Adiabatic bend approximation

In the adiabatic bend approximation (ABA) for the same reaction,18 the three radial coordinates are explicitly treated while an adiabatic approximation was used for the three angles. These reduced dimensional studies are dynamically approximate in nature, but nevertheless can provide important information characterizing polyatomic reactions, and they have been reviewed extensively by Clary,19 and Bowman and Schatz.20 However, quantitative determination of reaction probabilities, cross-sections and thermal reaction rates, and their relation to the internal states of the reactants would require explicit treatment of five or the full six degrees-of-freedom in these four-atom reactions, which TI methods could not handle. Other approximate quantum approaches such as the negative imaginary potential method16,21 and mixed classical and quantum time-dependent method have also been used.22... 

The agreement noted between the ground and excited bend-state reduced dimensionality resonances and the CS ones of Schatz is quite significant. It indicates that the adiabatic bend approximation for this reaction continues to be a realistic description of the dynamics even for total energies up to 1.20 eV. This should stimulate further inquiry about the realism of the adiabatic bend approximation in reactive scattering. 

Results of reduced dimensionality quantum calculations are displayed in the right panel of Fig. 5.1 the adiabatic bend approximation (ABA) of Bowman and coworkers [42] and the rotating bond approximation (RBA) of Clary [30]. Both approximations employ a three-dimensional description of the reaction process. For the H2+OH reaction the ABA approximation is in good agreement with the accurate result. In contrast, the RBA rate constants are too small at low temperatures (but, due to fortuitous cancelation of errors, agree well with experiment). However, this finding can not be generalized. For the H2+CN reaction, for example, RBA yields a better description than ABA (see Fig. 5.3). 

The quantities given by Eq. (56) represent the first-order approximation for the adiabatic bending potentials. If these potentials are known, V can be... 

Four-atom reactions came into focus with the development by Clary of the Rotating Bond Approximation (RBA)[10. 11] and Bowman s reduced-dimensionality adiabatic bend (RD-AB) calculations of four-atom reactions. In the latter three stretching vibrational motions are treated explicitly quantum dynamically while the bending degrees of freedom are treated adiabatically and one diatom is assumed to be a spectator[12, 13, 14, 15, 16], The RBA may be seen as an extension... 

The centrifugal sudden-adiabatic bend theory we jnst reviewed represents approximations to the partial wave vibrational state-to-state cumulative reaction probability Because that quan-... 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

In Table I, 3D stands for three dimensional. The symbol symbol in connection with the bending potentials means that the bending potentials are considered in the lowest order approximation as already realized by Renner [7], the splitting of the adiabatic potentials has a p dependence at small distortions of linearity. With exact fomi of the spin-orbit part of the Hamiltonian we mean the microscopic (i.e., nonphenomenological) many-elecbon counterpart of, for example, The Breit-Pauli two-electron operator [22] (see also [23]). 

First, let us note that the adiabatic potentials and V [Eq. (67)], even in the lowest order (harmonic) approximation, depend on the difference of the angles 4>j- and t >c this is an essential difference with respect to triatomics where the adiabatic potentials depend only on the radial bending coordinate p. The foims of the functions V, Vt, and Vc are determined by the adiabatic potentials via the following relations... 

It is of importance to note that we shall consider, in the present section, that the fast and bending modes are subject to the same quantitative damping. Indeed, the damping parameter of the fast mode yG and that of the bending mode y will be supposed to be equal, so that we shall use in the following a single parameter, namely y (= yG = y5). This drastic restriction cannot be avoided when going beyond the adiabatic approximation. 

We must stress that the use of a single damping parameter y supposes that the relaxations of the fast and bending modes have the same magnitude. A more general treatment of damping has been proposed [22,23,71,72] however, this treatment (discussed in Section IV.D) requires the use of the adiabatic approximation, so that its application is limited to very weak hydrogen bonds. 

Applying the adiabatic approximation, we restrict the representation of the Hamiltonian to the reduced base (89). Within this base, the Hamiltonian that describes an undamped H bond involving a Fermi resonance may be split into effective Hamiltonians whose structure is related to the state of the fast and bending modes ... 

Figure 7. The Fermi resonance mechanism within the adiabatic and exchange approximations. F, fast mode S, slow mode B, bending mode.

We shall give here a brief summary of our previous work [71,72] that was concerned with the introduction of the relaxation phenomenon within the adiabatic treatment of the Hamiltonian (77), as was done in the undamped case by Witkowski and Wojcik [74]. Following these authors, we applied the adiabatic approximation and then we restricted the representation of the Hamiltonian to the reduced base (89). Within this base, the Hamiltonian that describes a damped H bond involving a Fermi resonance may be split into effective Hamiltonians whose structure is related to the state of the fast and bending modes ... 

As shown in Section IV.D, it is possible, within the adiabatic approximation, to account for the general situation where the relaxation parameters of the fast mode y0 and of the bending mode y are not supposed to be equal. 

As a consequence of the two above points, there is presently no satisfactory theory able to incorporate the specific relaxation of the fast mode and of the bending modes in a model working beyond both the adiabatic and exchange approximations. 

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