Mulliken Symbols and Basis Functions
Look at any character table and you meet a strange alphabet: A₁, A₂, B₁, B₂, E, T₁, T₂. These are sometimes decorated with primes, double primes, and subscripts g or u. These are not arbitrary labels. They are a complete shorthand for how an orbital, vibration, or wavefunction behaves when a symmetry operation acts on it. Every letter, every subscript, every symbol encodes a specific transformation property. Once you can read that alphabet fluently, you can predict certain things without drawing a single picture. You can tell which d-orbital mixes with which ligand group orbital. You can tell which vibrational mode absorbs infrared light. You can also tell which electronic transition is symmetry-forbidden. This matters because the entire machinery of applying group theory to chemistry depends on one key step. That machinery includes building Symmetry-Adapted Linear Combinations, constructing MO Diagrams from Symmetry, assigning IR and Raman bands, and working out selection rules for UV-vis spectra. All of it depends on first correctly identifying which irreducible representation (irrep) a given atomic orbital, bond vector, or normal mode belongs to. That identification is done by reading off Mulliken symbols. You must match basis functions listed in the right-hand columns of a character table. Get the symbol wrong and the whole downstream analysis collapses. Get it right, and enormous chemical insight falls out almost mechanically. The system is due to Robert S. Mulliken. In 1933 he formalized a labeling convention that chemists still use today essentially unchanged. The letters A, B, E, T (and F in some older texts, G, H for icosahedral groups) tell you the dimension of the representation. This dimension shows how many degenerate partners travel together. The subscripts 1, 2, g, u, prime, and double-prime tell you the symmetry behavior. This behavior is relative to specific reference operations: rotation about the principal axis, the center of inversion, and reflection planes. Alongside each irrep, character tables list "basis functions." These are linear, quadratic, and sometimes cubic functions of x, y, z that transform according to that irrep. These functions are your Rosetta Stone. They connect an abstract label to a concrete atomic orbital. This article works through the labeling grammar systematically. It covers what A versus B means, and what the numerical subscripts mean. It also covers what g/u and prime/double-prime mean. Finally, it shows exactly how s, p, d, and f orbitals map onto these labels in each common point group. Building on the Anatomy of Character Tables, we derive the assignment rules from first principles. These principles include character under the identity, character under the principal rotation, and character under σₕ and i. We avoid asking you to memorize tables. Instead, we work three complete examples so the logic becomes second nature.
- A and B label one-dimensional irreps: A is symmetric (character +1) under the principal Cₙ rotation, B is antisymmetric (character −1).
- E labels two-dimensional (doubly degenerate) irreps; T (or F) labels three-dimensional (triply degenerate) irreps — the dimension equals χ(E).
- Subscripts 1/2 distinguish behavior under a C₂ perpendicular to the principal axis or under a vertical mirror plane σᵥ; subscript g/u (gerade/ungerade) records symmetry under inversion i; prime/double-prime records symmetry under σₕ.
- Basis functions listed beside each irrep — x, y, z (linear) and functions like z², x²−y², xy, xz, yz (quadratic) — tell you directly which p- and d-orbitals transform as which irrep.
- s orbitals are always totally symmetric (the A₁ or A₁g irrep); p orbitals transform as (x, y, z); d orbitals split according to how xy, xz, yz, x²−y², z² transform in that specific point group.
- Key idea: a Mulliken symbol is a compressed statement of a full row of characters, and matching an orbital's basis function to that row tells you the orbital's irreducible representation without ever drawing the molecule.
