Ligand Group Orbitals in Octahedral Complexes

Introduction

An octahedral complex has six σ-donor orbitals, one pointing toward the metal from each ligand. They should not be matched one by one to six metal orbitals in an arbitrary picture. First combine the donors into patterns that transform consistently under octahedral symmetry. These ligand group orbitals, or SALCs, reveal which metal s, p and d orbitals can mix and why the t₂g d set is nonbonding in a σ-only model.

Core explanation

Label the inward-pointing ligand σ donor functions σ(+x), σ(−x), σ(+y), σ(−y), σ(+z) and σ(−z). Their six-dimensional reducible representation decomposes in O h as A₁g+E g+T₁u. The dimensions check: A is onefold, E twofold and T threefold, so 1+2+3=6. This count assures that all six original donor functions are represented once in the new symmetry basis; it does not create or destroy orbitals.

The A₁g combination has all six donor lobes in phase with appropriate normalisation. It is totally symmetric and can overlap a metal ns orbital, also A₁g. The T₁u set consists of three directional patterns with opposite signs on opposite ligand pairs; these match the metal p x, p y and p z orbitals. The E g pair consists of two independent even combinations that distinguish axes while retaining the twofold symmetry of metal d(z²) and d(x²−y²). One pattern weights the z pair against x and y pairs, while the other distinguishes x from y.

Normalisation matters if numerical coefficients are used. An all-in-phase A₁g SALC has a factor 1/√6 multiplying the six equal donor functions if overlap between donors is neglected. A simple x-directed T₁u combination is [σ(+x)−σ(−x)]/√2 under the same approximation. An E g-like x²−y² combination can be proportional to σ(+x)+σ(−x)−σ(+y)−σ(−y); its normalisation differs from the first two. The exact phases depend on how individual donor orbital signs are defined, but the representation labels and allowed mixing do not.

No T₂g combination arises from six radial σ donors. Therefore the metal d(xy), d(xz) and d(yz) orbitals have no σ ligand group orbital of matching symmetry in this restricted model. They remain approximately metal-centred and nonbonding with respect to σ interactions. This is not a claim that t₂g never bonds: tangential ligand π orbitals can form T₂g SALCs and interact with it. Adding those orbitals is the next theoretical step.

Symmetry matching is necessary for interaction, not sufficient for a large interaction. A metal A₁g orbital and ligand A₁g SALC can mix if their energies and spatial overlap permit. If one is very far in energy or overlap is weak, the mixing may be small. The resulting MO pair contains a lower bonding and higher antibonding combination when appreciable interaction occurs. Thus character tables organize possible coupling, while quantum chemistry and experiment determine its strength.

The σ-only octahedral diagram usually places ligand-based bonding MOs below the predominantly metal d-like t₂g and antibonding e g set. Electron counting fills the ligand–metal bonding combinations with donor electrons before metal d electrons occupy the higher d-like levels. A formal dⁿ label remains useful, but actual occupied MOs have mixed metal–ligand character.

If ligands are not equivalent, exact O h symmetry disappears. The A₁g/E g/T₁u basis can still serve as a parent picture, but its exact degeneracies split and additional interactions become symmetry-allowed. One should not insist that mixed-ligand complexes have perfect threefold t₂g degeneracy.

Step-by-step reasoning

Place one σ donor on each of six axes. Form combinations that are symmetric or change sign predictably under O h operations, then verify the dimension sum 1+2+3=6. Match A₁g to metal s, T₁u to metal p and E g to metal d(e g). Note that no radial σ SALC has T₂g symmetry, then assess energetic and overlap conditions for actual mixing.

Visual explanation

Draw six arrows pointing from ligands toward a central metal. In one panel colour all arrows the same for A₁g. In three panels colour each opposite axial pair with opposite signs for T₁u. In two panels group axes with different weights for E g. Leave a separate metal t₂g card unmatched in the σ-only set.

Real-world analogy

Six singers can sing individual notes, but harmony is easier to analyse as coordinated patterns: all together, opposing pairs or axis-specific contrasts. The metal orbital can join only a ligand pattern with compatible symmetry, much as a musical part fits a matching harmony line.

Real-world example

In an octahedral hexaammine complex, each ammonia nitrogen offers a σ-donor pair. Their six radial donor functions combine into A₁g, E g and T₁u ligand group orbitals, enabling a compact MO diagram instead of six unrelated pairwise bonds.

Why?

Why does t₂g remain σ-nonbonding in this construction? The six radial ligand donors generate no SALC with T₂g symmetry. Without a symmetry-matched partner, direct σ mixing with metal t₂g is forbidden in ideal O h symmetry.

Common misconception

“Every ligand σ orbital overlaps a different metal d orbital.” Six donors combine collectively; symmetry-adapted patterns mix with metal s, p and e g d orbitals. t₂g lacks a radial σ partner.

Worked example

Check the six-donor decomposition. A₁g contributes one SALC, E g two and T₁u three: 1+2+3=6, matching six original donor orbitals. A metal d(z²) orbital is E g and can mix with one of the E g ligand combinations. A metal d(xy) orbital is T₂g and cannot mix with any of the six radial σ SALCs in this ideal model.

Quick check

1. Which ligand-group symmetry matches metal p orbitals in O h? Answer: T₁u, a threefold set matching p x, p y and p z.

Exam focus

State the decomposition A₁g+E g+T₁u and its 1+2+3 dimension check. Match symmetry before discussing overlap, and distinguish σ-nonbonding t₂g from universally nonbonding t₂g.

Advanced insight

The SALC construction is a change of basis, not an extra chemical assumption. It diagonalises the symmetry problem, making the Hamiltonian block-separated by irreducible representation and greatly reducing the number of orbital interactions that must be considered.

Summary

Six octahedral radial σ donors form A₁g, E g and T₁u ligand group orbitals. These match metal s, d(e g) and p functions; metal t₂g has no σ match until π donor or acceptor orbitals are included.

Practice questions

1. Write a normalised totally symmetric SALC if six donor functions are approximately orthogonal. Answer: (σ(+x)+σ(−x)+σ(+y)+σ(−y)+σ(+z)+σ(−z))/√6 has A₁g symmetry under the stated approximation. 2. Which metal functions can match an E g ligand σ SALC? Answer: The metal d(z²) and d(x²−y²) pair transforms as E g and can mix with E g ligand combinations if energies and overlap permit. 3. Does absence of a σ T₂g SALC mean metal t₂g can never interact with ligands? Answer: No. Ligand π orbitals can form matching T₂g combinations and shift those metal d-like levels. 4. Why is the dimension sum important? Answer: It confirms the SALCs account for all six initial ligand σ functions without omission or duplication.