The Franck-Condon Principle: Vertical Transitions on Molecular Potential Curves

The physical setup: two curves and a frozen snapshot

A diatomic molecule stores its energy in two nested potential energy curves — one for the ground electronic state, one for an excited state — each plotted as energy versus internuclear separation R. Every curve has a minimum at its equilibrium bond length Rₑ, and stacked inside each well are quantized vibrational levels v = 0, 1, 2, … whose wavefunctions χ_v describe where the nuclei are likely to be found.

The Franck-Condon principle rests on a simple physical fact: light-driven electronic rearrangement takes about 10⁻¹⁶–10⁻¹⁵ s, while one vibrational period is roughly 10⁻¹³–10⁻¹⁴ s — a hundred to a thousand times slower. During the photon absorption, the nuclei have no time to move or change momentum.

• On the diagram the transition is therefore vertical: R is conserved at the instant of the jump.
• The excited state usually has a longer bond (a bonding electron is promoted to an antibonding orbital), so its curve sits shifted to larger R.
• The vertical line lands not at the excited-state minimum but on the repulsive inner wall or outer branch, favoring an excited vibrational level.

From frozen nuclei to the overlap integral

Quantitatively, the transition probability comes from the electronic transition dipole moment integrated over both electrons and nuclei. Invoking the Born-Oppenheimer approximation — which separates fast electronic and slow nuclear motion — the total transition moment factorizes:

M = ⟨ψ_e' χ_v' | μ | ψ_e'' χ_v''⟩ ≈ μ_e · ⟨χ_v' | χ_v''⟩

Here μ_e = ⟨ψ_e'|μ|ψ_e''⟩ is the pure electronic transition dipole (assumed roughly constant with R, the Condon approximation), and ⟨χ_v'|χ_v''⟩ is the vibrational overlap integral between the initial level v'' and final level v'. The measurable line intensity scales as:

• I ∝ |μ_e|² · |⟨χ_v'|χ_v''⟩|²

The squared overlap |⟨χ_v'|χ_v''⟩|² is the Franck-Condon factor (FCF). Because the χ_v are normalized, the FCFs for a given lower level sum to one: Σ_v' |⟨χ_v'|χ_v''⟩|² = 1. Transitions with the biggest wavefunction overlap — where the nuclei can stay put and land on a classical turning point of the upper state — are the brightest. This is the modern, indeterminacy-safe version Condon published in 1928.

Characteristic numbers and a worked example

Consider molecular iodine, the textbook case. The B(³Π₀⁺u) ← X(¹Σg⁺) absorption spans roughly 500–650 nm (about 1.9–2.5 eV). The excited B state has a longer bond, Rₑ' ≈ 0.303 nm versus Rₑ'' ≈ 0.267 nm, a displacement ΔRe ≈ 0.036 nm. The vibrational spacing shrinks from ħω'' ≈ 214 cm⁻¹ (about 27 meV) in the ground state to ħω' ≈ 125 cm⁻¹ in the B state.

• Because ΔRe is finite, a vibrational progression appears: the vertical line from v''=0 lands near v' ≈ 20–30, so those bands are strongest, not the 0–0 line.
• The intensity envelope mirrors the ground-state probability density |χ₀''|², which is peaked at the classical turning points.

A useful shorthand is the Huang-Rhys factor S, roughly the average number of vibrational quanta excited. For a displaced harmonic oscillator the FCFs follow a Poisson distribution: |⟨v'|0⟩|² = e^(−S) S^v'/v'!. The 0–0 line carries weight e^(−S); when S = 3 that is only about 5%, so most intensity sits in higher members of the progression.

How it is observed and used

The Franck-Condon principle is read directly off any well-resolved electronic spectrum:

• UV-vis absorption shows the progression climbing from the ground state's v''=0; band spacings give ħω' of the excited state and, via a Birge-Sponer extrapolation, its dissociation energy De'.
• Fluorescence runs the process in reverse — vertical emission down from the relaxed excited-state minimum — producing a near mirror-image progression and the Stokes shift between absorption and emission maxima.
• The 0–0 line is the one transition common to both spectra and pins the true adiabatic energy gap.

Practically, FCFs govern the brightness of dyes and OLED emitters, the quantum yields of photochemistry, and the line strengths used in laser cooling of molecules, where researchers deliberately seek near-diagonal FCFs (⟨v'=0|v''=0⟩² ≈ 0.9+) so a molecule cycles thousands of photons without leaking into dark vibrational states. Franck-Condon factors also set the rates in Marcus electron-transfer theory and in vibrationally resolved photoelectron spectroscopy.

Related regimes and close cousins

The Franck-Condon principle is one link in a family of approximations, and it helps to see its boundaries:

• Born-Oppenheimer approximation is the parent: it justifies drawing fixed potential curves at all. Franck-Condon adds the further assumption that nuclei don't move during the electronic jump.
• Condon approximation assumes μ_e is independent of R. When it fails — near forbidden transitions or conical intersections — Herzberg-Teller coupling (vibronic borrowing) lets a nominally dark transition steal intensity, and the simple FCF picture breaks down.
• Kasha's rule complements it on the emission side: relaxation to the lowest excited vibrational level before emitting explains the mirror symmetry.
• For continuous absorption into a repulsive (dissociative) state, the upper 'level' is a continuum, the reflection approximation applies, and the band is structureless.

Compared with a purely classical picture, quantum FCFs also allow weak transitions into levels the classical vertical line would forbid, because wavefunctions have tails beyond turning points.

Significance, history, and open questions

James Franck, already a 1925 Nobel laureate for electron-atom collision work, argued in 1926 that the fastest, most probable excitations conserve nuclear positions and momenta. Edward Condon, then a young theorist, recast this in wave-mechanical language in 1926 and, in his definitive 1928 paper, replaced the classical 'fixed position' statement with the overlap integral, sidestepping any conflict with Heisenberg's uncertainty principle.

The principle remains foundational a century later, but active questions push past its assumptions:

• At conical intersections, electronic and nuclear motions couple strongly; the Born-Oppenheimer separation collapses and ultrafast (few-femtosecond) nonadiabatic dynamics take over — the frontier probed by attosecond and 2D electronic spectroscopy.
• Computing accurate multidimensional FCFs for large, floppy molecules (with Duschinsky rotation mixing the normal modes) is still numerically demanding.
• Engineering diagonal Franck-Condon factors is a live design problem for direct laser cooling of molecules like SrF, CaF, and YO — a route toward ultracold molecular quantum computing and precision tests of fundamental physics.

Franck-Condon behavior versus the geometry change between electronic states (displacement Δ of the excited-state minimum relative to the ground state).

Regime	Excited-state offset ΔRe	Most intense line	Band appearance
No geometry change	ΔRe ≈ 0	0–0 (v''=0 → v'=0)	Sharp, single dominant line
Small displacement	ΔRe ≈ 0.02–0.05 nm	v'=1 or 2	Short vibrational progression
Large displacement	ΔRe ≳ 0.1 nm	High v' (v'=10–30)	Long progression, intensity mirrors ground-state |χ₀|²
Displacement to dissociative wall	vertical hits above De'	Continuum (unbound)	Structureless absorption continuum