Electronic Transitions in Molecules

When an atom changes electronic state it emits or absorbs a single sharp line. Do the same to a molecule and that line explodes into a band system of thousands of components smeared across tens of nanometres — the violet glow of a comet, the blue-green of a candle flame's base, the deep purple of iodine vapour. A molecule is not a point: its nuclei vibrate along the bond and the whole frame tumbles, so every electronic state is not one level but a ladder of vibrational rungs, each fanned into rotational sublevels. Promote an electron and those quantum numbers change too — all at once. This article consumes the Born–Oppenheimer approximation wholesale. Because nuclei are thousands of times heavier than electrons, the electronic cloud follows any nuclear rearrangement essentially instantaneously, which lets us give each state its own potential-energy curve $U(R)$ and let the nuclei roll on it. We take those curves from aq-optics-chemical-bond-potential-energy-curves, and the ladders on them from aq-optics-harmonic-oscillator-vibration and aq-optics-rigid-rotor-model; the new physics is what happens when an electron jumps between two curves whose shapes differ. Two organizing ideas run through everything and are owned by the articles this one leads into. The Franck–Condon principle — the transition is vertical because the electron moves faster than the nuclei — sets which bands are bright; the selection rules, phrased through term symbols, decide which changes are allowed. Because electronic gaps are of order electron-volts, all of this lands in the visible and ultraviolet, making band spectra the richest optical fingerprint a molecule owns.

Key takeaways

A molecular electronic transition is typically accompanied by changes in vibrational and rotational quantum states, so one electronic transition can produce a vibronic band system containing many individual rovibronic lines rather than a single line.
The energy separates into three scales, $E \approx E_\mathrm{el}+E_\mathrm{vib}+E_\mathrm{rot}$ , of order eV, $10^{-2}$ – $10^{-1}$ eV, and $10^{-3}$ eV — a hierarchy you resolve by turning up a spectrometer.
Born–Oppenheimer gives each state its own potential curve $U(R)$ ; the equilibrium bond length usually shifts between the states, and that shift is what makes the spectrum interesting.
The Franck–Condon principle makes the transition vertical, so vibrational intensities are set by the squared vibrational overlap — the Franck–Condon factor $q_{v'v''}$ .
Because the states have different rotational constants ( $B'\neq B''$ ), one branch turns over into a sharp band head, the visual signature of an electronic band.
Key idea: an electronic transition carries simultaneous jumps in $v$ and $J$ ; Born–Oppenheimer assigns each state a potential curve, the Franck–Condon principle sets vibrational intensities by wavefunction overlap, and $\Delta J = 0,\pm1$ organizes the fine structure into P, Q, R branches that converge to a band head.

The core insight

An electronic transition in a molecule is vibronic-rotational: a change of electronic state carries with it simultaneous changes in vibrational ( $v$ ) and rotational ( $J$ ) quantum numbers. Born–Oppenheimer gives each state its own potential curve; the Franck–Condon principle sets vibrational intensities by wavefunction overlap; and $\Delta J = 0,\pm1$ organizes the fine structure into P, Q, R branches with a band head.

Where this lives today

Electronic band spectra remain among the most information-rich measurements in molecular and astrophysical physics. In combustion and plasma science, planar laser-induced fluorescence tunes a laser onto one vibronic-rotational line of a radical such as OH, CH, or NO and images the fluorescence across an engine cylinder; because rotational intensities follow $\propto(2J+1)e^{-F(J)hc/kT}$ , a single image maps concentration and temperature at kilohertz rates. In astrophysics the same bands are remote thermometers — the CN violet system, the C₂ Swan bands, and TiO bands in cool stars read for temperature and abundance — while JWST measures vibronic absorption in exoplanet atmospheres to hunt for CO₂, water, and disequilibrium chemistry. The frontier that would have astonished Herzberg is control. Laser cooling of molecules — once thought impossible, since a molecule scattering a photon can decay into a forest of vibrational levels and go dark — now works because some species (SrF, CaF, YO) have near-diagonal Franck–Condon factors: the excited state overlaps almost perfectly with a single ground level, so a few repump lasers close the cycle and the molecule is cooled to microkelvin and trapped in optical tweezers. The Franck–Condon factor, introduced here as what decides which bands are bright, now decides whether a molecule can be laser-cooled at all.

