Adiabatic channel method

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]. 

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. 

SACM (statistical adiabatic channel model) method for computing reaction rates... 

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. 

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. 

Thus the rate constant can be expressed in terms of the three quantities and Fc all of which are temperature dependent. These expressions, developed from Troe s adiabatic channel model, have the great virtue of sufficient complexity to express adequately the variation of k with T and p, but simple enough for ready programming, and hence, of being convenient for modelling purposes. It is quite possible to use other ways of expressing evaluated data for decomposition/combination reactions but none are so useful. For example, Tsang and Hampson have adopted a rather different approach, described in Section 3.4, but their methods do not lead to a simple analytical expression for k(T, [A/]). 

Qei and Qvtbrot denote electronic and rovibrational partition fimctions, respectively. In general, the contributions of the internal degrees of freedom of A and B cancel in g and gviiroXA)gv iro((B ), such that only contributions Irom the external rotations of A and B and the relative motion, summarized as "transitional modes", need to be considered. Under low temperature quantum conditions, these can be obtained by statistical adiabatic channel (SACM) calculations [9],[10] while classical trajectory (CT) calculations [11]-[14] are the method of choice for higher temperatures. CT calculations are run in the capture mode, i.e. trajectories are followed Irom large separations of A and B to such small distances that subsequent collisions of AB can stabilize the adduct. 

One version of the statistical adiabatic channel models, i.e. the maximum free energy method,20 was applied in the theoretical analysis of Fagerstrom et al,203 The calculated high-pressure limiting rate constant... 

The final stage in the adiabatic reduction is the solution of Eq. (4.24). Given the adiabatic potential of Eq. (4.26) this cannot be done analytically, but the resulting ordinary differential equation may be solved numerically using the finite difference method. As an example, we show in Fig. 20 a comparison between the even-parity adiabatic eigenvalues and the exact ones, obtained by solving the full coupled channels expansion, using the artificial channel method.69... 

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. 

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). 

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... 

A model developed earlier (4, ) used the collocation method to solve the equations for heat, mass and momentum transfer In a single, adiabatic channel of the monolith. The basic model Is the one described as Model II-A(5) a square duct with axial conduction of heat longitudinally In the solid walls, but with Infinitely fast conduction peripherally around the square, and Including the diffusion of heat and mass In the transfer direction In the fluid (See for a discussion of the Importance of Including this effec.) Nusselt and Sherwood numbers are not assigned priori, but are derived from the solution. The reaction rate expression P2 In (3) with a basic form... 

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. 

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