In an octahedral complex, explain the splitting of d-orbitals into t2g and eg sets — JEE Chemistry

In an octahedral field, ligands approach along the x, y, z axes.

• $e_g$ set: $d_{z^2}$ and $d_{x^2-y^2}$ — lobes point directly toward ligands → higher energy
• $t_{2g}$ set: $d_{xy}$, $d_{yz}$, $d_{xz}$ — lobes point between axes → lower energy

$$\Delta_0 = E(e_g) - E(t_{2g})$$

$$e_g raised by +\frac{3}{5}\Delta_0; \quad t_{2g} lowered by -\frac{2}{5}\Delta_0$$

$$e_g is higher in energy by \Delta_0$$