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When describing bonding in molecules with square pyramidal shape, like $\ce{BrF_5}$, the hybridisation scheme which is normally used is the $\ce{sp^3d^2}$ hybridisation. However, there is also another scheme of hybridisation—the $\ce{dsp^3}$ or $\ce{sp^3d}$ hybridisation. Usually, it produces molecules with trigonal bipyramidal geometry (when the d-orbital used in hybridisation is $\ce{d_{z^2}}$. When the $\ce{d_{x^2-y^2}}$ is used for $\ce{dsp^3}$ hybridisation, the shape of the molecule becomes square pyramidal.

The $\ce{\sigma}$-bonding framework in a square-pyramidal species may also be described in terms of an $\ce{sp^3d}$ hybridisation scheme. The change in spatial disposition of the five hybrid orbitals from trigonal pyramidal is a consequence of the the participation of a different d-orbital. [Inorganic Chemistry, 5th edition, Housecroft and Sharpe, Pearson] (bold letters are my addition, not in original source)

The only example of such a molecule that I could find in the book is $\ce{[Ni(CN)5]^3-}$. However, this complex ion also exists in trigonal bipyramidal shape, depending on the positive counterion. Both forms may also be present in the crystal, as in the case of $\ce{[Cr(en)3][Ni(CN)5]^3-}$.

So, does this mean that any compound which is square pyramidal can be described by the $\ce{dsp3}$ hybridisation state?

Is there any compound or ion which is only square pyramidal, and is best described by $\ce{dsp^3}$ state?

This question is quite similar, however, all I learnt from it is that hybridisation model is not always accurate.

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    $\begingroup$ You are well-intentioned, but I feel obliged to again say that in this context (basically anything involving d-orbitals), hybridisation is not a useful model. "Not always accurate" doesn't quite capture it: it often doesn't work, and when it does, it gives the right answers for the wrong reason. In this respect it is almost like the Bohr model, and you (presumably) wouldn't ask a question about how to use the Bohr model to describe a carbon atom, for example. $\endgroup$ – orthocresol Dec 23 '18 at 20:50
  • $\begingroup$ @orthocresol, I do understand that hybridisation model is quite outdated and not faithful, and there are now improved models of bonding like the Molecular Orbital theory, but since the hybridisation model is still taught in universities, colleges, and schools, they are important, at least for pedagogic reasons. $\endgroup$ – Shoubhik Raj Maiti Dec 23 '18 at 21:02
  • $\begingroup$ I don't fully agree with that, but it is off-topic for the comments section... oh well. At the very least, my comment can serve as a disclaimer to other readers, not all of which will have learnt the same model as you (I, for one, was never taught all this hybridisation stuff growing up). $\endgroup$ – orthocresol Dec 23 '18 at 21:04
  • $\begingroup$ Nothing is is "best described by dsp3 state". $\endgroup$ – Mithoron Dec 23 '18 at 22:41

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