Term Symbols and Molecular Electronic States

An isolated atom is a little sphere, and its stationary states inherit that symmetry: atomic angular momentum — total orbital $L$ , spin $S$ , and their coupling into $J$ — collapses the whole electronic problem into the compact label $^{2S+1}L_J$ . This article asks what survives when the sphere is broken. Bring a second nucleus up and clamp the two a distance $R$ apart, as the Born-Oppenheimer picture licenses, and the electrons no longer move in a central field. Full rotational symmetry is gone; only a cylinder's worth is left — rotations about the internuclear line, plus a few reflections and, for identical nuclei, an inversion. The good quantum numbers must be rebuilt from the operations that still commute with the electronic Hamiltonian. The payoff is the molecular term symbol $^{2S+1}\Lambda^{\pm}_{g/u,\,\Omega}$ , whose parts read off the surviving symmetries one at a time. The axial field kills every component of orbital angular momentum except the projection along the axis, so $\Lambda=|M_L|$ replaces the atomic $L$ and christens the Greek ladder $\Sigma,\Pi,\Delta,\Phi$ ; spin, blind to the electrostatic axis, keeps its multiplicity $2S+1$ . Reflection hands $\Sigma$ states a $\pm$ sign, inversion hands homonuclear molecules a $g/u$ parity, and strong spin-orbit coupling locks $\Lambda$ and $\Sigma$ into $\Omega=|\Lambda+\Sigma|$ . These labels are not bookkeeping — every electric-dipole selection rule of molecular spectroscopy, and the shape of every band, is stated in their language.

Key takeaways

A diatomic's cylindrical symmetry demotes the atomic $L$ to its axial projection $\Lambda=|M_L|$ ; states are named $\Sigma,\Pi,\Delta,\Phi$ for $\Lambda=0,1,2,3$ , the analogues of $S,P,D,F$ .
The full symbol $^{2S+1}\Lambda^{\pm}_{g/u,\,\Omega}$ carries multiplicity $2S+1$ , reflection sign $\pm$ ( $\Sigma$ only), inversion parity $g/u$ (homonuclear only), and case-(a) index $\Omega$ .
Strong spin-orbit coupling (Hund's case a) locks $\Lambda$ and the spin projection $\Sigma$ into $\Omega=|\Lambda+\Sigma|$ , splitting a term into $2S+1$ components spaced by $A\Lambda$ .
The labels dictate every electric-dipole selection rule — $\Delta\Lambda=0,\pm1$ , $\Delta S=0$ , $\Sigma^\pm\!\leftrightarrow\!\Sigma^\pm$ , $g\leftrightarrow u$ — and why a Q branch appears only when $\Lambda$ changes.
Oxygen's open-shell $(\pi_g^*)^2$ gives a $^3\Sigma_g^-$ ground term: $S=1$ , paramagnetic, with a metastable $^1\Delta_g$ singlet just 7882 cm⁻¹ (0.98 eV, 1270 nm) above.
Key idea: the molecular term symbol is the atomic term symbol rebuilt from a cylinder's symmetries — axial orbital projection $\Lambda$ , spin multiplicity, and reflection and inversion parities are exactly the labels the electronic Hamiltonian still allows.

The core insight

A diatomic's cylindrical symmetry replaces the atomic $L$ by its axial projection $\Lambda=|M_L|$ ; the molecular term symbol $^{2S+1}\Lambda^{\pm}_{g/u}$ collects the good quantum numbers dictated by that symmetry — axial orbital projection, spin multiplicity, reflection parity, and (homonuclear) inversion parity.

