Probabilities of Electronically Non-Adiabatic Processes

The interactions neglected in the adiabatic approximation should be considered in the region where the Massey parameter is not high, i.e. where the potential surfaces either cross or pseudo-cross. For these regions, the problem is formulated in terms of coupled equations for wave functions describing nuclear motions over several potential surfaces. If, however, the motion of nuclei can be considered as classical, these equations can be substantially simplified. The semiclassical version of an adiabatic function T (q, Q) (Eq. (9.3)) reads 

The breaking of the adiabatic approximation is manifested in that the state of the whole system cannot be described either by the adiabatic function or by superposition of such functions with constant coefficients. The exact wave function 0(q, t) corresponding to the given motion of the slow subsystem is to be determined from the time-dependent Schrodinger equation 

This wave function can be expanded in terms of the wave functions (9.5) but the coefficients ajx depend on time 

Therefore, the probabilities of finding the system in different adiabatic states of the fast subsystem ajx(t) are time dependent. This is the result of transitions between the quantum states of the fast subsystem which are called non-adiabatic transitions. The calculation of such transition probabilities is the basic problem of the theory of non-adiabatic transitions. 

The coefficients a satisfy the set of equations obtained by substituting Eq. (9.7) into Eq. (9.6) 

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