Electronic Transitions and the Franck-Condon Principle

The Franck–Condon principle is not a historical curiosity; it is a working design tool. In OLED and phosphor engineering, chemists tune the rigidity of an emitter's excited-state geometry specifically to minimize the displacement $\Delta R$ between ground and excited states, suppressing long vibrational progressions and non-radiative internal conversion pathways that compete with light emission — the practical payoff of understanding when a spectrum will show a sharp 0–0 line versus a broad, energy-wasting progression. In photodissociation dynamics and atmospheric chemistry, the reflection principle directly predicts the shape of continuous absorption bands such as the ozone Hartley band, letting atmospheric chemists connect measured UV absorption cross sections to the shape of a repulsive excited-state potential without ever resolving individual vibrational lines. In single-molecule and time-resolved spectroscopy, femtosecond pump–probe experiments literally watch a wavepacket launched vertically onto an excited-state surface begin to move — providing direct, real-time experimental confirmation of the frozen-nuclei assumption and its breakdown timescale. In photoreceptor and vision biochemistry, the vertical Franck–Condon transition in retinal's chromophore is the first step of the visual cycle, and the subsequent ultrafast cis–trans isomerization is a textbook example of what happens on the excited-state surface once the vertical transition has deposited the molecule away from its new equilibrium geometry. In quantitative fluorescence-based sensing and dye chemistry, the mirror-image absorption/emission relationship and the measured Stokes shift are routinely used as diagnostic fingerprints for solvent polarity, aggregation state, and excited-state proton transfer — all built on the vibronic-progression logic developed here and quantified further in Franck-Condon Factors and Vibronic Intensities.

Ask the tutor: How do I compute the full Franck–Condon factor progression (not just the 0–0 term) when both the displacement and the vibrational frequency change between two electronic states?

Picture two Born–Oppenheimer potential energy curves for a diatomic molecule plotted against internuclear distance $R$: a lower curve $V_g(R)$ for the ground electronic state and an upper curve $V_e(R)$ for an excited electronic state, each with its own equilibrium bond length ($R_g^0$ and $R_e^0$), its own force constant, and its own ladder of vibrational levels. In most cases $R_e^0 > R_g^0$, because promoting an electron to an antibonding or less-bonding orbital weakens the bond and stretches it — an expectation directly rooted in the orbital picture behind Molecular Term Symbols. Superimposed on each electronic curve is a set of vibrational wavefunctions $\chi_v(R)$: for low $v$ these are Gaussian-like and peaked at the potential minimum, but at higher $v$ they develop the classic anharmonic asymmetry and outer turning-point pile-up familiar from the Anharmonicity and the Morse Potential treatment, with probability density concentrated near the classical turning points. The Franck–Condon principle says: draw the electronic transition as a strictly vertical arrow, starting from wherever the nuclear probability density is largest in the initial vibronic state, and rising to the upper curve at the same $R$. The molecule's electrons rearrange in about $10^{-16}$ s (the light traversal timescale ties to the transition frequency, but the physically relevant comparison is that electronic wavefunction relaxation follows the light field essentially instantaneously on the nuclear time scale). Nuclear vibration periods are on the order of $10^{-14}$–$10^{-13}$ s. Because the ratio of these times is roughly $10^{2}$–$10^{3}$, the nuclei's position $R$ and momentum $P$ are unchanged during the absorption event — the transition is vertical in $R$, not diagonal. The consequence for intensity is immediate. The upper curve, evaluated at the fixed $R$ of the vertical arrow, may or may not be near its own minimum. If the two equilibrium geometries coincide ($R_e^0 \approx R_g^0$), the vertical arrow from $v''=0$ lands almost exactly on $v'=0$ of the excited state, and the spectrum is dominated by a single strong 0–0 line with weak overtones — this is typical of rigid aromatic chromophores. If the geometries differ substantially, the vertical arrow from the ground-state minimum lands on the side of the excited-state potential, at large $R$ relative to $R_e^0$, and by the correspondence-principle logic of vibrational wavefunctions, the excited-state level whose classical turning point sits at that $R$ is the one with the largest amplitude there — hence the largest overlap and largest intensity. This produces a vibrational progression: several vibronic peaks of increasing, then decreasing, intensity as $v'$ increases, with the intensity maximum located at whichever $v'$ satisfies the geometric matching condition. This is exactly the intuition later quantified as a set of Franck–Condon factors in Franck-Condon Factors and Vibronic Intensities.

