Point Groups and Symmetry Labels for d Orbitals

Introduction

The familiar t₂g/e g and e/t₂ names are not arbitrary labels for “low” and “high.” They are symmetry labels describing how groups of d orbitals transform under rotations and reflections of an ideal coordination polyhedron. Character tables organise those transformations. Learning their basic use makes molecular-orbital matching and optical selection rules much more systematic than judging orbital pictures by eye alone.

Core explanation

An ideal octahedral complex with six identical ligands belongs to the O h point group. Its symmetry operations include rotations, reflection-related operations and inversion through the metal. The five metal d functions split into two sets under O h: {d(z²), d(x²−y²)} transform as E g, a two-dimensional irreducible representation, and {d(xy), d(xz), d(yz)} transform as T₂g, a three-dimensional one. Lowercase e g and t₂g conventionally label orbital levels; uppercase E g and T₂g are often used for many-electron state terms. Both refer to symmetry, but an orbital and an electronic term are different objects.

The letters encode degeneracy in the ideal group: E is twofold and T is threefold. The subscript g means even under inversion: a d orbital has the same sign after r→−r because its angular function has even parity. For comparison, the metal s orbital transforms as A₁g, while the three p orbitals transform as T₁u, with u meaning odd under inversion. These labels help identify which metal and ligand combinations can mix: orbitals of different irreducible representations cannot form a simple symmetry-allowed combination in the ideal geometry.

An ideal tetrahedral complex belongs to T d. It lacks a centre of inversion, so g and u labels do not apply. The two d functions {d(z²), d(x²−y²)} transform as E, and the three {d(xy), d(xz), d(yz)} as T₂. The same grouping of orbital shapes occurs as a twofold plus threefold set, but the energy order reverses relative to octahedral CFT: tetrahedral e is lower and t₂ higher. Symmetry tells which orbitals remain degenerate; the interaction with ligands determines which set lies above the other.

A character table gives the trace, or character, of a representation for each class of symmetry operations. To build a metal–ligand MO diagram, one can form a reducible representation from ligand donor orbitals and decompose it into irreducible parts. Only ligand group orbitals with matching metal symmetry mix. For six octahedral σ donors, the relevant combinations include A₁g, E g and T₁u, matching metal s, d(e g) and p orbitals, respectively. The metal t₂g d set lacks a σ match in that restricted σ-only construction, so it remains nonbonding in the simplest diagram.

Symmetry labels also support selection rules. The electric dipole operator transforms as T₁u in O h. A transition between two purely g states is Laporte-forbidden in a centrosymmetric ideal complex because the dipole integral has the wrong parity. Vibrations can momentarily remove inversion symmetry and lend some intensity. Tetrahedral complexes have no g/u restriction, so their d–d bands are often stronger. Spin selection is separate from parity and must also be considered.

Real complexes with mixed ligands or distortions may have lower symmetry. Then a pair or triplet can split further and O h/T d labels become approximate parent labels. One should not keep exact degeneracy merely because a diagram was originally drawn as octahedral. Structural data determine whether the ideal point group is justified.

Step-by-step reasoning

Identify an idealised geometry and its point group. List the d-orbital shape basis functions from a character table, assigning E g+T₂g for O h or E+T₂ for T d. Use degeneracy and inversion labels correctly. For MO mixing, compare ligand-group symmetries with metal s, p and d symmetries; for spectra, check parity and spin rules separately.

Visual explanation

Draw five d orbitals as cards divided into a two-card group d(z²)/d(x²−y²) and a three-card group d(xy)/d(xz)/d(yz). Put O h labels E g and T₂g above the groups and T d labels E and T₂ below. Cross out the g/u suffix on the tetrahedral side because no inversion centre exists.

Real-world analogy

Choir voices can harmonise only when their parts fit the same musical pattern. Symmetry labels act like part labels: metal and ligand functions of compatible symmetry can combine, while mismatched patterns do not mix in an ideal structure.

Real-world example

An octahedral σ-only MO diagram leaves t₂g metal d orbitals without a matching σ ligand combination. When π-donor or π-acceptor orbitals are added, matching t₂g symmetry becomes available and those levels shift. The character-table label predicts which orbitals respond.

Why?

Why do tetrahedral labels omit the g subscript? A tetrahedron has no inversion centre, so there is no operation r→−r that leaves the full ligand arrangement unchanged. Even/odd parity under inversion is therefore not a valid symmetry classification for T d.

Common misconception

“E g means an excited state and T₂g a ground state.” E and T describe dimensions of symmetry representations, not an energy rank. In an octahedral d-orbital diagram E g happens to be higher than T₂g for the usual ligand field.

Worked example

Classify the five d orbitals of an ideal [ML₆] O h complex. The functions d(z²) and d(x²−y²) form the twofold E g set, while d(xy), d(xz) and d(yz) form the threefold T₂g set. Both are gerade. A σ ligand group orbital of E g symmetry can mix with the metal E g d pair, but a T₁u σ combination cannot mix directly with T₂g because their symmetries differ.

Quick check

1. What does the “g” in t₂g mean? Answer: Even parity under inversion through the centre in an O h complex.

Exam focus

Separate symmetry label from energy order and orbital from many-electron term. Use character tables to justify allowed mixing or parity arguments, not merely to decorate a diagram.

Advanced insight

Symmetry lowering correlates high-symmetry representations with smaller lower-symmetry ones. A T₂g triplet can split when an octahedron is distorted, and spectral or magnetic anisotropy may reveal that splitting even if average bond distances appear nearly octahedral.

Summary

O h d orbitals transform as E g+T₂g; T d d orbitals as E+T₂. Degeneracy and parity labels guide MO mixing and selection rules, while actual interaction energies set the order of levels.

Practice questions

1. Assign d(xy) and d(z²) in O h and T d symmetry. Answer: In O h, d(xy) belongs to T₂g and d(z²) to E g. In T d, they belong to T₂ and E, respectively. 2. Why is a pure O h d–d electric-dipole transition Laporte-forbidden? Answer: Both initial and final d-derived states are gerade, while the electric dipole operator is ungerade. The parity product cannot yield an allowed integral in ideal inversion symmetry. 3. Which octahedral metal d set has no matching ligand orbital in a σ-only six-donor MO construction? Answer: The T₂g d set; σ ligand combinations have A₁g, E g and T₁u symmetries in that simplest construction. 4. What are the corresponding tetrahedral d-orbital symmetry sets? Answer: A twofold E set and a threefold T₂ set, without g/u suffixes.