In terms of quantum mechanics, the probability of an electron transition is proportional to the squire of the overlap integral between the vibrational wavefunction and before and after the electronic transition.
To treat the problem quantitatively, the complete molecular wavefunction of the initial and the
final state are needed. Fortunately, it is often possible to present this wavefunction as a
product of the electronic wave function with depends on the coordinates of all
electrons and the vibrational wavefunction which is a solution of
the Schrödinger equation for nuclei and depend on the internuclear distance . This
approach is based on the Born-Oppenheimer approximation which argues with large difference in
mass for nuclei and electrons. In the Born-Oppenheimer approximation the probability of a
radiative transition is written as (see eq. (3)):
The integral over the electron coordinates does not dependent on the vibration of nuclei and it is identical for all pairs of v', v". The integral over represents overlap of the vibrational wave functions.
The quantities are called Franck-Condon factors. No selection rules exist for changes of the vibrational quantum number . This is because the vibrational wave functions of the initial and final states are in general not orthogonal to each other being the subject of Scrödinger equation with two different potentials and . Apart from that, the Franck-Condon-principle allows to calculate the probability of a transition from some vibrational level of the initial state to another vibrational level of the final state.
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