Nonadiabatic transition matrix

The connection between the amplitudes (ci,co)c and (ci,C2)k is given by the nonadiabatic transition matrix L... 

Tbe nonadiabatic transition matrix I, which represents the nonadiabatic transition at the avoided crossing point, is given by... 

As discussed by Miller and co-workers [52,53], it is worthwhile to develop theories that enable us to evaluate thermal reaction rate constants directly and not to rely on the calculations of the most detailed scattering matrix or the state-to-state reaction probabihty. Here, our formulation of the nonadiabatic transition state theory is briefly described for the simplest case in which the transition state is created by potential surface crossing [27]. 

The degree of nonadiabaticity was calculated using Landau-Zener formalism and involved a calculation of the transition matrix according to... 

As discussed earlier, matrix elements Ckm(t) of are supposed to be essentially different from zero near t = tp and to vanish as f — tp - oo. Let rkm be the characteristic time of changing Ckm near and htokm be the difference in adiabatic terms at tp. Then the N matrix will be quite different for the two limiting cases rkmiokm 1 and rkmwkm 1. If either of these conditions is fulfilled, then general methods can be suggested for the calculation of the matrix of nonadiabatic transitions. 

The new type of nonadiahatic transition between asymptotically degenerate states has been fully analyzed with use of the Morse j)Otential model, and the aiialytieal expression of the scattering matrix has been derived. The present analysis warns that we have to be emeful about this type of nonadiabatic transition whenever we encounter asymptotic deg( iieracy of potential (uiergy curves. Tlu n< w mechanism presented in this paper works whenever the limiting value at. x oo of the inverse Massey type parameter is a finite quantity ... 

In the NT case (Fig. 2b), the semiclassical idea of sequential events of nonadiabatic transition and adiabatic wave propagation can be utilized at energies higher than the bottom of the upper adiabatic potential. The 5-matrix in this case is... 

The same semiclassical idea of decomposing the whole process into sequential events of nonadiabatic transition and adiabatic wave propagation can be applied together with the complex phase integral for nonadiabatic transition. Thus the scattering matrix can be expressed in the same way as Eqs. (21)-(22b). The difference from the Landau-Zener case appears in the matrices Ix and Ox (=IX) as... 

The simplest model is the following the diabatic potentials are constant with V2 - Vx = A > 0 and the diabatic coupling is V e R where A = 2V0. Recently, Osherov and Voronin obtained the quantum mechanically exact analytical solution for this model in terms of the Meijer function (38). In the adiabatic representation this system presents a three-channel problem at E > V2 > Vu since there is no repulsive wall at R Rx in the lower adiabatic potential. They have obtained the analytical expression of a 3 X 3 transition matrix. Adding a repulsive potential wall at R Rx for the lower adiabatic channel and using the semiclassical idea of independent events of nonadiabatic transition at Rx and adiabatic wave propagation elsewhere, they derived the overall inelastic nonadiabatic transition probability Pl2 as follows ... 

Electron transfer is usually treated as a transition between two electronic states, 1> and 2>, in the diabatic representation. The nondiagonal matrix element V, which couples the two states, is assumed to be sufficiently small. The standard Golden Rule prescription for the mean rate constant ki2 of nonadiabatic transitions can be written as [4,21]... 

The nonadiabatically coupled system shown in Fig. 6.10 consists of two nonadiabatic transition processes one is cl <-> feO and the other is cO bl. Other processes, cO -o- bO and cl -o- bl, were omitted. This is because nonadiabatic coupling matrix elements between the two vibronic states with equal vibrational quantum numbers approximately give zero for the displaced harmonic potential model with dimensionless potential displacement A fcl < 1. 

A treatment of electronically nonadiabatic transitions in molecular collisions is possible on the basis of an extension of Miller s classical S-matrix method /106b/. This means using complex valued classical trajectories which go from one to another adiabatic surface through thisir (complex) crossing point. Quantum effects, such as tunneling through or reflection above a potential barrier also can be incorporated in the theory of nonadiabatic transitions. 

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