The metal's d-orbitals that are directly facing the ligands in K3[Co(CN)6] are:

🧠 Find the Orbitals That Point at Octahedral Ligands

In an octahedral complex, the 6 ligands sit on the ±x, ±y, ±z axes. The metal d-orbitals that have lobes pointing directly along these axes are:

• $d_{x^2-y^2}$ — lobes along ±x and ±y.
• $d_{z^2}$ — main lobes along ±z.

These two together form the $e_g$ set.

The other three — $d_{xy}, d_{xz}, d_{yz}$ — point between the axes (into the gaps between ligands), so they don't directly face the ligands. These are the $t_{2g}$ set.

🗺️ Why $e_g$ Is Higher in Energy

The two $e_g$ orbitals point at the ligands → maximum electron–lone-pair repulsion → destabilised relative to $t_{2g}$. That's the entire origin of $\Delta_o$ in CFT.

⚡ Quick Visual

| Set | Orbitals | Direction | Energy in oct field | |---|---|---|---| | $e_g$ | $d_{x^2-y^2}, d_{z^2}$ | along axes (at ligands) | higher | | $t_{2g}$ | $d_{xy}, d_{xz}, d_{yz}$ | between axes | lower |

⚠️ Don't Confuse $d_{z^2}$ Geometry

$d_{z^2}$ has a torus in the xy-plane and lobes along ±z, but the net density along z is what matters — those z-lobes do point at the axial ligands.

$\boxed{\text{Answer: (2) } d_{x^2-y^2}\text{ and } d_{z^2}}$