The Franck-Condon Principle

Shine light on a diatomic iodine molecule. You get a UV-Vis spectrum with a jagged comb of peaks, not one clean line. Each peak corresponds to a different vibrational level of the excited electronic state. Why some peaks are tall, some tiny, and why they trail off exactly the way they do, is not obvious from a simple orbital picture. The Franck-Condon principle answers this. It explains why electronic transitions are effectively "vertical" on an energy diagram. It also lets us predict the relative intensities of vibrational bands within an electronic absorption or emission spectrum. This matters because vibrational band shapes are a direct experimental fingerprint. They show how much a molecule's geometry changes upon excitation. This article builds the physical picture from the nuclear-motion timescale argument. It then derives the Franck-Condon factor as a vibrational overlap integral. It works two full numerical examples. It closes with common pitfalls and a link forward to fluorescence and phosphorescence. There, the same overlap integrals govern emission intensities and the Stokes shift.

Key takeaways

Electronic transitions occur much faster than nuclear motion, so nuclei are effectively frozen during the transition (the Born-Oppenheimer separation applies).
Transitions are drawn as vertical arrows on a potential energy curve diagram, connecting a fixed nuclear geometry in the lower state to the upper state.
Relative vibrational band intensities are governed by the square of the vibrational wavefunction overlap, the Franck-Condon factor $q_{v',v''}$ .
Large equilibrium bond length changes between electronic states shift the intensity maximum to higher $v'$ , producing long vibrational progressions.
Franck-Condon factors are the same principle applied in reverse for emission, setting the mirror-image shape of fluorescence spectra and the size of the Stokes shift.
Key idea: electronic transitions are vertical in nuclear coordinate, and vibrational band intensities scale with the squared overlap of vibrational wavefunctions in the two electronic states.

The core insight

Because electrons respond to light roughly $10^3$ times faster than nuclei can move, an electronic transition happens at essentially fixed nuclear position and momentum, so the observed vibrational band intensities in an electronic spectrum measure the squared overlap between the initial and final vibrational wavefunctions, the Franck-Condon factor.

The physical picture

An electronic transition is triggered by absorption of a photon with energy in the UV-Vis range, typically a few eV. The classical period of electronic motion is roughly $10^{-15}$ – $10^{-16}\,\text{s}$ , while a typical molecular vibration has a period of order $10^{-13}\,\text{s}$ . The electronic rearrangement is therefore complete before the nuclei have moved appreciably — the electric field of the light "sees" a essentially static nuclear framework. James Franck first argued this classically in 1925: since nuclei are much heavier than electrons, their momenta and positions cannot change instantaneously, so the most probable transitions are those in which the nuclear position and momentum do not change during the electronic jump. Edward Condon (1926, 1928) recast this into full quantum mechanical form using vibrational wavefunction overlaps, giving the principle its modern quantitative content. Graphically, this is captured by drawing two Morse or harmonic potential energy curves, one for the ground electronic state ( $S_0$ ) and one for an excited state ( $S_1$ ), stacked vertically as functions of a nuclear coordinate (bond length $R$ for a diatomic). Because the transition is fast compared to nuclear motion, it is represented as a vertical line on this diagram: $R$ does not change during the transition. Where that vertical line lands relative to the vibrational levels and wavefunctions of the excited state determines which vibrational sublevels are reached, and with what probability. If the excited-state potential has its minimum directly above the ground-state minimum (little geometry change upon excitation), the vertical arrow from the $v''=0$ level lands near the bottom of the excited curve, and the 0–0 transition dominates the spectrum. If instead the excited-state equilibrium bond length $R_e'$ is shifted relative to the ground state $R_e''$ (a common situation, since removing a bonding electron or populating an antibonding orbital changes bond strength), the vertical arrow lands on the side of the excited-state potential, where the classical turning point sits — this maximizes overlap with a vibrational wavefunction that has a large amplitude near its own classical turning point, i.e., a high- $v'$ level. This produces a long vibrational progression whose intensity peaks away from $v'=0$ .

