Quantum phase arising from nonadiabatic transitions

2 Quantum phase arising from nonadiabatic transitions 4.1.2.1 Phase in nonadiabatic interactions 

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). 

Child [83, 84] developed a uniform diagrammatic approach for onedimensional problems classical turning points, barrier penetration, and curve crossings are represented in terms of individual diagrams. These are further connected with link matrices of the semiclassical wavefunction coefficients. Child showed that a wide range of spectroscopic phenomena such as predissociation and restricted rotation are described as a combination of these elementary diagrams. The wavefunction is written in a semiclassical form as 

Chemical Theory Beyond the Born- Oppenheimer Paradigm 

A +(r)] /(27t), with r and r2 being the (complex) crossing points that exist on the anal3dically continued functions of the adiabatic potential surfaces, whereas the phase factor x is given as x = argr(fj/) — u ni/ - - u - - tt/4. Thus, a unified semiclassical theory has been established for a chemical reaction in which plural elementary processes may be involved. If one replaces V by the LZ exponent p =, Eqs. (4.11), (4.12) can be seen 

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