A transition-metal ion surrounded by ligands has its five degenerate d-orbitals split in energy by the electric field of the ligands — crystal field theory. The size and shape of the split depends on geometry (octahedral, tetrahedral, square planar) and on how strongly the ligand donates electron density, ranked by the spectrochemical series (I⁻ < Br⁻ < Cl⁻ < F⁻ < OH⁻ < H₂O < NH₃ < en < CN⁻ < CO).
place e⁻ one at a time, cheapest orbital first
cost(empty orbital) = ε(orbital)
cost(half-filled) = ε(orbital) + P (pairing energy)
Δ₀ ≫ P → low spin (electrons pair up in t2g)
Δ₀ ≪ P → high spin (electrons spread out, Hund's rule)
- Geometry — octahedral splits d-orbitals into t2g (3, lower) / eg (2, higher); tetrahedral gives the inverse pattern with a smaller gap (Δt ≈ 4/9 Δ₀), which is why tetrahedral complexes are almost always high-spin; square planar (common for d⁸ metals) splits into four distinct levels with dx²-y² pushed high.
- Ligand field strength Δ₀ — how strongly the chosen ligand splits the d-orbitals; strong-field ligands (CN⁻, CO) favour pairing electrons into lower orbitals (low spin) over climbing to a higher orbital.
- d-electron count — how many d-electrons the metal ion contributes; the ambiguity between high- and low-spin only exists for d⁴–d⁷ octahedral configurations.
- Complex colour — the sphere's hue is an illustrative approximation of the complementary colour of the light absorbed to promote an electron across Δ₀ (larger Δ₀ → higher-energy light absorbed → different visible colour transmitted) — the chemical basis of why transition-metal complexes are so often vividly coloured.
Real-world relevance: crystal field splitting explains catalytic activity (partially filled d-orbitals bind and activate substrates), the colour of gemstones and dyes, and why some coordination complexes (e.g. cisplatin, haemoglobin's iron centre) are central to medicine.