Ligand field theory: molecular orbital treatment of octahedral complexes {#key}

Intuition Beginner

Crystal field theory treats every ligand as a point charge that pushes on the metal d electrons. That picture explains the colour of many complexes, but it cannot explain why carbon monoxide — a neutral molecule — produces a bigger splitting than fluoride, a charged ion.

Ligand field theory fixes this by allowing the metal orbitals and the ligand orbitals to combine into molecular orbitals, just like you saw when two hydrogen atoms form H₂. The metal contributes its d, s, and p orbitals. Each ligand contributes orbitals that can overlap with the metal.

When a ligand orbital overlaps head-on with a metal orbital, the two form one bonding combination (lower energy) and one antibonding combination (higher energy). This is sigma bonding. In an octahedral complex, six ligand sigma orbitals combine with six metal orbitals (one s, three p, and two d), producing six bonding MOs and six antibonding MOs.

The three remaining metal d orbitals (t₂g) do not point toward any ligand. They are nonbonding in the pure sigma picture. The gap between these t₂g orbitals and the antibonding eₘg* level is the splitting Δₒ.

Now consider ligands that also have orbitals pointing sideways at the metal — pi orbitals. A pi-donor ligand such as F⁻ has filled p orbitals that donate electron density into the metal t₂g set. This raises the t₂g energy and shrinks Δₒ. A pi-acceptor ligand such as CO has empty π* orbitals that accept electron density from the metal t₂g set. This lowers the t₂g energy and enlarges Δₒ.

That is the spectrochemical series in one idea: ligands that only sigma-bond sit in the middle, pi-donors go to the weak-field end, and pi-acceptors go to the strong-field end.

Visual Beginner

The diagram shows the metal atomic orbitals on the left, the six symmetry-adapted ligand sigma orbitals on the right, and the resulting molecular orbitals in the centre. The t₂g set is nonbonding. The eₘg* set is antibonding. The splitting Δₒ is the energy gap between them.

Worked example Beginner

Why is CO a stronger-field ligand than H₂O?

Both CO and H₂O form sigma bonds to the metal. The difference is pi interaction.

Water has lone pairs in roughly pi-type orientations relative to the metal, acting as a weak pi donor. This raises t₂g slightly, giving a moderate Δₒ.

Carbon monoxide has empty π* antibonding orbitals. The filled metal t₂g orbitals donate electron density into these empty π* orbitals — this is called back-bonding. Back-bonding lowers the t₂g energy, widening Δₒ. The stronger the back-bonding, the larger the splitting, and the stronger the field.

So CO is a strong-field ligand not because of its sigma donation alone, but because pi back-bonding dramatically increases Δₒ.

Check your understanding Beginner

Formal definition Intermediate+

In ligand field theory the molecular orbitals of an octahedral complex ML₆ are constructed as linear combinations of the metal valence orbitals and symmetry-adapted linear combinations (SALCs) of the ligand orbitals. Under the Oₕ point group the metal orbitals transform as follows:

• s → a₁g
• (pₓ, pᵧ, pᶻ) → t₁u
• (dᵤ₂, dₓ²₋ᵧ²) → eₘg
• (dₓᵧ, dₓᶻ, dᵧᶻ) → t₂g

For sigma-only bonding, the six ligand sigma orbitals form SALCs of symmetry a₁g + eₘg + t₁u. These match the metal s, d(eₘg), and p orbitals. Overlap of matching symmetry produces:

• a₁g bonding / a₁g* antibonding
• t₁u bonding / t₁u* antibonding
• eₘg bonding / eₘg* antibonding

The metal t₂g orbitals have no sigma-match among the SALCs and remain nonbonding. The octahedral splitting parameter is:

$${\Delta }_o=E(e_g^{\ast })-E(t_{2g})$$

For pi-bonding, the twelve ligand pi orbitals (two per ligand) form SALCs of symmetry t₁g + t₂g + t₁u + t₂u. The t₂g SALCs overlap with the metal t₂g set. Whether the interaction raises or lowers the metal t₂g energy depends on the electron occupancy of the ligand pi orbitals:

Pi-donor ligands (F⁻, Cl⁻, H₂O): ligand pi orbitals are filled. The bonding combination (lowered) is predominantly ligand in character, and the antibonding combination (raised) is predominantly metal in character. The metal t₂g level is pushed up, reducing Δₒ.

Pi-acceptor ligands (CO, CN⁻, PR₃): ligand pi orbitals are empty. The bonding combination (lowered) is predominantly metal in character — the t₂g set drops in energy. The antibonding combination (raised) is predominantly ligand. This increases Δₒ.

The net splitting can be expressed as:

$${\Delta }_o={\Delta }_{\sigma }+{\Delta }_{\pi }$$

where Δσ is the sigma-only crystal-field contribution and Δπ encodes the pi effect (negative for pi donors, positive for pi acceptors).