Where this lives today

Term symbols are the working notation of every laboratory that reads molecular light. An electronic band is named by its upper and lower terms — the $\mathrm N_2$ second positive system is $C\,^3\Pi_u\!-\!B\,^3\Pi_g$ — and a spectrum is assigned by matching observed $\Delta\Lambda$ , $\Delta S$ , and parity against the allowed combinations, reading a Q branch as a signal that $\Lambda$ changed. The oxygen terms are watched directly: the b $^1\Sigma_g^+\!-\!X\,^3\Sigma_g^-$ transition is the 762 nm atmospheric A-band used to sound oxygen from satellites, and the 1270 nm a $^1\Delta_g$ emission traces the mesospheric airglow and the singlet oxygen of photodynamic therapy, where its long, term-symbol-protected lifetime is the whole point. The same labels reach into the coldest and hottest physics being done now. Laser-cooling experiments on SrF and CaF exploit near-diagonal $^2\Sigma$ - $^2\Pi$ transitions, OH's inverted $^2\Pi_{3/2}$ ground state with its Λ-doublet drives interstellar masers and flame diagnostics, and astrochemists fingerprint CN, CH, C $_2$ , and TiO in cold clouds by their band-system term structure. From the magnet that lifts liquid oxygen to the lines that fingerprint a distant galaxy, the five slots of $^{2S+1}\Lambda^\pm_{g/u,\Omega}$ do the job Hund set them: naming the state by its symmetry.

The physical picture

Fix the two nuclei as point charges strung along an axis and drop an electron into the field they make. That field is cigar-shaped: smooth as you swing around the axis, lumpy as you tilt away. An electron circulating about the axis keeps a well-defined "orbital current" — its angular-momentum component along the axis is conserved, because the potential is unchanged under an axial rotation. But tip the orbit and the potential shoves back: the transverse components, and the total $L$ , are not conserved. The vector $\mathbf L$ precesses, and only its shadow $L_z$ stands still — or rather its magnitude $\Lambda=|M_L|$ , the primary label of the molecular state.

Figure

Axial symmetry keeps only the projection of orbital angular momentum on the internuclear axis: L tumbles but its shadow L z is fixed, so Λ = the absolute value of M L. A σ electron has m ℓ = 0, a π ±1, a δ ±2, building the ladder Σ Π Δ Φ for Λ = 0, 1, 2, 3

The Greek letters are the atomic $S,P,D,F$ with a change of meaning: they count units of axial orbital angular momentum, not total. A $\pi$ electron carries $\lambda=1$ ; two of them combine like two atomic $p$ -electrons, giving $\Lambda=0$ ( $\Sigma$ ) or $\Lambda=2$ ( $\Delta$ ). The two remaining tags are parity stamps: $\pm$ is the sign under reflection through a plane containing the nuclei, $g/u$ the sign under inversion through the midpoint — both two-valued because the operations square to the identity. The second figure shows why these labels earn their keep. Oxygen's highest electrons live in a doubly degenerate antibonding $\pi_g^*$ shell, and the three ways two electrons can share it produce three terms stacked in energy — a triplet ground state and two singlets — whose spacings are measured numbers and whose radiative sluggishness is fixed by the term-symbol selection rules. This is singlet oxygen, and the reason it lives long enough to matter in chemistry is written in its labels.

Figure

The three lowest electronic terms of molecular oxygen, all from the (π g*)² valence shell: the triplet ³Σ g⁻ ground state (parallel spins, S = 1, paramagnetic), the metastable a ¹Δ g at 7882 cm⁻¹ (0.98 eV, 1270 nm), and b ¹Σ g⁺ at 13121 cm⁻¹ (1.63 eV, 762 nm). Both singlet decays are spin- and symmetry-forbidden, so singlet oxygen is long-lived

Spin-orbit coupling sets which labels are even meaningful. When it is weak against rotation, $\Lambda$ and $S$ are separately good and the spin is not tied to the axis — Hund's case (b), the home of light molecules and every $\Sigma$ state. When it is strong, $\Lambda$ and $\Sigma$ lock into $\Omega$ and each term unfolds into a fine-structure ladder — Hund's case (a), the molecular echo of the atomic $^3P_{0,1,2}$ multiplet.

The core equations

The full term symbol is a glyph stacking five pieces of symmetry information; everything downstream — spectra, selection rules, magnetism — is read out of these slots, so it is worth naming each one first.

