Adiabatic channel reaction systems

E.E.Nikitin and J.Troe, Correlation diagrams for accnrate adiabatic channel potentials of atom+linear molecule reaction system, J. Chem. Phys. 92, 6594... 

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 applications to a wide range of experimental results, one needs an efficient theoretical tool allowing for quick comparison of experiment and theory, perhaps followed by adjustment of theoretical parameters to experiment. With this goal in mind, the early formulation of the SACM included a simple empirical representation of the main features of the electronic potential by a very few adjustable parameters. Furthermore, the complicated calculation of adiabatic channels by a solution of the multidimensional clamped -q- rovibrational Schrbdinger equation, which is an exceedingly demanding task even today for larger than triatomic systems, was completely circumvented by a simple channel interpolation procedure. We shall present here a very brief description of this empirical approach for simple bond fission reactions. 

In contrast to H2/Pd, the vibrational effects in the adsorption of H2/Cu(l 0 0) are mainly caused by the curved reaction path. The basic mechanism can be discussed within a two-dimensional elbow plot shown in Fig. 2b. The PES corresponds to a so-called late barrier system which refers to the fact that the barrier is located after the curved region of PES. If the molecule is already initially vibrating, i.e. if it is oscillating back and forth in the d-direction, then the vibrational energy can be very efficiently used to make it around the curve and enter the dissociation channel. Nevertheless, adiabatic effects as just discussed in the context of the hydrogen dissociation on Pd(l 00) also contribute to the vibrational effects for H2/Cu(l 0 0). 

Extending the theory to interpret or predict the rovibrational state distribution of the products of the unimolecular dissociation, requires some postulate about the nature of the motion after the unimolecularly dissociating system leaves the TS on its way to form products. For systems with no potential energy maximum in the exit channel, the higher frequency vibrations will tend to remain in the same vibrational quantum state after leaving the TS. That is, the reaction is expected to be vibrationally adiabatic for those coordinates in the exit channel (we return to vibrational adiabaticity in Section 1.2.9). The hindered rotations and the translation along the reaction coordinate were assumed to be in statistical equilibrium in the exit channel after leaving the TS until an outer TS, the PST TS , is reached. With these assumptions, the products quantum state distribution was calculated. (After the system leaves the PST TS, there can be no further dynamical interactions, by definition.)... 

An alternative mixed quantum-classical evolution scheme which does not suffer from this limitation is Surface Hopping [1]. This method, although based on heuristic arguments applied to the coupled channel equation more than on a rigorous classical limit, maintains a multiconfiguration picture of the system, and is able to describe complex non-adiabatic phenomena such as proton transfer in solution, or the branching into different product channels of photo-excited chemical reactions in clusters and condensed phase environments [1-7]. 

We shall develop next a single-channel model that captures the key features of a catalytic combustor. The catalytic materials are deposited on the walls of a monolithic structure comprising a bundle of identical parallel tubes. The combustor includes a fuel distributor providing a uniform fuel/air composition and temperature over the cross section of the combustor. Natural gas, typically >98% methane, is the fuel of choice for gas turbines. Therefore, we will neglect reactions of minor components and treat the system as a methane combustion reactor. The fuel/air mixture is lean, typically 1/25 molar, which corresponds to an adiabatic temperature rise of about 950°C and to a maximum outlet temperature of 1300°C for typical compressor discharge temperatures ( 350°C). Oxygen is present in large stoichiometric excess and thus only methane mass balances are needed to solve this problem. 

Consider then an adiabatic well in the hyperspherical coordinate system. Classically, the motion of the periodic orbit at the well would be an oscillation from a point on the inner equipotential curve in the reactant channel to a point on the same equipotential curve in the product channel. This is qualitatively the motion of what are termed "resonant periodic orbits" (RPO s). For example the RPO s of the IHI system are given in Fig. 5. Thus, finding adiabatic wells in the radial coordinate system corresponds to finding RPO s and quantizing their action. Note that in Fig. 5 we have also plotted all the periodic orbit dividing surfaces (PODS) of the system, except for the symmetric stretch. By definition, a PODS is a periodic orbit that starts and ends on different equi-potentials. Thus the symmetric stretch PODS would be an adiabatic well for an adiabatic surface in reaction path coordinates. However, the PODS in the entrance and exit channels shown in Fig. 5 may be considered as adiabatic barrieres in either the radial or reaction path coordinate systems. Here, the barrier in radial coordinates, has quantally a tunneling path between the entrance and exit channels. 

Adiabatic, or as it has been termed vibrationally adiabatic,15 transition state theory has its origin in a paragraph in an article by Hirschfelder and Wigner.16 The treatment was developed further by a number of authors.17 In this type of transition state theory the eigenvalues of the system at each R, which are the vibrationally adiabatic eigenvalues, are plotted versus R. The N j in Eq. (2.1) then becomes the number of such states whose maximum energy on this plot does not exceed E, that is, N%j now denotes the sum of all open adiabatic reaction channels. 

We would like to complete this section by briefly describing some of the recent developments on electronically non-adiabatic reactions. From the standpoint of the coupled-channels method, there is in principle no added difficulty in treating more than one electronic state of the reactive system. This may be done, for example, by keeping electronic degrees of freedom in the Hamiltonian and expanding the total scattering wavefunction in the electronic states of reactants and products. In practice, however, some new difficulties may arise, such as non-orthogonality of vibrational states on different electronic potential surfaces. There is at present a lack of quantum mechanical results on this problem. 

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