Where this lives today
Mulliken symbols remain the working vocabulary of every symmetry-based spectroscopic technique in modern chemistry. Infrared and Raman spectroscopists routinely predict, before ever running an instrument, which vibrational modes of a new coordination complex or organometallic catalyst will appear in which spectrum, purely by decomposing the reducible representation of atomic motions into irreps and checking which ones match linear (IR-active) versus quadratic (Raman-active) basis functions — this is standard practice in characterizing everything from metal carbonyl clusters (where the number and symmetry of CO stretching bands directly reveals the coordination geometry) to catalytic intermediates observed by operando IR. In electronic spectroscopy and photophysics, Mulliken labeling underlies selection rules used to interpret UV-visible d–d and charge-transfer bands in transition-metal complexes, explaining why octahedral d–d transitions (g→g) are formally Laporte-forbidden and hence weak, while tetrahedral complexes (which lack an inversion center entirely) show much more intense d–d absorption — a rule of thumb every inorganic chemist uses to distinguish coordination geometries by color intensity alone. The same g/u bookkeeping governs the design of luminescent lanthanide and organometallic phosphors, where symmetry lowering (breaking the inversion center via an asymmetric ligand field) is deliberately engineered to relax Laporte forbiddenness and boost emission intensity. In materials science, the Eg/T₂g crystal-field splitting derived directly from Oₕ Mulliken labels explains magnetic and optical properties of oxide materials from ruby (Cr³⁺ in Al₂O₃) to colossal-magnetoresistance manganites, and the same logic (applied to D₄ₕ or D₃d subgroups) rationalizes Jahn–Teller distortions in copper(II) complexes central to bioinorganic active sites like plastocyanin and superoxide dismutase. Computational chemistry software (Gaussian, ORCA, VASP) prints Mulliken symmetry labels directly alongside computed molecular orbital energies, letting researchers assign frontier orbitals and predict reactivity, spectroscopic activity, and photochemical pathways without ever having to hand-derive the representation from scratch — the labeling convention from 1933 is, quite literally, still in the output files of every modern quantum chemistry calculation run today.
Ask the tutor: How do I build the reducible representation for a whole set of ligand orbitals (not just one atomic orbital) and reduce it to find which SALCs match the metal d-orbitals' Mulliken symbols?
The physical picture
A character table is a lookup device, and reading it correctly requires understanding that each row is simultaneously (i) a set of numbers (the characters), (ii) a compact name (the Mulliken symbol), and (iii) a list of functions that realize that behavior physically (the basis functions). The symbol is essentially mnemonic shorthand for the characters, and the basis functions are the bridge from abstract representation theory to actual atomic orbitals sitting on actual atoms. Start with dimension. The character under the identity operation E, written χ(E), always equals the dimension of the representation — the number of independent basis functions that mix among themselves under the group's operations. If χ(E) = 1, a single function transforms into ± itself under every operation, and the label is A or B. If χ(E) = 2, two functions transform into linear combinations of each other (never separately), and the label is E. If χ(E) = 3, three functions travel together, and the label is T (older texts write F). Icosahedral and higher-symmetry groups can have χ(E) = 4 or 5, labeled G and H, but these appear only in exotic contexts like fullerene photophysics. Within the one-dimensional world, A and B distinguish behavior under the principal rotation axis Cₙ. If the character under Cₙ is +1 (the function returns to itself after that rotation), the label is A. If the character under Cₙ is −1 (the function flips sign), the label is B. This is why in C₄ᵥ, for instance, you find A₁, A₂, B₁, B₂ — all four are one-dimensional, but A₁/A₂ have χ(C₄) = +1 while B₁/B₂ have χ(C₄) = −1. The numerical subscripts 1 and 2 resolve the further ambiguity within the A or B family. In groups with a C₂ axis perpendicular to the principal axis, subscript 1 means symmetric (χ = +1) under that C₂, subscript 2 means antisymmetric (χ = −1). In groups without perpendicular C₂ axes but with vertical mirror planes σᵥ, the same subscript convention applies to behavior under σᵥ instead. This is a genuinely important subtlety: which operation the subscript refers to depends on the group, so you read the convention off the character table itself, not from memory of a "universal" rule. The letters g and u — gerade (German: even) and ungerade (odd) — apply only in groups that possess a center of inversion i, such as D∞ₕ, Oₕ, D₄ₕ, and D₆ₕ. A representation is g if χ(i) = +1 (the function is unchanged by inversion through the center) and u if χ(i) = −1 (the function changes sign). This label is central to selection rules: the Laporte rule for centrosymmetric molecules forbids g→g and u→u electric-dipole transitions, permitting only g→u. Primes and double-primes appear in groups with a horizontal mirror plane σₕ but no inversion center, most commonly Cₛ, C₃ₕ, D₃ₕ. A′ (prime) means χ(σₕ) = +1; A″ (double prime) means χ(σₕ) = −1. Physically, prime representations describe functions symmetric with respect to reflection through the molecular plane (in-plane behavior), while double-prime representations describe functions antisymmetric to that reflection (out-of-plane, p_z-like behavior in planar molecules).
