Landau-Zener factor

Relation (12) describes a thermally activated transition with activation energy = ( j — A) /4 j equal to the potential at the point of intersection. The preexponential factor in Eqn. (12) is equal to the vibration frequency co/2n multiplied by the transmission coefficient P representing the Landau-Zener factor [16] for the transition between the terms in the region of their... 

Landau-Zener factor at particle rate v = when its energy is equal... 

Here the lower summation limit A Dmin = ( o -ex)/5 = 6 - i o is the activation barrier of formation of the complex N20 ( i +) without changing vibrational quantum number V (see Fig. 6-7) and ex is the kinetic energy of the relative O-N2 motion. The Landau-Zener factors don t change a lot with Ad, whereas the Frank-Condon factors abruptly decrease at higher differences A d in vibrational quantum numbers during transition. Summation (6-21) gives the formation probability of N20 ( i +) ... 

The problem of nonadiabatic tunneling in the Landau-Zener approximation has been solved by Ovchinnikova [1965]. For further refinements of the theory beyond this approximation see Laing et al. [1977], Holstein [1978], Coveney et al. [1985], Nakamura [1987]. The nonadiabatic transition probability for a more general case of dissipative tunneling is derived in appendix B. We quote here only the result for the dissipationless case obtained in the Landau-Zener limit. When < F (Xe), the total transition probability is the product of the adiabatic tunneling rate, calculated in the previous sections, and the Landau-Zener-Stueckelberg-like factor... 

Meyer and coworkers (18). In this case, we see from Figure 6 that the Weiner method predicts faster rates than the perturbation approach, the difference being about a factor of 5 at room temperature and more than an order of magnitude at low temperature. As might be expected, the two methods continue to diverge as increases. If gets so large that the transfer rate becomes comparable to v, the Landau-Zener correction ((5), eq 155) may be applied. c... 

The Landau-Zener expression is calculated in a time-dependent semiclassical manner from the diabatic surfaces (those depicted in Fig. 1) exactly because these surfaces, which describe the failure to react, are the appropriate zeroth order description for the long-range electron transfer case. As can be seen, in the very weak coupling limit (small A) the k l factor and hence the electron transfer rate constant become proportional to the absolute square of A ... 

A (>0) is the electronic factor. 1-P is the probability for continuing on the lower PES, which corresponds to ET. If the barrier disappears, the Landau-Zener model should not be used and it may be necessary to include the nuclear coordinates in a wave packet model. 

The transmission factor is related to the transition probability (P0) at the intersection of two potential energy surfaces, as given by the Landau-Zener theory.24... 

To understand the factors that are important in controlling the rates of ET reactions, it is best to refer to a specific theoretical model [35]. Choosing a model defines terms and allows us to analyze the results of experiments in precise ways. There are two main types of models classical and quantum mechanical. One way of smoothly moving from the use of a classical to a quantum mechanical model is provided by semiclassical (Landau-Zener) ET theory [36-38]. At high temperature, quantum mechanical models become equivalent in most respects to semiclassical ones. Thus the appropriate choice of model depends on the type of ET reaction that we are interested in studying. Ones in which the electron donor and acceptor have strong electronic interaction with each other prior to the ET event are well described by a classical model. In these systems an ET reaction always proceeds to products if the reactants reach the top of the reaction barrier (strong-... 

For cases of electron transfer between relatively weakly coupled reactants, the 2-state Landau-Zener model leads to the following expression for the electronic transmission factor, (as in... 

The calculation of the electronic-transmission factor currently involves three different methods, viz. the Landau-Zener formula, Fermi s golden mle [35], and electron tunneling formalism such as the Wentzel-Kramer-Brillouin method [36]. We used the Landau-Zener formula [37,38] to calculate it ... 

The nuclear frequency is related to the solvent and inner-shell reorganization energies as well as the corresponsing vibration frequencies. The electronic factor can be described on the basis of the Landau-Zener framework and is related to the electronic coupling matrix element... 

Newton determined the transmission coefficient, k, for the homogeneous electron transfer reaction between Fe " (H20)6 and Fe " (H20)6 redox couples in solution. The transmission coefficient was expressed in terms of the Landau-Zener " ° probability factor, P12, i.e.,... 

Stueckelberg also introduced two-state model but adopted time-independent formulation and used semiclassical approach for solution. The latter is in contrast to constant velocity assumptions in the treatment of Landau Zener, but is essential for analytical derivation of correct adiabatic phase factors. Semiclassical contom integral method and analysis of accompanying Stokes phenomena is used for deriving transition amplitude in time-independent formulation of this problem [395], which will be briefly mentioned in the next subsection (also see Ref. [99] for more details including corrections). 

Electronic nonadiabatic factor (Xeff) as a function of the ratio between nuclear and elfective electronic frequencies Vv/(XeiVei), or adiabaticity parameter (IkqsI ) as a function of the Landau-Zener classical transition probability (plz), illustrating their rapid convergence to the same as5miptotic limits. 

Expressions (1) and (2) are the basis for the Hush-Marcus model. They allow the construction of potential energy curves of parabolic shapes, when the energy is plotted as a function of a composite reaction coordinate. These curves in turn are the basis for an elementary description of the thermal and optical processes in mixed valences complexes. In principle, it is possible to compute a rate constant from this model, using the total reorganization energy as an activation energy and introducing an electronic transmission factor calculated by the Landau-Zener formula. However this procedure is now supplanted by the quantum models. 

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