Franck–Condon factors and electronic transitions

For two electronic surfaces a and b, integrate electronic coordinates first to obtain an electronic transition-dipole function of nuclear geometry. The remaining nuclear amplitude is the integral of this function between the initial and final vibrational wavefunctions. The Condon approximation replaces that electronic function by a constant over the nuclear region contributing significantly. The squared nuclear overlap is then the Franck–Condon factor. This approximation can fail when the transition dipole varies rapidly or vanishes by symmetry at the reference geometry.

The common vertical-transition picture means the electronic interaction is fast compared with nuclear rearrangement, so the initial nuclear coordinate distribution is projected onto a different surface before it relaxes. It is not a claim that a molecule follows a single classical coordinate with no quantum spread. The final vibrational distribution depends on wavefunction overlaps, including their signs and nodes. Rotational factors, electronic transition strength, photon-frequency factors, and initial populations must still be included to compare actual absorption or emission intensities.

Take one nuclear coordinate with the same reduced mass and harmonic angular frequency on both surfaces, but equilibrium positions separated by d. Start in the lower surface’s vibrational ground state. The displacement measured in oscillator-width units defines a dimensionless parameter $S=\mu\omega d^2/(2\hbar)$. Overlapping the two Gaussian ground states gives amplitude $e^{-S/2}$ and therefore zero-to-zero factor $e^{-S}$. A larger displacement suppresses that origin band without requiring a weaker electronic dipole.

Expanding the displaced ground state in the upper oscillator’s number basis gives weights $e^{-S}S^n/n!$. These form a Poisson distribution with mean S, and their sum over the complete infinite harmonic ladder is one. The derivation uses identical curvatures, one mode, an initial ground vibration, and a constant transition dipole. It does not justify a Poisson envelope for arbitrary anharmonic or many-mode molecules, where mode rotations and changed frequencies can reshape the overlaps.

When two surfaces have similar equilibrium geometries, low-vibrational overlaps can dominate. Larger displacement distributes weight among higher final vibrations, so the strongest observed transition need not be the electronic origin. This explains why a vertical absorption maximum and the relaxed electronic energy difference can differ. Emission after nuclear relaxation starts from a different initial nuclear state and can produce a shifted envelope, even before solvent or other environmental effects are considered.

Extracting geometry from intensities requires checking those assumptions rather than simply reading a peak number as a bond displacement. Non-Condon transition moments can borrow intensity, thermal occupation adds hot bands, and overlapping rotational structure or linewidths can obscure individual weights. Isotopes also change oscillator widths through reduced mass while leaving the leading electronic equilibrium displacement similar. The overlap calculation below is therefore an exact result for a declared illustrative model and a useful diagnostic starting point for richer spectroscopy.