Question

Using Crystal Field Theory explain how d orbital splitting arises in an octahedrarl field.

Answer 1

The d_z2 and d_{x2 - y2} orbitals lie along the three axis, while the three d_xy, d_xz, d_yz orbitals lie in the plane between the three major axis.(See picture)

5 d-orbitals of a free ion are all degenerate and have the same energy. In an octahedral field, 6 ligands approaches from the 6 vertices of a regular octahedron. As the 6 ligands approach the metal ions, the ligands will interact with the d-orbitals of the ion raising the energies of the orbitals. But the rise of the energies will not be the same. As d_z2 and d_{x2 - y2} orbitals lie along the axis, they will interact more with the ligands and energy rise will be higher than the other three orbitals(d_xy, d_xz, d_yz).   

d_z2 and d_{x2 - y2} (e_g set) will be degenerate and the other three d_xy, d_xz, d_yz (t2g set) will be a separate degenerate state.

With respect to a spherical field known as the barycenter (see picture), the t_2g set becomes lower in energy than the orbitals in the barycenter and the e_g orbitals are higher in energy than in the barycenter. If the splitting of the d-orbitals in an octahedral field is Δ_o, the three t_2g orbitals are stabilized relative to the barycenter by 0.4 Δ_o, and the e_g orbitals are destabilized by 0.6 Δ_o.