My previous post outlined the issue of the quadruple bond in C₂, and an interesting issue is, how do you visualize it? I think this is important because it permits the qualitative reasoning that may be more immediately useful to chemists, but there is something else. Sometimes, by looking at a problem in a different way, you get a different perspective. Thus through using the valence bond/hybridization perspective Shaik et al. (Nature Chem DOI:10.1038/NCHEM.1263) consider the bonds in C₂ as follows: a σ bond employing sp orbitals, accompanied by two π bonds arising from the pairing of two p electrons from each carbon atom, plus the additional electron in each carbon atom occurs in outward pointing hybrids, (i.e. with axes along the axis of the acetylenic cylinder). It would usually be considered that the directionality would prevent bond formation. Computations, however, show this is not the case.
 
In accord with the theme of this blog, is there an alternative way of looking at this? For me, yes, and it reminds me of my first theoretical paper! (Confession: not the best presented paper ever.) The problem then was that substituents adjacent to a cyclopropane ring experienced chemical effects different from those attached to an unstrained ring. These were being explained by partial charge delocalization from the cyclopropane ring, but my point was that the effects of strain were not properly considered. The changes in chemical effects were certainly explained in terms of an electric field at the substituent adjacent to a strained system that differed from that of a standard alkyl system, but the question was, was that due to charge delocalization, or to the strain energy? From Maxwell's electromagnetic theory, the work done moving electric charge behaves as if it is stored in an electric field derived from the accompanying polarization field. Charge was moved, but was the movement satisfactorily explained by constraining the movement to within the strained system?
 
To assess the strain energy, after a little mathematics and an assumption I subsequently did not find convincing, I came up with the strain energy being proportional to [sinθ/2]/√r, θ/2 being the angle that that bond deformed (θ the change of bond angle.) and r the new covalent radius. I was quite excited when I found out this was quite accurate; I was less so when I realized my "derivation" was simply too questionable. Accordingly, I simply placed it into the paper as an empirical proposition. However, if you take the bond energy scheme recently (then!) determined thermochemically by Cox, very good results were obtained for ethylene and acetylene. (That does not imply these systems did not delocalize electrons, but it did imply they did not if there was no adjacent unsaturation.)
 
Thus in this picture, acetylene was described as three bent sp³ bonds. As an aside, I could have made a prediction of the strain in [1,1,1] propellane, and I would have been the first to comment on it. Why didn't I? Partly because I never thought about it, but mainly because this part of the paper was an aside, to get the strain energy I needed. The objective was to calculate fields on adjacent substituents, and leaving aside the methylene carbons, there are no substituents adjacent to the junctions of a propellane, so that molecule was outside the scope of the paper.
 
Returning to C₂, the axial wave, containing the single electron is also inherently sp³ in this picture. Why that is relevant is that the sp³ wave has a primary wave with which everyone is familiar, but also a small lobe that is on the opposite side of the carbon atom. We get the fourth bond if these two small lobes constructively interfere. That argument says that a fourth bond is conceivable; what it does not show is whether the fourth bond has any net energy, and that requires calculation. It also requires a better definition of that small lobe. (Note that this picture is merely a different way of viewing the molecule. The concept of hybridization is simply one of combining component waves. When combining different waves of different energies, there are various ways that it can be done, provided the energies are properly accounted for.)
 
Does the different description offer anything? I think yes. In 2009 Wu et al. (Angew. Chem. Int. Ed. 48: 1407 –1410) describe the bond in [1,1,1] propellane as an inverted bond, i.e. they seem to consider the orbitals to have inverted and become directed inwards instead of outwards, however in my picture, it is a "normal" bond, derived from bicyclobutane, with all bonds more strained. However, it does explain why the "internal" bond in the propellane is relatively strong, and that in C₂ so weak. Of course, computations show this too, and in some ways more convincingly, nevertheless the qualitative view might at least show some experiments that might be worth doing. For example, if such an "sp³ orbital" were to invert, there would be a significant change in electric moment of the molecule, which, again from Maxwell's theory, would be promoted by the absorption of a photon. Thus suitable molecules should have a significant change in their UV spectra. Besides C₂, which in this picture should have relatively long-wavelength transitions, we might even consider something like 1,4-diazabicyclo[2.2.2]octane, even though it has two lone pairs with nowhere obvious to go. Tertiary amines usually have a weak UV absorption at about 215 nm, but for 1,4-diazabicyclo[2.2.2]octane I would expect the UV spectrum to show a significant spectral shift due to a new interaction between the nitrogen atoms. Would anyone care to run me a spectrum? I am curious now to see if this reasoning is correct.
 
Of course, the more thoughtful out there might argue that while the small lobe of an sp³ orbital might interfere, there is a reason they may not be capable of inverting in standard quantum mechanics. Can you see it?