octahedral field splitting
/ ok-tuh-HEE-dral /
The commonest shape a metal ion takes when six groups bind it is the octahedron: imagine the metal at the center of a cube with one ligand on each of the six faces, or equivalently one along each end of the x, y, and z axes. This is the arrangement of [Co(NH3)6]3+, [Fe(H2O)6]2+, and countless others. Octahedral field splitting is the specific way the five d orbitals split apart in this six-ligand cage.
Here is the pattern, and it is worth memorizing. The two orbitals that point straight along the axes — dz2 and dx2-y2 — aim directly at all six ligands and are pushed UP. Together they form the upper set, labeled eg. The three orbitals that point between the axes — dxy, dxz, dyz — slip into the gaps and are pushed down, forming the lower set, labeled t2g. So in an octahedral field t2g lies BELOW eg, and the energy gap between them is delta-o (the o is for octahedral), also called 10Dq. By the barycenter rule the three t2g orbitals sit 2/5 delta-o below the average and the two eg orbitals sit 3/5 delta-o above it.
Everything flows from this single diagram: the d-electron count plus the size of delta-o decides whether a complex is high-spin or low-spin, the color comes from electrons absorbing a photon to jump from t2g to eg, and the crystal field stabilization energy is computed by counting electrons in each set. Because the octahedron is so common, the octahedral splitting diagram is the single most-used picture in all of transition-metal chemistry.
In [Ti(H2O)6]3+, the lone d electron occupies a t2g orbital. Shining light on it, the electron absorbs a photon of about 243 kJ/mol (around 500 nm, green) to climb from t2g to eg. That single t2g-to-eg jump is the value of delta-o for this complex, and it is exactly why the solution is violet.
delta-o is literally the energy of the t2g-to-eg jump, readable straight off the absorption spectrum.
The labels t2g and eg come from group theory (they describe how the orbitals transform under the octahedron's symmetry), not from the orbitals' shapes; the 'g' tag, for gerade, only appears in octahedral and other centrosymmetric fields, which is why tetrahedral labels drop it.