Inorganic chemistry

Crystal field theory and ligand field theory

Why do complexes with the same metal ion have different colours, magnetic moments and preferred shapes? The surrounding ligands split the energies of the metal d orbitals.

1 Lower secondary 2 Upper secondary 3 Undergraduate 4 Advanced 5 Research

IntuitionIntuition: ligands lift the fivefold d-orbital degeneracy

Imagine five equal-energy orbital shapes around an isolated metal ion. Ligands approach from particular directions and repel d-electron density unevenly; orbitals pointing toward ligands rise more in energy than those between them.

Select octahedral or tetrahedral geometry and compare the splitting and d-electron occupancy.

SchoolSchool level: two common splitting patterns

Definition: Crystal-field splitting energy

In an octahedral complex, six ligands lie on the Cartesian axes: the d orbitals divide into lower t₂g (dxy, dxz, dyz) and higher eg (dz², dx²−y²). Their separation is Δo.

\Delta_o = E(e_g)-E(t_{2g}),\qquad E(t_{2g})=-\frac{2}{5}\Delta_o,\quad E(e_g)=+\frac{3}{5}\Delta_o

In tetrahedral coordination no ligands lie on the Cartesian axes, so the order reverses: e is lower and t₂ higher. The splitting Δt is usually smaller than Δo, often approximated as Δt ≈ 4Δo/9 for related ligands.

Ideal d-level splitting
Geometry	Lower set	Upper set
Octahedral	t₂g (3 orbitals)	eg (2 orbitals)
Tetrahedral	e (2 orbitals)	t₂ (3 orbitals)

Example: High-spin d⁶ in an octahedron

For [Fe(H₂O)₆]²⁺, Fe²⁺ is d⁶ and water gives a weak field. Fill the octahedral levels under Hund’s rule.

Solution

$t_{2g}^{4}e_g^{2}$ ; four electrons remain unpaired (S = 2). This is a high-spin configuration because pairing in t₂g costs more than promoting an electron across Δo.

\Delta_o = 10Dq

UndergraduateUndergraduate: ligand identity, pairing and geometry

The spectrochemical series is empirical: I⁻ < Br⁻ < Cl⁻ < F⁻ < OH⁻ < H₂O < NH₃ < en < CN⁻ ≈ CO for many metal ions. Oxidation state, metal row, bond covalency and geometry also change Δ; the series is not an absolute universal scale.

\text{high spin if }\Delta_o < P;\qquad \text{low spin if }\Delta_o > P

Here P is the pairing energy, including the cost of putting two electrons in the same orbital. For d⁴–d⁷ octahedral ions, the balance between Δo and P determines high- or low-spin occupancy; d⁰, d¹⁰ and several other counts have no such choice.

Definition: Crystal-field stabilisation energy (CFSE)

CFSE is the sum of d-electron energies relative to the barycentre of the unsplit d set. In an octahedron, each t₂g electron contributes −0.4Δo and each eg electron +0.6Δo; add pairing terms separately when comparing configurations.

Example: Compare two d⁶ configurations

Ignoring a common baseline, compare high-spin $t_{2g}^{4}e_g^{2}$ with low-spin $t_{2g}^{6}$ . Which is favoured when pairing is counted?

Solution

High spin has CFSE $4(-0.4)+2(0.6)=-0.4\Delta_o$ with 1 paired electron set; low spin has $6(-0.4)=-2.4\Delta_o$ with 3 pairs. Low spin gains $2.0\Delta_o$ in orbital stabilisation but pays $2P$ for the two extra pairings, so low spin is favoured when $\Delta_o > P$ .

Ligand field theory extends the electrostatic picture by treating metal–ligand bonding with molecular orbitals. σ donation and possible π donation/back-donation shift the metal-based levels; π acceptors such as CO often enlarge the splitting, while π donors can reduce it.

AdvancedBeyond the point-charge picture

Real complexes are not spherical ionic environments. Covalency, orbital overlap, distortions, spin–orbit coupling and vibronic interactions mix states. Angular overlap models assign ligand-specific σ and π interactions to individual d orbitals; electronic-structure calculations can resolve the resulting orbital character.

For ideal octahedral symmetry, an uneven occupancy of degenerate orbitals can drive a Jahn–Teller distortion. The classic d⁹ Cu(II) case often elongates axial bonds, lowering symmetry and splitting the t₂g/eg-derived levels further; the distortion is a consequence of electronic energy gain balanced against elastic cost.

The observed ligand-field splitting can be measured from electronic absorption bands, but assigning a single band to Δ is safest for simple d¹ octahedral ions. For multi-electron ions, Tanabe–Sugano diagrams account for interelectronic repulsion as well as the field strength.

References

Inorganic Chemistry · C. E. Housecroft, A. G. Sharpe, 2018
Ligand Field Theory and Its Applications · B. N. Figgis, M. A. Hitchman, 2000