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How do ligands split the energies of a metal ion’s d orbitals in an octahedral complex?

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Structural diagram of the vitamin B12 coenzyme molecule, highlighting its organometallic core with five nitrogen-cobalt bonds and one cobalt-carbon bond at the center, surrounded by complex organic groups. — coordination compound Coordination compounds contain a central metal atom surrounded by nonmetal atoms or groups of atoms, called ligands. For example, vitamin B₁₂ is made up of a central metallic cobalt ion bound to multiple nitrogen-containing ligands.

In an octahedral complex, ligands split the energies of a metal ion’s five d orbitals into two distinct subsets due to the electrostatic repulsion between the electrons in the metal's d orbitals and the negative charges (or lone pairs) of the surrounding ligands.12

When six ligands are arranged at the vertices of an octahedron, the spatial orientation of the d orbitals relative to these ligands determines their energy levels:

$e_{g}$ orbitals: The $d_{z^{2}}$ and $d_{x^{2} - y^{2}}$ orbitals point directly toward the ligands, experiencing greater electrostatic repulsion and thus occupying a higher energy level.12
$t_{2 g}$ orbitals: The $d_{x y}$ , $d_{x z}$ , and $d_{y z}$ orbitals are directed between the ligands, experiencing less repulsion and occupying a lower energy level.12

The energy difference between these two sets is known as the crystal field splitting energy, denoted as $Δ_{o}$ (where o stands for octahedral). The magnitude of $Δ_{o}$ depends on the nature of the ligands; strong-field ligands cause a large splitting, while weak-field ligands result in a small splitting. This splitting is fundamental to understanding the magnetic, spectroscopic, and thermodynamic properties of coordination compounds.32