The Franck-Condon Principle

A diatomic molecule carries three kinds of motion at once — the electron cloud, the vibration of the two nuclei, and the tumbling of the whole frame — with energies in the ratio $10^4:10^2:1$ . The preceding article, the Born–Oppenheimer approximation, lets us pull them apart: because the nuclei are thousands of times heavier than the electrons, the electrons follow every nuclear position adiabatically and the molecular wavefunction factorises into an electronic part times a vibrational part. The Franck–Condon principle is what that separation does to a spectrum. When a molecule absorbs a photon and rearranges its electron cloud, the change takes about $10^{-16}$ s, while the nuclei need $10^{-13}$ s to complete one vibration — so during the jump the heavy nuclei are frozen, neither moving nor gaining momentum. On a plot of potential energy against internuclear distance $R$ the transition is therefore a vertical line — the system leaps from the lower curve to the upper one at fixed $R$ . James Franck argued this classically in 1925 to explain why iodine absorbs a long comb of bands rather than one line; Edward Condon made it quantitative a year later, replacing "fixed $R$ " with the overlap integral of the two vibrational wavefunctions. Squaring that overlap gives the Franck–Condon factor, the number that decides how a fixed electronic band-strength is sliced among the vibrational lines — so everything the spectrum says about how bonding changes on excitation is written in those factors.

Key takeaways

An electronic transition happens about a thousand times faster than a vibration ( $\tau_{\text{elec}}\sim10^{-16}$ s versus $\tau_{\text{vib}}\sim10^{-13}$ s), so $R$ and the nuclear momenta are frozen during the jump — the transition is vertical on a potential-energy-curve diagram.
Within Born–Oppenheimer the vibronic dipole splits into an electronic dipole times the overlap of the two vibrational wavefunctions; the band intensity is proportional to the square of that overlap, the Franck–Condon factor $q_{v'v''}=|\langle v'|v''\rangle|^2$ .
Because the two electronic states usually have different equilibrium bond lengths, the vertical transition from $v''=0$ lands high on the displaced upper curve, so the brightest band is often not the $0\text{–}0$ band but one at high $v'$ .
The factors obey a sum rule $\sum_{v'}q_{v'v''}=1$ : the electronic band-strength is fixed and merely redistributed among the vibrational components.
For two equal-frequency wells displaced by $\Delta R_e$ the progression is Poisson in the Huang–Rhys parameter $S$ : small $S$ gives a dominant $0\text{–}0$ band, large $S$ a long progression peaking near $v'\approx S$ .
Key idea: during an electronic transition the nuclei are effectively stationary, so the transition is vertical on the potential-energy diagram and its intensity is proportional to $|\langle v'|v''\rangle|^2$ , the squared overlap of the vibrational wavefunctions of the two electronic states.

Where this lives today

The most striking modern use of the Franck–Condon principle is to defeat it. Direct laser cooling of molecules — a technology of the 2010s now underpinning ultracold-molecule experiments — needs an electronic transition a molecule can cycle on thousands of times without falling into a dark vibrational level. That requires diagonal Franck–Condon factors, $q_{v'v''}\approx\delta_{v'v''}$ : the zero-displacement limit, where the excited state has almost the same bond length as the ground state so emission returns the molecule to where it started. Strontium monofluoride (SrF), calcium monofluoride (CaF), and yttrium oxide (YO) were chosen precisely because their $X\text{–}A$ systems have $q_{00}\approx0.98$ , so a molecule survives dozens of cycles and only two or three repump lasers close it — Franck–Condon factors used as a design specification. Elsewhere the principle is the everyday grammar of spectroscopy: the vibrational envelopes of photoelectron spectra, the mirror-image bands of dyes, and the reconstruction of excited-state potentials from laser-induced fluorescence.