The physical picture

Stack two potential-energy curves, ground and excited, each a bowl in the bond length $R$ . Promoting an electron — often from a bonding into an antibonding orbital — weakens the bond, so the upper bowl usually sits at a larger separation $R_e'$ . Now absorb a photon. The electron rearranges in about $10^{-16}$ s, but a vibrational period is roughly $10^{-14}$ s, so the nuclei cannot move during the jump. On the diagram the transition is a vertical arrow rising from the lower bowl at fixed $R$ — and it lands not at the bottom of the upper bowl but on its wall, depositing several quanta of vibration. That geometric mismatch, read straight off the curves, is the physical origin of the Franck–Condon progression.

Figure

Two potential-energy curves of I₂ for the B ← X system the excited bond is longer Re′ ≈ 3.05 Å versus Re″ = 2.666 Å so the vertical Franck-Condon transition of ≈ 15725 cm⁻¹ near 636 nm leaves v″ = 0 and lands high on the upper wall

Each vibrational rung is itself a fan of rotational levels, so the molecule also acquires rotational fine structure. For the common electric-dipole case, $\Delta J=\pm1$ gives the $P$ and $R$ branches, while a $Q$ branch ( $\Delta J=0$ ) may occur for transitions with the appropriate angular-momentum selection rules. Because the two electronic states can have different rotational constants ( $B'\ne B''$ ), the line spacing can change with $J$ and, for the appropriate sign of $B'-B''$ , a branch can crowd together and form a band head. The spectrum is a hierarchy of zoom levels: far out, one electronic transition of a few eV; zoom in, a comb of vibrational bands; zoom again, hundreds of rotational lines converging on their head.

Figure

Anatomy of one vibronic band the R-branch lines ΔJ = +1 crowd together and reverse at a band head while the P-branch lines ΔJ = −1 fan out only because B′ ≠ B″ the lower panel is the Fortrat parabola of line position versus J

The core equations

The whole subject rests on one act of separation. Because the electrons respond instantly to the nuclei, a state's total energy splits into an electronic part (the curve minimum), a vibrational part (motion along the bond), and a rotational part (tumbling) — three wildly different scales, which is why the spectrum reads at three resolutions.

E = E_\mathrm{el} + E_\mathrm{vib} + E_\mathrm{rot} \qquad (1)

where: $E_\mathrm{el}$ — electronic energy, order eV; $E_\mathrm{vib}$ — vibrational energy, order $10^{-2}$ – $10^{-1}$ eV; $E_\mathrm{rot}$ — rotational energy, order $10^{-3}$ eV. The ratios $\sim1:10^{-2}:10^{-3}$ are why a coarse spectrometer sees a blur and a great one resolves rotational lines. A single line records a jump in all three books at once, from a lower state to an upper. Writing each contribution as a term value in cm $^{-1}$ , the line position is upper minus lower in each ledger.

\tilde\nu = \big(T_e' - T_e''\big) + \big(G'(v') - G''(v'')\big) + \big(F'(J') - F''(J'')\big) \qquad (2)

where: $\tilde\nu$ — transition wavenumber (cm $^{-1}$ ); $T_e',\,T_e''$ — electronic term values (curve minima); $G(v)$ — vibrational term value at $v$ ; $F(J)$ — rotational term value at $J$ . Note the convention (a common trap): upper single-primed, lower double-primed, written upper $\leftarrow$ lower for absorption. The vibrational ladder is not evenly spaced: a real bond is anharmonic — its potential softens as it stretches — so rungs crowd toward dissociation. The leading correction is quadratic in $(v+\tfrac12)$ .

G(v) = \omega_e\left(v+\tfrac{1}{2}\right) - \omega_e x_e\left(v+\tfrac{1}{2}\right)^2 + \cdots \qquad (3)

where: $\omega_e$ — harmonic vibrational constant (cm $^{-1}$ ); $\omega_e x_e$ — first anharmonicity constant, positive, that closes the ladder; $v$ — vibrational quantum number. Extrapolating the shrinking spacings to zero (a Birge–Sponer plot) estimates the dissociation energy from the spectrum. The rotational ladder is approximately that of a rigid rotor, with the leading correction accounting for centrifugal distortion as rotational excitation stretches the molecule.