The vibrational-progression picture has a semiclassical cousin, the reflection principle, most useful when the upper state is dissociative (no bound minimum at all, as when a photon promotes an electron to a strongly antibonding orbital). Here there is no discrete vibrational ladder above — the excited curve rises or falls monotonically and the nuclear "wavefunction" above threshold is a continuum scattering state. The classical version of the Franck–Condon idea says: the probability of absorption at photon energy $h\nu$ is proportional to the ground-state vibrational probability density $|\chi_{v''}(R)|^2$ at the particular $R$ where the vertical gap $V_e(R)-V_g(R)$ equals $h\nu$. Because $|\chi_0(R)|^2$ for the ground vibrational level is a Gaussian centered at $R_g^0$, and the vertical energy gap is a smooth, usually monotonic function of $R$ near $R_g^0$, mapping ("reflecting") this Gaussian through the gap function onto the photon-energy axis produces a smooth, broad, structureless absorption band — exactly what is observed for many photodissociative continua (e.g., the visible absorption of Br₂ or I₂ vapor near their dissociation limits, and the Hartley band of ozone).

A third essential picture is emission. After absorption, the excited molecule usually relaxes vibrationally (collisional or radiative cascade down the excited-state vibrational ladder) faster than it emits a photon — this is Kasha's rule territory, developed fully in Fluorescence and Phosphorescence: The Quantum Picture. By the time fluorescence occurs, the molecule sits in $v'=0$ of the excited electronic state, with its own equilibrium geometry $R_e^0$. The emission step is again vertical, now downward, landing on whichever ground-state vibrational level $v''$ has the best overlap with $\chi_{v'=0}^{excited}$. Because the same displacement $R_e^0 - R_g^0$ that produced an absorption progression above $v''=0$ also produces an emission progression above $v''=0$ on the way down, the emission spectrum is often an approximate mirror image of the absorption spectrum, reflected about the 0–0 line — and because emission always occurs from the relaxed, lower-energy $v'=0$ level while absorption originates from $v''=0$ and rises to higher, unrelaxed vibrational sublevels of the excited state, the emitted photon is systematically lower in energy than the absorbed one. This gap is the Stokes shift.

The starting point is the Born-Oppenheimer Approximation factorization of the full molecular wavefunction into an electronic part (parametric in nuclear coordinates $R$) and a nuclear part:

where: $\psi_{\text{el}}(r;R)$ is the electronic wavefunction for fixed nuclear geometry $R$, and $\chi_v(R)$ is the vibrational wavefunction on the potential curve generated by that electronic state.

Reading: electrons see the nuclei as fixed points while the nuclei move on a potential surface built from the electronic energy — the separation that makes the vertical transition concept meaningful in the first place. The transition dipole moment integral, built from the Transition Dipole Moment framework, connects an initial vibronic state $\psi_g\chi_{v''}$ to a final vibronic state $\psi_e\chi_{v'}$:

where: $\hat{\boldsymbol{\mu}} = -e\sum_i \mathbf{r}_i + e\sum_A Z_A \mathbf{R}_A$ is the total electric dipole operator (electronic plus nuclear contributions), and integration runs over all electronic coordinates $r$ and the nuclear coordinate(s) $R$.

Reading: this is the full quantum mechanical transition moment that decides absorption intensity; the Franck–Condon principle emerges when this double integral is factored using the Condon approximation below. Since $\hat{\boldsymbol\mu}$ acts predominantly through its electronic part when the electronic transition itself is dipole-allowed, and since $\psi_e, \psi_g$ vary slowly with $R$ compared to their strong $R$-dependence through energy, the Condon approximation replaces the $R$-dependent electronic transition moment $\boldsymbol\mu_{eg}(R) = \int \psi_e^*\hat{\boldsymbol\mu}_{\text{el}}\psi_g\,dr$ with its value at a single representative geometry (often $R_g^0$):

where: $\boldsymbol{\mu}_{eg}(R_g^0)$ is the purely electronic transition dipole moment evaluated at fixed geometry, and $\int \chi_{v'}^{*}\chi_{v''}\,dR$ is the vibrational overlap integral, generally denoted $\langle \chi_{v'}|\chi_{v''}\rangle$.