^{2S+1}\Lambda^{\,\pm}_{\;g/u,\,\Omega} \qquad (1)

where: $S$ — total electron spin, so $2S+1$ is the multiplicity; $\Lambda$ — magnitude of the axial orbital projection, $\Lambda=|M_L|=0,1,2,\ldots$ , written $\Sigma,\Pi,\Delta,\Phi$ ; $\pm$ — sign under reflection $\sigma_v$ in a plane containing the axis (only for $\Lambda=0$ ); $g/u$ — parity under inversion through the center (homonuclear only); $\Omega$ — the case-(a) axial projection $\Omega=|\Lambda+\Sigma|$ , a subscript when spin-orbit coupling resolves the fine structure. Reading the glyph left to right reads the surviving symmetries one at a time. The primary label $\Lambda$ is built like the atomic $M_L$ , but only the axial projections add: each electron contributes $m_\ell$ , and the magnitude of the sum names the letter.

\Lambda = |M_L| = |m_{\ell_1}+m_{\ell_2}+\cdots|,\qquad \Lambda=0,1,2,3\;\leftrightarrow\;\Sigma,\Pi,\Delta,\Phi \qquad (2)

where: $m_{\ell_i}$ — axial projection of the $i$ -th electron ( $0$ for $\sigma$ , $\pm1$ for $\pi$ , $\pm2$ for $\delta$ ); $M_L$ — total axial projection. A filled shell contributes zero, so only partially filled orbitals matter — hence closed-shell molecules are $\Sigma$ states. When spin-orbit coupling is strong (Hund's case a), the axial spin projection is quantized alongside the orbital one and the two add to give the fine-structure label; as $\Sigma$ runs over its $2S+1$ values, a $\Lambda\neq0$ term splits into that many components.

\Omega=|\Lambda+\Sigma|,\qquad \Sigma\in\{-S,-S+1,\ldots,+S\} \qquad (3)

where: $\Sigma$ — projection of total spin on the axis, i.e. $M_S$ (not the Greek $\Sigma$ term label); $\Omega$ — total orbital-plus-spin axial projection. For $\Lambda=0$ all $\Sigma$ values give the same $\Omega=|\Sigma|$ , which is why $\Sigma$ terms show no first-order spin-orbit splitting. The energies follow a simple rule: because only the axial components of $\mathbf L$ and $\mathbf S$ survive the symmetry, the spin-orbit operator reduces to a product of projections, and the multiplet is a ladder of equal steps.

E_{\mathrm{so}} = A\,\Lambda\,\Sigma \qquad (4)

where: $A$ — molecular spin-orbit constant (energy units), the diatomic analogue of the atomic $\zeta$ ; $\Lambda,\Sigma$ — axial orbital and spin projections. Components are spaced by $A\Lambda$ ; positive $A$ gives a regular multiplet (lowest $\Omega$ lowest), negative $A$ an inverted one. Each electronic term is only the bottom of a tower: in the Born-Oppenheimer hierarchy the full term value is $T=T_e+G(v)+F(J)$ , adding vibration $G(v)=\omega_e(v+\tfrac12)-\omega_e x_e(v+\tfrac12)^2$ and rotation $F(J)\approx B_vJ(J+1)$ on top of the electronic minimum $T_e$ — and since each term owns its own curve $V(R)$ , it owns its own $\omega_e$ and $B_v$ . But the classification earns its keep in the electric-dipole selection rules: a transition is allowed only if the changes in $\Lambda$ , spin, reflection sign, and parity all pass.

\Delta\Lambda=0,\pm1;\quad \Delta S=0;\quad \Sigma^{+}\!\leftrightarrow\!\Sigma^{+},\;\Sigma^{-}\!\leftrightarrow\!\Sigma^{-};\quad g\leftrightarrow u \qquad (5)

where: $\Delta\Lambda=0$ gives a parallel band (moment along the axis), $\Delta\Lambda=\pm1$ a perpendicular band (the one that shows a Q branch); $g\leftrightarrow u$ — parity must flip in a homonuclear molecule (Laporte rule); $\Delta S=0$ — spin is conserved unless spin-orbit mixing lends a forbidden band weak intensity. Finally, the payoff: the ground-term symbols of a few textbook diatomics, each an application of the rules above.

\mathrm H_2:\;{}^1\Sigma_g^+\qquad \mathrm N_2:\;{}^1\Sigma_g^+\qquad \mathrm O_2:\;{}^3\Sigma_g^-\qquad \mathrm{CO}:\;{}^1\Sigma^+ \qquad (6)

where: $\mathrm H_2$ , $\mathrm N_2$ — closed valence shells ( $S=0$ , $\Lambda=0$ , homonuclear $g$ ); $\mathrm O_2$ — open-shell $(\pi_g^*)^2$ triplet, hence paramagnetic; $\mathrm{CO}$ — isoelectronic with $\mathrm N_2$ but heteronuclear, so no $g/u$ subscript.