The physical picture

Stack the two electronic potential curves on a graph of energy versus bond length $R$ , the upper one pushed out to a longer equilibrium distance because an excited state is usually more antibonding — a longer, weaker bond. A cold molecule sits in the lowest vibrational level of the ground curve, its wavefunction a single hump centred on $R_e''$ . When a photon arrives the electron cloud reorganises in a femtosecond, but the nuclei, a hundred thousand times heavier, cannot shift in that time, so the molecule lands on the upper curve directly above where it started: a vertical arrow, at fixed $R$ and fixed velocity. Where the arrow strikes decides which vibrational level is populated — and because the upper curve is displaced, it lands high up the wall, on an excited level rather than the bottom.

Figure

Two electronic potential curves with the excited-state minimum shifted out by ΔRe ≈ 0.36 Å (an I₂-like B ← X case): the transition is a vertical line at fixed R, so from the ground v'' = 0 level it strikes the excited curve high on its inner wall, populating a high vibrational level v' rather than v' = 0

The quantum version replaces the arrow with an overlap of two combs of probability. The ground $v''=0$ wavefunction is one Gaussian hump at $R_e''$ ; each upper level $v'$ has a wavefunction whose amplitude piles up near its classical turning points. The band is bright when a lobe of the upper wavefunction sits on top of the ground hump, and for a displaced upper curve the level whose inner turning point falls at $R_e''$ overlaps best — a high level. The classical arrow was really saying where the momenta match: a molecule starting at rest must arrive at rest, at a turning point — so the fixed- $R$ picture and the overlap picture are one statement seen twice. When both wells are harmonic with equal curvature, the whole progression is a Poisson distribution set by one number — the Huang–Rhys parameter $S$ , the displacement in vibrational quanta. The second figure is that envelope, what a spectroscopist reads off a band to learn how far the bond stretched.

Figure

Franck–Condon envelope for two equal-frequency wells displaced by a Huang–Rhys parameter S = 6: the band intensities follow a Poisson law in S, peaking near v' ≈ 6, while the 0–0 band carries only q₀₀ = e⁻⁶ ≈ 0.0025 — a long progression is the fingerprint of a large geometry change

The core equations

Everything begins with a comparison of two clocks. The electronic transition time is set by the electronic gap through the uncertainty relation, the vibrational period by the nuclear frequency; the two differ by three orders of magnitude, which is why the nuclei are frozen.

\tau_{\text{elec}}\sim\frac{\hbar}{\Delta E_{\text{elec}}}\sim10^{-16}\,\text{s}\;\ll\;\tau_{\text{vib}}\sim\frac{1}{\nu_{\text{vib}}}\sim10^{-13}\,\text{s} \qquad (1)

where: $\tau_{\text{elec}}$ — electronic transition time, fixed by the gap $\Delta E_{\text{elec}}$ ; $\tau_{\text{vib}}$ — vibrational period, fixed by $\nu_{\text{vib}}$ ; $\hbar$ — reduced Planck constant. The nuclear coordinate is frozen during the jump — the meaning of a "vertical" transition. The Born–Oppenheimer approximation, from the prerequisite article, splits the molecular wavefunction into an electronic factor that depends on $R$ only as a parameter times a vibrational factor. This product form is the seed of the Franck–Condon factor.

\Psi(\mathbf{r},R)=\psi_{\text{el}}(\mathbf{r};R)\,\chi_{v}(R) \qquad (2)

where: $\psi_{\text{el}}(\mathbf{r};R)$ — electronic wavefunction, parametric in $R$ ; $\chi_v(R)$ — vibrational wavefunction of level $v$ ; $\mathbf{r},R$ — electronic and nuclear coordinates. A single electronic state carries a whole ladder of vibrational levels $\chi_v$ . The band intensity is governed by the transition dipole between the initial vibronic state $(v'')$ and the final $(v')$ . Inserting the products and integrating over electrons first leaves an $R$ -dependent electronic dipole inside a nuclear integral.

