Reaction mechanisms phase space

The quantum mechanical phase space theory has also been developed [75—77] and applied to ion—molecule reactions, including some endothermic reactions [78, 79]. While results again seem to show that the theory is promising, the apparent future problem would be how to extend the treatment to systems involving more than three atoms. 

The description of chemical reactions as trajectories in phase space requires that the concentrations of all chemical species be measured as a function of time, something that is rarely done in reaction kinetics studies. In addition, the underlying set of reaction intennediates is often unknown and the number of these may be very large. Usually, experimental data on the time variation of the concentration of a single chemical species or a small number of species are collected. (Some experiments focus on the simultaneous measurement of the concentrations of many chemical species and correlations in such data can be used to deduce the chemical mechanism [7].)... 

The beauty of the prior approximations is that by assuming a mean-field influence of solvation we can continue to work in a phase space having the same dimensionality as that for the gas phase that being the case, analysis using the tools of TST is mechanically identical for the two phases. When the solvent is not fully equilibrated with the complete reaction path, however, the reacting system can no longer legitimately be described exclusively in terms of solute coordinates. 

As in the case ml = 1, in accordance with the above properties of Jacobian matrix (160), it follows that, under the assumption of the oriented connectivity for the reaction mechanism involving no intermediate interactions, the time shift is the phase space (or balance polyhedron) compression in the metric... 

Thus the mechanism formed by steps (l)-(4) can be called the simplest catalytic oscillator. [Detailed parametric analysis of model (35) was recently provided by Khibnik et al. [234]. The two-parametric plane (k2, k 4/k4) was divided into 23 regions which correspond to various types of phase portraits.] Its structure consists of the simplest catalytic trigger (8) and linear "buffer , step (4). The latter permits us to obtain in the three-dimensional phase space oscillations between two stable branches of the S-shaped kinetic characteristics z(q) for the adsorption mechanism (l)-(3). The reversible reaction (4) can be interpreted as a slow reversible poisoning (blocking) of... 

The basic assumption in statistical theories is that the initially prepared state, in an indirect (true or apparent) unimolecular reaction A (E) —> products, prior to reaction has relaxed (via IVR) such that any distribution of the energy E over the internal degrees of freedom occurs with the same probability. This is illustrated in Fig. 7.3.1, where we have shown a constant energy surface in the phase space of a molecule. Note that the assumption is equivalent to the basic equal a priori probabilities postulate of statistical mechanics, for a microcanonical ensemble where every state within a narrow energy range is populated with the same probability. This uniform population of states describes the system regardless of where it is on the potential energy surface associated with the reaction. 

Some of the initial work dealt with the formation of proton-bound dimers in simple amines. Those systems were chosen because the only reaction that occurs is clustering. A simple energy transfer mechanism was proposed by Moet-Ner and Field (1975), and RRKM calculations performed by Olmstead et al. (1977) and Jasinski et al. (1979) seemed to fit the data well. Later, phase space theory was applied (Bass et al. 1979). In applying phase space theory, it is usually assumed that the energy transfer mechanism of reaction (2 ) is valid and that the collisional rate coefficients kx and fc can be calculated from Langevin or ADO theory and equilibrium constants. 

The statistical theories provide a relatively simple model of chemical reactions, as they bypass the complicated problem of detailed single-particle and quantum mechanical dynamics by introducing probabilistic assumptions. Their applicability is, however, connected with the collisional mechanism of the process in question, too. The statistical phase space theories, associated mostly with the work of Light (in Ref. 6) and Nikitin (see Ref. 17), contain the assumption of a long-lived complex formation and are thus best suited for the description of complex-mode processes. On the other hand, direct character of the process is an implicit dynamical assumption of the transition-state theory. 

With this brief overview of classical theories of unimolecular reaction rate, one might wonder why classical mechanics is so useful in treating molecular systems that are microscopic, and one might question when a classical statistical theory should be replaced by a corresponding quantum theory. These general questions bring up the important issue of quantum-classical correspondence in general and the field of quantum chaos [27-29] (i.e., the quantum dynamics of classically chaotic systems) in particular. For example, is it possible to translate the above classical concepts (e.g., phase space separatrix, NHIM, reactive islands) into quantum mechanics, and if yes, how What is the consequence of... 

Figure 22. Schematic mechanism of reaction including intramolecular energy transfer. The phase-space of state A is partitioned into Aj and A2. Si is a representation of an intramolecular energy transfer dividing surface, and S2 is the A-state separatrix.

The master equation will be derived by looking at the catalyst and its adsorbates in phase space. This is, of course, a classical mechanics concept, and one might wonder if it is correct to look at the reactions on an atomic scale and use classical mechanics. The situation here is the same as for the derivation of the rate equations for gas phase reactions. The usual derivations there also use classical mechanics. [5-9] Although it is possible to give a completely quantum mechanical derivation formalism, [10-13] the mathematical complexity hides much of the important parts of the chemistry. Besides, it is possible to replace the classical expressions that we will get by semi-quantum mechanical ones, in exactly the same way as for gas phase reactions. However, these expressions will not account for quantum efliects as tunneling and interference. 

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