# Does >2e contribution of an atomic orbital (AO) to a hybrid AO of a molecule violate Pauli exclusion?

This question is inspired by On the Role of d Orbital Hybridization in the Chemistry Curriculum J. Chem. Educ., 2007, 84 (5), p 783.

The article calculates 6 bonding sulfur atomic orbitals in SF6 as each having a 0.768s term (s meaning 3s). This implies $6\times0.768^2 = 3.5$ of the 6 valence electrons corresponding to 3s character.

Is it valid to say there is that much 3s character?

• I would agree that this looks strange, but without details of the calculations it is difficult to understand what all this means. Suddenly, "Readers requiring further details concerning theoretical methods are asked to contact the author directly." Without details we could only speculate... – Wildcat Aug 2 '16 at 14:48
• Just a random thought: IIRC, a bonding orbital does not necessarily implies a filled one. So, while there might exist 6 such hybrid orbitals, they should not necessarily be all filled (with 6 electrons). – Wildcat Aug 2 '16 at 14:50
• @Wildcat later the article does say "Thus, the six bonds in SF6 can be considered to be made up of four bonding orbitals and two nonbonding orbitals located primarily on the fluorines resulting in a bond order of 2/3 for each SF bond", although even 2/3 of 3.5 is greater than 2. – DavePhD Aug 2 '16 at 15:00
• I also agree that this look incredibly strange, but I need to look at the article which I cannot currently do. It would be interesting to see what the rest of the orbitals look like. What happened to the p orbitals? And how did they calculate the value. – Martin - マーチン Aug 2 '16 at 15:22
• @Martin-マーチン The 6 sulfur hybrid atomic orbitals (disregarding the small 3d terms) are: 0.768s +0.636pz, 0.768s -0.636pz, 0.768s +0.636py, 0.768s -0.636py, 0.768s +0.636px, 0.768s -0.636px. – DavePhD Aug 2 '16 at 15:41

The answer to your question is yes, such a contribution violates the Pauli principle, and is hence not a meaningful wave function.

The demonstration of this calculation (paper) aimed at a different statement and is unfortunately very thin on the calculation details. Furthermore it is a bit difficult to distinguish between the different models they use.
However, I believe the key is actually given in the first sentence of the "Results and Discussion" section (see emphasis below). As far as I understand it, the formula presented in equations (2) are not normalised. Furthermore it neglects any ionic contribution and/or occupation numbers. This further manifests itself that the structure is about 600 kcal/mol higher in energy than the resonance hybrid.

When the $\ce{3s}$, $\ce{3_{$px,y,z$}}$, and $\ce{3d_{$x^2−y^2$}}$ sulfur AOs are free to mix in any proportion to form six bonding orbitals on the central sulfur, the resulting energy is 1305.8 kcal mol1 (entry 1, Table 1) with sulfur orbital mixing as described by [..., equation (2)].
John Morrison Galbraith, J. Chem. Educ. 2007, 84 (5), 783-787. (emphasis mine)

While the wave function constructed in equation (3) seems to obey the criteria of a physically meaningful wave function, but it produces an energy even higher.

After that they suddenly jump into the MO picture of the argument, but not really explaining that. In this description one can clearly see in figure 3, that the s orbital only contributes to one molecular orbital.

While I completely agree with the basic premise of the paper, from my point of view it completely fails to present a compelling argument. For example: how did they come up with an energy of >300 kcal/mol of $\ce{SF6}$ relative to the $\ce{S + 6F}$ fragmentation? (And who uses Gaussian98 in 2007?)

I personally would rather recommend our own discussion:

Last but not least I would like to recommend some educational papers that I find more enlightening on the matter (no order):

• Martin, what do you mean by "formula presented in equations (2) are not normalized". (0.768)^2 + (0.635)^2 + (0.079)^2 = 0.999. The sum of the squares of the coefficients are 1 for each of the 6 equations. Do you mean something else by normalized? – DavePhD Aug 4 '16 at 12:20
• @DavePhD at the time I wrote it, it made perfect sense to me. I can't puzzle it together on my phone now and I'm going on vacation soon. I have another go at it soon. – Martin - マーチン Aug 5 '16 at 17:00
• Thanks Martin, enjoy your vacation. Maybe you could eventually add more about in principle whether or not a linear combination of atomic orbitals (as an approximation of the actual best orbital approximation of the wavefunction) is strictly limited to a 2 electron contribution from a single atomic orbital. Does the fact that the quantum numbers for the atomic orbitals are not actual quantum numbers of the molecule loosen the requirement at all? – DavePhD Aug 5 '16 at 17:41