Electronic Spectra in Diatomics

It was emphasized in the introduction to this chapter that molecular electronic transitions are generally accompanied by simultaneous changes in vibrational and rotational states. A calculation of the transition energy of a particular spectroscopic line thus requires knowledge of the rovibrational energy for a diatomic with vibrational and rotational quantum numbers v and J in the lower diatomic state 

In the Born-Oppenheimer approximation, the total diatomic wave functions in the upper and lower states are 

Since changes in rotational and vibrational as well as electronic state are possible, we must consider both the r- and R- dependence in the total dipole moment operator when calculating El transition probabilities, 

These expressions ignore rotation, which is considered in Section 4.6. The term proportional to eil ei) vanishes due to orthogonality of the electronic states. Note that A eil eiX.- does not factor into /ieil ei Z lx., , since the 

The Franck-Condon factors do obey a sum rule, however. If one sums the Franck-Condon factors for transitions from a particular vibrational level v in the lower electronic state to the complete set of levels v in the upper electronic state, one obtains 

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