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I know that, although many elementary books say that d-orbitals are involved in the bonding of hypervalent main group compounds period 3 or higher such as the sulfate ion but in reality they are only barely, if at all, involved in bonding.

The question is- is it true that "the d-orbitals are not directly involved in bonding but rather in polarisation" and what does that exactly mean? I saw that statement on the comment sections of a post of mine but don't know the proof.

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Well, please be more specific about the molecule(s) in question for that quotation. d-orbitals are certainly involved in some bonding, especially for the transition metals, but you are right that the participation of d-orbitals in the bonding of main group covalent molecules and polyatomic ions has been overstated in the past and is in fact fairly minimal.

As you mentioned, there is very little, if any, d-orbital participation in the bonding of sulfate. As for d-orbitals being involved in polarization but not bonding, that literally does not make sense for something like sulfate. Sulfur's d-orbitals are empty, so it is impossible for them to polarize without bonding. Only when there are occupied d-orbitals (as in transition metals) can non-bonding d-orbitals affect things like the shape and polarity of the compound (as in crystal field theory/LFT).

So, long story short, I think that comment you read is simply wrong, and your original understanding is correct. See the below references if you want to read more about it.

References

  1. Reed, A. E.; Schleyer, P. v. R. Chemical Bonding in Hypervalent Molecules. The Dominance of Ionic Bonding and Negative Hyperconjugation over d-Orbital Participation. J. Am. Chem. Soc. 1990, 112 (4), 1434–1445. https://doi.org/10.1021/ja00160a022.
  2. Cunningham, T. P.; Cooper, D. L.; Gerratt, J.; Karadakov, P. B.; Raimondi, M. Chemical Bonding in Oxofluorides of Hypercoordinate Sulfur. Faraday Trans. 1997, 93 (13), 2247–2254. https://doi.org/10.1039/a700708f.
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    $\begingroup$ I am certain the original comment about d orbitals makes sense in a way. In order to have a p orbital polarised (mathematically) you need d orbital like functions. In approximate solutions of the Schrödinger equation, like HF, MP2, KS-DFT, etc. these would show up as d orbital contributions. These functions (or orbitals) then also do partake in binding. $\endgroup$ – Martin - マーチン Dec 3 '19 at 10:42
  • $\begingroup$ I concur with Martin. In computational Chemistry, orbitals of higher angular momentum are included in extended basis sets, and these polarisation functions add directionality to the bonding (for example, they allow the p orbitals in the antibonding π* orbital of a diatomic to be polarised away from each other, better representing the actual situation as compared to a minimal basis set). $\endgroup$ – ANZGC FlyingFalcon Dec 3 '19 at 17:19

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