Q. Do set of d orbitals split equally w.r.t the average energy of d orbitals in Crystal Field?

No. The reason is that the total energy must be constant. If we consider a d’° ion such as Zn²⁺ in an octahedral field, the destabilisation caused by the four electrons in the dx²−y² and dz² orbitals must be offset by the stabilisation gained by the six electrons in the dxy, dzx and dz² orbitals. This requires that

(6 × the energy of dxy, dyz, dzx orbitals) + (4 × the energy of dx²−y², dz² orbitals) = 0

That is, the energy gained equals the energy lost. Also, the energy difference between the orbitals is:

(the energy of dx²−y², dz²) − (the energy of dxy, dyz, dzx) = Δₒ

To satisfy these two relationships, it is necessary that

the energy of dxy, dyz, dzx = −0·4 Δₒ

the energy of dx²−y², dz² = −0·6 Δₒ

The splitting is as shown:

The splitting of the two groups of orbitals is not equal with respect to the average energy of the d orbitals.