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According to crystal field theory (CFT), $\mathrm{d}$-orbitals of central metal atom splits due to non uniform repulsion of ligands around the metal atom. Why isn't $\mathrm{p}$-orbital splitting discussed?

Argument: As in case of square planar complexes $\mathrm{(dsp^2)},$ $\mathrm{p}$-orbitals are involved which are facing different conditions of repulsion. So, $\mathrm{p}_{xy}$ must have higher energy than $\mathrm{p}_{xz}$ or $\mathrm{p}_{yz}.$ This happens in case of trigonal pyramidal structure too.

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According to Crystal Field Theory, d-orbitals of central metal atom splits due to non uniform repulsion of ligands around the metal atom. Why isn't p-orbital splitting discussed?

To be blunt, the reason nobody mentions p-orbital splitting or non-splitting is that neither the teacher nor the learner has ever understood the crystal field theory properly in its true depth (including myself). They just propagate what they saw in standard textbooks, repeat the same in class, the students pass never to see CFT again. The story ends. If that student ever becomes a teacher, the story continues.

You are mixing apples and oranges. Crystal field theory has nothing to do with hybridization. The original work of Hans Bethe, the man behind crystal field theory, wrote a 72-paged highly theoretical paper in Annalen der Physik in German. The translations are available and I quote the abstract where he does clarify what happens to p-orbitals. How many can claim that that they read this paper completely before teaching CFT. I cannot. The abstract clarifies your misconception.

The influence of an electric field of prescribed symmetry (crystalline field) on an atom is treated wave - mechanically . The terms of the atom split up in a way that depends on the symmetry of the field and on the angular momentum I (or J) of the atom . No splitting of s terms occurs , and p terms are not split up in fields of cubic symmetry . For the case in which the individual electrons of the atom can he treated separately ( interaction inside the atom turned off) the elgenfunctions of zeroth approximation are stated for every term in the crystal; from these there follows a concentration of the electron density along the symmetry axes of the crystal which is characteristic of the term. - The magnitude of the term splitting is of the order of some hundreds of cm-1. - For tetragonal symmetry, a quantitative measure of the departure from cubic symmetry can be defined, which determines uniquely the most stable arrangement of electrons In the crystal.

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    $\begingroup$ Nobody ever needs to read about CFT ever again! It is outdated and deprecated. It should not be taught anymore, especially not in the truncated form that is so popular. The successor, ligand field theory, is somewhat easier understood and more robust. $\endgroup$ – Martin - マーチン Apr 14 at 15:00
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    $\begingroup$ You keep telling me that I am outdated! All right, I am, and so are school textbooks. But these concepts have their importance too. My textbook has mentioned about LFT, but not explained it. Can you add a link that explains it sufficiently enough? $\endgroup$ – B.Anshuman Apr 14 at 16:22
  • $\begingroup$ CFT is a part and parcel of most undergraduate chemistry curricula $around$ the world - Martin. Pray tell me which university has completely outcast it? $\endgroup$ – M. Farooq Apr 14 at 16:24
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    $\begingroup$ Ligand field theory $depends$ on crystal field theory or uses the knowledge of the principles developed in CFT. Focus on CFT first and then move on the ligand field theory. One thing at a time. I said mixing of apples and oranges, there is no need to mix two independent ideas of hybridization, with CFT. The person who developed it (Hans Bethe), did not need the ideas to invoke hybridization. $\endgroup$ – M. Farooq Apr 14 at 16:35
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    $\begingroup$ Crystal field theory is to me a good rule of thumb for how orbitals are split in a complex. When it is time to render precise predictions with this splitting then we go to its successores $\endgroup$ – Oscar Lanzi Apr 20 at 18:55

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