Zener Formulation

Landau and Zener made the first attempts to formulate the probability of transition of the reacting system from a lower electronic state to an upper one, and thus, to find the adiabaticity and nonadiabaticity of a reaction. 

Landau and Zener independently gave the first mathematical formulation of the condition of adiabatic and nonadiabatic reactions in terms of probability factors. 

The probability that a reaction will occur adiabatically is expressed 

The probability that the reaction will occur nonadiabatically can be expressed as 

Vr is the relative velocity of approach of the one reactant to another during reaction, and (Si - S2) is the net force exerted on the system tending to restore to its original state or take it to a final state. Hence, the probability of the adiabatic transition is dominant. If V12 is small, the probability of a nonadiabatic transition is dominant. If the relative kinetic energy corresponding to relative velocity of approach of the reaction systems,Vr,is large compared to V12, the reaction becomes nonadiabatic. 

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According to the Wert-Zener formulation, Do = yalfj cxpi-AS lR), where y is a geometrical factor, f the correlation factor, oo the lattice parameter, v a characteristic vibration frequency usually the Debye frequency and AS =... 

The well-known Landau-Zener [155-158] formula relating to the probability of an electronic jump near the crossing point of two potential-energy curves or surfaces has been seriously critiqued [4, 154], New treatments of greater validity have been formulated [154, 159, 160]. 

It has been realized recently that the parameter O is of importance in calculating differential scattering. This, in turn, stimulated calculation of for the earliest model of nonadiabatic coupling, the Landau-Zener-Stueckel-berg model [28-30]. The model is formulated for a Hamiltonian with AH linear in R ... 

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). 

To the best of our knowledge, most of the general theories of nonadia-batic transition has been formulated to treat one-dimensional nonadiabatic systems of an isolated avoided crossing, the Landau Zener theory being an example. On the other hand, the theory of PSANB (and SET as well as its approximation) is not limited to those specific cases. Nevertheless, we need to examine how practically PSANB works by comparing its results with the full quantum calculations in a couple of one-dimensional... 

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 concept of diabatic states dates back to the beginnings of molecular quantum mechanics, since it is implicit in the work of Landau and Zener on the electronic transitions occurring at a curve crossing (see Valence Bond Curve Crossing Models). Similar concepts underlie the drawing of correlation diagrams, first introduced by Hund and Mulliken around 1930, and subsequently exploited, among others, by Woodward and Hoffmann, to formulate their famous rules (see Photochemistry). 

The most simple description for the electronically diabadc processes was formulated independently by Landau and by Zener. Following the model of these authors, the probability of passing from one surface to the other, considered as unidimensional, in the avoided crossing region, is given by... 

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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