Molecular Term Symbols

Molecular term symbols are the working vocabulary of combustion diagnostics: the CN violet system ($B\,^2\Sigma^+\to X\,^2\Sigma^+$) and C₂ Swan bands ($d\,^3\Pi_g \to a\,^3\Pi_u$) are routinely used to extract flame temperatures from rotational-line intensity ratios in laser-induced fluorescence and chemiluminescence diagnostics used in engine and jet-combustion research. Atmospheric and photochemical science depend on O₂'s term-symbol structure directly: the metastable $^1\Delta_g$ "singlet oxygen" state, forbidden from radiating back to the $^3\Sigma_g^-$ ground state by both $\Delta S\neq0$ and orbital symmetry, survives long enough to diffuse through tissue and drive photodynamic-therapy cytotoxicity, oxidative damage in photoaged materials, and much of atmospheric ozone photochemistry. Astrochemistry leans on exactly this notation to identify interstellar and cometary radicals from rotationally resolved emission bands — CN, C₂, CH, and CO⁺ term symbols are read directly off telescope spectra to infer chemical composition and excitation temperature in comet comae and diffuse interstellar clouds. In laser physics, the term symbols of N₂ and N₂⁺ excited states ($C\,^3\Pi_u \to B\,^3\Pi_g$) define the gain transition of the nitrogen laser, a workhorse pump source, while excimer lasers (ArF, KrF, XeCl) are built entirely around term-symbol-labeled bound excited states paired with dissociative ground states of noble-gas halides. Finally, cold-molecule and quantum-control experiments — laser cooling of diatomics like SrF and CaF, precision spectroscopy tests of fundamental symmetries — depend on carefully chosen $\Sigma\to\Sigma$ or near-diagonal Franck-Condon-favorable term-symbol transitions to build closed optical cycling loops, linking this labeling scheme directly to the vibronic intensity questions taken up in Electronic Transitions and Franck-Condon and to the intersystem-crossing physics of Fluorescence and Phosphorescence: The Quantum Picture.

Ask the tutor: How would the term symbol and selection rules change if we added spin-orbit coupling explicitly, moving from Hund's case (a) to case (c) for a heavy diatomic like IBr?

An atom lives in a spherically symmetric potential (to a first approximation, the nuclear Coulomb field), so total orbital angular momentum $L$ and total spin $S$ are good quantum numbers, and atomic term symbols $^{2S+1}L_J$ record them. A diatomic molecule's electronic Hamiltonian, by contrast, is symmetric only under rotation about the internuclear axis (call it $z$) plus, for identical nuclei, inversion through the midpoint and reflection in any plane containing the axis. Rotation about an arbitrary axis no longer commutes with the electronic Hamiltonian — only rotation about $z$ does. That collapses $\hat L^2$ as a good observable; only $\hat L_z$ survives, with eigenvalues $\lambda\hbar$, $\lambda = 0,\pm1,\pm2,\dots$ for each electron. Electrons occupying molecular orbitals are classified by $|\lambda|$: a $\sigma$ orbital has $\lambda=0$ (no angular momentum about the axis, cylindrically symmetric electron density), a $\pi$ orbital comes in a degenerate pair with $\lambda = +1$ and $\lambda=-1$ (nodal plane containing the axis), a $\delta$ orbital pair has $\lambda=\pm2$, and so on — precisely the atomic $s,p,d,f$ analogy rotated onto the bond axis. This is the "molecular orbital angular momentum" idea assumed as background here: it is what makes $\pi$ and $\delta$ orbitals doubly degenerate in linear molecules. For a many-electron configuration, the total projection is the algebraic sum $M_L = \sum_i \lambda_i$, and because $+z$ and $-z$ senses of circulation are degenerate in energy (no external field breaks that symmetry), states are labeled by the magnitude