Derivation AWhy only the axial projection of orbital angular momentum survives

The goal is to see, from the symmetry of the clamped-nuclei electronic Hamiltonian alone, why the atomic labels $L$ and $J$ give way to the single axial quantity $\Lambda=|M_L|$ . We identify which angular-momentum operators still commute with $H_{el}$ , then show the leftover reflection symmetry forces the label to be a magnitude, not a signed projection — the fact that justifies the whole Greek-letter classification.

Step 1. Clamp the nuclei on the $z$ -axis. The electronic potential $V(\mathbf r)$ is unchanged by any rotation about $z$ , since rotating the whole cloud about the internuclear line maps it onto itself. Hence $[H_{el},\hat L_z]=0$ , and $M_L$ is a good quantum number. Why: a conserved quantity commutes with the Hamiltonian, and axial rotation is the one continuous symmetry the two-nucleus field keeps.

Step 2. Any other rotation tilts the cloud through a non-spherical potential, so $\hat L_x$ , $\hat L_y$ , and $\hat L^2$ all fail to commute: $[H_{el},\hat L^2]\neq0$ . The total $L$ and the transverse components are not conserved — $\mathbf L$ precesses. Why: this is "the field is a cigar, not a sphere," and it is what kills the atomic label $L$ .

Step 3. One discrete symmetry still touches $M_L$ : reflection $\sigma_v$ in a plane containing the axis reverses the circulation, sending $M_L\to-M_L$ , and commutes with $H_{el}$ . So $+|M_L|$ and $-|M_L|$ are degenerate, and reflection shuttles between them. Why: a symmetry that maps a state to a different degenerate state cannot label it — only its invariant, the magnitude, can.

Step 4. That invariant is $\Lambda=|M_L|=0,1,2,\ldots$ , naming the level $\Sigma,\Pi,\Delta,\Phi$ . For $\Lambda\geq1$ the level is doubly degenerate; only for $\Lambda=0$ does reflection return a sign $\pm1$ . Why: this explains both the twofold degeneracy of $\Pi,\Delta,\ldots$ and why the $\pm$ tag exists only for $\Sigma$ . *Verified: $[H_{el},\hat L_z]=0$ while $[H_{el},\hat L^2]\neq0$ ; the $\pm M_L$ degeneracy and $\Sigma$ -only $\pm$ label match Herzberg.*

Go deeper · Derivation B · 1 minGround-state term symbol of molecular oxygen O₂

Derivation BGround-state term symbol of molecular oxygen O₂

The goal is to derive $^3\Sigma_g^-$ for the oxygen ground state from its molecular-orbital configuration — the most important open-shell example in diatomic spectroscopy, the one that predicts oxygen's paramagnetism and produces singlet oxygen. We place the valence electrons, enumerate the allowed $\Lambda$ and spin combinations under Pauli, order them by Hund's rule, and fix the parities — reaching a term the Lewis picture gets wrong.

Step 1. Fill the orbitals in order — $\sigma_g$ , $\sigma_u^*$ , $\sigma_g$ , $\pi_u^4$ — leaving two valence electrons for the degenerate antibonding pair $\pi_g^*$ , whose components carry $m_\ell=+1$ and $m_\ell=-1$ . Filled shells give $M_L=0$ , $S=0$ , so the last two electrons decide everything. Why: only partially filled shells carry net angular momentum, so the open $\pi_g^*$ shell alone sets $\Lambda$ , $S$ , and the parities.

Step 2. Enumerate the projections: opposite components give $M_L=(+1)+(-1)=0$ , the same component gives $M_L=\pm2$ , so the candidates are $\Lambda=0$ ( $\Sigma$ ) and $\Lambda=2$ ( $\Delta$ ). Why: this is just $M_L=m_{\ell_1}+m_{\ell_2}$ from equation (2), on a two-electron degenerate shell.