Basis functions are the payoff of all this bookkeeping. Once you know a function's transformation behavior — the row of characters — you can ask "which irrep does this row match?" and read the Mulliken symbol straight off the table. Character tables save you the trouble by listing, in a column usually to the right of the characters, the common linear (x, y, z), quadratic (x², y², z², xy, xz, yz, x²−y²), and sometimes cubic or higher combinations that transform as each irrep. Because atomic p orbitals transform exactly like the coordinates x, y, z, and d orbitals transform exactly like the quadratic combinations, this column is a direct dictionary from orbital shape to symmetry label — no further calculation needed once you trust the table.
This last figure is worth dwelling on because it is precisely the group-theoretic backbone of crystal field theory: the familiar eg/t2g splitting of d-orbitals in an octahedral field is nothing but the statement that the five d-orbitals span a reducible 5-dimensional representation of Oₕ that decomposes into Eg ⊕ T₂g. The letters and subscripts are not decoration; they are the entire content of the splitting pattern.
The core ideas
The reduction of a Mulliken symbol to a definite meaning rests on four defining relationships, each tying a symbol feature to a specific character.
where: χ(E) is the character of the representation Γ under the identity operation, and the dimension equals the number of basis functions that transform together (mixing among each other but not with functions outside the set).
Reading: χ(E) is always the largest number in any row of a character table for a real chemical group, because every other operation can only permute or partially cancel components, never increase the trace above the dimension.
where: Cₙ is the principal rotation axis that defines the point group (the axis of highest order), and χ(Cₙ) is the corresponding character for one-dimensional representations only.
Reading: A representation is A if it is symmetric under the defining rotation, B if it flips sign every time you apply that rotation — geometrically, an A function looks the same after rotating by 360°/n, while a B function looks inverted.
where: C₂⊥ is a C₂ axis perpendicular to the principal axis (used when such axes exist, e.g. D-type groups), and σᵥ is a vertical mirror plane (used in C-type groups lacking perpendicular C₂ axes).
Reading: the subscript 1 vs 2 resolves a residual ambiguity that the Cₙ character alone cannot: two different one-dimensional irreps can both be "A" (symmetric under Cₙ) yet differ in how they behave under a secondary symmetry element.
where: i is the inversion operation (present only in centrosymmetric groups) and σₕ is the horizontal mirror plane (present in groups with a σₕ but no i, such as Cₙₕ or Dₙₕ with odd n in certain cases, and simple Cₛ).
Reading: g/u is a parity label under point inversion through the molecular center; prime/double-prime is a parity label under reflection through a horizontal plane — a molecule cannot carry both labeling schemes simultaneously in the same table, because a group either has i (uses g/u) or has σₕ without i (uses ′/″), never a redundant mixture in the same symbol.
Goal. Show explicitly, using only the reduction formula and the Oₕ character table, why the five d orbitals split into the Eg and T₂g labels seen in every crystal-field diagram — deriving the labels rather than quoting them.
Step 1 — Set up the reducible representation of the d-orbital basis. Treat the five d orbitals (dz², dx²−y², dxy, dxz, dyz) as a basis set and find the character each Oₕ operation produces by counting how many of the five orbitals map into themselves (with sign) under that operation, using the standard formula for the trace of a d-orbital transformation under a rotation by angle θ:
This is the general formula for l = 2 angular momentum functions under a proper rotation by θ (with χ(i) = +1 always for d-orbitals since they are even functions, and improper rotations handled via Sₙ = σₕ·Cₙ combined with parity).
Step 2 — Evaluate at each class of operation in Oₕ. The classes are E, 8C₃, 6C₂, 6C₄, 3C₂(=C₄²), i, 6S₄, 8S₆, 3σₕ, 6σd. Using θ = 0° for E, 120° for C₃, 180° for C₂, 90° for C₄:
Parity makes all g-type: χ(i) = +5 for these even functions times the parity factor appropriate to l=2 gives χ(i)=5·(+1)=5 is wrong for a reducible trace under inversion of individual functions — correctly, since each d-function is unchanged in sign under inversion (x→−x, y→−y, z→−z leaves products like xy, x², z² invariant), χ(i) = +5. Continuing with the improper operations using the general formula for Sₙ acting on l=2 functions:
Step 3 — Reduce using the reduction formula. The number of times irrep Γᵢ appears is
where h = 48 is the order of Oₕ and g_R is the size of each class. Carrying out this sum against the Eg row (characters 2, −1, 2, 0, 0, 2, 0, −1, 2, 0) gives n(Eg) = 1, and against the T₂g row (characters 3, 0, −1, −1, 1, 3, −1, 0, −1, 1) gives n(T₂g) = 1. All other irreps give zero.
