Franck-Condon Principle

We can calculate according to Franck-Condon principle the intensities of transitions between different vibrational states of different electronic states: since the electron mass is quite small an electronic transition occurs very fast comparing with nuclear motion. It means that nuclei suddenly "feels" the change of a potential after an electronic transition (time duration of about 10-16s) and they move then in the field of the new potential. It means that there is vertical transition between the two states which is shown on the figure below. It prefers to have such transitions when nuclei adapts to a new potential in a very fast way. On the language of quantum mechanics it means that the most possible transition changes the vibrational wavefunction just a bit. One can see the following three transitions on figure below. The transition from the ground vibrational state v'' = 0 of the lower electronic state into vibrational state v' = 5 of the upper electronic state; certainly, there will be transitions into v' = 4 and v' = 6 since the corresponding wavefunctions of the two states finely overlap with the ground state v'' = 0 wavefunction.

Electronic transitions according to Franck-Condon principle.

In order to make quantitative calculation of these transitions one needs to have total wavefunctions of initial and resultant states. Fortunately, we can represent the wavefunction of states as the product of electronic wavefunction yel (r), which depends only on the electron coordinates (r), and vibrational wavefunction yv (R), which depends on the nuclear coordinates (R). (We can apply the Born-Oppenheimer approximation here because of the great difference between electron and nuclear masses).

Quantum mechanical calculation of the transition dipole moment µ = -er gives(el', v' ← el'', v''):

µ = -e ∫ [yel'(r)yv'(R)]*·r [yel''(r)yv''(R)] dτElec dτNucleus

= -e òyel'*(r)r yel''(r) dτElec · òyv'*(R)yv''(R) dτNucleus

The first term doesn't depend on the nuclear vibrations and thus it's equal for all (v', v''). The second integral depends on the overlapping of both wavefunctions for vibration. Since the transition intensity depends on the (sum) squared transition dipole moment we will have the following (v', v'')-dependence:

½ò yv'* · yv'' · dt½2

If both potential curves have the same or at least very similar behavior, i.e. re' ≈ re'' and we' »we'' then there will be the strongest transition v' = v'' between the two curves (i.e. Δv = 0). But it's quite often case when re' > re'' and we' <we'' as shown on the figure above. For higher v' values one will have two intensive transitions (a and b). For v' = 0 it is only possible to have (b). The transition (c) leads to dissociation of molecule.