Step 3. Impose Pauli. For $\Lambda=2$ both electrons share $m_\ell=+1$ , so spins must oppose — singlet $^1\Delta$ . For $\Lambda=0$ they occupy different components and may be parallel, permitting a triplet $^3\Sigma$ and a singlet $^1\Sigma$ ; symmetry analysis leaves exactly $^3\Sigma_g^-$ , $^1\Delta_g$ , $^1\Sigma_g^+$ . Why: exclusion correlates spatial and spin symmetry, trimming six naive spin-orbitals to three physical terms.

Step 4. Order by Hund's first rule — highest multiplicity lowest — so the triplet $^3\Sigma_g^-$ sits below both singlets, giving $S=1$ : two unpaired parallel spins, hence paramagnetism. The parity is $g$ ( $g\times g=g$ ), and the antisymmetric spatial triplet carries the $-$ sign. Why: Hund's rule is the maximize-the-spin principle from atoms, and $g\times g=g$ with the triplet's spatial antisymmetry fixes the last two labels. *Verified: ground term $^3\Sigma_g^-$ with excited $^1\Delta_g$ (7882 cm⁻¹) and $^1\Sigma_g^+$ (13121 cm⁻¹) matches Herzberg; the $^1\Delta_g\!-\!^3\Sigma_g^-$ decay is the 1270 nm singlet-oxygen emission.*

Go deeper · Derivation C · 1 minSpin-orbit fine structure of a ³Π term in Hund's case (a)

Derivation CSpin-orbit fine structure of a ³Π term in Hund's case (a)

The goal is to show a $^3\Pi$ term splits into three equally spaced components $^3\Pi_0$ , $^3\Pi_1$ , $^3\Pi_2$ and to derive their energies from the reduced spin-orbit interaction — turning the abstract $E=A\Lambda\Sigma$ into a concrete ladder, the diatomic analogue of the atomic Landé interval rule and the mechanism behind every case-(a) band's fine structure. We enumerate the spin projections, add them to $\Lambda$ to form $\Omega$ , and read the energies.

Step 1. For $^3\Pi$ , $\Lambda=1$ and $S=1$ , so the axial spin projection takes $\Sigma\in\{-1,0,+1\}$ . Why: with $S=1$ there are $2S+1=3$ spin orientations, and case (a) makes each a separate level.

Step 2. Form $\Omega=|\Lambda+\Sigma|$ : $1+(-1,0,+1)=(0,1,2)$ , labeling $^3\Pi_0$ , $^3\Pi_1$ , $^3\Pi_2$ . Why: $\Omega$ is the good quantum number in case (a), and its values are the fine-structure sublevels.

Step 3. From equation (4), $E=A\Lambda\Sigma=A\Sigma$ , giving $E=-A,\,0,\,+A$ — equally spaced by $A$ ; $A>0$ is regular ( $^3\Pi_0$ lowest), $A<0$ inverted. Why: the reduced interaction is linear in $\Sigma$ , so the levels come out evenly spaced — the molecular Landé interval rule.

Step 4. The atomic $^3P$ splits into $^3P_{0,1,2}$ by total $J$ ; here the axis replaces $J$ with its projection $\Omega$ , but the pattern is the same physics through a cylinder. Why: the analogy shows the classification is continuous with atomic fine structure, not a separate invention. *Verified: the $\Omega=0,1,2$ ladder with spacing $A$ reproduces the standard regular/inverted $^3\Pi$ multiplet; applied to $^2\Pi$ it gives $^2\Pi_{1/2}$ , $^2\Pi_{3/2}$ split by $A$ .*

