Ligand Field & Molecular Orbital Theory
Crystal field theory works far better than a model that pretends bonds are electrostatic has any right to. This part explains why it works, by replacing the point charges with real orbitals. Building the ML₆ molecular orbital diagram properly turns Δo from an unexplained parameter into the gap between a bonding and an antibonding set — and once π interactions are added, the spectrochemical series stops being a list to memorise and becomes something you can derive. Two layers on every section: a slow, hand-held beginner path and a research-grade advanced/reference path.
The 10 sections in Part 4
- 1The evidence that the bonding is covalent Free below
- 2Symmetry-adapted combinations of the six ligand σ orbitals
- 3The σ-only ML₆ diagram, and what Δo really is
- 4π-donor ligands — why halides give small splittings
- 5π-acceptor ligands, and the series derived
- 6Synergic bonding in metal carbonyls, and the infrared proof
- 7The nephelauxetic effect — covalency, measured
- 8Non-innocent ligands and the collapse of oxidation state
- 9The 18-electron rule from the MO diagram, and the 16-electron square plane
- 10Where this leads — a pointer to Part 5
The evidence that the bonding is covalent
Free extractSection F.1 of Part 4, reproduced in full from the book — figures and all. No sign-in, no paywall on this section.
Three experiments that a point-charge model cannot survive. None of them is subtle — each one directly detects metal electrons sitting on the ligands.
Crystal field theory makes a very specific claim: the d electrons belong to the metal and nowhere else. The ligands supply an electrostatic field and no electron density. If that is true, a metal d electron can never be found on a ligand atom. Three independent experiments say it can.
Evidence 1 — EPR hyperfine coupling to ligand nuclei
An electron paramagnetic resonance (EPR) spectrum splits when the unpaired electron feels the magnetic moment of a nearby nucleus. The size of that hyperfine splitting is proportional to how much of the unpaired electron’s density sits on that nucleus. This is a direct measurement, not an inference.
The classic case is the hexachloroiridate(IV) ion, [IrCl₆]²⁻ — a low-spin d⁵ ion with one unpaired electron, studied as a dilute dopant in a diamagnetic host lattice. Its EPR spectrum shows hyperfine structure from the chlorine nuclei. The unpaired electron, which crystal field theory says is a pure iridium t2g electron, is demonstrably spending part of its time on the chloride ligands. Analysis of the splitting pattern puts a few per cent of the spin density on each chloride, so something like a fifth to a third of the unpaired electron is off the metal altogether.
There is no way to explain a chlorine hyperfine coupling with point charges. Point charges have no orbitals for an electron to occupy.
Evidence 2 — the nephelauxetic effect
The repulsion between two d electrons on the same metal ion is measured by the Racah parameter B (Part 5 develops this properly). B can be extracted from the electronic spectrum, and it can also be measured for the free gaseous ion. The comparison is devastating for the electrostatic model: B is always smaller in the complex than in the free ion, typically by 10–40%.
Smaller repulsion between the d electrons means they are further apart on average, which means the orbital they occupy is bigger. The Greek for ‘cloud expanding’ gives the effect its name. A d orbital that has expanded has expanded onto the ligands — it has acquired ligand character. Crystal field theory has no mechanism for this at all: in CFT the ligands cannot change the size of a metal orbital, because they cannot touch it.
Evidence 3 — the spectrochemical series is in the wrong order
This is the argument that usually lands hardest, because the student has already memorised the series. Look at the ends of it:
If Δo were an electrostatic effect, the strongest field would belong to the ligand with the largest negative charge sitting closest to the metal. Test that prediction three ways and it fails three times.
- CO is neutral and it is at the top. Carbon monoxide has a dipole moment of about 0.11 D — essentially the least polar ligand in the whole series — and it produces the largest Δo known. A point charge of zero cannot split anything.
- OH⁻ is below H₂O. Hydroxide carries a full negative charge; water carries none. Electrostatics says OH⁻ must be far stronger. It is weaker.
- Halide order runs the wrong way for size. F⁻ is the smallest and hardest halide and gives the largest halide Δo; I⁻ is the largest and softest and gives the smallest. That much a point-charge model can rationalise. But the same model then has to explain why the neutral, bulky, soft ligand CO beats F⁻ by a wide margin, and it cannot.
The quantitative form of the same three arguments, and the reason CFT nevertheless works — which is a symmetry argument, not a lucky accident.
Each of the three qualitative arguments above has a quantitative version, and the quantitative versions are what appear in Part C questions.
The orbital reduction factor
If the ‘metal’ t2g orbital is really a mixture with ligand character, its orbital angular momentum is partly quenched, because the ligand-based part of the wavefunction is not centred on the metal nucleus. Stevens introduced an orbital reduction factor k to describe this: the effective orbital contribution to the magnetic moment is scaled by k, with k = 1 for a purely ionic complex and k measurably less than 1 for real ones.
Why an obviously wrong model gives obviously right answers
The deepest question in this section is not why CFT fails but why it succeeds. The answer is that the results CFT gets right are the ones fixed by symmetry alone, and symmetry is the one thing the point-charge model gets exactly right.
In Oh the five d orbitals span the representations eg + t2g. That is a statement about the octahedron, not about the bonding. Any perturbation with octahedral symmetry — a set of point charges, a set of real ligand orbitals, an anisotropic solvent cage, anything — must split the d set into one doubly degenerate level and one triply degenerate level, and can do nothing else. CFT therefore gets the pattern of the splitting right for free. What it cannot get right is the magnitude and the ordering across ligands, because those depend on the interaction, not on the symmetry. And magnitude and ordering are precisely where the spectrochemical series lives.
There is also an honest arithmetical failure worth knowing. If you actually evaluate the octahedral splitting from a point-charge electrostatic integral, using realistic metal–ligand distances and formal ligand charges, the number that comes out is far too small to account for the observed Δo, and for neutral ligands it comes out near zero. Some texts patch this by treating the ligand as a point dipole rather than a point charge, which helps a little and still fails for CO. The model is not merely incomplete; on this quantity it is numerically wrong.
A student argues: ‘CN⁻ gives a larger Δo than F⁻ because CN⁻ is bigger and gets closer.’ Dismantle the argument. Medium
Sources for the covalency evidence in this section:
- Figgis and Hitchman, Ligand Field Theory and its Applications — the standard monograph treatment of covalency parameters, orbital reduction and the nephelauxetic effect.
- Cotton and Wilkinson, Advanced Inorganic Chemistry, and Huheey, Keiter and Keiter, Inorganic Chemistry — textbook statements of the evidence against the purely electrostatic model.
Read the rest of Part 4
The remaining 9 sections of this part — Symmetry-adapted combinations of the six ligand σ orbitals, The σ-only ML₆ diagram, and what Δo really is, π-donor ligands — why halides give small splittings, π-acceptor ligands, and the… — and all nine parts of Coordination Chemistry are part of ChemVidya Full Access, along with the other books, 55 Study Notes and 6,000+ practice questions.
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