3.3 Franck-Condon principle

The Franck-Condon principle explains why certain vibronic transitions in a molecule are strong while others are weak. It connects the speed of electronic transitions to the resulting vibrational structure in absorption and emission spectra, making it essential for interpreting UV-Vis spectroscopic data.

Fundamentals of the Franck-Condon Principle

Franck-Condon principle in electronic transitions

The core idea rests on the Born-Oppenheimer approximation: electrons move so much faster than nuclei that, during an electronic transition (~$10^{-15}$ s), the nuclei essentially don't move. Because the nuclear positions and momenta stay frozen, the transition is drawn as a straight vertical line on a potential energy diagram. That's why these are called vertical transitions.

But which vibrational level does the molecule land in after the transition? That depends on the Franck-Condon factor, which is the square of the overlap integral between the vibrational wavefunction of the initial state and the vibrational wavefunction of the final state:

$\text{FC factor} = |\langle \chi_{v'} | \chi_{v''} \rangle|^2$

The larger the overlap, the more probable (and more intense) that particular vibronic transition will be. The pattern of Franck-Condon factors across all accessible vibrational levels is what shapes the intensity envelope you see in a UV-Vis spectrum.

Vibrational structure of spectra

A vibronic transition involves a simultaneous change in both electronic and vibrational quantum numbers. In a spectrum, these appear as a series of peaks called a vibrational progression, where each peak corresponds to a transition ending in a different vibrational level.

How the intensity is distributed across that progression depends on how much the equilibrium geometry changes between the two electronic states:

• Small geometry change: The best overlap is between $v = 0$ in the ground state and $v' = 0$ in the excited state. The 0-0 transition dominates, and the progression is short.
• Large geometry change: The $v = 0$ wavefunction of the ground state overlaps best with higher vibrational levels in the excited state. The 0-0 transition is weak, and the progression extends over many peaks.

The Huang-Rhys factor ($S$) quantifies this displacement between the two potential energy curves. A larger $S$ means a bigger geometry change and a longer vibrational progression. The most probable transition lands near vibrational level $v' \approx S$.

Emission spectra typically originate from the lowest vibrational level of the excited state ($v' = 0$), following Kasha's rule. The resulting emission progression is often an approximate mirror image of the absorption spectrum, reflected about the 0-0 transition. Deviations from this mirror-image relationship occur when the excited-state potential energy surface has a significantly different shape than the ground state.

Advanced Concepts and Applications

Vertical transitions and Franck-Condon principle

Because the transition is vertical, the molecule arrives in the excited electronic state with its nuclei still in the ground-state equilibrium geometry. If the excited state has a different equilibrium bond length, the molecule is now in a non-equilibrium nuclear configuration and carries excess vibrational energy.

What happens next follows a predictable sequence:

1. The vertical transition populates a distribution of vibrational levels in the excited state, weighted by Franck-Condon factors.
2. Vibrational relaxation rapidly dissipates excess vibrational energy (typically on a $10^{-12}$ s timescale), funneling population down to $v' = 0$.
3. Emission then occurs vertically from $v' = 0$, landing on various vibrational levels of the ground state.
4. The ground state undergoes its own vibrational relaxation back to $v = 0$.

The energy lost to vibrational relaxation in both states means the emitted photon has less energy than the absorbed photon. This energy gap between the absorption maximum and the emission maximum is the Stokes shift. A larger geometry change between states produces a larger Stokes shift.

Factors in vibronic transition intensity

Several factors beyond the basic Franck-Condon overlap influence what you actually observe in a vibronic spectrum:

• Equilibrium geometry changes: As discussed, the magnitude and direction of bond length or angle changes between states control the breadth and peak position of the vibrational progression.
• Symmetry considerations: Molecular symmetry imposes selection rules on which vibronic transitions are allowed. In benzene, for example, the $S_0 \rightarrow S_1$ electronic transition is symmetry-forbidden but gains intensity through vibronic coupling with certain vibrational modes.
• Temperature effects: At higher temperatures, excited vibrational levels of the ground state become thermally populated. Transitions originating from these levels ($v > 0$) produce hot bands on the low-energy side of the absorption spectrum.
• Solvent interactions: The solvent stabilizes different electronic states to different extents, shifting and reshaping the potential energy surfaces. This leads to solvatochromism, where absorption and emission maxima shift with solvent polarity.
• Vibronic coupling: Interaction between electronic and vibrational motions can cause a formally forbidden electronic transition to "borrow" intensity from a nearby allowed transition. This is why some symmetry-forbidden transitions still appear in spectra, though with reduced intensity.
• Duschinsky rotation: The normal modes of vibration can mix (rotate in normal-mode space) between the ground and excited states. This complicates the Franck-Condon analysis because you can no longer treat each mode independently; multidimensional overlap integrals become necessary.