Step 1: Crystal Field Theory treats the metal-ligand bond as a purely electrostatic attraction between the positive metal ion and the negative (or lone-pair) ends of the ligands. We only track what this field does to the five d orbitals.
Step 2: Isolated metal ion: all five d orbitals are equal in energy (degenerate).
Step 3: Place six ligands at the corners of an octahedron, i.e. along the coordinate axes. The d orbitals split into two groups depending on their spatial direction:
\[ e_g = \{d_{x^2-y^2},\, d_{z^2}\} \;\text{(point at ligands)} \]
\[ t_{2g} = \{d_{xy},\, d_{yz},\, d_{zx}\} \;\text{(point between ligands)} \]
Step 4: Electrons in the \(e_g\) orbitals sit head-on with the ligand electrons, so repulsion is large and these orbitals go up in energy. The \(t_{2g}\) orbitals avoid the ligands, so they are relatively stabilised (go down).
Step 5: The vertical gap between the \(t_{2g}\) and \(e_g\) levels is the crystal field stabilisation gap \(\Delta_o\) (also written \(10\,Dq\)). Keeping the mean energy fixed, three \(t_{2g}\) orbitals drop by \(0.4\Delta_o\) and two \(e_g\) orbitals rise by \(0.6\Delta_o\), so total energy is conserved: \(3(0.4\Delta_o) = 2(0.6\Delta_o)\).
Step 6 (simple level picture):
Higher level (two orbitals, \(e_g\)) — nope, stated as: two orbitals up, three orbitals down, separated by \(\Delta_o\).
Step 7: The size of \(\Delta_o\) (set by the ligand's position in the spectrochemical series) controls the colour absorbed and whether electrons pair up (low spin) or spread out (high spin).
\[\boxed{\Delta_o = E(e_g) - E(t_{2g})}\]