Back-bonding is the synergic process in which sigma donation from ligand to metal is accompanied by pi donation from metal to ligand. In metal carbonyls, the metal donates electron density from filled t₂g orbitals into the empty π* orbitals of CO. This strengthens the metal-ligand bond while weakening the C–O bond (observable as a red-shift in the CO stretching frequency in IR spectroscopy).

Key result Intermediate+

The molecular orbital treatment of octahedral complexes provides a unified explanation for the spectrochemical series. Sigma-only interactions produce the basic t${}_{2g}$/e${}_g^{\ast }$ splitting. Pi-donor ligands (F${}^{-}$, Cl${}^{-}$, H${}_2$O) raise the t${}_{2g}$ energy and shrink ${\Delta }_o$, placing them low in the spectrochemical series. Pi-acceptor ligands (CO, CN${}^{-}$, phosphines) lower the t${}_{2g}$ energy and expand ${\Delta }_o$, placing them high. This orbital picture explains why CO produces the largest crystal field splitting despite being neither strongly charged nor small.

Exercises Intermediate+

1. Construct the qualitative sigma-only MO diagram for [Ti(H₂O)₆]³⁺ (d¹). Identify the HOMO and LUMO and estimate the energy of the d–d transition.

2. Explain why [CoF₆]³⁻ is high-spin while [Co(CN)₆]³⁻ is low-spin, using the MO description of pi-donor versus pi-acceptor effects on Δₒ.

3. The CO stretching frequency in free CO is 2143 cm⁻¹. In Ni(CO)₄ it drops to 2057 cm⁻¹. Explain this observation using the back-bonding model.

4. Draw the pi-bonding SALCs for an octahedral complex and identify which interact with the metal t₂g set. Why do the t₁g and t₂u SALCs remain nonbonding?

5. For a tetrahedral complex, the metal e set (dᵤ₂, dₓ²₋ᵧ²) interacts with ligand sigma SALCs while the t₂ set (dₓᵧ, dₓᶻ, dᵧᶻ) remains nonbonding in the sigma-only picture. Sketch the qualitative MO diagram and explain why Δₜ < Δₒ for the same metal and ligand.

6. The complex [Fe(CN)₆]⁴⁻ is diamagnetic but [Fe(H₂O)₆]²⁺ is paramagnetic. Rationalize both observations using ligand field MO theory.

Angular overlap model and advanced topics Master

The angular overlap model (AOM) provides a quantitative parameterization of metal-ligand orbital interactions. It decomposes each metal-ligand interaction into sigma and pi contributions governed by angular geometry:

$$e_{\sigma }=\frac{S_{\sigma }^2\cdot H_{LL}}{1-S_{\sigma }^2},e_{\pi }=\frac{S_{\pi }^2\cdot H_{LL}}{1-S_{\pi }^2}$$

where Sσ and Sπ are overlap integrals and H_LL is the effective ligand orbital energy. For an octahedral complex, the crystal field splitting is:

$${\Delta }_o=3e_{\sigma }-4e_{\pi }$$

This shows directly that pi donors (positive eπ) reduce Δₒ while pi acceptors (negative eπ, because the acceptor orbital is above the metal d level) increase it.

Charge transfer transitions. In addition to d–d transitions, coordination complexes exhibit charge transfer (CT) bands, which are fully allowed (ε ~ 10⁴ M⁻¹cm⁻¹) and therefore much more intense.

• Ligand-to-metal charge transfer (LMCT): An electron is promoted from a ligand-based orbital (typically a filled p or pi orbital) into an empty or partially filled metal d orbital. LMCT is common in high-oxidation-state metals with reducible ligands — for example, MnO₄⁻ (Mn(VII), d⁰) shows an intense purple LMCT band.

• Metal-to-ligand charge transfer (MLCT): An electron is promoted from a filled metal d orbital into an empty ligand π* orbital. MLCT dominates in complexes with strong pi-acceptor ligands and low-oxidation-state metals — for example, [Ru(bpy)₃]²⁺ shows a prominent MLCT band in the visible region.

The energy of CT transitions depends on the redox potentials of the metal centre and the ligand, providing a spectroscopic handle on electronic structure.

Metal-metal bonding in dimers. When two metal centres are close enough (typically < 3 Å), their d orbitals overlap directly. In a dinuclear complex, each pair of d orbitals forms a bonding and an antibonding combination. The resulting MO scheme determines the metal-metal bond order.

The classic example is [Re₂Cl₈]²⁻, which has a Re–Re bond length of 2.24 Å — shorter than in metallic rhenium. The bonding involves one σ bond (dᵤ₂–dᵤ₂ overlap), two π bonds (dₓᶻ and dᵧᶻ overlap), one δ bond (dₓᵧ–dₓᵧ overlap), and corresponding antibonding levels, giving a quadruple bond (σ²π⁴δ² configuration). The δ bond is responsible for the eclipsed conformation of the chloride ligands, because δ overlap is maximized when the two ReCl₄ units are aligned.