Step 4 — Assign basis functions to confirm the labels. The Oₕ table lists (2z²−x²−y², x²−y²) under Eg and (xy, xz, yz) under T₂g. These are exactly the five d-orbital combinations, split 2 + 3 = 5, matching the crystal-field result that the Eg pair (dz², dx²−y²) points directly at octahedral ligands (higher energy) while the T₂g trio (dxy, dxz, dyz) points between them (lower energy). The Mulliken symbols therefore are the crystal-field splitting pattern, derived purely from group theory with no electrostatic calculation required.
Go deeper · Derivation B · 2 minWhy p orbitals are always T1u (or A2u + Eu, or T2, etc.) depending only on the point group — the general rule for linear functions
Goal. Establish the general principle that (x, y, z) transform as a specific well-defined representation in every point group, and show how the same three functions split differently depending on symmetry, using D₄ₕ (where z is distinguished from x, y) as the worked case.
Step 1 — Recognize that z transforms differently from (x, y) whenever there is a unique principal axis. In any group with a Cₙ (n ≥ 3) principal axis chosen along z, the coordinate z is invariant in direction under every rotation about that axis (it may only change sign under σₕ or C₂⊥, or under S_n improper rotations), so z by itself spans a one-dimensional representation. Meanwhile x and y rotate into linear combinations of each other under Cₙ, so (x, y) together span a two-dimensional representation (unless the group has no true 2D irreps, e.g. purely abelian groups like C₂ᵥ, D₂ₕ, in which case x and y each become separate one-dimensional irreps).
Step 2 — Apply this to D₄ₕ specifically. Look up (or derive) the characters of z under the D₄ₕ operations E, 2C₄, C₂, 2C₂′, 2C₂″, i, 2S₄, σₕ, 2σᵥ, 2σd:
This row matches exactly the A₂ᵤ row in the D₄ₕ table. So z transforms as A₂ᵤ — antisymmetric under inversion (u, as expected for a linear odd function) and symmetric under C₄ (character +1, hence "A" not "B"), but antisymmetric under C₂′ (hence subscript 2, not 1).
Step 3 — Determine the representation of (x, y) together. Because x and y mix into each other under C₄ (a 90° rotation sends x → y, y → −x, which cannot be represented by two separate 1×1 characters consistently across the whole group), (x, y) must span a genuine 2-dimensional irrep. Computing the trace of the 2×2 rotation/reflection matrices for each class gives
matching the Eᵤ row of D₄ₕ.
Step 4 — State the general result and its limiting case. In D₄ₕ, therefore, (x, y, z) split as A₂ᵤ ⊕ Eᵤ, not as a single three-dimensional T-type irrep — because D₄ₕ is a lower-symmetry group than Oₕ or Tᵈ and does not treat all three Cartesian directions equivalently. This is the general rule: (x, y, z) transform as T₁ᵤ (Oₕ) or T₂ (Tᵈ) only in cubic/octahedral/tetrahedral groups where x, y, z are symmetry-equivalent; in any group with a unique principal axis of order 3 or higher but lower overall symmetry (Cₙᵥ, Dₙₕ, Dₙ, Sₙ), z separates from the (x, y) pair. As a limiting-case check, setting n → ∞ (the point group C∞ᵥ for a linear molecule like HCl or D∞ₕ for CO₂) reproduces the familiar spectroscopic result directly: z transforms as A₁ (Σ⁺) or A₂ᵤ (Σᵤ⁺), while (x, y) transforms as the doubly-degenerate E₁ (Π) representation — exactly the σ/π orbital classification used throughout diatomic and linear-molecule spectroscopy.