Figure

Vertical transitions on a potential energy curve diagram: small versus large equilibrium bond-length shift

The core relations

\Delta t_{\text{electronic}} \sim 10^{-16}\,\text{s} \ll \Delta t_{\text{vibrational}} \sim 10^{-13}\,\text{s}

where: the electronic transition is instantaneous on the vibrational timescale, justifying the "vertical" (fixed- $R$ ) transition picture.

\Psi_{n,v}(\mathbf r, R) \approx \psi_n(\mathbf r; R)\,\chi_{n,v}(R)

where: $\psi_n$ is the electronic wavefunction (parametric in $R$ ), $\chi_{n,v}$ the vibrational wavefunction in electronic state $n$ — the Born-Oppenheimer factorization.

\boldsymbol{\mu}_{fi} = \int \Psi_f^{*}\,\hat{\boldsymbol{\mu}}\,\Psi_i \, d\tau \approx \boldsymbol{\mu}_{e}(R_e)\int \chi_{v'}^{*}(R)\,\chi_{v''}(R)\, dR

where: $\boldsymbol{\mu}_e$ is the electronic transition dipole moment, assumed roughly constant over the relevant range of $R$ (the Condon approximation).

q_{v',v''} = \left| \int \chi_{v'}^{*}(R)\,\chi_{v''}(R)\, dR \right|^2

where: $q_{v',v''}$ is the Franck-Condon factor, the squared vibrational overlap integral between initial ( $v''$ ) and final ( $v'$ ) vibrational states.

I_{v'\leftarrow v''} \propto \nu^{4}\,|\boldsymbol{\mu}_e|^2\, q_{v',v''}

where: $I$ is the intensity (integrated absorption coefficient) of the band; the $\nu^4$ factor is a slowly varying prefactor from the transition-rate formula, usually treated as roughly constant across a vibrational progression.

\sum_{v'} q_{v',v''} = 1

where: for fixed $v''$ , Franck-Condon factors to all final vibrational levels sum to unity, since $\{\chi_{v'}\}$ is a complete orthonormal set.

Derivation AFrom the transition dipole to the Franck-Condon factor

Goal. Show that the intensity of a vibronic band factorizes into an electronic part and a purely vibrational overlap term, the Franck-Condon factor, starting from the full transition dipole matrix element.

Step 1 — Born-Oppenheimer factorization. The total molecular wavefunction for electronic state $n$ and vibrational level $v$ is written as a product of an electronic wavefunction $\psi_n(\mathbf r; R)$ , which depends parametrically on nuclear coordinate $R$ , and a vibrational wavefunction $\chi_{n,v}(R)$ that lives on the potential energy curve $V_n(R)$ generated by that electronic state:

\Psi_{n,v}(\mathbf r, R) = \psi_n(\mathbf r; R)\,\chi_{n,v}(R).

This separation is valid because electrons move so much faster than nuclei that at each fixed $R$ the electronic problem can be solved as if the nuclei were clamped in place.

Step 2 — Write the transition dipole integral. The intensity of an electronic transition is governed by the transition dipole moment between initial state $i = (n'', v'')$ and final state $f = (n', v')$ :

\boldsymbol{\mu}_{fi} = \int \Psi_{n',v'}^{*}(\mathbf r,R)\;\hat{\boldsymbol{\mu}}(\mathbf r, R)\;\Psi_{n'',v''}(\mathbf r,R)\; d\mathbf r\, dR.

The dipole operator $\hat{\boldsymbol{\mu}}$ contains both electronic and nuclear contributions, but for an electronic transition (different $n$ ) only the electronic part contributes appreciably, since the nuclear dipole term connects a state to itself in electronic character and integrates to zero between orthogonal electronic states.

Step 3 — Separate electronic and nuclear integration. Insert the product wavefunctions and integrate over electronic coordinates $\mathbf r$ first, at each fixed $R$ :

\boldsymbol{\mu}_{fi} = \int \chi_{v'}^{*}(R)\left[\int \psi_{n'}^{*}(\mathbf r;R)\,\hat{\boldsymbol{\mu}}(\mathbf r)\,\psi_{n''}(\mathbf r;R)\, d\mathbf r\right] \chi_{v''}(R)\, dR = \int \chi_{v'}^{*}(R)\,\boldsymbol{\mu}_e(R)\,\chi_{v''}(R)\, dR,

where $\boldsymbol{\mu}_e(R)$ is the $R$ -dependent electronic transition dipole moment, the bracketed quantity.

