Landau-Zener-Stueckelberg

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... 

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... 

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. 

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). 

From now on we will assume that we are dealing with a set of two crossing diabatic states for all R and an interaction matrix element Hl2(R) = <(/>, Hel 02), at least at R = Rc. In the next section we will discuss the well-known Landau-Zener-StUeckelberg (henceforth abbreviated as LZS-) model in order to solve the coupled equations (4). 

Transitions Due to Curve Crossings Complete Solutions of the Landau-Zener-Stueckelberg Problems and Their Applications. 

According the standard arguments of the Landau-Zener-Stueckelberg theory, this difference emerges mostly from passing the point Q x ) = Qc where the adiabaticity is violated. In the vicinity of this point eqs. (B.9) simplify to... 

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... 

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. ... 

Already in 1932, independently of each other, Landau, Zener and Stueckelberg proposed an expression for the inelastic transition probability... 

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). 

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