Splitting of d orbitals in an octahedral crystal field:

In an octahedral crystal field, the d orbitals of a transition metal ion are split into two sets of orbitals with different energies. This splitting is a result of the electrostatic interaction between the metal ion and the surrounding ligands. The ligands exert a repulsive force on the d orbitals, causing the energy levels to shift.

The splitting of the d orbitals in an octahedral crystal field can be represented by a diagram known as the crystal field splitting diagram. In this diagram, the five d orbitals (dxy, dyz, dzx, dx^2-y^2, and dz^2) are shown as five lines, with the energy increasing from bottom to top.

Actual configuration of split d orbitals in an octahedral crystal field:

The actual configuration of the split d orbitals in an octahedral crystal field is determined by the relative values of Δ₀ (the crystal field splitting parameter) and the pairing energy (P).

1. Δ₀ (Crystal field splitting parameter):
- Δ₀ is a measure of the energy difference between the two sets of d orbitals.
- If Δ₀ is large, the splitting between the two sets of d orbitals is significant.
- A large Δ₀ results in a larger energy gap between the lower energy set of orbitals (eg) and the higher energy set of orbitals (t₂g).

2. Pairing energy (P):
- Pairing energy refers to the energy required to place two electrons in the same orbital.
- If P is small, it is energetically favorable to have electrons in different orbitals rather than paired in the same orbital.
- A small P value favors a high-spin configuration, where the electrons occupy the higher energy set of orbitals (t₂g) before pairing up.

Effect of relative values of Δ₀ and P:

1. When Δ₀ > P:
- The energy gap between eg and t₂g is large.
- It is energetically favorable for the electrons to occupy the lower energy set of orbitals (t₂g) before pairing up.
- This results in a low-spin configuration, where the electrons occupy the t₂g orbitals first and then pair up in the eg orbitals.

2. When P > Δ₀:
- The energy gap between eg and t₂g is small.
- It is energetically favorable for the electrons to occupy different orbitals rather than pairing up.
- This results in a high-spin configuration, where the electrons occupy the higher energy set of orbitals (eg) before pairing up.

In summary, the actual configuration of the split d orbitals in an octahedral crystal field is determined by the relative values of Δ₀ and P. If Δ₀ is larger than P, a low-spin configuration is favored, whereas if P is larger than Δ₀, a high-spin configuration is favored.