Worked examples
Example 1 (Foundational): Assigning Mulliken symbols to the C₂ᵥ character table by hand. C₂ᵥ has four one-dimensional irreps and operations E, C₂, σᵥ(xz), σᵥ′(yz). Consider the row with characters (1, 1, 1, 1). Since χ(E) = 1, this is one-dimensional: A or B. Since χ(C₂) = +1, it is A (not B). There is no perpendicular C₂ in C₂ᵥ, so the 1/2 subscript in this group is assigned by behavior under σᵥ(xz): here χ(σᵥ) = +1, so subscript 1. This row is A₁, and indeed the character table lists basis functions z, x², y², z² alongside it — the totally symmetric functions, matching physical intuition that z (along the C₂ axis) and any purely quadratic combination should be invariant under every operation of C₂ᵥ. Now take the row (1, 1, −1, −1). One-dimensional (χ(E)=1), symmetric under C₂ (χ(C₂)=+1) so "A," but antisymmetric under σᵥ(xz) (χ=−1) so subscript 2. This is A₂, and the basis function listed is xy (and sometimes Rz, the rotation about z) — no linear x, y, or z function transforms this way, which is chemically important: A₂ is IR-inactive in C₂ᵥ because no dipole component (x, y, or z) has A₂ symmetry, but it can be Raman-active because xy does appear. Take the row (1, −1, 1, −1): one-dimensional, antisymmetric under C₂ (χ=−1) so "B," and its behavior under σᵥ(xz) is +1, giving subscript 1: this is B₁, with basis function x (and xz). Finally (1, −1, −1, 1) is B₂, basis function y (and yz). Notice the elegant closure: between A₁, A₂, B₁, B₂ the three functions x, y, z are fully accounted for (x→B₁, y→B₂, z→A₁), and all six independent quadratics are likewise distributed (x², y², z²→A₁; xy→A₂; xz→B₁; yz→B₂) — every possible orbital shape in a C₂ᵥ molecule (like H₂O or SO₂) has a definite, derivable home.
Example 2 (Applied): Symmetry labels of the frontier orbitals in trans-N₂F₂ (C₂ₕ) and prediction of IR activity. trans-N₂F₂ (or any planar trans-disubstituted diazene) belongs to C₂ₕ, with operations E, C₂, i, σₕ. This group is centrosymmetric, so labels use g/u, and since there is only one nontrivial rotation axis (order 2) and no perpendicular C₂ or σᵥ, the only distinguishing subscript beyond g/u is nothing further — C₂ₕ has just four one-dimensional irreps: Ag, Bg, Au, Bu. Assign: χ(E)=1 always. For Ag: χ(C₂)=+1, χ(i)=+1 (gerade). Basis functions: x², y², z², xy. For Bg: χ(C₂)=−1, χ(i)=+1. Basis functions: xz, yz. For Au: χ(C₂)=+1, χ(i)=−1. Basis function: z. For Bu: χ(C₂)=−1, χ(i)=−1. Basis functions: x, y. Now apply the Laporte/parity selection rule directly from these labels. Infrared activity requires the vibrational mode to transform as x, y, or z (so that it can couple to the oscillating electric field of light, which is a vector, transforming as x/y/z). From the table above, only Au (matches z) and Bu (matches x, y) carry linear basis functions — both are ungerade. Any vibration of Ag or Bg symmetry, though it may be perfectly real and physically present, is IR-silent because no g-type function can couple to the dipole operator (which is itself ungerade, being linear in x, y, z). Conversely, Raman activity requires matching a quadratic function (x², xy, etc.), and only Ag and Bg carry those — so Ag/Bg modes are Raman-active and IR-inactive, while Au/Bu modes are IR-active and Raman-inactive. This mutual exclusion (a mode active in IR cannot be active in Raman, and vice versa) is a direct, exact consequence of the centrosymmetric g/u labeling and holds for every molecule with an inversion center — N₂O₄, ethylene's out-of-plane modes in its D₂ₕ form, benzene's ring-breathing modes, and octahedral MX₆ complexes all obey it for exactly this group-theoretic reason.
Example 3 (Exam/competition level): Determining the correct Mulliken symbol and orbital assignment for the metal dz² orbital in a square-planar D₄ₕ complex, and predicting its spectrochemical role. Consider [PtCl₄]²⁻, a square-planar complex of D₄ₕ symmetry, with the C₄ axis (and the unique z direction) perpendicular to the molecular plane. We are asked: what Mulliken symbol does the metal dz² orbital carry, and how does this differ from its behavior in an octahedral field?
