MO Theory of Inorganic Molecules

Symmetry-based molecular-orbital theory builds the bonding of inorganic molecules and complexes by combining metal orbitals with symmetry-adapted ligand combinations of matching symmetry.

Definition

MO theory of inorganic molecules is the application of symmetry and molecular-orbital theory to construct the bonding, nonbonding, and antibonding orbitals of inorganic molecules and complexes from metal and symmetry-adapted ligand orbitals.

Scope

This topic covers the construction and interpretation of molecular-orbital diagrams for inorganic molecules and coordination complexes using group theory: forming ligand group orbitals as symmetry-adapted linear combinations, matching them to metal s, p, and d orbitals of the same symmetry, building sigma- and pi-bonding diagrams for octahedral and other geometries, and recovering the ligand-field splitting as a molecular-orbital result. It applies the representation theory of the previous topics to bonding.

Core questions

• How are ligand group orbitals formed and matched to metal orbitals?
• How does a molecular-orbital diagram of an octahedral complex arise?
• How does the molecular-orbital picture recover ligand-field splitting?
• How do pi-donor and pi-acceptor ligands change the diagram?

Key concepts

• Ligand group orbitals
• Symmetry matching of orbitals
• Sigma and pi bonding in complexes
• Molecular-orbital diagrams
• Recovery of ligand-field splitting
• Pi-donor and pi-acceptor effects

Key theories

Ligand group orbitals and symmetry matching

Ligand orbitals are combined into symmetry-adapted group orbitals that transform as irreducible representations; only metal orbitals of the same symmetry can interact with them, which dictates the bonding pattern.

Molecular-orbital view of complexes

Building the diagram for an octahedral complex places the metal eg orbitals in sigma-antibonding combinations and the t2g orbitals as nonbonding (or pi-interacting), reproducing the d-orbital splitting of ligand-field theory from molecular orbitals.

Pi-bonding and the spectrochemical series

Including ligand pi orbitals shows that pi-donor ligands raise the t2g set and reduce the splitting while pi-acceptor ligands lower it and increase the splitting, giving a molecular-orbital rationale for the spectrochemical series.

Clinical relevance

Molecular-orbital diagrams explain the bonding, magnetism, colour, and reactivity of inorganic molecules and complexes and underpin the rational interpretation of their spectra and the design of catalysts and materials.

History

Molecular-orbital theory, developed by Mulliken and others, was extended to inorganic molecules and complexes in the mid-twentieth century, when symmetry methods were used to build ligand-field molecular-orbital diagrams. The work of Gray, Hoffmann, and others made these diagrams a standard description of inorganic bonding.

Key figures

• Robert Mulliken
• Harry Gray
• Roald Hoffmann

Seminal works

• cottongrouptheory1990
• weller2018
• albright2013

Frequently asked questions

How does molecular-orbital theory improve on crystal-field theory for complexes?

Crystal-field theory treats ligands as point charges and ignores covalency, whereas molecular-orbital theory explicitly mixes metal and ligand orbitals; it reproduces the same d-orbital splitting but also explains covalent effects, pi-bonding, and the spectrochemical series.

What is a ligand group orbital?

A ligand group orbital is a symmetry-adapted linear combination of the individual ligand orbitals that transforms as one of the complex's irreducible representations, so that it can be matched with a metal orbital of the same symmetry to form molecular orbitals.