Group Theory Applications in Inorganic Chemistry

Core Idea

Group theory provides a systematic method for constructing molecular orbital diagrams, predicting spectroscopic selection rules, and analyzing vibrational modes of coordination compounds. By identifying the point group of a complex, generating reducible representations for ligand orbital sets, and decomposing them into irreducible representations using character tables, you can determine which metal and ligand orbitals can interact — all without solving any integrals.

Explainer

In physical chemistry, you learned group theory as a framework for predicting IR and Raman activity of molecular vibrations. In inorganic chemistry, group theory becomes an even more powerful tool because coordination compounds have high symmetry, and the electronic structure of d-orbital complexes is exquisitely sensitive to that symmetry. The central application is constructing MO diagrams using symmetry-adapted linear combinations (SALCs) rather than guessing which orbitals interact.

The procedure is systematic. First, assign the point group of the complex (Oh for octahedral, Td for tetrahedral, D₄h for square planar, etc.). Second, identify the basis set of ligand orbitals you want to analyze — for sigma bonding, these are the ligand lone pairs pointing at the metal. Third, determine how this basis set transforms under each symmetry operation of the group, generating a reducible representation. Fourth, decompose the reducible representation into irreducible representations using the reduction formula. The resulting irreducible representations tell you exactly which metal orbitals can form bonding and antibonding combinations with the ligand set. For Oh sigma bonding, the decomposition gives a₁g + eg + t₁u — meaning metal s, d(eg), and p orbitals can form sigma bonds, while the t₂g d-orbitals are left nonbonding.

The same procedure applies to pi bonding. The twelve pi-donor (or pi-acceptor) orbitals on six octahedral ligands generate their own reducible representation, which decomposes into t₁g + t₁u + t₂g + t₂u. Only the t₂g component has a metal orbital counterpart (the d_xy, d_xz, d_yz set), so only the t₂g ligand pi-orbitals participate in metal-ligand pi bonding. This is the group-theoretic proof that pi interactions affect only the t₂g level in octahedral complexes — the foundation of ligand field theory's explanation of the spectrochemical series.

Selection rules for electronic transitions follow from the same orthogonality principle. The transition dipole moment integral ⟨ψ_f|μ|ψ_i⟩ is nonzero only if the direct product of the irreducible representations of ψ_f, μ, and ψ_i contains the totally symmetric representation. For d-d transitions in Oh, both initial and final states are gerade, while the dipole operator is ungerade (t₁u) — the product is ungerade and cannot contain a₁g, so the transition is Laporte-forbidden. This group-theoretic derivation replaces hand-waving arguments about parity with a rigorous mathematical proof. The same framework predicts which vibrations are IR-active, which are Raman-active, and whether specific electronic transitions can be observed in polarized spectra of oriented single crystals.