Step 1: Recall (or re-derive) the transformation of dz² under D₄ₕ operations. The function z² depends only on z, and under any rotation about z (C₄, C₂), z itself is unchanged (θ-independent for rotations about its own axis), so z² is invariant: χ(C₄) = +1, χ(C₂) = +1. Under C₂′ (a twofold axis in the xy-plane, along x or y), z → −z, but z² → (−z)² = +z², so χ(C₂′) = +1 as well. Under σₕ (the xy-plane), z → −z again, and z² → +z², so χ(σₕ) = +1. Under i, z → −z, again z² unaffected, χ(i) = +1. This makes z² symmetric under literally every operation of D₄ₕ.
Step 2: Match to the D₄ₕ character table. The all-+1 row (with χ(E)=1) is by definition the totally symmetric irrep A₁g. This is confirmed by consulting the basis-function column: D₄ₕ tables list z² (often combined as "z², x²+y²") explicitly under A₁g.
Step 3: Compare with dx²−y². This function is not invariant under C₄: rotating 90° sends x → y, y → −x (or similar, depending on convention), so x² − y² → y² − (−x)² = y² − x² = −(x² − y²). Hence χ(C₄) = −1. Under C₂ (180°) it is invariant (χ=+1), under σₕ invariant (χ=+1), under i invariant (χ=+1, quadratics are always gerade). This matches the row (1, −1, 1, 1, −1, 1, −1, 1, 1, −1) — i.e., B₁g in the standard D₄ₕ table, with explicit basis function x² − y².
Step 4: Chemical consequence. In octahedral symmetry, dz² and dx²−y² are degenerate partners inside the single Eg irrep — group theory requires them to have identical energy in a perfectly octahedral field. The moment the complex distorts to square planar (a common consequence of a strong Jahn–Teller distortion or of d⁸ metal centers like Pt(II), Pd(II), Au(III) preferring 4-coordination), the symmetry drops from Oₕ to D₄ₕ, and the correlation table shows Eg(Oₕ) → A₁g ⊕ B₁g(D₄ₕ). The former degenerate pair splits into two non-degenerate orbitals of different energy: dx²−y² (B₁g), pointing directly at the four in-plane ligands, rises sharply in energy (strong σ-antibonding overlap), while dz² (A₁g), pointing at the now-vacant axial positions, drops substantially in energy. This splitting pattern — universally drawn in every inorganic chemistry course for square-planar d⁸ complexes — is entirely a consequence of tracking which Mulliken symbol the dz² and dx²−y² basis functions carry as the symmetry is lowered from Oₕ to D₄ₕ, with no further calculation needed beyond the correlation table.
Limiting cases, scope & common traps
| Situation | What governs the label | Common error |
|---|---|---|
| Group has no perpendicular C₂ or σᵥ (e.g., C₃, C₄) | Subscripts 1/2 do not appear at all; only A, B (and possibly complex-conjugate E pairs) | Trying to force a "1/2" subscript where the group genuinely has none |
| Centrosymmetric group (has i) | Use g/u exclusively | Using primes (′/″) in a group that has i — primes belong only to groups with σₕ but no i |
| Group has σₕ but no i (Cₛ, C₃ₕ, D₃ₕ) | Use ′/″ exclusively | Confusing ′ (symmetric, in-plane-like) with g, or ″ with u — they are different operations (σₕ vs i) and not interchangeable labels |
| Doubly/triply degenerate irreps (E, T) | χ(E) alone fixes dimension; never split an E or T pair into two separate one-dimensional labels | Writing "E₁, E₂" as if they were independent A-type levels; E is a single irreducible 2D block whose components cannot be labeled or split further within the group |
| Linear molecules (C∞ᵥ, D∞ₕ) | Uses Σ, Π, Δ, Φ… (spectroscopic term-like symbols) instead of A/B/E/T | Trying to force A/B/E labeling onto a linear molecule; the ∞-fold axis requires the continuous-group Σ/Π/Δ notation |
| High-symmetry cubic/icosahedral groups (Oₕ, Tᵈ, Iₕ) | x, y, z are symmetry-equivalent, giving one T-type irrep for all three coordinates together | Assuming (x,y,z) always split as in D₄ₕ; in Oₕ/Tᵈ they do not split — they form a single 3D irrep (T₁ᵤ or T₂) |
The chemist behind it
Origins · 1 minRobert Sanderson Mulliken and how this was discovered
Mulliken was one of the principal architects of molecular orbital theory itself, not merely its notation. Trained originally in chemical engineering and then physics under Robert Millikan (no relation, despite the similar name), Mulliken spent his career translating the new quantum mechanics of the 1920s and 1930s into a working language chemists could use to describe bonding in molecules — coining the very term "molecular orbital" and developing population analysis, electronegativity scales, and charge-transfer complex theory along the way. His 1933 papers systematizing the labeling of molecular electronic states and symmetry species, building on earlier spectroscopic term symbols and the mathematical representation theory of finite groups, produced the A/B/E/T/g/u/prime notation still used verbatim in every modern character table.