States with $\Lambda = 0,1,2,3$ are called $\Sigma,\Pi,\Delta,\Phi$ respectively — the Greek-letter echo of $S,P,D,F$. All $\Lambda>0$ states are doubly degenerate (clockwise/counterclockwise circulation), just as $\pi$ and $\delta$ orbitals are doubly degenerate. Total spin $S$ is built the same way as in atoms — pairwise vector addition of individual electron spins — and multiplicity $2S+1$ is written as a left superscript, exactly as for atoms. For centrosymmetric molecules (homonuclear diatomics, or more generally any molecule with an inversion center), the electronic wavefunction's behavior under the inversion operator $\hat i$ gives a further label: $g$ (gerade, even, wavefunction unchanged) or $u$ (ungerade, odd, wavefunction sign-flipped). This is a purely orbital-parity story, built by multiplying together the individual-orbital parities. Finally, for $\Sigma$ states only, there is a subtlety with no atomic analog: reflection through any plane containing the internuclear axis ($\hat\sigma_v$) can either leave the total electronic wavefunction unchanged ($\Sigma^+$) or reverse its sign ($\Sigma^-$). This distinction only bites for $\Lambda=0$ states because $\Pi,\Delta,\dots$ states are doubly degenerate and $\hat\sigma_v$ simply maps one member of the degenerate pair into the other (or into a combination), so the $+/-$ label would be ambiguous and is simply not used there. Putting all four pieces together gives the general molecular term symbol format:

with the $g/u$ subscript present only for centrosymmetric species and the $\pm$ superscript present only when $\Lambda=0$.

Configurations built entirely from closed-shell $\sigma$ and $\pi$ orbitals (all $\lambda$ paired to zero) automatically give $\Lambda=0$, $S=0$: a $^1\Sigma$ ground state, the diatomic default. Open-shell configurations — one or two unpaired electrons in $\pi$ or $\delta$ levels — are where the interesting term symbols (and the interesting chemistry, like O₂'s paramagnetism) come from.

The total axial angular-momentum projection is

where: $\lambda_i$ is the signed angular-momentum projection ($0,\pm1,\pm2,\dots$) of electron $i$ in its molecular orbital, and the sum runs over all electrons outside filled, angular-momentum-paired closed shells. Reading: $\Lambda$ behaves exactly like the atomic $L$ but restricted to the one surviving direction (the bond axis); the Greek-letter naming convention is a direct alphabetic transliteration of $S,P,D,F$. Total spin follows ordinary vector-coupling rules,

where: individual electron spins $s_i=\tfrac12$ couple by the same Clebsch–Gordan-type addition used for atomic terms; multiplicity is written as the pre-superscript, e.g. singlet (1), doublet (2), triplet (3). Overall parity for a centrosymmetric species is the product of individual orbital parities,

where: each doubly-occupied orbital contributes $g\times g=g$ or $u \times u=g$ automatically (any orbital squared is gerade), so parity is decided entirely by the unpaired electrons' orbital parities and, subtly, by how many electrons occupy $u$-type orbitals in total. Reading: parity multiplies like a $\mathbb{Z}_2$ charge — filled shells never change it, only the number of electrons in ungerade orbitals (mod 2) matters, since $g$ acts as the identity. The reflection symmetry label for $\Sigma$ states is obtained from the eigenvalue of $\hat\sigma_v$ acting on the total electronic wavefunction,

where: $\hat\sigma_v$ is reflection through a plane containing the internuclear axis; this label exists only for $\Lambda=0$ states because for $\Lambda\neq0$, $\hat\sigma_v$ interconverts the two degenerate components rather than acting diagonally. The electric-dipole selection rules that make term symbols operationally useful are

where: these arise from the vanishing or non-vanishing of the transition-dipole integral $\langle \Psi_f|\hat\mu|\Psi_i\rangle$ under the symmetry operations of the point group; the dipole operator itself transforms as $\sigma_u^+$-like (odd under inversion, $\Delta\Lambda=0,\pm1$ pieces along $z$ and in-plane). Reading: a transition is spectroscopically "allowed" only if every one of these labels is compatible between the two states — this is the direct payoff of building term symbols in the first place, and it is exactly the machinery invoked when analyzing which bands appear in Electronic Transitions and Franck-Condon spectra.

