Step 1: Before splitting. An isolated metal ion has all five d-orbitals at one energy (five-fold degenerate).
Step 2: Bring in an octahedral field. Six ligand lone pairs point straight at the metal along \(\pm x, \pm y, \pm z\). Electrons in metal d-orbitals feel repulsion from these lone pairs, but not equally.
Step 3: Sort the orbitals.
- Along-axis orbitals \(d_{z^2}\) and \(d_{x^2-y^2}\) (called \(e_g\)) point right at the ligands, so they are destabilised and go up in energy.
- Between-axis orbitals \(d_{xy}, d_{yz}, d_{zx}\) (called \(t_{2g}\)) avoid the ligands, so they are stabilised and go down.
Step 4: Picture and magnitude. The result is a two-tier pattern: a lower triple \(t_{2g}\) and an upper double \(e_g\), split by \(\Delta_o\). Keeping the average energy fixed, the \(e_g\) pair rises \(+0.6\,\Delta_o\) and the \(t_{2g}\) trio falls \(-0.4\,\Delta_o\).
Energy diagram (up = higher energy):
\(\;\;e_g:\ d_{z^2},\ d_{x^2-y^2}\ \ (+0.6\,\Delta_o)\)
\(\;\;\text{— barycentre —}\)
\(\;\;t_{2g}:\ d_{xy},\ d_{yz},\ d_{zx}\ \ (-0.4\,\Delta_o)\)
\[\boxed{\text{lower } t_{2g}(3) \text{ and upper } e_g(2),\ \text{split } \Delta_o}\]