Reading: the electronic "switch" (allowed or forbidden, and how strongly allowed) is decided once, at fixed geometry, exactly as in ordinary Molecular Term Symbols-based selection rules; everything about the vibrational fine structure and relative intensities of the vibronic peaks is then carried entirely by the nuclear overlap integral — no further electronic physics enters. Squaring gives the observable intensity, proportional to the Franck–Condon factor:

where: the first factor sets the overall band strength (allowed vs. forbidden electronic transition), and the second factor, the Franck–Condon factor, redistributes that total strength among the vibronic sublevels $v'=0,1,2,\dots$

Reading: this single line is the master intensity formula of vibronic spectroscopy; a full quantitative treatment of $|\langle \chi_{v'}|\chi_{v''}\rangle|^2$ for displaced harmonic and Morse oscillators is the subject of Franck-Condon Factors and Vibronic Intensities, but the qualitative content — vertical arrow, overlap-controlled intensity — is fully contained here.

Example 1 — Foundational: locating the vertical transition and reading off qualitative intensity for CN's B–X system. Take a diatomic with ground-state equilibrium bond length $R_g^0 = 1.17$ Å and excited-state equilibrium bond length $R_e^0 = 1.23$ Å (loosely modeled on a violet CN band system, chosen with an asymmetric, non-round displacement). Model both curves harmonically near their minima, with ground-state vibrational frequency $\tilde\nu_g = 2069$ cm⁻¹ and excited-state frequency $\tilde\nu_e = 1743$ cm⁻¹ (again deliberately unequal, non-round numbers). The displacement is

Convert this geometric displacement into the dimensionless Huang–Rhys-type displacement parameter used for harmonic oscillators,

where $\mu$ is the reduced mass of the diatomic (for CN, $\mu = \dfrac{12.000\times14.003}{12.000+14.003}\ \text{amu} = 6.462\ \text{amu} = 1.073\times10^{-26}\ \text{kg}$) and $\omega_e = 2\pi c\,\tilde\nu_e$. Compute $\omega_e$:

Then

Taking the square root,

With $\Delta R = 0.06\ \text{Å} = 6.0\times10^{-12}$ m,

This dimensionless displacement of about $1.1$ (not small, not enormous) predicts, by the standard harmonic Franck–Condon formula $q_{v'0} = e^{-S}S^{v'}/v'!$ with Huang–Rhys parameter $S = d^2/2 = 0.605$, a vibrational progression peaking near

so the $v'=0$ and $v'=1$ peaks should be comparably strong with $v'=0$ slightly favored, and a visible but decaying tail out to $v'=2,3$. This matches the qualitative expectation that a modest 0.06 Å bond-length change gives a short, recognizable vibrational progression rather than either a single sharp 0–0 line ($d\to 0$) or a very long progression merging into a continuum ($d\gg 1$, relevant to dissociative upper states). Limiting case check: if $R_e^0 \to R_g^0$ (no geometry change at all), then $\Delta R\to 0$, so $d\to 0$ and $S\to 0$; the Poisson-like distribution $q_{v'0}=e^{-S}S^{v'}/v'!$ collapses to $q_{00}=1$, $q_{v'>0,0}=0$ — a single 0–0 line with no progression, exactly the rigid-chromophore limit described qualitatively above.

Example 2 — Applied: Stokes shift and the mirror-image rule in a fluorescent dye. A conjugated dye absorbs with a vibronic maximum at $\tilde\nu_{abs} = 22{,}700$ cm⁻¹ (440.5 nm) and emits with a vibronic maximum at $\tilde\nu_{em} = 19{,}200$ cm⁻¹ (520.8 nm) — asymmetric, realistic numbers for a moderately Stokes-shifted dye (larger shift than a rigid rod like anthracene, smaller than a highly floppy TICT dye). Suppose spectroscopic analysis assigns the 0–0 (purely electronic) transition at $\tilde\nu_{00} = 20{,}900$ cm⁻¹ from the crossing point of the normalized absorption and emission curves. The Stokes shift is