Example 1 — Foundational: term symbol of the CN radical ground state. CN has 13 valence electrons (4 from C, 5 from N valence, wait — count properly: C contributes 4 valence electrons, N contributes 5, total 9 valence electrons; core 1s² on each nucleus are omitted). The valence MO filling order for CN (isoelectronic-ish with N₂⁺ patterns, and experimentally established) is

using 2+2+4+1 = 9 valence electrons. All orbitals except the last are fully paired: $(\sigma_{2s})^2$ contributes $\lambda=0$ paired, $(\sigma_{2s}^*)^2$ contributes $\lambda=0$ paired, $(\pi_{2p})^4$ has all four $\pi$ slots ($\lambda=+1$ up/down, $\lambda=-1$ up/down) filled, so $\sum\lambda = 0$ from that shell too. The single electron in $\sigma_{2p}$ carries $\lambda=0$ and spin $s=\tfrac12$.

CN is heteronuclear (no inversion center), so there is no $g/u$ label. For the $+/-$ label on this $\Sigma$ state: a single electron in a $\sigma$ orbital is symmetric under $\hat\sigma_v$ (a $\sigma$ orbital has no nodal plane containing the axis, cylindrically symmetric), so the overall wavefunction is symmetric,

This matches the spectroscopically established ground term of the CN radical, $X\,^2\Sigma^+$, whose violet-system emission ($B\,^2\Sigma^+ \to X\,^2\Sigma^+$) is a classic combustion and astrochemical diagnostic (it is how comet tails and Bunsen-flame edges are identified spectroscopically). As a limiting-case check: if instead this last electron sat in a $\pi$ orbital (one $\pi$ electron only, half-filled $\pi$ shell), the same bookkeeping would give $\Lambda=1$ (a $^2\Pi$ state, no $+/-$ label needed since $\Lambda\neq0$) — confirming the formula correctly distinguishes shell type, not merely electron count.

Example 2 — Applied: term symbols of the C₂ Swan-band states relevant to combustion diagnostics. The Swan bands of C₂ ($d\,^3\Pi_g \to a\,^3\Pi_u$), central to flame emission spectroscopy and comet-tail chemistry, connect two triplet $\Pi$ states of opposite parity. Build the lower ($a\,^3\Pi_u$) term from the C₂ valence configuration (8 valence electrons, C contributing 4 each):

(This ordering, with $\pi_{2p}$ below $\sigma_{2p}$, is the well-known C₂/B₂/N₂-type "reversed" MO ordering for light homonuclear diatomics, an important instance of the general diatomic MO ordering rules.) The closed shells $(\sigma_{2s})^2(\sigma_{2s}^*)^2$ contribute nothing to $\Lambda$ or $S$. The open shell here has three electrons in $\pi_{2p}$ (out of four slots) plus one in $\sigma_{2p}$. Three electrons in the four $\pi$ slots ($\lambda=+1,+1,-1,-1$ for the four spin-orbitals) is equivalent to one hole in an otherwise filled $\pi^4$ shell: the missing electron determines $\Lambda$ and $S$ contributions just as a single "hole" does in atomic $p^5$ configurations. A single hole in $\pi_{2p}$ gives $\Lambda=1$ from the $\pi$ shell (a $\pi$-hole carries the same $|\lambda|=1$ as a $\pi$-electron) and contributes spin $\tfrac12$. Coupling this with the additional unpaired electron in $\sigma_{2p}$ ($\lambda=0$, spin $\tfrac12$), total

so $\Lambda=1$, triplet: a $^3\Pi$ term. Parity: bonding $\pi_{2p}$ is $u$; a hole in a $u$-type shell still carries net $u$ character in the product sense used for open shells (an odd number of electrons in $u$-type orbitals overall gives $u$); the extra electron in $\sigma_{2p}^{}$ here is gerade in this MO scheme (bonding $\sigma_{2p}$ built from $2p_z$ AOs is $g$). Combining $u$ (from the $\pi$ hole) $\times$ $g$ (from the $\sigma$ electron) $= u$:

