 Displaced-oscillator Franck-Condon progression

For equal-frequency harmonic ground and excited molecular potentials displaced by $\Delta Q$, the low-temperature vibrational overlap has the Poisson weights in the title, with $S=M{\omega }_v(\Delta Q{)}^2/(2\hslash )$. Optical lines occur at $E_{00}+\nu \hslash {\omega }_v$, apart from broadening and slowly varying electronic or optical prefactors. The normalized weights sum to one and have mean vibrational excitation $S$. Unequal frequencies, several vibrations, hot initial states and intermolecular coupling modify this simple Franck-Condon principle model.