Electronic structure of metal carbonyl clusters. Polyatomic metal carbonyl clusters such as Os₃(CO)₁₂ or [Rh₆(CO)₁₆] require extended MO treatments. The Wade-Mingos rules, derived from MO theory, relate the cluster electron count to the cluster geometry: closo (n vertices, 2n + 2 skeletal electrons), nido (2n + 4), arachno (2n + 6), analogous to borane chemistry. The total valence electron count for a cluster is:

$$\text{TVE}=\text{metal valence electrons}+\text{CO donations}-\text{charge}$$

Each terminal CO donates two electrons via the lone pair on carbon. Bridging COs donate two electrons shared between two metals. The framework electron count determines whether the cluster adopts a tetrahedral, octahedral, or larger polyhedral geometry.

Connections Master

Ligand field MO theory connects to several major areas of inorganic and physical chemistry:

• Photochemistry and photophysics: MLCT excited states in polypyridyl complexes (Ru(bpy)₃²⁺, Ir(ppy)₃) are the basis of dye-sensitized solar cells, photocatalytic water splitting, and OLED emitters. The lifetime and energy of the MLCT state depend directly on the ligand field parameters.

• Organometallic catalysis: Back-bonding into π* orbitals of CO, alkenes, and alkynes activates these substrates for insertion, migratory insertion, and reductive elimination — the elementary steps of homogeneous catalysis. Understanding the orbital interactions is essential for rational catalyst design.

• Bioinorganic chemistry: Metalloenzymes exploit specific ligand field effects. The intense blue colour of type I copper proteins (blue copper proteins) arises from an intense LMCT band (S(Cys) → Cu(II)) at ~600 nm. The geometry of the copper centre is constrained by the protein scaffold, modifying the ligand field relative to a simple aqueous complex.

• Solid-state chemistry: The band structure of transition metal oxides (e.g., NiO, La₂CuO₄) can be understood as an extended ligand field problem. The Mott-Hubbard insulator behaviour of NiO, where the on-site Coulomb repulsion U exceeds the bandwidth W, is a direct consequence of the ligand field splitting in the solid-state limit.

• Molecular magnetism: The magnetic exchange coupling constant J in polynuclear complexes depends on the overlap between magnetic orbitals through bridging ligands — a problem that reduces to orbital symmetry and energy matching, exactly the toolkit of ligand field theory.

Historical notes Master

Crystal field theory was developed by Hans Bethe in 1929, who showed how the electrostatic field of surrounding ions splits the d-orbital energies in a crystal lattice. Bethe's treatment was purely electrostatic — ligands were point charges with no orbitals.

In the 1930s and 1940s, John Van Vleck and others recognized that the electrostatic model could not account for covalency. Spectroscopic evidence, particularly the covalent character of metal-ligand bonds revealed by electron spin resonance and nephelauxetic effect data, demanded an MO-based approach. Van Vleck proposed combining crystal field theory with MO theory, giving rise to what is now called ligand field theory.

Carl Ballhausen's 1962 textbook Introduction to Ligand Field Theory systematized the MO approach for coordination complexes and became the standard reference. Ballhausen treated sigma and pi interactions in octahedral, tetrahedral, and square-planar geometries with full symmetry analyses.

The angular overlap model was developed independently by Schaffer and Jorgensen (1965) and by Lever (1968), providing a semi-empirical parameterization that could be fitted to spectroscopic data. This model remains widely used in interpretive inorganic chemistry.

The discovery of quadruple metal-metal bonds in [Re₂Cl₈]²⁻ by F. Albert Cotton in 1964 opened the field of multiple metal-metal bonding. Cotton's MO analysis of the δ bond was a landmark application of ligand field principles to dinuclear systems and established the importance of d-orbital symmetry in determining molecular geometry.

Modern computational chemistry — density functional theory and post-HF methods — has largely superseded hand-constructed MO diagrams for quantitative work. However, the qualitative ligand field framework remains indispensable for interpreting spectra, predicting magnetic properties, and teaching the principles that govern the bonding in coordination compounds.

Bibliography Master

• Miessler, G. L., Fischer, P. J. & Tarr, D. A. — Inorganic Chemistry, 5th ed. (Pearson, 2014), Ch. 9.
• Shriver, D. F. & Atkins, P. W. — Inorganic Chemistry, 5th ed. (Oxford, 2010), Ch. 9.
• Ballhausen, C. J. — Introduction to Ligand Field Theory (McGraw-Hill, 1962), Ch. 2–3.
• Cotton, F. A. — "Metal-Metal Bonding in [Re₂Cl₈]²⁻ and Other Metal Atom Clusters," Inorg. Chem. 1965, 4, 334–336.
• Schaffer, C. E. & Jorgensen, C. K. — "The Angular Overlap Model," Mol. Phys. 1965, 9, 401–412.
• Lever, A. B. P. — "The Electrochemical Parameterization of Metal-Ligand Interactions," Inorg. Chem. 1990, 29, 1271–1285.