matching the spectroscopically assigned metastable $a\,^3\Pi_u$ state of C₂ (long-lived, itself the reservoir state feeding the Swan system, and one of the strongest sources of visible flame and comet emission). The upper state $d\,^3\Pi_g$ arises from promoting an electron so the odd-parity count flips, giving overall $g$ parity while preserving $\Lambda=1,S=1$. The transition $d\,^3\Pi_g \to a\,^3\Pi_u$ obeys $\Delta\Lambda=0$ (allowed, parallel-type in the $\Lambda$ sense though it is actually a $\Pi$–$\Pi$ transition with both perpendicular and parallel components structurally), $\Delta S=0$ (triplet–triplet, allowed), and $g\leftrightarrow u$ (Laporte-allowed) — all selection rules satisfied, consistent with the Swan system being one of the most intense combustion emissions used to estimate flame temperatures.

Example 3 — Exam/competition level: determining all term symbols from an excited O₂ configuration, and checking a non-round-number energy ordering. Consider a hypothetical (but pedagogically standard) exercise: an electron is promoted from $(\pi_{2p})^4$ into $(\sigma_{2p}^{*})^1$ in O₂'s valence framework, giving the excited configuration

(core and $2s$ shells omitted as always, since they are closed and paired). This configuration has three open subshells to handle: a $\pi_{2p}$ hole (3 of 4 slots filled), a full $\pi_{2p}^{*2}$ (which by Example-1-style analysis of Section "core ideas" derivation gives its own sub-terms $^1\Delta,\ ^3\Sigma^-,\ ^1\Sigma^+$ exactly as derived for ground-state O₂), and a single $\sigma_{2p}^*$ electron. Focus on just the spin/orbital bookkeeping for the overall lowest-energy term (Hund's-rule ground member of this excited manifold), since a full multi-term enumeration is a much longer group-theoretic reduction. Take the $\pi_{2p}$ hole (contributes $|\lambda|=1$, spin $\tfrac12$) coupled with the $\pi_{2p}^{*2}$ pair in its own triplet-$\Sigma^-$ substate ($\Lambda=0$ contribution, $S=1$ from that pair) and the lone $\sigma_{2p}^*$ electron ($\lambda=0$, spin $\tfrac12$). Coupling orbital parts,

and coupling all spins for maximum multiplicity (Hund's rule favors the highest achievable $S$ among these three unpaired-spin carriers: the hole spin $\tfrac12$, the $\pi^{*2}$ pair's triplet spin $1$, and the lone $\sigma^*$ spin $\tfrac12$),

This gives a $^5\Pi$ state as the highest-multiplicity term reachable from this configuration — a useful check because it demonstrates that promoting an electron out of a closed shell can, via Hund's-rule spin-stacking across independently-open subshells, generate multiplicities well beyond simple doublet/triplet expectations, exactly the kind of non-obvious, non-round-number result (quintet, not the "expected" triplet or singlet) that a rigorous subshell-by-subshell derivation — rather than a memorized shortcut — is needed to catch. Parity follows the same product rule: $\pi_{2p}$ hole is $u$-type (bonding $\pi$ is $u$), $\pi_{2p}^{*2}$ pair is $g\times g=g$ (gerade squared), $\sigma_{2p}^*$ single electron is $u$-type (antibonding $\sigma$ is $u$); total parity $u\times g\times u = g$. So this particular high-spin term is $^5\Pi_g$ — an experimentally real, if weakly populated, excited state family in the O₂ manifold above the well-known $^3\Sigma_g^-, {}^1\Delta_g, {}^1\Sigma_g^+$ triad.

The formal grammar of molecular term symbols was assembled through the joint efforts of molecular spectroscopists who translated Russell–Saunders atomic coupling into the reduced symmetry of linear molecules; the individual most closely tied to systematizing diatomic electronic-state notation and its use in band-spectra analysis is Robert S. Mulliken, whose extensive 1920s–1930s papers on molecular orbital theory and electronic states of diatomic molecules established the Greek-letter labeling scheme, the $g/u$ and $+/-$ conventions, and the selection rules connecting them to observed band systems.