How do the d orbitals of a transition metal ion become non degenerate?

The d orbitals of a transition metal ion become non-degenerate primarily due to the presence of ligands around the metal ion, which leads to a phenomenon known as crystal field splitting.

In a free transition metal ion, the five d orbitals (d_xy, d_yz, d_zx, d_x²-y², d_z²) are degenerate, meaning they have the same energy level. However, when the metal ion is placed in a crystal field created by surrounding ligands, the arrangement and types of ligands can affect the energy levels of these orbitals.

In an octahedral field, for instance, the ligands approach the central metal ion along the axes. This causes the d_x²-y² and d_z² orbitals, which point directly towards the ligands, to be raised in energy due to increased electron-electron repulsion. Conversely, the d_xy, d_yz, and d_zx orbitals, which lie between the axes, experience less repulsion and thus have lower energy. This results in a splitting of the d orbitals into two sets of different energy levels: the lower-energy triply degenerate orbitals (t_2g) and the higher-energy double degenerate orbitals (e_g).

Similarly, in a tetrahedral field, the d_xy, d_yz, and d_zx orbitals experience higher energy, while the d_x²-y² and d_z² orbitals become lower in energy, further demonstrating how the environment of the metal ion leads to non-degenerate d orbitals.

Ultimately, the specific arrangement and the strength of the ligands influence how the d orbitals split. This is critical in determining the electronic structure and, consequently, the chemical properties of the transition metal complexes.