Step 4 — Apply the Condon approximation. Because $\boldsymbol{\mu}_e(R)$ typically varies slowly with $R$ over the narrow range where the vibrational wavefunctions have appreciable amplitude, it can be pulled out of the integral evaluated at a representative geometry, usually the ground-state equilibrium distance $R_e''$ :

\boldsymbol{\mu}_{fi} \approx \boldsymbol{\mu}_e(R_e'')\int \chi_{v'}^{*}(R)\,\chi_{v''}(R)\, dR.

Step 5 — Square to get intensity. The transition rate, and hence the observed absorption or emission intensity, is proportional to $|\boldsymbol{\mu}_{fi}|^2$ :

I_{v'\leftarrow v''} \;\propto\; |\boldsymbol{\mu}_e(R_e'')|^2 \left|\int \chi_{v'}^{*}(R)\,\chi_{v''}(R)\, dR\right|^2.

Step 6 — Identify the Franck-Condon factor. The purely vibrational, purely geometric piece is defined as the Franck-Condon factor:

\boxed{q_{v',v''} = \left|\int \chi_{v'}^{*}(R)\,\chi_{v''}(R)\, dR\right|^2, \qquad I_{v'\leftarrow v''} \propto \nu^4\,|\boldsymbol{\mu}_e|^2\, q_{v',v''}}

The entire vibrational band structure of an electronic spectrum — which bands are strong, which are weak — is controlled by these overlap integrals between vibrational wavefunctions belonging to two different potential energy curves.

Worked examples

Example 1 — Foundational: which band is strongest, and why? Consider a diatomic molecule with ground-state equilibrium bond length $R_e'' = 1.20\,\text{Å}$ and an excited electronic state with $R_e' = 1.20\,\text{Å}$ as well — i.e., essentially no geometry change upon excitation (a common situation for a non-bonding-to-Rydberg or weakly perturbing $n \to \pi^*$ transition). Model both vibrational manifolds as harmonic oscillators. Because $\Delta R_e = R_e' - R_e'' \approx 0$ , the classical vertical transition from the bottom of the ground-state well ( $v''=0$ , whose wavefunction $\chi_{0}''$ is a Gaussian centered at $R_e''$ ) lands almost exactly on the minimum of the excited-state curve. The excited-state vibrational wavefunction with the largest overlap with a Gaussian centered at its own minimum is $\chi_{0}'$ , since $\chi_0'$ is itself a Gaussian peaked at $R_e'$ . Quantitatively, for identical force constants (same vibrational frequency in both states) and zero equilibrium shift, the Franck-Condon factor for the $0$ – $0$ band approaches its maximum possible value:

q_{0,0} = \left|\int \chi_0'^{*}(R)\chi_0''(R)\,dR\right|^2 \to 1

with $q_{v'\geq 1, 0} \approx 0$ . The absorption spectrum is dominated by a single sharp origin band at the 0–0 energy gap, with very weak hot and cold vibrational satellites. This is exactly what is observed, for example, in the sharp near-UV absorption of many aromatic amines with minimal excited-state distortion. Sanity check: the sum rule $\sum_{v'} q_{v',0} = 1$ is satisfied by essentially all the intensity residing in $q_{0,0} \approx 1$ , consistent with a compact, near-atomic-like spectrum rather than a long progression.

Example 2 — Applied: estimating the vibrational progression maximum for I₂ Molecular iodine, $\mathrm{I_2}$ , shows the textbook example of a long Franck-Condon progression in its visible absorption spectrum (the violet-to-green $B\,{}^3\Pi_{0u^+} \leftarrow X\,{}^1\Sigma_g^+$ system). Take the following experimental parameters: Ground state ( $X$ ): $R_e'' = 2.666\,\text{Å}$ , vibrational wavenumber $\tilde\nu_e'' = 214.5\,\text{cm}^{-1}$ . Excited state ( $B$ ): $R_e' = 3.025\,\text{Å}$ , vibrational wavenumber $\tilde\nu_e' = 125.7\,\text{cm}^{-1}$ .

Step 1: quantify the geometry change.

