Non-Condon corrections of the FC values

NON-CONDON CORRECTIONS OF THE FC VALUES 6.1. Basic theory of the non-Condon corrections 

The above analysis is based on the Born-Openheimer approximation in which the adiabatic electron wave functions are frozen at the bottom of the corresponding minimum. An important advantage of this approach is that we work with the limited size for the electron basis functions and not with the infinite basis of the vibrational states. This makes the problem solvable in simple terms. 

Now we want to introduce the non-Condon corrections to the electron wave functions by lifting the restriction of the rigid-frame approximation and provide the electronic wave functions with an additional flexibility, namely the ability to follow the nuclear vibrations. We return to the expression (8) for the second-order RF. The numerator of this expression consists of three factors namely the bra ( T/ p I, the operator 8 Cm), and the ket I P/q). In order to find the second-order correction 

As we are interested in the low energy states close to the bottom of the wells, the amplitude of nuclear motion is small compared to the overall average value of the nuclear displacement. Thus the criterion for smallness comes from the small deviation qp of the displacement from the bottom of the minimum point. Ultimately, we should include nuclear motion as a part of the dynamic problem so that the parameter qp will become a dynamic variable associated with the ground harmonic oscillator state 10) in well p. However, this will not be considered further here. 

It is necessary next to relate these small displacements to the collective coordinates g, from a Taylor expansion of the general potential energy U(Q). In the adiabatic approximation, the nuclear coordinates Q are free parameters and can be used as a basis for the Taylor expansion. Thus we write IJ(Q) in the general form (Ref. [2], Chapter 3)  

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