F(J) = B\,J(J+1) - D\,[\,J(J+1)\,]^2 \qquad (4)

where: $B$ — rotational constant, $B=\hbar/(4\pi c\,I)$ in cm $^{-1}$ ; $I=\mu R_e^2$ — moment of inertia; $D$ — centrifugal distortion constant (tiny, positive); $J$ — rotational quantum number. Because $B\propto1/I$ and, for a diatomic molecule, $I=\mu R_e^2$ , measuring $B$ provides a direct measure of the equilibrium bond length $R_e$ when the reduced mass $\mu$ is known. Not every band is equally bright: the $v'\!-\!v''$ band's intensity is set by how well the two vibrational wavefunctions overlap — the squared overlap integral, or Franck–Condon factor.

q_{v'v''} = \left|\int \chi_{v'}^{*}(R)\,\chi_{v''}(R)\,\mathrm{d}R\right|^2 \qquad (5)

where: $q_{v'v''}$ — Franck–Condon factor for the $v'\!\leftarrow\!v''$ band; $\chi_{v'},\,\chi_{v''}$ — upper and lower vibrational wavefunctions. If the curves sit at the same $R_e$ the overlap is diagonal ( $q_{v'v''}=\delta_{v'v''}$ ) and only $0\!-\!0$ shows; the more displaced the curves, the longer the progression. Within a band the rotational lines split by $\Delta J$ . Combining $F'$ and $F''$ for $\Delta J=+1$ (R) and $-1$ (P) gives two quadratics in the lower-state $J$ , sharing a band origin $\tilde\nu_0$ ; the coefficient $B'-B''$ decides whether and where a head forms.

\tilde\nu_{R}(J)=\tilde\nu_0 + (B'+B'')(J+1) + (B'-B'')(J+1)^2,\qquad \tilde\nu_{P}(J)=\tilde\nu_0 - (B'+B'')J + (B'-B'')J^2 \qquad (6)

where: $\tilde\nu_0$ — band origin, vibronic energy at $J=0$ ; $B',\,B''$ — upper and lower rotational constants; $J$ — lower-state quantum number. When $B'<B''$ the R branch turns over (head on the high- $\tilde\nu$ side, shaded red); when $B'>B''$ the P branch does (shaded violet). Finally, only certain changes are dipole-allowed — the principal rules constrain angular momentum along and about the axis, and the spin.

\Delta J = 0,\ \pm 1;\qquad \Delta S = 0;\qquad \Delta\Lambda = 0,\ \pm 1 \qquad (7)

where: $\Delta J$ — total rotational change, giving P ( $-1$ ), Q ( $0$ ), R ( $+1$ ) branches; $\Delta S$ — spin change, zero when spin-allowed; $\Delta\Lambda$ — change in the axial projection of orbital angular momentum. Parity rules ( $g\leftrightarrow u$ , $+\leftrightarrow+$ ) supplement these; a violated $\Delta S=0$ gives a weak intercombination band via spin–orbit coupling.

Derivation AThe Franck-Condon factor from the transition dipole

The goal is to show why band intensities are governed by a pure overlap of nuclear wavefunctions, with the electronic physics factored cleanly out front. We start from the full vibronic transition-dipole matrix element, integrate over the electrons first, then invoke the Condon approximation — that the electronic transition moment barely varies over the range the nuclei explore. What survives is the Franck–Condon factor, the quantitative form of "the transition is vertical."

Step 1. Write the vibronic dipole $\boldsymbol{\mu}_{fi}=\int \psi_\mathrm{el}'^{*}\chi_{v'}^{*}\,\boldsymbol{\mu}\,\psi_\mathrm{el}''\chi_{v''}\,\mathrm{d}\mathbf{r}\,\mathrm{d}R$ , over electrons $\mathbf{r}$ and nuclei $R$ . Why: the intensity is $|\boldsymbol{\mu}_{fi}|^2$ , so which bands are bright is buried in this integral.

Step 2. Integrate over the electrons at fixed $R$ to define $\boldsymbol{\mu}_\mathrm{el}(R)=\int\psi_\mathrm{el}'^{*}\boldsymbol{\mu}\,\psi_\mathrm{el}''\,\mathrm{d}\mathbf{r}$ , leaving $\boldsymbol{\mu}_{fi}=\int \chi_{v'}^{*}\,\boldsymbol{\mu}_\mathrm{el}(R)\,\chi_{v''}\,\mathrm{d}R$ . Why: the Born–Oppenheimer factorization — electrons first, nuclei second — isolates the only $R$ -dependent piece.

