Landau-Zener classical transition

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 transition described by (2.62) is classical and it is characterized by an activation energy equal to the potential at the crossing point. The prefactor is the attempt frequency co/27c times the Landau-Zener transmission coefficient B for nonadiabatic transition [Landau and Lifshitz 1981]... 

It is instructive to examine further the approximate semi-classical form for R7 shown above because, when viewed as a rate of transition between two intersecting energy sur ces, one anticipates that connection can be made with the well known Landau-Zener theory (10). For a non-linear molecule with N atoms, the potentials (Q) depend on 3N-6 internal degrees of fi eedom (for a linear molecule, Vj f depend on 3N-5 internal coordinates). The subspace S... 

This is, beyond all doubt, the most important process and the only one which has been already tackled with theoretically. Nevertheless, the prediction given by the classical overbarrier transition model is not correct for this collision [9] and the modified multichaimel Landau-Zener model developed by Boudjema et al. [34] caimot explain the experimental results for collision velocities higher than 0.2 a.u.. With regard to the collision energy range, we have thus performed a semi-classical [35] collisional treatment... 

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 situation is radically different in the inverted region, as well as in certain cases of nonequihbrium back transfer (see below), which are always nonadiabatic whatever the coupling strength is. For large V, the ET rate is no longer controlled by transport to the transition region but rather by nonadiabatic transitions between adiabatic states (see Fig. 9.1). Therefore, one should expect a decrease of the ET rate with increasing V to follow the solvent-controlled plateau. Usually, the Landau-Zener formula is used for the description of nonadiabatic transitions in the classical limit [162,163]. 

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 ... 

Thus this book describes the recent theories of chemical dynamics beyond the Born-Oppenheimer framework from a fundamental perspective of quantum wavepacket dynamics. To formulate these issues on a clear theoretical basis and to develop the novel theories beyond the Born-Oppenheimer approximation, however, we should first learn a basic classical and quantum nuclear dynamics on an adiabatic (the Born-Oppenheimer) potential energy surface. So we learn much from the classic theories of nonadiabatic transition such as the Landau-Zener theory and its variants. 

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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