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

Question

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?

Answer 1

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.

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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:

• Why do compounds like SF6 and SF4 exist but SH6 and SH4 don't?

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

• Ronald J Gillespie, Bernard Silvi, Coord. Chem. Rev. 2002, 233-234, 53-62. The octet rule and hypervalence: two misunderstood concepts;
• Ronald J. Gillespie, Edward A. Robinson, Inorg. Chem. 1995, 34 (4), 978–979. Hypervalence and the Octet Rule;
• Thomas A. Halgren, Leo D. Brown, Daniel A. Kleier, William N. Lipscomb, J. Am. Chem. Soc. 1977, 99 (21), 6793–6806. Apparent octet rule violations, fractionality, and the interpretation of localized molecular orbital structures. Polarization and hybridization functions in chemical bonding.