Step 3. Since $\boldsymbol{\mu}_\mathrm{el}(R)$ varies slowly across the narrow overlap window, pull it out at $R_e$ : $\boldsymbol{\mu}_{fi}\approx \boldsymbol{\mu}_\mathrm{el}(R_e)\int \chi_{v'}^{*}\chi_{v''}\,\mathrm{d}R$ , so intensity $\propto |\boldsymbol{\mu}_\mathrm{el}|^2\,q_{v'v''}$ . Why: the Condon approximation is exactly "the jump is vertical"; the surviving squared nuclear overlap is the Franck–Condon factor.

Step 4. With the upper curve at larger $R_e'$ , the $v''=0$ wavefunction overlaps best not with $v'=0$ but with an excited $v'$ whose turning point sits above $R_e''$ , so the progression peaks away from $0\!-\!0$ — which is why iodine vapour absorbs a long comb rather than one line, and why the brightest band reveals the bond-length change at a glance. Two limiting checks confirm the result: coincident curves give $q_{v'v''}=\delta_{v'v''}$ (only the $0\!-\!0$ band survives), and summing over all upper levels gives $\sum_{v'}q_{v'v''}=1$ by the completeness of the vibrational basis.

Go deeper · Derivation B · 1 minFormation and position of the band head

Derivation BFormation and position of the band head

The aim is to prove that the sharp edge terminating an electronic band is not an accident of intensity but a genuine extremum of line position, and to locate it. The R and P branch formulas of equation (6) are two halves of one parabola once a signed running index absorbs the branch label; because the coefficient $B'-B''$ is nonzero, that parabola turns over, and lines pile up where its slope vanishes.

Step 1. Set $m=J+1$ for R and $m=-J$ for P. Both formulas merge into one quadratic, the Fortrat parabola $\tilde\nu(m)=\tilde\nu_0+(B'+B'')\,m+(B'-B'')\,m^2$ , R on $m>0$ , P on $m<0$ . Why: unifying the branches gives one smooth curve whose vertex is the head.

Step 2. Differentiate and set to zero: $(B'+B'')+2(B'-B'')\,m=0$ , so $m_\mathrm{head}=-\dfrac{B'+B''}{2(B'-B'')}$ . Why: where the slope vanishes, successive integer lines coincide, producing the abrupt edge the eye reads as a head.

Step 3. If $B'<B''$ then $m_\mathrm{head}>0$ : the head lies in the R branch, band red-degraded; if $B'>B''$ it lies in the P branch, band violet-degraded. In $J$ : $J_\mathrm{head}\approx\dfrac{B'+B''}{2\,|B''-B'|}-\dfrac12$ . Why: which way a band shades fixes the sign of the bond-length change.

Step 4. The head sits off the origin by $\tilde\nu_\mathrm{head}-\tilde\nu_0=-\dfrac{(B'+B'')^2}{4(B'-B'')}$ ; measuring it and $J_\mathrm{head}$ returns $B'-B''$ , hence $\Delta R_e$ . Why: a feature you find with a ruler on a plate delivers an excited-state bond length. *As a limiting check: when $B'=B''$ the quadratic term vanishes, no head forms, and equation (6) reduces to an ordinary vibration–rotation band.*

Go deeper · Derivation C · 1 minCombination differences that isolate each rotational constant

Derivation CCombination differences that isolate each rotational constant

The goal is to extract the ground- and excited-state rotational constants separately and cleanly, without ever knowing the band origin or the vibrational terms. The method — combination differences — subtracts pairs of lines chosen to share a common upper or lower level, so everything except one state's rotational structure cancels, turning a tangle of band lines into two bond lengths.

Step 1. Take the R line from $J-1$ and the P line to $J+1$ : both equal $\tilde\nu_0+F'(J)-F''(J\mp1)$ and share the same upper level $J'=J$ , so $\tilde\nu_0$ and $F'(J)$ are identical. Subtract: $\Delta_2F''(J)\equiv R(J-1)-P(J+1)=4B''\big(J+\tfrac12\big)$ . Why: sharing the upper level cancels the origin and upper term, so a plot against $J+\tfrac12$ gives $B''$ from its slope $4B''$ .