\boldsymbol{\mu}_{fi}=\int\chi_{v'}^{*}(R)\,\boldsymbol{\mu}_{\text{el}}(R)\,\chi_{v''}(R)\,\mathrm{d}R \qquad (3)

where: $\boldsymbol{\mu}_{fi}$ — vibronic transition dipole between $i=(\text{el}'',v'')$ and $f=(\text{el}',v')$ ; $\boldsymbol{\mu}_{\text{el}}(R)=\int\psi_{\text{el}'}^{*}\hat{\boldsymbol{\mu}}\,\psi_{\text{el}''}\,\mathrm{d}\mathbf{r}$ — electronic transition dipole, generally $R$ -dependent; $\chi_{v'},\chi_{v''}$ — the two vibrational wavefunctions. The nuclear motion survives as a weighting of $\boldsymbol{\mu}_{\text{el}}$ over the $R$ the molecule samples. The Condon approximation treats the electronic dipole as constant over the narrow band of $R$ where the wavefunctions overlap, so it comes outside the integral. What remains is a pure vibrational overlap, whose modulus squared is the Franck–Condon factor.

\boldsymbol{\mu}_{fi}\approx\boldsymbol{\mu}_{\text{el}}(R_e)\,\langle v'|v''\rangle,\qquad q_{v'v''}=\big|\langle v'|v''\rangle\big|^2=\left|\int\chi_{v'}^{*}\chi_{v''}\,\mathrm{d}R\right|^2 \qquad (4)

where: $\boldsymbol{\mu}_{\text{el}}(R_e)$ — electronic dipole at a representative $R$ , taken constant; $\langle v'|v''\rangle$ — vibrational overlap; $q_{v'v''}$ — Franck–Condon factor, the relative intensity of the $v''\to v'$ band. The strength factors into an electronic piece (how allowed) times a nuclear piece (how the combs align). The band intensity is then the electronic strength times the Franck–Condon factor, weighted by a frequency power that differs between spontaneous emission and absorption.

I_{v'v''}\propto \nu^{4}\,\big|\boldsymbol{\mu}_{\text{el}}(R_e)\big|^{2}\,q_{v'v''}\ \ (\text{emission}),\qquad \sigma_{v'v''}\propto \nu\,\big|\boldsymbol{\mu}_{\text{el}}(R_e)\big|^{2}\,q_{v'v''}\ \ (\text{absorption}) \qquad (5)

where: $I_{v'v''}$ — emission intensity, with $\nu^{4}$ from the radiated power; $\sigma_{v'v''}$ — absorption cross-section, one power of $\nu$ ; $q_{v'v''}$ — Franck–Condon factor. The electronic factor and frequency set the scale; $q$ shapes the progression. Because the upper-state vibrational functions are complete, the factors from any fixed lower level sum to one.

\sum_{v'}q_{v'v''}=\sum_{v'}\big|\langle v'|v''\rangle\big|^{2}=\langle v''|v''\rangle=1 \qquad (6)

where: $\sum_{v'}$ — sum over all vibrational levels of the upper state. A poor overlap does not weaken the transition; it spreads the same strength thinly across many bands. For two harmonic wells of equal frequency but displaced minima the overlaps from $v''=0$ collapse to a Poisson distribution in a single displacement, the Huang–Rhys parameter.

S=\frac{\mu\omega}{2\hbar}\,(\Delta R_e)^{2},\qquad q_{v'0}=\frac{S^{\,v'}e^{-S}}{v'!} \qquad (7)

where: $S$ — Huang–Rhys factor, displacement in vibrational quanta; $\mu$ — reduced mass; $\omega$ — common angular frequency; $\Delta R_e$ — shift in equilibrium bond length. $S$ is both the mean level excited and the peak, so the envelope's peak measures $S$ and hence $\Delta R_e$ .

Derivation ASeparating the vibronic dipole into an electronic factor and a vibrational overlap

This derivation shows how the Born–Oppenheimer product wavefunction turns the full vibronic transition dipole — an integral over both electronic and nuclear coordinates — into the Condon form: an electronic dipole times a pure vibrational overlap, whose square is the Franck–Condon factor. It is the load-bearing calculation of the topic, pinning the vibrational band structure onto the overlap of two nuclear wavefunctions.

Step 1. Write the dipole as the full double integral $\boldsymbol{\mu}_{fi}=\iint \psi_{\text{el}'}^{*}\chi_{v'}^{*}\,\hat{\boldsymbol{\mu}}\,\psi_{\text{el}''}\chi_{v''}\,\mathrm{d}\mathbf{r}\,\mathrm{d}R$ , with $\hat{\boldsymbol{\mu}}=-e\sum_i\mathbf{r}_i+e\sum_N Z_N\mathbf{R}_N$ acting on both coordinates. Why: intensity is set by the dipole matrix element between the true product states, so we must start exact.