\Delta R_e = R_e' - R_e'' = 3.025 - 2.666 = 0.359\,\text{Å}

This is a substantial shift — about 13% of the bond length — because the transition promotes an electron from a bonding-character orbital to one with more antibonding character in the excited configuration, weakening and lengthening the bond (consistent with $\tilde\nu_e'$ dropping to 59% of $\tilde\nu_e''$ ).

Step 2: locate the classical turning point of the $v''=0$ level. The vertical transition originates near $R_e''$ (since $\chi_0''$ peaks there), and lands on the excited-state curve at $R \approx R_e'' = 2.666\,\text{Å}$ , which lies to the inner side of $R_e' = 3.025\,\text{Å}$ .

Step 3: estimate which excited vibrational level has its classical turning point near $R = 2.666\,\text{Å}$ . Treat the excited state as harmonic with force constant

k' = 4\pi^2 c^2 (\tilde\nu_e')^2 \mu

Using the reduced mass of $^{127}\mathrm{I}_2$ , $\mu = \dfrac{126.90}{2}\,\text{u} = 63.45\,\text{u} = 1.054\times10^{-25}\,\text{kg}$ , and $\tilde\nu_e' = 125.7\,\text{cm}^{-1} = 1.257\times10^{4}\,\text{m}^{-1}$ :

k' = 4\pi^2 (3.00\times10^{8}\,\text{m s}^{-1})^2 (1.257\times10^4\,\text{m}^{-1})^2 (1.054\times10^{-25}\,\text{kg}) = 172\,\text{N m}^{-1}

The vibrational amplitude at level $v'$ (classical turning point displacement from $R_e'$ ) satisfies

\tfrac12 k' \, x_{v'}^2 = \left(v'+\tfrac12\right)hc\,\tilde\nu_e'

Set the turning point at the displacement needed to reach $R = 2.666\,\text{Å}$ , i.e., $x_{v'} = |R_e' - 2.666\,\text{Å}| = 0.359\,\text{Å} = 3.59\times10^{-11}\,\text{m}$ :

v' + \tfrac12 = \frac{k' x_{v'}^2}{2hc\,\tilde\nu_e'} = \frac{(172)(3.59\times10^{-11})^2}{2(6.626\times10^{-34})(3.00\times10^{8})(1.257\times10^4)}

Numerator: $172 \times 1.289\times10^{-21} = 2.217\times10^{-19}\,\text{J}$ . Denominator: $2 \times 6.626\times10^{-34} \times 3.00\times10^{8} \times 1.257\times10^{4} = 5.00\times10^{-21}\,\text{J}$ .

v' + \tfrac12 \approx \frac{2.217\times10^{-19}}{5.00\times10^{-21}} \approx 44.3 \implies v' \approx 44

Step 4: interpret. This (deliberately simplified harmonic) estimate places the Franck-Condon intensity maximum around $v' \approx 20$ – $25$ once anharmonicity is properly included (the real $\mathrm{I_2}$ $B$ -state potential is strongly anharmonic and dissociates near $v' \approx 65$ ), but even the crude harmonic estimate correctly captures the qualitative result: the absorption maximum sits far from $v'=0$ , at a high vibrational quantum number, because the equilibrium bond length shift is large. This reproduces the well-known observation that gaseous $\mathrm{I_2}$ shows a long, resolvable vibrational progression converging toward a dissociation continuum near 20,000 cm⁻¹, exactly the qualitative behavior predicted by the Franck-Condon principle for a state with significant $\Delta R_e$ . Sanity check: a small $\Delta R_e$ (Example 1) gave $v'_{\max}=0$ ; a large $\Delta R_e$ (this example) gives $v'_{\max} \gg 0$ . The direction and rough scale of the shift are physically consistent with $\Delta R_e$ nearly tripling relative to Example 1.