Step 2. Choosing lines that share a common lower level instead gives the mirror result $R(J)-P(J)=4B'\big(J+\tfrac12\big)$ , slope $4B'$ . Why: the same trick isolates the upper-state constant $B'$ , so one band yields both.

Step 3. Convert each with $B=\hbar/(4\pi cI)$ , $I=\mu R_e^2$ ; the difference $R_e'-R_e''$ shapes the Franck–Condon envelope and fixes the head. Why: two slopes reconstruct the geometry of a state that may live only nanoseconds. *The slopes $4B'$ and $4B''$ are the standard combination-difference relations, independent of $\tilde\nu_0$ and the vibrational terms.*

Limiting cases

Case	Result	Meaning
Vertical (Franck–Condon) limit	$R=\mathrm{const}$ during the jump	Electronic time ( $\sim10^{-16}$ s) $\ll$ vibrational period ( $\sim10^{-14}$ s), so nuclei freeze; intensities are pure vibrational overlaps.
Coincident curves ( $R_e'=R_e''$ )	$q_{v'v''}=\delta_{v'v''}$	Only the $0\!-\!0$ band appears; the progression is entirely due to a geometry change between the states.
Equal constants ( $B'=B''$ )	no band head	The quadratic term in equation (6) vanishes; lines spread monotonically — the head is a fingerprint of $B'\neq B''$ .
Dissociative upper state	$E>D_e'$	A vertical transition above the upper dissociation limit ends unbound; the spectrum becomes a continuum (photodissociation).
High temperature	high $J$ , $v''$ populated	Hot bands and long rotational envelopes as $\propto(2J+1)e^{-F(J)hc/kT}$ fills high $J$ ; the band becomes a thermometer.

Common trapAssuming the two electronic states share the same bond length, vibrational spacing, and rotational constant

They need not. Promotion of an electron can change the bonding character and therefore the equilibrium geometry, rotational constant, and vibrational frequency of the molecule. Excitation from a bonding to an antibonding orbital often weakens the bond, giving $R_e'>R_e''$ and hence $B'<B''$ , but other electronic transitions can shorten the bond or produce smaller changes. That inequality is not a nuisance: it is the whole story. The curve displacement stretches a single line into a Franck–Condon progression, and $B'-B''$ makes a branch turn over into a band head. Set $R_e'=R_e''$ and you erase the progression; set $B'=B''$ and you erase the head. (CN is the famous exception where the bond shortens on excitation, so $B'>B''$ and the head jumps to the P branch.) And mind the primes: upper single-primed, lower double-primed, written upper $\leftarrow$ lower.

Worked examples

1 · The three energy scales in one band system (confidence builder)

For the iodine B $^3\Pi_{0u}\leftarrow$ X $^1\Sigma_g^+$ system, take $T_e=15769$ cm $^{-1}$ , $\omega_e''=214.5$ , $\omega_e''x_e''=0.61$ cm $^{-1}$ (ground) and $\omega_e'=125.7$ , $\omega_e'x_e'=0.76$ cm $^{-1}$ (excited). Find the $0\!-\!0$ band-origin wavelength and compare the three energy scales.

Step 1. Zero-point terms from equation (3): $G''(0)=214.5(0.5)-0.61(0.25)=107.10$ cm $^{-1}$ , $G'(0)=125.7(0.5)-0.76(0.25)=62.66$ cm $^{-1}$ — the upper state vibrates more slowly.

Step 2. Band origin from equation (2) with $J'=J''=0$ : $\tilde\nu_{00}=15769+62.66-107.10=15724.6$ cm $^{-1}$ , so $\lambda=10^7/15724.6=\mathbf{636\ nm}$ — deep red, at the foot of the band that colours iodine vapour purple.

Step 3. The scales: electronic $\tilde\nu_{00}\times1.240\times10^{-4}=\mathbf{1.95\ eV}$ ; vibrational spacing $G''(1)-G''(0)=213.3$ cm $^{-1}=\mathbf{0.026\ eV}$ ; rotational, with $B''\approx0.037$ cm $^{-1}$ at $J=10$ , $F(10)=4.1$ cm $^{-1}=\mathbf{5\times10^{-4}\ eV}$ . These are the nested hierarchy of equation (1) — electronic ≈ 1.95 eV, vibrational ≈ 0.026 eV, rotational ≈ 5×10⁻⁴ eV, nearly four orders of magnitude in one spectrum.

