Q.Describe how the five degenerate $d$-orbitals of a free metal ion split when six ligands approach it along the Cartesian axes to form an octahedral complex. Name the two resulting sets of orbitals, state their relative energies, and define the crystal field splitting energy $\Delta_o$.

[!TLDR] $e_g$ ($d_{z^2}, d_{x^2-y^2}$) point at the 6 ligands → raised by $0.6\Delta_o$; $t_{2g}$ ($d_{xy},d_{yz},d_{xz}$) point between them → lowered by $0.4\Delta_o$. [!ANSWER] Two orbital sets, $t_{2g}$ (lower, $ …

In an octahedral complex, six ligands approach the metal ion symmetrically along the $\pm x$, $\pm y$, $\pm z$ axes. Two d-orbitals, $d_{z^2}$ and $d_{x^2-y^2}$ (together the $e_g$ set), have their lobes of electron density pointing directly along these same axes, so they experience the strongest possible electrostatic repulsion from the six approaching ligand charges, raising their energy. The other three orbitals, $d_{xy}$, $d_{yz}$, and $d_{xz}$ (together the $t_{2g}$ set), have their lobes pointing into the gaps between the axes, away from the direct line of ligand approach, so they experience comparatively weaker repulsion, lowering their energy relative to the $e_g$ set. Measured relative to the hypothetical average ('barycentre') energy the five orbitals would share in a perfectly spherical field, the three $t_{2g}$ orbitals are lowered by $0.4\Delta_o$ and the two $e_g$ orbitals are raised by $0.6\Delta_o$ — values chosen so the total energy change across all five orbitals sums to zero if all were equally populated ($3\times(-0.4) + 2\times(0.6) = 0$). The overall energy gap between the two sets …

Identify which d-orbitals point directly along the axes of ligand approach (experience strongest repulsion, raised) versus which point between the axes (weaker …

Mixing up which set is which — the $e_g$ set (only 2 orbitals) is HIGHER in energy in an octahedral field, and the $t_{2g}$ set (3 orbitals) is LOWER; this is the opposite orde …