Collinear triatomic reaction

A qualitatively appealing feature of the RPH is that the kinematic coupling between motion along the reaction coordinate and the vibrational modes is made explicit in the matrix elements. From the early work of Polanyi and co-workers on collinear triatomic reactions, we know that from the position of the saddle point relative to the corner in the reaction path, qualitative conclusions can be drawn about the effect of the energy distribution in the reactants on the rate and about the distribution of energy in the products [20, 22]. The Bj appear to contain quite analogous and quite detailed information about curvature effects for polyatomic systems, identifying which modes should couple to (and... 

In this section we focus on the QNF theory as a tool for the computation of the CRP and the thermal reaction rate constant. To demonstrate the efficiency and capability of the QNF method and to compare it with other existing methods we follow Ref. [31] and address the CRP and the thermal rate in a collinear triatomic reaction, namely the nitrogen exchange reaction... 

In the above equations, E, and are the thermal energies of the reactants and of the transition state, respectively. Such a thermodynamic integration was used within a discrete variable representation of QI approximation to compute the rate constant for several collinear triatomic reactions [33]. In Ref. [46], it is generalized and presented in a form suitable for a path integral evaluation. Unlike gr and Caa, the energies are normalized quantities because they can be written as logarithmic derivatives ... 

Collinear triatomic reaction coordinates. Now we want to be more specific (the next section is taken as an extraction from Kuppermann [10]). Let be... 

Fig. 5.1. Coordinates and regions of scaled configuration space for integrating the Schrodinger equation for collinear triatomic reactions. (Reprinted, by permission, from Kuppermann, A, in Theoretical Chemistry and Perspectives, 6A, Henderson, D. (Ed.) (Academic, NY, 1981), p. 79. Copyright 1981 by Academic Press, Inc.)...

Figure 6 Schematic potential energy surface for a symmetric collinear triatomic system in skewed coordinates (panel a). The dashed and dotted lines correspond respectively to the valley bottoms and ridges, which are represented in panel c as a function of p. In panel b a conventional view of the minimum energy path is sketched as a function of a generic reaction coordinate s.

Figure 9.11 illustrates time evolution of the nonequilibrium rate constant ki2(t) and the electronic population in the donor state Pi(f) in the course of the back-transfer reaction in a model ET complex (rigid collinear triatomic molecule with equivalent donor and acceptor sites separated by a neutral spacer) in a polar solvent. The long-time value of k 2(t) is seen to be much smaller than the maximum values it achieves during the relaxation process. Hence, it is the evolution of ki2(t) that dominates the electronic state population evolution at short times. After this rapid nonequilibrium stage of nuclear relaxation is over, with its eharaeteristie sequence of plateaus and dips, ET proceeds further very slowly in its usual way with a small (activated) equilibrium rate eonstant. 

Figure 9.11. Time evolution of (a) nonequilibrium rate constant k 12(f)and (b) electronic population in the donor state Pi(t) for V= 120cm . The parameter Icijlt) was obtained from the molecular dynamics simulation data for the model back ET reaction in a rigid collinear triatomic molecule with equivalent donor and acceptor sites separated by a neutral spacer in a polar solvent. The parameter Pfi) is a result of solution of the master equation. (Reproduced from [62c] with permission. Copyright (1996) by the American Institute of Physics.)...

The classical counterpart of resonances is periodic orbits [91, 95, 96, 97 and 98]. For example, a purely classical study of the H+H2 collinear potential surface reveals that near the transition state for the H+H2 H2+H reaction there are several trajectories (in R and r) that are periodic. These trajectories are not stable but they nevertheless affect strongly tire quantum dynamics. A study of tlie resonances in H+H2 scattering as well as many other triatomic systems (see, e.g., [99]) reveals that the scattering peaks are closely related to tlie frequencies of the periodic orbits and the resonance wavefiinctions are large in the regions of space where the periodic orbits reside. 

An example of a potential surface of a reacting system in Figure 22.1. This kind of plot shows the potential energy for a special (collinear or bent) arrangement of atoms during a bimolecular reaction in a triatomic system ... 

Figure 8 Schematic view of the various arrangements for a triatomic collinear reaction (a) and cuts through the potential at different p values (b) as a function of the kinematic angle.

Band and Freed have criticized the quasi-diatomic approximation and emphasized that any complete theory of dissodation must involve the use of the correct sets of normal modes Q and O of the molecule in the initial and final states (> and /> respectively. The two sets are not independent, but are related by a co-ordinate transformation. A detailed, quantum mechanical description has been developed in which the set Q in state ( > are taken to be the normal modes of the unexdted parent molecule for direct photodissodation, or the metastable photo-excited molecule for indirect predissociation, and the set Q ) in the state /> are separated into QUQi, wh e Qi is the reaction co-ordinate on the final repulsive surface and IQi are the normal modes in the photofragments. For a linear, triatomic molecule, Qi is simply the vibrational mode of the diatomic fragment and Q) indudes the symmetric and antisymmetric stretching modes (if collinearity is preserved). The matrix elements for the transition from... 

The procedure we have outlined, a classical mechanical trajectory analysis based upon a chosen potential-energy surface, has been applied in a far more sophisticated fashion to a number of triatomic systems. The restriction to collinear collision geometries has been lifted and more realistic potential-energy surfaces have been used. We consider two examples, the reactions H + Dg HD + D and Ar+ + Dg ArD+ + D, to demon-... 

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