Crystal field theory treats the ligands around a transition-metal ion as point negative charges. The five d-orbitals of the free metal ion start out at equal energy, but as ligands approach along the geometry's characteristic directions, orbitals pointing toward a ligand are pushed up in energy more than orbitals pointing between ligands — the five-fold degeneracy splits into groups.
Octahedral: t2g (3, low) vs eg (2, high) Δo
Tetrahedral: e (2, low) vs t2 (3, high) Δt ≈ 4/9 Δo
Sq. planar: 4 distinct levels, widest total spread
Linear: 3 levels, smallest total spread
- Geometry — sets how many ligands coordinate the metal and which d-orbitals feel the strongest push, i.e. the shape of the splitting diagram.
- Ligand strength — moves the complex along the spectrochemical series, scaling Δ. It is the single biggest factor in whether electrons pair up or spread out.
- d-electron count — how many electrons the metal ion contributes; they fill the split orbitals following the aufbau principle and Hund's rule.
- High-spin vs low-spin — when Δ is smaller than the pairing energy P, an electron would rather occupy an empty higher orbital than pair up (high-spin, more unpaired electrons, usually weak-field ligands). When Δ exceeds P, pairing in the lower group is cheaper (low-spin). This only produces two distinct outcomes for octahedral/tetrahedral fields; square-planar fields are split widely enough that they are almost always low-spin.
- Complex colour — the sphere's colour is a simplified stand-in for what you'd see in a lab: a complex absorbs light whose energy matches Δ and appears the complementary colour. Small Δ absorbs low-energy (red/orange) light and looks green/cyan; large Δ absorbs high-energy (blue/violet) light and looks orange/yellow.
Real-world relevance: crystal field / ligand field theory explains why transition-metal complexes are coloured, why some are magnetic (paramagnetic, unpaired electrons) and others are not (diamagnetic, all paired), and underlies catalysis, pigments, and the chemistry of metalloenzymes such as haemoglobin.