Crystal Field Theory: How Ligands Split the d Orbitals

Why is a solution of titanium(III) purple, while zinc(II) is colourless? Why does cobalt(III) with ammonia give a compound with no unpaired electrons, when cobalt(III) with fluoride has four? A single, surprisingly simple model answers both questions. Crystal field theory (CFT) treats the ligands around a metal ion as nothing more than points of negative charge, and asks one question: what do those charges do to the energies of the metal’s five d orbitals?

The answer is that they stop being equal. The pattern in which they separate, and the size of the gap, explain much of the colour, magnetism and stability of transition metal complexes.

Start with an isolated ion

In a free, gaseous transition metal ion, the five d orbitals (dxy, dxz, dyz, dx²−y² and dz²) all have the same energy. They are degenerate. The electrons can sit in any of them with no preference except for the usual rule of spreading out with parallel spins first (see Hund’s rule).

Now surround the ion with a sphere of negative charge. Every d orbital is pushed up in energy by the same amount, because electrons in all of them are repelled equally. Still no splitting.

Splitting only happens when the negative charge is not spread evenly, but concentrated in particular directions. That is exactly what ligands do.

The octahedral case

Place six ligands on the x, y and z axes, at +x, −x, +y, −y, +z and −z. This is an octahedral complex, like [Ti(H₂O)₆]³⁺.

Now look at the shapes of the d orbitals (see atomic orbital shapes):

• dx²−y² has its lobes along the x and y axes, pointing straight at four ligands.
• dz² has its main lobes along the z axis, pointing straight at two ligands.
• dxy, dxz and dyz have their lobes between the axes, pointing into the gaps between ligands.

An electron in dx²−y² or dz² sits right where the ligands’ lone pairs are, so it is strongly repelled. An electron in dxy, dxz or dyz is further from the ligands and is repelled less. So the five orbitals split into two sets:

• a lower set of three, called t₂g (dxy, dxz, dyz);
• a higher set of two, called eg (dx²−y², dz²).

The energy gap between them is the octahedral splitting energy, written Δo (or sometimes 10Dq).

An analogy

Think of five seats in a small room where six people are standing against the walls, one on each wall, floor and ceiling. Two seats face people directly; three face the corners. If you dislike being close to others, the corner-facing seats are more comfortable. The electrons “choose” the same way.

Where the average sits

The splitting keeps the average energy constant, a rule called the barycentre rule. Because there are three t₂g orbitals and two eg orbitals:

• each t₂g orbital is 0.4 Δo below the average;
• each eg orbital is 0.6 Δo above the average.

Check: 3 × (−0.4) + 2 × (+0.6) = 0. The energy lost by three orbitals exactly balances the energy gained by two.

Crystal field stabilisation energy (CFSE)

Because the lower orbitals fill first, a complex is often more stable than it would be without splitting. The crystal field stabilisation energy measures this:

CFSE = (number of t₂g electrons × −0.4 Δo) + (number of eg electrons × +0.6 Δo)

A few examples for octahedral complexes:

Ion	d count	Configuration	CFSE
Ti³⁺	d¹	t₂g¹	−0.4 Δo
Cr³⁺	d³	t₂g³	−1.2 Δo
Mn²⁺ (high-spin)	d⁵	t₂g³ eg²	0
Co³⁺ (low-spin)	d⁶	t₂g⁶	−2.4 Δo (before pairing costs)
Ni²⁺	d⁸	t₂g⁶ eg²	−1.2 Δo
Zn²⁺	d¹⁰	t₂g⁶ eg⁴	0

Two practical consequences follow:

• Cr³⁺ (d³) complexes are unusually inert. A large CFSE means that distorting the complex to swap ligands costs energy, so substitution is slow.
• Hydration enthalpies show a “double hump”. Across the first-row M²⁺ ions from Ca²⁺ to Zn²⁺, measured hydration enthalpies are more exothermic than a smooth trend would predict, except at d⁰, high-spin d⁵ and d¹⁰, where CFSE is zero. Those three ions sit on the smooth line; the others sit below it.

For d⁴ to d⁷, electrons have a choice between entering eg singly or pairing up in t₂g. That choice creates high-spin and low-spin complexes, which get their own treatment in high-spin vs low-spin complexes.

The tetrahedral case

In a tetrahedral complex, four ligands sit at alternate corners of a cube around the metal. No ligand lies on an axis. The orbitals that point between the axes (dxy, dxz, dyz) now point closer to the ligands than dx²−y² and dz² do. The pattern inverts:

• a lower pair, called e (dx²−y², dz²);
• a higher trio, called t₂ (dxy, dxz, dyz).

The g subscript disappears because a tetrahedron has no centre of symmetry.

The tetrahedral splitting, Δt, is much smaller than Δo for the same metal and ligands, roughly 4/9 of Δo. There are fewer ligands, and none of them points directly at any d orbital. As a result, tetrahedral complexes are almost always high-spin, and their colours can differ strongly from their octahedral cousins. The blue [CoCl₄]²⁻ ion and the pink [Co(H₂O)₆]²⁺ ion are a classic pair, used in cobalt chloride moisture indicators.

