Franck Condon Principle

Using the Franck-Condon principle, we are able to calculate the intensities of transitions between vibrational states of the electronic energy levels. Due to their low mass, an electronic transition occurs fast (within 10-16s). In turn, the potential around the nuclei changes and they will adopt another vibration. In a diagram that relates two electronic states and nuclear coordinates with the energy of a molecule, the first step is reflected by a vertical line, i.e. the initial and final positions of nuclei are identical. Nevertheless, as we can imagine cases where the nuclear coordinates allow to adopt more easily some new vibrational state, these states are of relevance. Such transistions are preferred and the molecules will continue to vibrate with another amplitude and equilibrium position re. In terms of quantum mechanics, we find that the probability of a transition correlates with the overlap between the vibrational wavefunction Ψv"(R) and Ψv'(R) before and after electronic excitation respectively. For example, in the energy diagram below, arrow a connects wavefunction Ψv"=0(R) with Ψv'=5(R). Transitions towards v" = 4 and v" = 6 will occur too, as, to some extent, there is as well an overlap with the wavefunction for the ground state.
Franck Condon Principle
The arrows indicate transitions which are favoured by the Franck Condon principle.

To treat the problem quantitatively, the complete wavefunction for the initial and final state are needed. Fortunately, it is possible to replace these wavefunctions by a product of the electronic wave function Ψel with a variable r that represents the position of the electrons and the vibrational wavefunction Ψv(R) which is solely a function of a variable R, the coordinates of the nuclei. This approach is based on the Born-Oppenheimer approximation which argues with large difference in mass for nuclei and electrons.

The quantum mechanical calculation starts with the transition dipole moment µ = -er (el', v' ← el", v"):

µ = -e ∫ [ψel'(rv'(R)]*·rel"(rv"(R)]dτelecnuclei

   = -e ∫ψel'*(r)rψel"(r) dτelec  ·  ∫ψv'*(Rv"(R) dτnuclei

The first integral is not dependent on the vibration of nuclei and therefore identical for all pairs of v', v". The second represents the overlap of the two vibrational wave functions. Eventually, we find the intensity of transitions being proportional to the square of the magnitude of the transition dipole moment.

|∫ ψv'*ψv" dτ|2

In cases where both potential curves do not differ greatly, i.e. with only slight changes for equilibrium distances re and frequency ωe, the most intense transition is found for v' = v", i.e. Δv = 0. But generally, we have re' > re" and ωe' < ωe". Under these circumstances depicted in the above diagram, the two transitions indicated by arrow a and b are especially probable. For v' = 0 there is only one transition (arrow b). The excitation represented by arrow c initiates the dissociation of the molecule.


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