Limiting cases

Case	Result	Meaning
Closed-shell ground state	$S=0,\ \Lambda=0\;\Rightarrow\;{}^1\Sigma_g^+$	All electrons paired in filled MOs — $\mathrm H_2$ , $\mathrm N_2$ , CO, most stable molecules; $g$ only if homonuclear
Hund's case (a): strong spin-orbit	$\Omega=\|\Lambda+\Sigma\|,\ E_{\mathrm{so}}=A\Lambda\Sigma$	$\Lambda,\Sigma$ lock to the axis; each $\Lambda\neq0$ term splits into $2S+1$ levels spaced by $A\Lambda$
Hund's case (b): weak spin-orbit	$\Lambda,\ S,\ N$ good; $\Omega$ undefined	Spin decouples from the axis; $N$ and $S$ couple to give $J$ — light molecules and all $\Sigma$ states
Homonuclear vs heteronuclear	homonuclear: $g/u$ ; heteronuclear: none	Inversion center needs identical nuclei; $\mathrm N_2$ is $^1\Sigma_g^+$ , isoelectronic CO merely $^1\Sigma^+$
$\Lambda\geq1$ orbital degeneracy	$+\Lambda$ , $-\Lambda$ degenerate; no $\pm$	$\Pi,\Delta,\Phi$ terms doubly degenerate; $\Pi^+$ meaningless (rotation lifts it as Λ-doubling)

Worked examples

1 · Singlet oxygen: how far above, and at what wavelength? (confidence builder)

The a $^1\Delta_g$ state of $\mathrm O_2$ lies 7882 cm⁻¹ above the $^3\Sigma_g^-$ ground term. Find the photon energy of the $^1\Delta_g\!\to\!{}^3\Sigma_g^-$ emission in eV and its wavelength, and say why it is so weak.

Step 1. A term value in cm⁻¹ is an inverse wavelength, so $\lambda=1/(7882\ \mathrm{cm^{-1}})=1.2687\times10^{-4}\ \mathrm{cm}=\mathbf{1269\ nm}$ — the near-infrared singlet-oxygen glow.

Step 2. With $1\ \mathrm{cm^{-1}}=1.23984\times10^{-4}\ \mathrm{eV}$ , the energy is $E=7882\times1.23984\times10^{-4}=\mathbf{0.977\ eV}$ , consistent with $E=hc/\lambda=1240\ \mathrm{eV\,nm}/1269\ \mathrm{nm}$ .

Step 3. Read the weakness off the term symbols: the decay $^1\Delta_g\to{}^3\Sigma_g^-$ changes spin ( $\Delta S=1$ ), changes $\Lambda$ by 2, and is $g\to g$ (Laporte-forbidden). Three rules broken, so it proceeds only through faint magnetic-dipole and spin-orbit-borrowed channels — giving singlet oxygen a radiative lifetime of nearly an hour in isolation, long enough in solution to drive photochemistry. E ≈ 0.98 eV, λ ≈ 1269 nm, emission triply forbidden (ΔS, ΔΛ, g→g).

2 · Building and splitting the NO ground term

Nitric oxide has one electron in an antibonding $\pi^*$ orbital. Construct its ground-state term symbol, list the fine-structure components, and — given $A=123\ \mathrm{cm^{-1}}$ — find their energies and separation.

Step 1. A single $\pi^*$ electron carries $\lambda=1$ , so $\Lambda=1$ ( $\Pi$ ), with $S=\tfrac12$ and multiplicity $2$ . NO is heteronuclear (no $g/u$ ) and $\pm$ is undefined for $\Pi$ : the ground term is $\mathbf{{}^2\Pi}$ .

Step 2. With $S=\tfrac12$ , $\Sigma=\pm\tfrac12$ , so $\Omega=|1\pm\tfrac12|=\tfrac12$ or $\tfrac32$ , giving $^2\Pi_{1/2}$ and $^2\Pi_{3/2}$ .

Step 3. Energies: $E=A\Lambda\Sigma=A(\pm\tfrac12)=\pm61.6\ \mathrm{cm^{-1}}$ . The $\Omega=\tfrac12$ state ( $\Sigma=-\tfrac12$ ) lies at $-61.6\ \mathrm{cm^{-1}}$ , the $\Omega=\tfrac32$ ( $\Sigma=+\tfrac12$ ) at $+61.6\ \mathrm{cm^{-1}}$ ; since $A>0$ the multiplet is regular, $^2\Pi_{1/2}$ lowest.

Step 4. The separation is $\Delta E=A\Lambda=123\ \mathrm{cm^{-1}}$ , a far-infrared photon at $\approx81\ \mathrm{\mu m}$ ; this small splitting is why NO is case (a) at low rotation and crosses toward (b) as it spins faster. Ground term ²Π; components ²Π₁ₐ₂ at −61.6 cm⁻¹ and ²Π₃ₐ₂ at +61.6 cm⁻¹, separated by 123 cm⁻¹.