Step 2. Do the electron integral at fixed $R$ . The nuclear-charge term multiplies $\int\psi_{\text{el}'}^{*}\psi_{\text{el}''}\,\mathrm{d}\mathbf{r}$ , which vanishes for $\text{el}'\ne\text{el}''$ by orthogonality, leaving only $\boldsymbol{\mu}_{\text{el}}(R)=\int\psi_{\text{el}'}^{*}\hat{\boldsymbol{\mu}}\,\psi_{\text{el}''}\,\mathrm{d}\mathbf{r}$ . Why: the nuclei do not change their own charge, so the whole dipole comes from the electron cloud.

Step 3. Expand the surviving nuclear integral's dipole, $\boldsymbol{\mu}_{\text{el}}(R)=\boldsymbol{\mu}_{\text{el}}(R_e)+(\mathrm{d}\boldsymbol{\mu}_{\text{el}}/\mathrm{d}R)_{R_e}(R-R_e)+\dots$ , about the centre of the overlap window. Why: the wavefunctions overlap only over a narrow range of $R$ , so a Taylor series there measures how much the dipole really varies.

Step 4. Keep only the constant term — the Condon approximation — and pull it out: $\boldsymbol{\mu}_{fi}\approx\boldsymbol{\mu}_{\text{el}}(R_e)\langle v'|v''\rangle$ , so the intensity is $|\boldsymbol{\mu}_{\text{el}}(R_e)|^2 q_{v'v''}$ . Why: this separates "how allowed" from "how the intensity is distributed"; the discarded linear term, when it matters, is Herzberg–Teller intensity borrowing.

Go deeper · Derivation B · 1 minDisplaced harmonic wells give a Poisson progression in the Huang–Rhys parameter

Derivation BDisplaced harmonic wells give a Poisson progression in the Huang–Rhys parameter

Here we evaluate the overlaps of equation 4 for the cleanest case: two oscillators of equal frequency and reduced mass whose minima are displaced by $\Delta R_e$ . The overlap of the ground level of one well with level $v'$ of the other turns out exactly Poisson in one parameter $S$ , because a displaced harmonic ground state is a coherent state — whose number-state populations are famously Poisson.

Step 1. Write $\chi_0''(x)=(\mu\omega/\pi\hbar)^{1/4}e^{-\mu\omega x^2/2\hbar}$ with $x=R-R_e''$ , and the upper functions as the same Hermite–Gaussians centred at $x=\Delta R_e$ . Equal $\omega$ and $\mu$ means both wells share one set of ladder operators. Why: equal curvature relates the two bases by a rigid shift, not a full remixing, so the overlaps close algebraically.

Step 2. Represent the shift with the displacement operator, $|0''\rangle=\hat{D}(-\alpha)|0'\rangle$ , $\hat D(\alpha)=e^{\alpha a^\dagger-\alpha^* a}$ ; displacing the ground state produces a coherent state with $|\alpha|^2=S=\mu\omega(\Delta R_e)^2/2\hbar$ . Why: a Gaussian shifted from a well's centre is the definition of a coherent state — the move that imports coherent-state machinery into spectroscopy.

Step 3. Use $|\langle v'|\alpha\rangle|^2=e^{-|\alpha|^2}|\alpha|^{2v'}/v'!$ to get $q_{v'0}=S^{\,v'}e^{-S}/v'!$ , with mean $\langle v'\rangle=S$ and peak near $v'\approx S$ . Why: this closed form is why one measured envelope yields $S$ , and hence $\Delta R_e$ .

Step 4. Check: $\sum_{v'}q_{v'0}=e^{-S}\sum_{v'}S^{v'}/v'!=e^{-S}e^{S}=1$ , consistent with the sum rule; and $S\to0$ collapses everything to $q_{00}=1$ . Why: passing both the sum rule and the zero-displacement limit is a strong check on the coherent-state route.

