Landau-Zener transition probability

The Landau-Zener transition probability is derived from an approximation to the frill two-state impact-parameter treatment of the collision. The single passage probability for a transition between the diabatic surfaces H, (/ ) and R AR) which cross at is the Landau-Zener transition probability... 

If S Sx the avoided crossing is traversed diabatically and if S Sx it is traversed adiabatically. Between the two extremes of purely adiabatic and purely diabatic traversals, the probability of making the diabatic transition is given by the Landau-Zener transition probability, as has been demonstrated by Rubbmark et... 

The problem of two crossing points was considered by Shimshoni and Gefen [1991]. In this case the resulting value of the transition probability is dominated by an interference effect, which becomes important when the separation between crossing points is comparable with the characteristic length of interaction region for a single Landau-Zener transition. 

In any problem involving avoided crossings, there are two modes of variation. A physical system will evolve adiabatically, i.e. it will remain on one or other branch if the intensity A is varied slowly. If. however, the variation is rapid then diabatic or Landau-Zener transitions may occur [508]. The probability of a transition from one branch to the other at the avoided crossing is given by... 

Stuckelberg did the most elaborate analysis (15). He applied the approximate complex WKB analysis to the fourth-order differential equation obtained from the original second-order coupled Schrodinger equations. In the complex / -plane he took into account the Stokes phenomenon associated with the asymptotic solutions in an approximate way, and finally derived not only the Landau-Zener transition probability p but also the total inelastic transition probability Pn as... 

Figure 6 Landau-Zener transition probability in the linear potential model of the LZ type with a2 = 1.0. a2 and b2 are defined by Eq. (19). Landau-Zener means the result of Eq. (10).

The Landau-Zener transition probability pZN is as simple as the famous Landau-Zener formula of Eq. (10) but much better. Note that p is expressed in terms of a2 and b2 as... 

Figure 12 The Landau-Zener transition probability p in the LZ case. Dashed line exact, solid line Eqs. (170) and (183). (From Ref. 23.)...

The Landau-Zener transition probability yZN can be explicitly given as... 

Other reactions of spin-forbidden processes include a study by Manna and Chabalowski on the reaction, NO (X II) + CO (X S ) N (" S) + C02 (X Sg), using the SA-MCSCF/SOCI approach to locate the MEXP (60 kcal/mol above the reactants). A similar study was carried out on CH3 (X A2 ) + N( S) HCN (X S+) -P H2 (X S+), where it was found that the MEXP is located below (8 kcal/mol) the reactants. The barrierless transition occurs at the MEXP in 1 out of 208 vibrational periods, as deduced from the Landau-Zener transition probability using a 30 cm spin-orbit matrix element. 

Table 1 Parameters and Landau-Zener Transition Probabilities for the Neutral - Ionic Avoided Crossing in LiF and NaCl (au)...

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... 

The influence of the fluctuational motion along the reaction coordinates on the probability of the electron transition has been considered recently in the framework of the Landau-Zener method.102 A Hamiltonian of the form... 

Section 6.5.1), two avoided crossings arise in Figure 6.10c at t = +10 fs between states /) and 1 3), the first of which is marked by a gray circle. Due to the highly nonadiabatic time evolution, diabatic transitions between these dressed states are likely to occur. The Landau-Zener model [48, 104, 105] estimates the probability for a diabatic transition at the avoided crossings as... 

The probability of transition from one diabatic term to another when passing the point q, can be calculated using the Landau-Zener model [10]... 

The process may then be described by classical trajectories and by transition probabilities for changing the potential surfaces when these trajectories come close to a crossing. The transition probability for changing the diabatic curves at a crossing line R, is given by a Landau-Zener... 

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