Born-Oppenheimer approximation vibrational selection rules

Hence, according to the symmetry selection rule, n —> n transitions are allowed but n —> ti transitions are forbidden. However, in practice the n —> it transition is weakly allowed due to coupling of vibrational and electronic motions in the molecule (vibronic coupling). Vibronic coupling is a result of the breakdown of the Born-Oppenheimer approximation. 

So far, this discussion of selection rules has considered only the electronic component of the transition. For molecular species, vibrational and rotational structure is possible in the spectrum, although for complex molecules, especially in condensed phases where collisional line broadening is important, the rotational lines, and sometimes the vibrational bands, may be too close to be resolved. Where the structure exists, however, certain transitions may be allowed or forbidden by vibrational or rotational selection rules. Such rules once again use the Born-Oppenheimer approximation, and assume that the wavefunctions for the individual modes may be separated. Quite apart from the symmetry-related selection rules, there is one further very important factor that determines the intensity of individual vibrational bands in electronic transitions, and that is the geometries of the two electronic states concerned. Relative intensities of different vibrational components of an electronic transition are of importance in connection with both absorption and emission processes. The populations of the vibrational levels obviously affect the relative intensities. In addition, electronic transitions between given vibrational levels in upper and lower states have a specific probability, determined in part... 

When discussing symmetry selection rules it was mentioned that vibrational motion can influence both the shape and the intensity of electronic absorption bands. In the usual Born-Oppenheimer approximation with molecular wave functions written as products as in Equation (1.12) this can be understood as follows. 

In Section 4.4 we worked out the El electronic and vibrational selection rules for electronic band spectra, and it remains for us to determine the selection rules that govern the rotational fine structure. We have seen that no symmetry selection rule exists for Ay, but that the vibrational band intensities are proportional to Franck-Condon factors in the Born-Oppenheimer approximation. To understand the selection rules for simultaneous changes in electronic and rotational state, we must find how l eiZrot) = transforms under... 

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