Questions: Group Theory Applications in Inorganic Chemistry

Question 1

For an octahedral ML₆ complex, the six sigma-donor ligand orbitals generate a reducible representation that decomposes into which irreducible representations of the Oh point group?

• A. a₁g + eg + t₂g
• B. a₁g + eg + t₁u
• C. t₂g + t₁u + a₂u
• D. a₁g + t₁u + t₂g + eg

Answer: (B)

The six sigma-donor orbitals (one on each ligand, pointing at the metal) form a basis set for a reducible representation in Oh symmetry. Applying the reduction formula gives: Γ_sigma = a₁g + eg + t₁u. The a₁g combination matches the metal s orbital, the eg pair matches the metal d_z² and d_x²−y² orbitals, and the t₁u set matches the metal p_x, p_y, p_z orbitals. Crucially, no ligand sigma combination has t₂g symmetry — this is why the metal t₂g orbitals (d_xy, d_xz, d_yz) are nonbonding in a sigma-only octahedral complex, and it is the group-theoretic origin of the crystal field splitting.

Question 2

The projection operator in group theory generates symmetry-adapted linear combinations (SALCs) from a single ligand orbital by applying all symmetry operations of the group and weighting the results by the characters of the target irreducible representation.

• T. True
• F. False

Answer: True

The projection operator is the practical tool for constructing SALCs. You pick one ligand orbital as a generating function, apply each symmetry operation to see where it maps, multiply the result by the character of the target irreducible representation for that operation, and sum. The result is a SALC that transforms as the specified irreducible representation. For the a₁g SALC of six sigma orbitals in Oh, all characters are +1, so you get the sum of all six (properly normalized). For eg SALCs, the characters give specific linear combinations that match the shapes of d_z² and d_x²−y². This systematic procedure replaces guesswork with algorithm.

Question 3

Group theory can predict whether a d-d electronic transition is symmetry-allowed but cannot predict its intensity.

• T. True
• F. False

Answer: False

Group theory predicts both whether a transition is symmetry-allowed (does the direct product of initial state × operator × final state irreducible representations contain the totally symmetric representation?) AND provides qualitative intensity information. Symmetry-forbidden transitions (those where the direct product excludes the totally symmetric representation) have zero intensity by symmetry — the transition dipole integral vanishes exactly. For allowed transitions, group theory identifies which polarization component (x, y, or z) carries the intensity, enabling polarized absorption measurements on oriented crystals. What group theory cannot predict is the absolute magnitude of an allowed transition — that requires calculating the actual integral.

Question 4

Using group theory, explain why the d-d transitions of an octahedral complex are Laporte-forbidden and describe the mechanism by which they gain weak intensity.

Model answer: In Oh symmetry, the d-orbitals belong to gerade (even) representations: t₂g and eg. The electric dipole operator transforms as t₁u (ungerade, odd). The direct product of two gerade representations with an ungerade operator is always ungerade, which cannot contain the totally symmetric a₁g representation. Therefore the integral ⟨ψ_initial|μ|ψ_final⟩ = 0 by symmetry — the transition is Laporte-forbidden. Intensity is gained through vibronic coupling: molecular vibrations of ungerade symmetry (t₁u, t₂u) temporarily distort the complex away from perfect Oh symmetry, mixing ungerade character into the d-orbitals. The transition becomes weakly allowed through the combined electronic-vibrational (vibronic) mechanism, typically giving molar absorptivities of 1-100 M⁻¹cm⁻¹.

This analysis also explains why tetrahedral complexes have stronger d-d bands: Td has no inversion center, so the Laporte rule does not apply. The d-orbitals can mix with p-orbitals of the same irreducible representation (both t₂ representations contain d and p orbitals in Td), giving the transitions partially allowed character.