2 · A band head and the excited-state bond length

The CN violet system B $^2\Sigma^+\leftarrow$ X $^2\Sigma^+$ has its $0\!-\!0$ head near 388.3 nm. With $B''=1.900$ , $B'=1.973$ cm $^{-1}$ ( $\mu_{CN}=6.46$ u), find which branch forms the head, its $J$ , its displacement, and the bond lengths.

Step 1. Since $B'=1.973>B''=1.900$ , $B'-B''=+0.073$ cm $^{-1}>0$ : by the derivation the head forms in the P branch and the band shades to the violet — the CN bond contracts on excitation.

Step 2. $J_\mathrm{head}=\dfrac{B'+B''}{2|B''-B'|}-\dfrac12=\dfrac{3.873}{0.146}-0.5=\mathbf{26}$ . Displacement $\tilde\nu_\mathrm{head}-\tilde\nu_0=-\dfrac{(B'+B'')^2}{4(B'-B'')}=-\dfrac{(3.873)^2}{0.292}=\mathbf{-51\ cm^{-1}}$ — the head sits 51 cm $^{-1}$ red of the origin, lines fanning to the violet.

Step 3. Bond lengths from $R_e=\sqrt{\hbar/(4\pi c\mu B)}$ , with $\mu=1.073\times10^{-26}$ kg: $B''=1.900\Rightarrow R_e''=\mathbf{1.172\ \text{\AA}}$ ; $B'=1.973\Rightarrow R_e'=\mathbf{1.150\ \text{\AA}}$ . The bond shortens by 0.022 Å on excitation — read straight off one band's rotational structure.

Physics of the portrait

Molecular electronic spectroscopy is a science of reading — extracting bond lengths, vibrational frequencies, dissociation energies, and the very shapes of potential curves from a smear of band lines. No one did more to codify that craft than the man whose atlases still sit on every shelf.

Origins · 1 minGerhard Herzberg and how this was discovered

Gerhard Herzberg

1904–1999 · molecular spectroscopy and the structure of diatomic molecules

Gerhard Herzberg turned the reading of band spectra from an art into a systematic discipline. Trained in Germany, he emigrated in 1935 — his wife Luise, herself a spectroscopist, was Jewish — settling at the National Research Council of Canada in Ottawa, where he built the world's leading molecular-spectroscopy laboratory. His three-volume Molecular Spectra and Molecular Structure (the first, Spectra of Diatomic Molecules, 1950) became the field's bible, laying out term symbols, selection rules, Fortrat parabolas, combination differences, and Franck–Condon analysis with a completeness never superseded. A master of the free radical, he obtained the first laboratory spectra of species such as CH₂ and identified molecular ions and radicals in comets and interstellar space, winning the 1971 Nobel Prize in Chemistry for his work on the electronic structure and geometry of molecules. His genius was a way of seeing: given a sufficiently resolved band system, he could use its vibrational and rotational structure to infer molecular constants and reconstruct the underlying potential-energy curves—the inverse problem at the heart of this article.

History

1887

— Deslandres orders band-spectrum lines into regular tables, exposing their vibrational structure

1925

— Franck argues that an electronic transition happens at fixed nuclear positions

1927

— Born and Oppenheimer justify separating electronic from nuclear motion

1928

— Condon recasts the Franck principle quantitatively as a vibrational overlap integral

1950

— Herzberg's Spectra of Diatomic Molecules systematizes the entire field

Going further

Texts. Herzberg, Molecular Spectra and Molecular Structure I: Spectra of Diatomic Molecules — the definitive treatment of band structure, Fortrat parabolas, and combination differences · Demtröder, Laser Spectroscopy, Vol. 1 · Atkins & Friedman, Molecular Quantum Mechanics — Franck–Condon factors · Bransden & Joachain, Physics of Atoms and Molecules · Foot, Atomic Physics · Hecht, Optics. Key numerical anchors: the I₂ $0\!-\!0$ origin is $15724.6$ cm $^{-1}\Rightarrow636$ nm (from $T_e,\omega_e,\omega_ex_e$ ); the energy-scale ratios are $\sim1:10^{-2}:10^{-3}$ ; the CN head falls at $J\approx26$ with displacement $-51$ cm $^{-1}$ , giving bond lengths $R_e''=1.172$ Å and $R_e'=1.150$ Å from $B''=1.900$ , $B'=1.973$ cm $^{-1}$ , $\mu=6.46$ u. The Fortrat band-head condition is $m_\mathrm{head}=-(B'+B'')/[2(B'-B'')]$ and the combination differences are $\Delta_2F''=4B''(J+\tfrac12)$ . Molecular constants are standard Herzberg/NIST values.