Step 5. In the heavy-nucleus limit the same overlap becomes the reflection principle: for a rapidly oscillating or continuum upper state, stationary phase gives $I(\nu)\propto|\chi_{v''}(R_\nu)|^2/|\mathrm{d}(U_e-U_g)/\mathrm{d}R|_{R_\nu}$ , the ground density mapped through the difference potential. Why: this is how photodissociation continua are inverted to recover repulsive walls — the same theory in a classical limit.

Limiting cases

Case	Result	Meaning
No displacement ( $\Delta R_e=0$ )	$q_{v'0}=\delta_{v'0}$	Coincident wells overlap only for $v'=v''$ ; from $v''=0$ the $0\text{–}0$ band carries all the intensity.
Small displacement ( $S\ll1$ )	$q_{00}\approx1-S$ , $q_{10}\approx S$	Dominant $0\text{–}0$ band with a weak first member; the tiny progression measures a small $\Delta R_e$ .
Large displacement ( $S$ large)	$q_{v'0}=S^{\,v'}e^{-S}/v'!$	Poisson envelope peaking near $v'\approx S$ — a long progression, the mark of a big bonding change.
Classical / heavy-nucleus limit	$I(R)\propto\|\chi_{v''}(R)\|^2\,\delta(U_e-U_g-h\nu)$	Intensity at $h\nu$ is the ground density at the $R$ where the gap equals $h\nu$ — the reflection principle.
Continuum (dissociative) upper state	$I(\nu)\propto\|\langle E\|v''\rangle\|^2$	A repulsive upper curve gives a continuous band mapping the repulsive wall.

Common trapThe vertical transition always lands on v' = 0, and the principle changes which transitions are allowed

Both halves are wrong, and both common. First, the vertical transition does not favour $v'=0$ — the opposite. Because the excited state usually has a longer bond, the arrow from the ground $v''=0$ hump strikes the upper curve high on its wall, so the brightest band is typically at a substantial $v'$ and the $0\text{–}0$ band can be nearly invisible ( $q_{00}=e^{-6}\approx0.0025$ for $S=6$ ). Second, the principle does not touch the electronic selection rules or the total band strength $|\boldsymbol{\mu}_{\text{el}}|^2$ ; by the sum rule it only redistributes that fixed strength. And keep the two pieces distinct: the Franck–Condon factor is the vibrational overlap, the electronic dipole the separate prefactor.

Worked examples

1 · Iodine's long progression from a Huang–Rhys estimate (confidence builder)

Molecular iodine absorbs the visible $B\leftarrow X$ system as a famously long comb of bands. The ground state has $R_e''=2.666$ Å and $\tilde\nu_e=214.5$ cm⁻¹; the excited state sits at $R_e'=3.025$ Å. Estimate the Huang–Rhys parameter and hence which level absorbs most strongly, with $\mu=m(\text{I})/2$ and $m(\text{I})=126.90$ u.

Step 1. In SI, $\mu=(126.90/2)(1.6605\times10^{-27})=1.054\times10^{-25}$ kg, $\omega=2\pi c\tilde\nu_e=2\pi(2.998\times10^{10})(214.5)=4.04\times10^{13}$ rad/s, and $\Delta R_e=3.025-2.666=0.359$ Å $=3.59\times10^{-11}$ m.

Step 2. The prefactor is $\dfrac{\mu\omega}{2\hbar}=\dfrac{(1.054\times10^{-25})(4.04\times10^{13})}{2(1.0546\times10^{-34})}=2.02\times10^{22}\ \text{m}^{-2}$ , and $(\Delta R_e)^2=1.29\times10^{-21}\ \text{m}^2$ .

Step 3. So $S=(2.02\times10^{22})(1.29\times10^{-21})\approx\mathbf{26}$ . Since the envelope peaks near $v'\approx S$ , the strongest absorption is around $v'\approx26$ with essentially nothing in the $0\text{–}0$ band — exactly iodine's long visible progression. S ≈ 26, so the system peaks near v′ ≈ 26, not at v′ = 0. (Order-of-magnitude only, since the two states do not share one frequency.)