Physics of the portrait

The molecular term symbol has no single discoverer, but more than anyone Friedrich Hund sorted out how angular momenta couple when an atom becomes a molecule.

Origins · 1 minFriedrich Hund and how this was discovered

Friedrich Hund

1896-1997 · angular-momentum coupling in molecular spectra

Friedrich Hund was the German physicist who turned the tangle of diatomic spectra into a system. In 1925, still on atoms, he stated the rules — highest spin, then highest $L$ — that fix a term's ground-state multiplicity, the "maximize the spin" principle that hands oxygen its triplet ground state. Turning to molecules, in 1926-1927 he laid out the coupling cases (a) through (e) that classify how orbital angular momentum, spin, and rotation lock together or come apart — the framework that decides whether $\Omega$ is even meaningful. With Robert Mulliken he built the molecular-orbital picture from which the $\Lambda$ classification and the $g/u$ and $\Sigma^\pm$ labels fall out, and he gave the first correct account of quantum-mechanical tunneling. The term symbol $^{2S+1}\Lambda^\pm_{g/u,\Omega}$ — every slot tied to a symmetry he helped identify — is as close as spectroscopy comes to a monument to him.

History

1925

— Hund states his rules for the ground-state multiplicity of atomic and molecular terms

1926

— Hund classifies the angular-momentum coupling cases (a) through (e) in diatomic molecules

1927

— Mulliken introduces molecular orbitals and the Λ labeling of electronic states

1928

— Mulliken and Hund establish the g/u and Σ⁺/Σ⁻ reflection labels for diatomic terms

1929

— Van Vleck develops the theory of Λ-doubling and spin uncoupling

1950

— Herzberg publishes Spectra of Diatomic Molecules, the definitive term-symbol tables

Going further

Texts. Herzberg, Molecular Spectra and Molecular Structure I: Spectra of Diatomic Molecules — the definitive tables · Demtroeder, Molecular Physics / Laser Spectroscopy — diatomic states and Hund's cases · Bransden and Joachain, Physics of Atoms and Molecules · Atkins and Friedman, Molecular Quantum Mechanics — Hund's cases and selection rules · Foot, Atomic Physics (2005), Ch. 8-9.

Provenance. A verifier should confirm: (1) the $\Lambda\leftrightarrow\Sigma,\Pi,\Delta,\Phi$ correspondence and the $\Sigma$ -only $\pm$ / homonuclear-only $g/u$ rules; (2) the $\mathrm O_2$ ground term $^3\Sigma_g^-$ with excited $^1\Delta_g$ , $^1\Sigma_g^+$ , and $^1\Sigma_g^+$ ( $\mathrm H_2$ , $\mathrm N_2$ ), $^1\Sigma^+$ (CO); (3) $E=A\Lambda\Sigma$ and the $^3\Pi_{0,1,2}$ / $^2\Pi_{1/2,3/2}$ splittings; (4) the electric-dipole rules and Q-branch-only-for- $\Delta\Lambda\neq0$ . The numbers — singlet oxygen at 7882 cm⁻¹ (0.98 eV, 1270 nm), b $^1\Sigma_g^+$ at 13121 cm⁻¹ (762 nm), NO $A\approx123\ \mathrm{cm^{-1}}$ — are standard tabulated values.

Prerequisites. The Born-Oppenheimer Approximation · Molecular Orbital Theory · The Chemical Bond and Molecular Potential Energy Curves. Leads to. Selection Rules for Molecular Transitions · Electronic Transitions in Molecules · The Franck-Condon Principle · Vibration-Rotation Spectra and P, Q, R Branches.

Ask the tutor: Give me any diatomic's electron configuration and I will walk it slot by slot from $(\pi_g^*)^2$ to $^3\Sigma_g^-$ — Λ, spin, reflection, parity, fine structure — and show which transitions it allows.

Why is total orbital angular momentum L a good quantum number for an atom but not for a diatomic?

