Stueckelberg

The motivation comes from the early work of Landau [208], Zener [209], and Stueckelberg [210]. The Landau-Zener model is for a classical particle moving on two coupled ID PES. If the diabatic states cross so that the energy gap is linear with time, and the velocity of the particle is constant through the non-adiabatic region, then the probability of changing adiabatic states is... 

Stueckelberg derived a similar fomiula, but assumed that the energy gap is quadratic. As a result, electronic coherence effects enter the picture, and the transition probability oscillates (known as Stueckelberg oscillations) as the particle passes through the non-adiabatic region (see [204] for details). 

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... 

For example, the ZN theory, which overcomes all the defects of the Landau-Zener-Stueckelberg theory, can be incorporated into various simulation methods in order to clarify the mechanisms of dynamics in realistic molecular systems. Since the nonadiabatic coupling is a vector and thus we can always determine the relevant one-dimensional (ID) direction of the transition in multidimensional space, the 1D ZN theory can be usefully utilized. Furthermore, the comprehension of reaction mechanisms can be deepened, since the formulas are given in simple analytical expressions. Since it is not feasible to treat realistic large systems fully quantum mechanically, it would be appropriate to incorporate the ZN theory into some kind of semiclassical methods. The promising semiclassical methods are (1) the initial value... 

Stokes theorem, geometric phase theory, eigenvector evolution, 14-17 Stueckelberg oscillations, direct molecular dynamics, trajectory surface hopping, 398-399... 

Stueckelberg, E. C. G. (1932). Theorie der unelastischen Stosse zwischen Atomen. Helv. Phys. Acta, 5 369. 

Then fie may be thought of as the phase accumulated by the function C](r) during the period /3. To find B in (C.7) we should compare the phase < (/6 + t,t) to that calculated in the adiabatic approximation 4>sd. According to the standard arguments of Landau-Zener-Stueckelberg theory, this difference arises mostly from passing the point Q(r ) = Qc where the adiabaticity is violated. In the vicinity of this point Eqs. (C.9) simplify to... 

The quasiclassical trajectory method disregards completely the quantum phenomenon of superposition (13,18,19) consequently, the method fails in treating the reaction features connected with the interference effects such as rainbow or Stueckelberg-type oscillations in the state-to-state differential cross sections (13,17,28). When, however, more averaged characteristics are dealt with (then the interference is quenched), the quasiclassical trajectory method turns out to be a relatively universal and powerful theoretical tool. Total cross-sections (detailed rate constants) of a large variety of microscopic systems can be obtained in a semiquantitative agreement with experiment (6). 

Equation (96) differs from those found earlier by Stueckelberg [15] and Bates [89], Stueckelberg [15] used the perturbation theory to calculate PAB (b) at b > jPc and applied a Landau-Zener type of formula for b < Re. It was then found that the main contribution to distant collisions with b > PC. ... 

In connection with the two-state approximation we mention paper [107] giving detailed analysis of the Landau-Zener-Stueckelberg model in which transition probabilities have been calculated for a wide set of parameters of the model, and also papers devoted to extensions of the model [108-110]. The exponential model with the Hamiltonian given by equation (26) was discussed in [111] and extended in [112],... 

The figure also includes calculations by van den Bos conducted in 1969 based on the Landau-Zener-Stueckelberg dss approximation. The significantly higher cross section can be primarily attributed to the smaller coupling of 0.356 eV calculated by van den Bos. 

E.E.Nikitin and A.I.Reznikov, Stueckelberg phase calculations for the two-state exponential model, J.Phys.B. 11, 695 (1978)... 

Electron-jump in reactions of alkali atoms is another example of non-adiabatic transitions. Several aspects of this mechanism have been explored in connection with experimental measurements (Herschbach, 1966 Kinsey, 1971). The role of vibrational motion in the electron-jump model has been investigated (Kendall and Grice, 1972) for alkali-dihalide reactions. It was assumed that the transition is sudden, and that reaction probabilities are proportional to the overlap (Franck-Condon) integral between vibrational wavefunctions of the dihalide X2 and vibrational or continuum wave-functions of the negative ion X2. Related calculations have been carried out by Grice and Herschbach (1973). Further developments on the electron-jump mechanism may be expected from analytical extensions of the Landau-Zener-Stueckelberg formula (Nikitin and Ovchinnikova, 1972 Delos and Thorson, 1972), and from computational studies with realistic atom-atom potentials (Evans and Lane, 1973 Redmon and Micha, 1974). 

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