What's the actual difference between an A label and a B label?
Both are one-dimensional (non-degenerate) irreps, so they describe a single basis function that maps into ±itself under every operation of the group. The distinguishing test is the character under the principal rotation axis, Cₙ: if that character is +1 (the function is unchanged by the defining rotation), the label is A; if it is −1 (the function flips sign), the label is B. Everything else — subscripts, g/u, primes — is a secondary refinement layered on top of this basic A/B split, resolving ambiguity when more than one A-type or B-type irrep exists in the same group.
Why do d-orbitals never carry a "u" label, only "g"?
Because d-orbital basis functions are quadratic combinations of x, y, z (like xy, x²−y², z²), and any quadratic function is automatically even under inversion: replacing (x,y,z) with (−x,−y,−z) leaves every product of two coordinates unchanged in sign, since (−x)(−y) = xy. This is a purely mathematical fact about parity and multiplication of odd functions, independent of which specific point group you're in. So in any centrosymmetric group, all d-orbital irreps are necessarily gerade — you will never correctly assign a d-orbital to a "u" label in Oₕ, D₄ₕ, D∞ₕ, or any other group containing an inversion center.
Does every point group use g/u labeling?
No — only point groups that actually contain an inversion center i as one of their symmetry operations use g/u. This includes Cᵢ, C₂ₕ, C₄ₕ, D₂ₕ, D₄ₕ, D₆ₕ, D∞ₕ, Oₕ, and a few others. Groups without an inversion center, like C₂ᵥ, C₃ᵥ, D₃ₕ, D₃d(actually this one does have i), Td, use different subscript schemes entirely (plain 1/2 numerals, or prime/double-prime if there's a σₕ but no i). You always determine the correct scheme by checking the actual symmetry elements listed at the top of that group's character table, not by assuming any convention transfers across groups.
Why does E always mean "doubly degenerate" and never anything else?
Because χ(E) — the character under the identity operation — equals the dimension of the representation by definition (the identity operation is represented by the identity matrix, whose trace equals its size), and the Mulliken letter E is reserved specifically for the case where that dimension equals 2. This is slightly confusing notation because "E" is used both as the name of the identity operation (present in every group) and as the label for two-dimensional irreps — they are unrelated coincidences of notation, not the same symbol reused for two different reasons. Context always disambiguates: E in a list of operations across the top of a table is the identity; E in the leftmost column listing irreps is the doubly-degenerate label.
How do I know whether to use the "1/2" subscript convention or the "prime/double-prime" convention for a molecule I've just assigned a point group to?
Check the actual symmetry elements present in that specific point group, which is exactly the information already established once you've correctly assigned the group using standard flowchart procedures. If the group contains an inversion center i, use g/u. If it lacks i but contains a horizontal mirror plane σₕ (and is a Cₙₕ- or Dₙₕ-type group, or Cₛ), use prime/double-prime. If neither i nor σₕ is present (pure Cₙ, Cₙᵥ, Dₙ, Sₙ, Td groups), use the plain numeral subscripts 1/2 tied to whatever secondary element (C₂⊥ or σᵥ) the group does possess. Never guess — the character table itself, once you look it up, makes the applicable scheme immediately unambiguous.
Can a molecule's dxy and dx2−y2 orbitals ever belong to the same Mulliken symbol?
Yes, but only in specific point groups — this happens precisely when the group's symmetry treats these two functions as equivalent partners in a degenerate irrep. In Oₕ they do not pair with each other (dxy joins dxz, dyz in T₂g, while dx²−y² joins dz² in Eg), but in D₄ₕ, dxy actually becomes its own separate one-dimensional irrep (B₂g), distinct from dx²−y² (B₁g) — they are not paired there either. In Tᵈ symmetry, however, all three of dxy, dxz, dyz group together as T₂ while dz² and dx²−y² pair as E, the same overall pattern as octahedral. The pairing is entirely dictated by which functions the specific group's rotations mix into each other; there is no universal answer independent of point group.