explicit specification of all good quantum numbers and symmetries for individual channels, throughout the correlation between all reactant and product levels (see Figure 3). At the time of introducing the adiabatic channel model this was a new feature in statistical theory, which can be systematically used both for obtaining simple computational results and for obtaining a deeper understanding of some of the basic properties of the reaction processes involved. We shall summarize here some of the ideas and results. 

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