Band shapes in UV/Visible spectra: Franck-Condon Principle

Let us consider an electronic transition from state $A$ to state $B$. The intensity of the pure electronic transition is governed by the transition dipole moment ${\mu }_{AB}$:

$$I\propto |{\mu }_{AB}{|}^2.$$

In reality, electronic transitions are accompanied by changes in the vibrational state of the molecule (vibronic transitions). Assuming only transitions from the ground vibrational state of electronic state $A$, $n=0$, (i.e., zero temperature) the intensity of vibronic transition from ${\chi }_{A0}$ to ${\chi }_{Bm}$ is in the first approximation:

$$F_{m0}={\mu }_{AB}|⟨{\chi }_{Bm}|{\chi }_{A0}⟩{|}^2,$$

where ${\mu }_{AB}$ is the electronic transition dipole moment, whereas $|⟨{\chi }_{Bm}|{\chi }_{A0}⟩|$ is the Franck-Condon factor equal to the overlap of the vibrational wave functions in different electronic states. Thus an electronic band (for example in the UV/Visible absorption spectrum) consists of vibronic lines, which may or may not be resolved in the experiment. This shape is in general asymmetric unlike Lorentz and Gauss profiles.

References

• Franck, J. & Dymond, E. G. "Elementary processes of photochemical reactions," Trans. Faraday Soc. 21, 536-542 (1926). DOI: 10.1039/TF9262100536
• Condon, E. U. "A theory of intensity distribution in band systems," Phys. Rev. 28, 1182-1201 (1926). DOI: 10.1103/PhysRev.28.1182