Landau-Zener approximation

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 simplicity we will continue to assume that the reaction path is mapped out by a single coordinate Q. In the Landau-Zener approximation [23,24] we need the distance between the upper and lower PES. This energy may be calculated variationally as an excitation energy at the top of the barrier when the symmetric system has full symmetry. In the case of an asymmetric system, the transition state occurs when the relevant MO s are localized half on donor and half on acceptor. 

In this study we first examine the systematics that exist in the RKR potentials of the state to establish that the potential curves are strongly ionic for R R. (R. is the distance of the pseudocrossing point). Next we evaluate an essentially experimental value of the parameters relevant to the cross section for the charge transfer process in the Landau-Zener approximation, We also construct a model ionic potential which can be used to describe the charge transfer process in the ionic region. 

Stancil and Zygelman (21) have derived a simple formula, based on the Landau-Zener approximation, for the influence of a kinematic isotope effect on low-energy total charge transfer cross sections. Their predictions have since been verified for various systems. Taking as a representative example the electron capture into... 

In the Landau-Zener approximation for the transition probability (Landau and Lifshitz, 1965 Zener, 1933 Nikitin, 1974 Hirst, 1985 Nesbitt and Hynes, 1986), P = 1 - Pj is given by... 

The probability for passing from the lower PES to the upper is expressed in the Landau-Zener approximation. We will use this approximation to calculate k in Equation 10.51. The Landau-Zener probability P for crossing from the ground state PES to the upper state may be expressed in terms of the slopes of the matrix elements, Hji and H22, just outside the crossing region and the coupling H,2 = 1/2A ... 

To define the conpUng, we take the pragmatic approach that the gap A is the quantity that goes into the Landau-Zener approximation in other words, the gap marked ont in Eigure 10.10. The gap is thus at the same time the minimum gap between two PES. Koopmans theorem may be used to obtain the gap, no matter whether this is the minimum gap. This gap is simply equal to IH22 - Hjj, but we cannot know beforehand where the minimum is situated along the Q-axis. On the other hand, we know that when the gap attains the minimnm valne, the wave functions are equally distributed on the donor D and acceptor A. 

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