Diatomic Molecular Term Symbols

Introduction

Atomic term symbols use total orbital L because an isolated atom is spherically symmetric. A diatomic molecule has only cylindrical symmetry about its internuclear axis, so total orbital angular momentum perpendicular to that axis is generally not a good conserved label. Diatomic state symbols instead use the projection Λ along the bond. Their superscripts and subscripts also encode spin, reflection and inversion behaviour. These are labels of the whole electronic state, not of one orbital taken in isolation.

Core explanation

For the many-electron electronic state, Λ is the magnitude of the sum of signed orbital-angular-momentum projections along the internuclear axis. Λ=0 is written Σ, Λ=1 is Π, Λ=2 is Δ and Λ=3 is Φ. The Greek capital letters distinguish state terms from lower-case one-electron orbital labels σ, π and δ. Occupation of a π orbital does not automatically make the total state Π: two electrons in π functions can have opposite signed projections that add to Λ=0.

The left superscript 2S+1 gives spin multiplicity, with S the total electronic spin. A state with S=0 is singlet, S=1/2 doublet and S=1 triplet. For a closed-shell diatomic ground state with all electrons paired, S=0 and Λ often equals zero, yielding a ^1Σ-type term. Open-shell configurations can yield multiple terms because electron spins and signed orbital projections can combine in different allowed ways subject to Pauli exclusion.

A Σ state can carry a superscript + or − according to its behaviour under reflection in a plane containing the internuclear axis. A plus state is unchanged; a minus state changes sign. For Λ>0, reflection generally interchanges +Λ and −Λ partner functions, so the simple Σ-style reflection sign is not assigned in the same way to a Π or Δ term. This ± sign is not electric charge or bonding character.

If the diatomic molecule is homonuclear and centrosymmetric, the total electronic state also has inversion parity g or u. The parity of a simple product of occupied orbital factors follows the product of their parities, but antisymmetry and a full state construction must be respected. Heteronuclear molecules lack an exact inversion centre, so their terms do not carry g/u as exact molecular symmetry labels. A complete weak-coupling symbol may look like ^3Σ g^−, with spin multiplicity, axial Λ, inversion parity and reflection behaviour all specified.

Electronic spin also has a projection Σ spin along the molecular axis. When spin–orbit coupling is important, Ω = Λ + Σ spin is often used as a more appropriate state label in Hund's case descriptions. The notation Σ for a Λ=0 term and Σ spin for spin projection are related but distinct quantities. The degree to which Λ, S and Ω are individually useful depends on coupling strength and molecular rotation. A term symbol is an organised approximation to state symmetry, not a replacement for the actual Hamiltonian.

As a simple case, H₂⁺ has one electron in a bonding σg orbital in its ground electronic state. One electron gives S=1/2 and multiplicity 2; the σ orbital has zero axial projection, giving a Σ term. Inversion is g and reflection in a plane containing the axis leaves the usual σ state unchanged, so its idealised term is ^2Σ g^+. Neutral H₂'s closed-shell ground state is ^1Σ g^+. These examples connect orbital labels to full-state labels only because their electron arrangements are especially simple.

Term symbols guide selection rules, but a transition requires checking the full set of angular, spin, inversion and reflection conditions appropriate to the radiation interaction. A g-to-u change is necessary for an ideal electric-dipole transition in a centrosymmetric diatomic but not sufficient for every line to be allowed. Rotational and vibrational structure adds further quantum numbers not shown in the simple electronic term.

Step-by-step reasoning

Determine the total electronic configuration and couple electron spins to S. Sum signed one-electron axial projections to find possible Λ values, respecting Pauli restrictions. For Λ=0, assign reflection + or − by transforming the whole state. Add g/u only for a homonuclear centrosymmetric molecule. If spin–orbit coupling is relevant, examine Ω and state the coupling approximation.

Visual explanation

Draw a horizontal internuclear axis. Put two small arrows circling around it in opposite senses to represent +λ and −λ orbital projections. If they cancel, label the state Σ; if one unit remains, label Π. Beside that show spin arrows combining to singlet or triplet multiplicity, and a midpoint inversion arrow for a homonuclear molecule.

Real-world analogy

Several rowers can contribute clockwise and counterclockwise turning tendencies; the net tendency may cancel even though individual rowers are active. Similarly, electrons occupying π orbitals can produce a total Σ state if their signed projections cancel. The analogy is limited because quantum angular momenta and antisymmetrised states follow discrete coupling rules.

Real-world example

The ground electronic state of O₂ is commonly labelled X ^3Σ g^−. The triplet multiplicity is consistent with its observed paramagnetism, while the additional symbols specify axial, inversion and reflection properties. One must derive this state from the open-shell electronic configuration and Pauli constraints; simply seeing π orbitals in an MO diagram is not enough to name the term.

Why?

Why replace atomic L with molecular Λ? The diatomic nuclear framework selects one axis and breaks full spherical rotational symmetry. Only the component of orbital angular momentum along that axis has the appropriate symmetry status in a simple electronic model. A state with total atomic-like L need not retain that value as a good quantum number once two nuclei define a direction.

Common misconception

The lower-case orbital π and upper-case state Π are not interchangeable. A state may contain electrons in π orbitals yet have Λ=0 and a Σ label. The superscript + or − on a Σ state denotes reflection behaviour, not positive or negative electrical charge, and g/u is valid only when inversion is a molecular symmetry.

Worked example

For the one-electron H₂⁺ ground configuration σg¹, spin is S=1/2, so 2S+1=2. The occupied orbital has λ=0, giving Λ=0 and a Σ state. The whole state is even under inversion and even under reflection in a plane containing the bond. Combining the labels gives ^2Σ g^+. This simple derivation would need extra coupling and antisymmetry analysis for a multi-open-shell configuration.

Quick check

1. Can two occupied π orbitals produce a Σ electronic state? Answer: Yes. Their signed axial projections can cancel, leaving total Λ=0. 2. When is a g/u state label valid? Answer: When inversion through the molecular centre is a valid symmetry, as in an ideal homonuclear diatomic.

Exam focus

Separate orbital and state notation and distinguish spin multiplicity from reflection signs. State the molecule's symmetry before assigning g/u. For open shells, couple electron angular momenta and spins under Pauli constraints rather than reading a term directly from one orbital label.

Advanced insight

Molecular rotation and spin–orbit interaction lead to different Hund coupling cases. In a case where spin is strongly tied to the internuclear axis, Ω is especially useful; in weaker coupling, Λ and S can remain more transparent labels. Spectroscopic line patterns can reveal which coupling description works best, so term symbols are part of a model tested against observation.

Summary

Diatomic term symbols classify many-electron states by spin multiplicity and total axial orbital projection Λ, giving Σ, Π, Δ and higher labels. Σ states can have reflection + or −, while centrosymmetric homonuclear states also have g/u parity. Ω can describe spin–orbit-coupled projection. These labels must be derived for the whole state, not copied from one occupied orbital.

Practice questions

1. Why is a heteronuclear AB diatomic's electronic state not given an exact g/u label? Answer: Inversion through the bond midpoint exchanges unlike nuclei, so the molecular Hamiltonian lacks inversion symmetry and parity is not an exact state label. 2. A state has total S=1 and Λ=1. What are its multiplicity and Greek-letter term core? Answer: Multiplicity is 2S+1=3, and Λ=1 gives Π, so its core notation is ^3Π before any further labels. 3. What does the minus sign in ^3Σ g^− describe? Answer: It means the Σ electronic state changes sign under reflection through a plane containing the internuclear axis; it is neither electrical charge nor an antibonding star.