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I thought that d-orbitals were used to "extend the octet rule" in these molecules until I looked it up on Google and saw that they actually contributed only a little to the molecule. A little, but not zero.

  1. Is there a hypervalent period-3-element molecule that has mathematically been proved not to involve d-orbitals at all?

  2. Do they have a role, any role, in the chemical properties of the final molecules?

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    $\begingroup$ Answer to question 1: yes, pretty much all of them. $\endgroup$ – Jan Oct 25 at 13:35
  • $\begingroup$ @ regulars: I wonder if this can serve as a canonical d-orbital (non)-participation dupe target =D $\endgroup$ – Jan Oct 25 at 13:36
  • $\begingroup$ Partial answer to 2: They have an effect on the polarisation of the electron density. || This could well be a canonical question; @Jan you're a regular, I'm counting on you ;) $\endgroup$ – Martin - マーチン Oct 25 at 13:47
  • $\begingroup$ @Jan really? Because one Chemistry.SE post I saw (Google the words "d-orbital SO3", quote marks and all) said that d-orbitals contributed to 1% of the mathematical hybrid, 1% but still not 0%, in sulfur trioxide... $\endgroup$ – Just A Young Artist Oct 25 at 13:50
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    $\begingroup$ The explanation for this is that these are most likely polarisation functions, which do not correspond to actual orbitals (even in the MO framework not). If you would remove those functions when you attempt to solve for the wave function, you will still get a very, very good approximation. And since orbitals are a mere mathematical concept, you can include or exclude them at will; we don't know what is right. $\endgroup$ – Martin - マーチン Oct 25 at 14:34
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The conclusion is left to the reader. I believe that there is no useful value in involving d-orbitals in the description of "hypervalent" molecules while remaining on the level of the octet rule.


A typical example for when the octet rule seems to fail is sulfur hexafluoride.

The naive approach

Let's do a quantum chemistry calculation. Using a PBE-D3/def2-TZVP geometry for $\ce{SF6}$, we will take a look at the orbital coefficients of the basis functions centered on $\ce{S}$ from a HF/def2-SVP single point calculation:

                11        12        13        14        15        16   
             -1.81184  -1.68415  -1.68415  -1.68415  -1.63226  -1.63226
              2.00000   2.00000   2.00000   2.00000   2.00000   2.00000
             --------  --------  --------  --------  --------  --------
0S   1s      0.039494 -0.000000 -0.000000 -0.000000  0.000000 -0.000000
0S   2s      0.243289 -0.000000 -0.000000 -0.000000  0.000000 -0.000000
0S   3s     -0.378889  0.000000  0.000000  0.000000 -0.000000  0.000000
0S   4s      0.060237 -0.000000 -0.000000 -0.000000  0.000000 -0.000000
0S   1pz    -0.000000 -0.083520 -0.002820 -0.008492 -0.000000  0.000000
0S   1px     0.000000 -0.008939  0.022821  0.080342 -0.000000  0.000000
0S   1py    -0.000000  0.000390 -0.080789  0.022991  0.000000  0.000000
0S   2pz    -0.000000 -0.137305 -0.004635 -0.013960 -0.000000  0.000000
0S   2px     0.000000 -0.014696  0.037518  0.132082 -0.000000  0.000000
0S   2py    -0.000000  0.000641 -0.132816  0.037798  0.000000 -0.000000
0S   3pz     0.000000  0.015799  0.000533  0.001606  0.000000 -0.000000
0S   3px    -0.000000  0.001691 -0.004317 -0.015198  0.000000 -0.000000
0S   3py    -0.000000 -0.000074  0.015282 -0.004349 -0.000000  0.000000
0S   1dz2   -0.000000 -0.000000  0.000000  0.000000  0.091291 -0.003993
0S   1dxz   -0.000000 -0.000000 -0.000000 -0.000000  0.000000  0.000000
0S   1dyz    0.000000  0.000000 -0.000000  0.000000  0.000000  0.000000
0S   1dx2y2  0.000000  0.000000 -0.000000 -0.000000  0.003993  0.091291
0S   1dxy    0.000000 -0.000000  0.000000  0.000000  0.000000  0.000000

Alright, on MO 15 there is a d-orbital coefficient of 0.09. That's a d-orbital contribution. End of story.