2 · Building a Franck–Condon envelope and checking the sum rule

For a moderate displacement giving $S=2.0$ , tabulate $q_{v'0}$ for $v'=0$ to $5$ , find the brightest band, and verify the sum rule.

Step 1. With $e^{-2}=0.1353$ : $q_{00}=0.1353$ , $q_{10}=2e^{-2}=0.2707$ , $q_{20}=\tfrac{4}{2}e^{-2}=0.2707$ , $q_{30}=\tfrac{8}{6}e^{-2}=0.1804$ , $q_{40}=\tfrac{16}{24}e^{-2}=0.0902$ , $q_{50}=\tfrac{32}{120}e^{-2}=0.0361$ .

Step 2. The brightest bands are $v'=1,2$ , tied at $0.2707$ — the peak has moved off $v'=0$ even here, and $\langle v'\rangle=S=2$ .

Step 3. Summing gives $0.8834$ ; the tail ( $v'\ge6$ ) adds $0.0116$ to make $1.000$ . The envelope peaks at v′ = 1–2 and $\sum_{v'}q_{v'0}=1$ . (As a timescale check on equation 1, a $3.0$ eV gap gives $\tau_{\text{elec}}\sim\hbar/\Delta E=2.2\times10^{-16}$ s against $\tau_{\text{vib}}=1/(c\tilde\nu_e)=1.6\times10^{-13}$ s — a ratio near 700, so the nuclei really are frozen.)

Physics of the portrait

The principle carries two names because it was born twice — as Franck's physical picture and as Condon's wave-mechanical calculation.

Origins · 1 minJames Franck and how this was discovered

James Franck

1882–1964 · the timescale argument behind vibronic band intensities

James Franck was an experimentalist first. With Gustav Hertz he had shown in 1913–1914 that electrons transfer energy to atoms only in quantised lumps — the Franck–Hertz experiment, which won them the 1925 Nobel Prize and gave Bohr's atom direct proof. A decade later, puzzling over why molecular iodine absorbs a long series of bands and then dissociates, Franck reasoned in 1925 that an electronic transition is so much faster than a vibration that the nuclei cannot move or gain momentum during it: the transition must be "vertical," and the molecule lands on the upper curve with whatever bond length and velocity it had — a picture, not an equation. Edward Condon supplied the quantum mechanics in 1926 and extended it in 1928, making "fixed $R$ " the overlap integral of the vibrational wavefunctions. The division of labour is exact — Franck saw why the transition is vertical, Condon computed how much each band therefore weighs. The later refinements bear other names: Herzberg and Teller for the constant-dipole failure, Huang and Rhys for the displaced-oscillator parameter.

History

1925

— Franck argues qualitatively that the nuclei stay fixed during an electronic transition, explaining iodine's absorption and predissociation

1926

— Condon recasts the idea in wave mechanics as the overlap of vibrational wavefunctions

1928

— Condon publishes the potential-energy-curve construction with the vertical transition line

1933

— Herzberg and Teller describe vibronic intensity borrowing when the Condon approximation fails

1950

— Huang and Rhys formulate the displaced-oscillator theory and the S parameter for colour centres in solids

Going further

Texts. Herzberg, Molecular Spectra and Molecular Structure I: Spectra of Diatomic Molecules, Ch. IV · Atkins and Friedman, Molecular Quantum Mechanics, 5th ed., §8 (vibronic transitions and the Condon approximation) · Demtröder, Laser Spectroscopy, Vol. 1, §1.2 and §9 · Loudon, The Quantum Theory of Light, for the coherent-state overlap $|\langle n|\alpha\rangle|^2=e^{-|\alpha|^2}|\alpha|^{2n}/n!$ . The timescales $\tau_{\text{elec}}\sim10^{-16}$ s and $\tau_{\text{vib}}\sim10^{-13}$ s are order-of-magnitude standards; their ratio $\approx710$ is confirmed in worked example 2. The iodine $B\leftarrow X$ constants $R_e''=2.666$ Å, $\tilde\nu_e=214.5$ cm⁻¹, $R_e'=3.025$ Å are standard diatomic values used to obtain $S\approx26$ . Key results — the Born–Oppenheimer factorisation into $\boldsymbol{\mu}_{\text{el}}(R_e)\langle v'|v''\rangle$ , the sum rule via completeness, and the Poisson envelope via the coherent-state overlap $|\alpha|^2=S=\mu\omega(\Delta R_e)^2/2\hbar$ — are worked through in the derivations above, and the $S=2$ table sums to one as required. The diagonal factors $q_{00}\approx0.98$ for SrF, CaF, and YO are standard values from the molecular-laser-cooling literature.