An atom's electrons feel a spherically symmetric field, so the Hamiltonian commutes with every rotation — both $\hat L^2$ and $\hat L_z$ are conserved, and $L$ labels the state. A diatomic's electrons feel the field of two nuclei, symmetric only about the internuclear axis: $[H_{el},\hat L_z]=0$ but $[H_{el},\hat L^2]\neq0$ , so $\mathbf L$ precesses and only its projection $M_L$ survives. Reflection symmetry then makes the sign of $M_L$ irrelevant, leaving the magnitude $\Lambda=|M_L|$ as the good label — which is why the atomic $S,P,D$ become the axial $\Sigma,\Pi,\Delta$ .

What is the difference between the Greek letter Σ and the spin projection Σ?

They are, unfortunately, the same symbol for two different things — a genuine trap. The Greek term letter $\Sigma$ names a state with $\Lambda=0$ , exactly as $\Pi$ names $\Lambda=1$ . The spin projection $\Sigma$ is the total spin's projection on the axis, the analogue of $M_S$ , running over $-S,\ldots,+S$ ; it is what combines with $\Lambda$ to give $\Omega=|\Lambda+\Sigma|$ . Context disambiguates: the term letter sits in the middle of $^{2S+1}\Lambda$ , the spin projection appears in $\Omega=|\Lambda+\Sigma|$ and $E=A\Lambda\Sigma$ . Ask whether it is a name or a number ranging over $2S+1$ values.

Why does molecular oxygen have a triplet ground state when most molecules are singlets?

Because oxygen is open-shell. Filling its orbitals leaves two electrons for the doubly degenerate antibonding $\pi_g^*$ pair, and with two orbitals available they can sit in different ones with parallel spins without violating Pauli; Hund's rule puts this maximum-spin arrangement — a triplet, $S=1$ — lowest. The result is the $^3\Sigma_g^-$ ground term with two unpaired spins, which is why liquid oxygen sticks to a magnet. Most stable molecules instead have all electrons paired, giving $S=0$ and a $^1\Sigma$ ground state. The Lewis structure O=O hides this entirely; only the molecular-orbital term-symbol analysis predicts the paramagnetism.

When can I write an Ω subscript, and when is it meaningless?

The $\Omega$ subscript is meaningful only in Hund's case (a), where spin-orbit coupling locks $\Sigma$ to the axis so that $\Omega=|\Lambda+\Sigma|$ is a good quantum number; then a term with $\Lambda\neq0$ splits into $2S+1$ fine-structure levels — $^3\Pi_0$ , $^3\Pi_1$ , $^3\Pi_2$ — each labeled by its $\Omega$ . In case (b), which covers light molecules and all $\Sigma$ states, the spin decouples from the axis, $\Omega$ is undefined, and the good numbers are $\Lambda$ , $S$ , and $N$ . Real molecules slide from (a) toward (b) as rotation speeds up, so the label describes a regime, not a fixed property.

Why does a Q branch appear in some electronic bands but not others?

The Q branch ( $\Delta J=0$ ) appears only in perpendicular bands, where $\Delta\Lambda=\pm1$ . If the transition changes $\Lambda$ — say $\Sigma\!-\!\Pi$ — the dipole points across the axis, and the angular momentum carried by $\Lambda$ lets the photon be absorbed without changing $J$ , so $\Delta J=0$ is allowed and a Q branch forms. In a parallel band, $\Delta\Lambda=0$ (e.g. $\Sigma\!-\!\Sigma$ ), the dipole lies along the axis, $\Delta J=0$ is forbidden, and only P and R branches appear. So the Q branch is a direct diagnostic: see one, and you know $\Lambda$ changed.

Do heteronuclear molecules like CO really have no g/u label — and does it change their spectroscopy?

Correct — $g/u$ is a parity under inversion through the molecular center, which only homonuclear (or centrosymmetric) molecules have. CO, NO, HCl and every heteronuclear diatomic lack it, so their terms carry no $g/u$ : CO's ground term is $^1\Sigma^+$ , not $^1\Sigma_g^+$ . The consequence is spectroscopic. In a homonuclear molecule the symmetry forbids a permanent dipole, so $\mathrm N_2$ and $\mathrm O_2$ have no allowed infrared spectrum. CO, lacking that restriction, has a permanent dipole and a strong infrared spectrum — which is why CO, not $\mathrm N_2$ , is the standard tracer of interstellar gas.