Landau-Zener classical transition probability

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. 

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... 

If we further assume that the diabatic coupling V(t) is constant, then Eq. (12) can be solved exactly in terms of the Weber function. Then the final transition probability is exactly equal to Eq. (10). The linearity in time t is very much different from the linearity in coordinate R and the effects of turning points are completely neglected in the former approximation. In Landau s treatment this corresponds to the assumption of the common straightline classical trajectory with constant velocity. Thus, the Landau-Zener formula Eq. (10) is valid only at collision energies much higher than the crossing point. 

Another practical method is TSH (18), in which ordinary classical trajectories are run until they come close to the surface crossing region where the trajectories are jumped to the other surface with probability given by the Landau-Zener formula. This method is simple and convenient, but suffers from the following drawbacks all phases are completely neglected and only the probabilities (not the probability amplitudes) are handled the detailed balance is not necessarily satisfied and nonadiabatic transitions at energies... 

The Landau Zener formula is a scattering solution to the problem of crossing between two diabatic cmves, derived by assmning that the nuclei follow a classical trajectory. The diabatic curves are linearized around the crossing point, which translates into an electronic energy gap varjung linearly with time. The transition probability P is ... 

The probability of the Pj transition depends essentially on the form of the potential surfaces , and E. Its correct calculation for real multidimensional PES s constitutes a complex mathematical problem, which is why it is a common practice to perform in this case one-dimensional approximation (the reaction coordinate is approximated by one parameter) and make use of the classical expression for the probability of a transition between the PES s ( and known as the Landau-Zener formula. 

The classical probability of transition probability was first formulated by Landau and Zener [23,24], and was presented in eq. 5.53. Using the notation of this chapter... 

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