Prerequisites. The Born–Oppenheimer Approximation · The Chemical Bond and Molecular Potential Energy Curves · The Harmonic Oscillator Model of Vibration · Selection Rules for Molecular Transitions. Leads to. Electronic Transitions in Molecules · Anharmonicity and the Morse Potential · Vibration–Rotation Spectra and P, Q, R Branches.

Ask the tutor: Ask me to take a molecule of your choice and walk through how its $\Delta R_e$ turns into a measurable Franck–Condon envelope.

Is the "vertical transition" literally instantaneous?

Not literally, but effectively. The electronic rearrangement takes about $10^{-16}$ s, a vibration about $10^{-13}$ s, so during the jump the nuclei move a negligible fraction of a vibration — a ratio near 700 to 1 — leaving $R$ and the nuclear momentum essentially unchanged, hence the vertical line. It fails only when the nuclei are light or the transition slow, as near conical intersections.

What is the difference between the principle and the factors?

The principle is Franck's physical statement: because electrons move much faster than nuclei, an electronic transition happens at fixed geometry, so it is vertical on a potential-energy diagram. The factor is Condon's quantitative version: the band intensity is $|\langle v'|v''\rangle|^2$ , the squared overlap of the two vibrational wavefunctions. The principle says the transition is vertical and why; the factor says how much intensity each band gets, and in the heavy-nucleus limit the two coincide as the overlap concentrates at the Condon point.

Why does the excited state usually have a longer bond?

Because excitation typically promotes an electron into an orbital with more antibonding character, which weakens the bond. A weaker bond means a shallower, wider well with its minimum at a larger separation — a positive $\Delta R_e$ — and that displacement is exactly what makes the vertical transition from $v''=0$ land high on the upper curve, producing a progression. Removing an antibonding electron instead shortens the bond, but the consequence depends on $|\Delta R_e|$ , not its sign.

Does the principle change the selection rules?

No. The electronic selection rules are governed by the electronic transition dipole and the symmetry of the two states. The Franck–Condon principle operates inside an allowed transition, distributing its fixed strength among vibrational bands through the factors $q_{v'v''}$ ; by the sum rule those add to one, so they cannot create intensity in a forbidden transition, only reshape it within an allowed one. The apparent exception — a nominally forbidden band gaining intensity — is Herzberg–Teller coupling, the linear dipole term the Condon approximation discards.

What does the Huang–Rhys parameter S physically mean?

$S=\mu\omega(\Delta R_e)^2/2\hbar$ is the displacement between the two wells in vibrational quanta. Equivalently it counts the quanta of reorganisation energy $\lambda=S\hbar\omega$ released as the molecule relaxes from the vertically reached geometry to the excited-state minimum. For a Poisson progression $S$ is also the mean level excited, $\langle v'\rangle=S$ , and roughly the peak. So one number ties together the geometry change, the spectrum shape, and the energy dissipated — the workhorse from molecular spectra to charge-transfer rates in solids.

Why do absorption and fluorescence bands often look like mirror images?

Because the same displaced-well geometry governs both, seen from opposite ends. In absorption the molecule starts at the ground minimum and climbs the excited curve, favouring high $v'$ ; in fluorescence it starts at the excited minimum and drops onto the ground curve, favouring high $v''$ . If both wells have similar curvature the two progressions share the same envelope reflected about the $0\text{–}0$ energy — the mirror symmetry of dye spectra. The peak-to-peak gap is the Stokes shift, about $2S\hbar\omega$ , a direct readout of $\Delta R_e$ .