Well, no. Let's do a few more calculations. Same as above, but let's dispense with the d-functions on $\ce{S}$ (HF/def2-SVP-d).

                11        12        13        14        15        16   
             -1.88534  -1.73860  -1.73860  -1.73860  -1.67047  -1.67047
              2.00000   2.00000   2.00000   2.00000   2.00000   2.00000
             --------  --------  --------  --------  --------  --------
0S   1s      0.043781  0.000000  0.000000 -0.000000 -0.000000  0.000000
0S   2s      0.271803  0.000000  0.000000 -0.000000 -0.000000  0.000000
0S   3s     -0.432659 -0.000000 -0.000000  0.000000  0.000000 -0.000000
0S   4s      0.055247  0.000000 -0.000000 -0.000000  0.000000 -0.000000
0S   1pz    -0.000000  0.091235  0.002771 -0.009073  0.000000 -0.000000
0S   1px     0.000000  0.009479 -0.023157  0.088248  0.000000 -0.000000
0S   1py    -0.000000 -0.000375  0.088712  0.023319 -0.000000 -0.000000
0S   2pz    -0.000000  0.153047  0.004648 -0.015220  0.000000 -0.000000
0S   2px     0.000000  0.015901 -0.038846  0.148037  0.000000 -0.000000
0S   2py    -0.000000 -0.000629  0.148816  0.039118 -0.000000  0.000000
0S   3pz     0.000000 -0.004655 -0.000141  0.000463 -0.000000  0.000000
0S   3px    -0.000000 -0.000484  0.001181 -0.004503 -0.000000  0.000000
0S   3py    -0.000000  0.000019 -0.004526 -0.001190  0.000000 -0.000000

So the orbital order and symmetry are identical and the orbital energies are very similar. Thus: Are the d-orbitals of vital importance to this system? Let's do the same thing after adding a f and a g-function to $\ce{S}$ (HF/def2-SVP+fg). Turns out there are maximum f and g orbital coefficients in the bonding orbitals of $0.024$ and $0.017$, respectively. Does that mean we now have to consider spdfg-hybrid orbitals?

Let's do the same kind of calculations for $\ce{CH4}$. For def2-SVP (one d-function on $\ce{C}$), we find a maximum d-orbital coefficient of $0.022$. d-orbitals are now important for $\ce{C}$? (For def2-SVP+fg, the maximum f and g-orbital coefficient are actually below $0.01$.) It turns out just looking at numerical results is not going to help us getting very far.

Stepping back

We need to remember that hybridization and nice MO diagrams are cleaned up, idealized versions of quantum chemistry calculations. They are mainly educational tools for undergrad students and zeroth-order approximations to be kept readily available in a chemist's head. I cannot think of a purpose for d-orbitals on $\ce{S}$ in that regard. If you can, continue to use them. I will cite the (unfortunately German-only) book of Riedel$^{[1]}$, which states in the 5th edition (p. 114, translated by me):

In conclusion: The significance of d-orbitals in nonmetal [main group] chemistry is not settled. The pros and cons of d-orbital participation are being theoretically debated.

In the 6th edition, however (p. 111):

For a long time, it was supposed that in nonmetals of the 3rd row and below, d-orbitals participate in hybridization and that these hybrid orbitals (such as d$^2$sp$^3$ or dsp$^3$) form bonds.

This reflects the shift in the interpretative thinking on results similar to the ones above. Unfortunately, no sources, primary or otherwise, are being cited in Riedel's book.

A few words on hybridization

We also need to remember that hybridization does not occur except in our minds. It is not part of the quantum chemical calculations themselves or of observable phenomena. Rather, it is a posteriori approach to making quantum chemical results more accessible to other chemists. As such, it is useful and there is a place for it. It introduces a language that helps in the description of a lot of (organic) chemistry. But it cannot succeed as a standalone theory and application beyond the 2nd row of the periodic table fails rather easily. (Consider the bonding angles of $\ce{H2O}$ (102°) and $\ce{H2S}$ (92°). The former fits nicely with sp$^3$ hybridization, the latter looks more like a result of p-s $\sigma$-bonding.)


$^{[1]}$ Erwin Riedel: Anorganische Chemie. De Gruyter, New York & Berlin. 5th edition 2002, 6th edition 2004. With contributions by Christoph Janiak.

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