Using the vibronic-progression picture, this total shift splits into a ground-state reorganization contribution and an excited-state reorganization contribution around the shared 0–0 line:

Numerically,

which checks against the directly measured Stokes shift, confirming self-consistency of the assigned 0–0 line. The near-equality of $1{,}800$ cm⁻¹ (ground-state vibrational reorganization energy probed in absorption) and $1{,}700$ cm⁻¹ (excited-state-probed reorganization energy seen in emission) is the quantitative signature of the approximate mirror-image relationship: both numbers reflect essentially the same nuclear displacement $\Delta R$, sampled once going up (absorption, on the excited curve's shape) and once coming down (emission, on the ground curve's shape), and they need not be identical because the two curves generally have somewhat different curvatures (different vibrational frequencies), which is exactly why real mirror symmetry is only approximate. Limiting case check: if the ground and excited potential curves had identical curvature and identical equilibrium geometry ($\Delta R = 0$), both reorganization energies would vanish, $\tilde\nu_{abs}=\tilde\nu_{em}=\tilde\nu_{00}$, and the Stokes shift would be exactly zero — the textbook definition of a "zero-zero" limit realized in some rigid, symmetric rare-gas-matrix-isolated species.

Example 3 — Competition level: computing a Franck–Condon factor between two Morse-potential vibrational ground states with different force constants and a finite bond-length displacement. Consider two electronic states of a diatomic modeled as harmonic oscillators (a legitimate approximation restricted to the $v=0$ levels, where anharmonic corrections from the Anharmonicity and the Morse Potential treatment are smallest), with

• ground state: $\tilde\nu_g'' = 1580$ cm⁻¹, equilibrium at $R_g^0 = 1.098$ Å (a diatomic like NO's ground state, values chosen asymmetric),
• excited state: $\tilde\nu_e' = 890$ cm⁻¹ (softer, as expected for a more weakly bound excited state), equilibrium at $R_e^0 = 1.258$ Å,
• reduced mass $\mu = 7.466$ amu $= 1.240\times10^{-26}$ kg (representative of an N–O-like diatomic).

We want the Franck–Condon factor $q_{00} = |\langle \chi_0^{e}|\chi_0^{g}\rangle|^2$ for the purely electronic 0–0 transition, i.e., the overlap between the $v=0$ harmonic oscillator wavefunctions of two oscillators with different force constants and different centers — the general (non-symmetric) overlap formula, deliberately more demanding than the equal-frequency Poisson-distribution special case used in Example 1. For two 1-D Gaussian ground-state oscillator wavefunctions

with $\alpha = \mu\omega/\hbar$, the exact overlap integral (a standard Gaussian-integral result) is

Step 1 — compute $\alpha_g$ and $\alpha_e$.

Then

Step 2 — compute the prefactor.

Step 3 — compute the exponential (displacement) factor.

Step 4 — assemble the overlap and square it.

So only about 3.8% of the absorption intensity from this electronic transition goes into the purely electronic 0–0 vibronic line; the remaining ~96% is distributed among $v'=1,2,3,\dots$ — a strongly displaced, long-progression case consistent with the fairly large $\Delta R = 0.16$ Å and the substantial frequency change between the two curves. Limiting case check: if the two curves had the same force constant ($\alpha_g=\alpha_e=\alpha$) and zero displacement ($\Delta R=0$), the formula gives prefactor $\to 1$ and exponential $\to 1$, so $q_{00}\to 1$ exactly — full overlap, the expected result when nothing distinguishes the two oscillators at all, and a useful sanity check on the general formula used here.

James Franck was an experimentalist who built his intuition from electron-collision experiments on diatomic gases; Edward Condon, a theorist steeped in the newly minted quantum mechanics of 1926, recast Franck's qualitative collision picture into the rigorous overlap-integral formalism used today. Their names are now permanently attached to the principle, but the two men arrived at it from opposite directions — one from scattering experiments on mercury vapor, one from Schrödinger's equation applied to diatomic vibration — and the synthesis is a small case study in how classical intuition and quantum formalism can converge on the same physical law.