Limiting cases & common traps

Situation	Franck-Condon consequence
$\Delta R_e \approx 0$ , similar force constants	Strong 0–0 band, short progression, spectrum resembles a single sharp line plus weak satellites
$\Delta R_e$ moderate	Progression with intensity maximum at intermediate $v'$ , several resolvable bands
$\Delta R_e$ large	Long progression, intensity maximum at high $v'$ , possible diffuseness near dissociation limit
Excited state repulsive (no minimum)	Continuous absorption band, no vibrational structure — vertical transition lands directly on a repulsive wall
Emission (fluorescence)	Mirror-image progression built on $v''$ levels of ground state, governed by the same $q_{v',v''}$ evaluated from the relaxed excited-state geometry

The chemist behind it

The principle bears the names of James Franck, a German experimental physicist working on photodissociation and collision spectroscopy of diatomic molecules, and Edward Condon, an American theorist who gave the qualitative picture its rigorous quantum-mechanical (overlap-integral) formulation just a few years later. Franck's original 1925 argument was built purely on classical mechanics: he reasoned that since electronic transitions are fast and nuclei are heavy, the most probable outcome of a transition is one that leaves nuclear position and velocity essentially unchanged, immediately predicting vertical transitions on an energy diagram. Condon showed in 1926–1928 that this same conclusion falls directly out of the overlap of vibrational wavefunctions, converting a physically intuitive picture into a calculable quantity — the Franck-Condon factor — that remains standard in every modern treatment of vibronic spectra, from diatomic absorption bands to polyatomic fluorescence spectra and photoelectron spectroscopy.

Origins · 1 minJames Franck and how this was discovered

James Franck

1882-1964 · German experimental physicist; shared the 1925 Nobel Prize in Physics with Gustav Hertz for the Franck-Hertz experiment on electron impact excitation, and later gave the classical argument for vertical electronic transitions

Ask the tutor: Want to see how Franck-Condon overlaps translate into the mirror-image shape and Stokes shift of a real fluorescence spectrum?

Is the Franck-Condon principle a strict selection rule, like a symmetry selection rule?

No. Symmetry selection rules (from the electronic transition dipole and point-group symmetry) determine whether a transition is allowed at all — whether $\boldsymbol{\mu}_e$ is nonzero. The Franck-Condon principle operates one level down: given that the electronic transition is allowed, it distributes the total intensity among the various vibrational bands according to the squared vibrational overlap integrals $q_{v',v''}$ . A "forbidden" electronic transition stays weak regardless of how favorable the Franck-Condon factors are.

Does the Franck-Condon principle apply only to diatomic molecules?

No, though diatomics make the clearest teaching example because the potential energy curve is a simple function of one coordinate, the bond length. For polyatomic molecules the same logic applies along each normal mode, with a multidimensional vibrational overlap integral (often approximated as a product over normal modes). Geometry changes along totally symmetric modes, such as ring-breathing or bond-stretching coordinates, typically dominate the vibronic progressions seen in UV-Vis spectra of aromatic and conjugated molecules.

How does the Franck-Condon principle connect to fluorescence and the Stokes shift?

After absorption, the molecule relaxes vibrationally in the excited state down to its new equilibrium geometry before emitting (Kasha's rule). Emission then occurs as a vertical transition downward from that relaxed geometry, governed by the same Franck-Condon factors evaluated with the roles of the two states reversed. Because absorption and emission originate from different, relaxed geometries, the emission spectrum is red-shifted relative to absorption — the Stokes shift — and often appears as an approximate mirror image of the absorption band.

Why does a repulsive excited state give a smooth, structureless absorption band instead of sharp vibrational lines?

If the upper electronic state has no potential minimum (it is dissociative), there are no discrete vibrational levels to land on. The vertical transition from each point of the ground-state vibrational wavefunction projects onto a continuum of nuclear momentum states on the repulsive curve, producing a broad, featureless absorption band whose width and shape map the ground-state vibrational wavefunction and the slope of the repulsive curve, rather than a comb of resolved lines.

Can Franck-Condon factors be calculated exactly, or only estimated qualitatively?

They can be calculated exactly once both potential energy curves (or surfaces) are known, typically modeled as Morse or harmonic potentials fit to spectroscopic constants ( $R_e$ , $\omega_e$ , $\omega_e x_e$ ). The overlap integrals $\int \chi_{v'}^* \chi_{v''}\, dR$ are evaluated numerically or, for harmonic oscillators with a displaced origin, in closed form using Hermite polynomial recursion relations. Programs used in modern computational spectroscopy compute full Franck-Condon-Herzberg-Teller intensity envelopes this way for polyatomic vibronic spectra.