Square planar complexes

Remove the two ligands on the z axis from an octahedron and you get a square planar complex. dx²−y², still pointing at four ligands, becomes very high in energy. The orbitals with z character drop. d⁸ ions with strong-field ligands, such as Ni²⁺ with CN⁻ or Pt²⁺ with almost anything, prefer this geometry: all eight electrons fit into the four lower orbitals, leaving the highly unfavourable dx²−y² empty. That is why cisplatin and [Ni(CN)₄]²⁻ are square planar.

What sets the size of Δ?

Four factors, each with a physical reason:

1. The ligand. Some ligands split d orbitals much more than others. Ordered from small to large Δ, this is the spectrochemical series: I⁻ < Br⁻ < Cl⁻ < F⁻ < OH⁻ < H₂O < NH₃ < en < bipy < CN⁻ < CO. Our list of common ligands explains where each sits.
2. The metal’s oxidation state. A higher charge pulls the ligands closer and increases Δ. Complexes of M³⁺ ions have noticeably larger splittings than the corresponding M²⁺ ions.
3. The row of the periodic table. 4d and 5d orbitals are larger and reach further towards the ligands than 3d orbitals, so Δ increases down a group. This is why complexes of ruthenium, rhodium, palladium and platinum are nearly always low-spin.
4. The geometry. Octahedral splitting is larger than tetrahedral splitting for the same metal and ligands.

Colour: measuring Δ with light

If a photon’s energy matches Δo, a t₂g electron can absorb it and jump to eg. That is a d–d transition. The relationship between the gap and the absorbed wavelength is

ΔE = hc / λ

Take [Ti(H₂O)₆]³⁺, a d¹ ion. It has one broad absorption band with a maximum close to 500 nm, in the green-yellow region. Using h = 6.626 × 10⁻³⁴ J s, c = 2.998 × 10⁸ m s⁻¹ and λ = 500 × 10⁻⁹ m:

• energy per photon = (6.626 × 10⁻³⁴ × 2.998 × 10⁸) / (500 × 10⁻⁹) ≈ 3.97 × 10⁻¹⁹ J;
• multiplying by the Avogadro constant (6.022 × 10²³ mol⁻¹) gives about 239 kJ mol⁻¹.

So Δo for this ion is roughly 240 kJ mol⁻¹, comparable in size to a weak chemical bond. The solution absorbs green-yellow light and transmits the rest, so it looks purple. You can repeat the calculation for other wavelengths with the photon energy calculator, and UV-visible spectroscopy explains how such spectra are measured.

The model also explains why some ions are colourless. Zn²⁺ (d¹⁰) has a full eg set, so there is nowhere for an electron to jump. Sc³⁺ (d⁰) has no d electrons to jump. Mn²⁺ (high-spin d⁵) is very pale pink, because every possible d–d transition would require an electron to flip its spin, which makes those absorptions extremely weak.

Where the model falls short

Crystal field theory is powerful, but it is built on a fiction: ligands are not point charges. Three problems show this:

• Neutral CO and anionic I⁻. A pure charge model predicts that anions should split orbitals more than neutral molecules. In reality, neutral CO is the strongest-field ligand and I⁻ is one of the weakest.
• Covalency is ignored. Metal–ligand bonds share electrons. Magnetic and spectroscopic measurements show that d electrons spread partly onto the ligands.
• π bonding is missing. Ligands like CO and CN⁻ accept electron density from the metal’s t₂g orbitals into empty π* orbitals, lowering t₂g and widening the gap.

Ligand field theory, which combines CFT with molecular orbital theory, fixes these problems while keeping the same t₂g/eg language. For most school and first-year problems, the simple crystal field picture gives the right answer.

Common mistakes

• Getting the octahedral pattern backwards. In octahedral complexes t₂g is lower because those orbitals point between the ligands. It is the eg orbitals that point at them.
• Using the octahedral pattern for tetrahedral complexes. The pattern inverts, and Δt is only about 4/9 of Δo.
• Confusing the colour absorbed with the colour seen. A complex that absorbs green-yellow light appears purple, the complementary colour.
• Forgetting the barycentre. CFSE uses −0.4 Δo and +0.6 Δo per electron, not −1 and +1.
• Thinking ligand charge alone decides Δ. The spectrochemical series is set by bonding, especially π effects, not by charge.
• Assuming colour always comes from d–d transitions. Intense colours such as the purple of permanganate come from charge-transfer transitions, which CFT doesn’t describe.

Key takeaways

• Ligands split the five degenerate d orbitals because their lone pairs point at some orbitals more than others.
• Octahedral: t₂g lower (−0.4 Δo each), eg higher (+0.6 Δo each). Tetrahedral: the pattern inverts and Δt ≈ 4/9 Δo.
• CFSE explains the inertness of Cr³⁺ complexes and the double hump in hydration enthalpies.
• Δ grows with a stronger-field ligand, a higher metal oxidation state and a heavier (4d, 5d) metal.
• d–d transitions absorb light of energy Δ, giving complexes their colour; d⁰ and d¹⁰ ions have no d–d colour.
• CFT is a point-charge model. Ligand field theory adds covalency and π bonding to explain the spectrochemical series properly.