Prerequisites. The Born–Oppenheimer Approximation · The Chemical Bond and Molecular Potential Energy Curves · Molecular Orbital Theory · The Harmonic Oscillator Model of Vibration · The Rigid Rotor Model. Leads to. The Franck–Condon Principle · Selection Rules for Molecular Transitions · Vibration–Rotation Spectra and P, Q, R Branches · Molecular Term Symbols.

Ask the tutor: Want to trace a real band system end to end — from two potential curves to the Franck–Condon envelope, the P/R branches, and the band head — for a molecule you choose?

Why does an electronic transition make a band while an atomic one makes a line?

An atom has only electronic degrees of freedom, so changing its state gives one line. A molecule carries two more storehouses — vibration of the bond and rotation of the frame — each with a full ladder. When the electron jumps, $v$ and $J$ generally change too, so the one electronic gap is dressed by hundreds of $(v',v'',J',J'')$ combinations, each a separate line; together they form the band system. The band is the electronic line resolved into the internal motions the atom does not have.

What exactly makes the transition "vertical" on the potential curves?

Timescales. The electronic rearrangement takes about $10^{-16}$ s, a vibrational period around $10^{-14}$ s — a hundred times longer. During the instant the electron redistributes, the heavy nuclei cannot move, so $R$ is frozen; on a plot of energy versus $R$ , "constant $R$ " is a vertical line. This is the Franck–Condon principle stated geometrically, and it is why a transition from the bottom of the lower well lands on the wall of the upper well whenever the wells sit at different bond lengths, depositing vibrational energy and lighting up a progression.

Why does one branch form a sharp band head and the other does not?

In equation (6) both branches carry the same quadratic coefficient $B'-B''$ but opposite-sign linear terms. When $B'<B''$ the R branch's lines first spread, then slow, stop, and reverse — piling up at a head — while the P branch spreads without limit; when $B'>B''$ the roles swap. Only the branch whose linear and quadratic terms fight turns over. If $B'=B''$ the quadratic term vanishes and neither reverses, so no head forms — as in an ordinary infrared band.

How can I measure an excited-state bond length if the state lives for nanoseconds?

Through the rotational constant $B'$ — you never catch the molecule sitting still. Combination differences subtract band lines sharing a common lower level, leaving $R(J)-P(J)=4B'(J+\tfrac12)$ ; a straight-line fit gives $B'$ , free of the origin and vibrational terms. Since $B'=\hbar/(4\pi c\mu R_e'^2)$ , inverting gives $R_e'$ . The information is imprinted on the photons the molecule emits, so a plate exposed for seconds records the geometry of a state that individually survives only nanoseconds. The CN example above yields $1.150$ Å.

What is the difference between the band origin and the band head?

The band origin $\tilde\nu_0$ is the position of the $J=0$ line — the pure vibronic energy $T_e'-T_e''+G'-G''$ with no rotational term. It is a bookkeeping reference; there may be no actual line there. The band head is a real, visible feature: the wavenumber where a branch's lines pile up as their spacing shrinks to zero. The two do not coincide — the head is displaced by $-(B'+B'')^2/[4(B'-B'')]$ , about $51$ cm $^{-1}$ for CN. Spectroscopists report the easily measured head, but the meaningful energy differences live at the origin.

Do Franck-Condon factors ever help rather than complicate?

Enormously — modern AMO physics depends on it. Normally the spread of Franck–Condon factors is a headache: an excited molecule can decay into many ground vibrational levels, so one you try to laser-cool quickly falls into a "dark" level the laser cannot reach. But a few species — SrF, CaF, YO — have curves almost coincident ( $R_e'\approx R_e''$ ), giving near-diagonal factors: the excited state overlaps overwhelmingly with a single ground level, so one main laser plus a repump or two closes the optical cycle. The overlap integral introduced here as an intensity rule has become the figure of merit for whether a molecule is coolable at all.