Henry Rzepa's blog

In this follow-up to the previous post, I will try to address the question what is the nature of the bonds in penta-coordinate carbon?

This is a difficult question to answer with any precision, largely because our concept of a bond derives from trying to define what the properties of the electrons located in the region between any two specified atoms are. Such a local picture is somewhat at variance with the idea of electrons being delocalized across the entire molecule. Two procedures for analyzing the local electronic behaviour which we have been using recently are AIM (Atoms-in-Molecules) and ELF (the topology of the Electron localization function). There are many useful published articles which elaborate these concepts; if you want to read some of them, start at DOI 10.1021/ct8001915 and follow the cited articles.

Firstly, the AIM analysis of the system below, where X=cyclopentadienyl anion and Y=CN.

This is shown below. If you click on the image, you will see a rotatable version of this diagram. The coloured (red, yellow and green) dots represent so-called critical points in the curvature of the electron density function ρ(r). The red dots are known as bond critical points, or BCPs. These (almost) always are found along the line connecting two atoms which we tend to refer to as a bond. You will see two that have been circled in the diagram below, and these appear to show a bond connecting the central 5-coordinate carbon atom and a carbon of each of the cyclopentadienyl rings (which themselves are revealed as rings by the presence of a yellow dot). Indeed, that central carbon atom does seem to have five red dots radiating out along lines connecting it to five carbon atoms.

So is the case proven for pentavalent carbon? Well, no. Firstly, one has to inspect the value of ρ(r) at the circled red dot. This has a (calculated) value of 0.022 au and a calculated bond length of ~2.7Å. We need to calibrate this against a real system as reported in DOI: 10.1021/ja710423d (below):

Here, the electron density ρ(r) was actually measured using X-ray diffraction, and found to be ~0.017 for bond critical points found connecting the central carbon and each of the four oxygen atoms. The length of these “bonds” was measured as  ~2.7Å. The agreement with our frozen transition state is quite striking.

One can go a little further and inspect the (2nd) derivative of the electron density at the bond critical point, termed the Laplacian, or ∇²ρ, which tells what kind of “bond” one might have. The measured value of ∇²ρ for the system above was ~0.06 au, and the calculated value for our pentacoordinate system is 0.04 au, which again suggests we are dealing with a very similar interaction in both systems (one hypothetical and calculated, the other real and measured). The use of the term  interaction was deliberate.  It is less loaded than the term  bond. Thus the value of ρ(r) for an undisputed C-C single bond is around 0.28 au, around ten times higher than our putative bonds. Since we do not really wish to grace a ρ(r) value of 0.022 with the term decibond (or any other fraction of a single bond) perhaps it is best to call it just an interaction, and leave open the question of how strong that interaction is! So, despite the  AIM analysis  finding a bond critical point, we shall settle for interpreting that merely as an interaction, and not a bond!  Well, is an interaction (or come to that, a decibond) worthy of counting towards a coordination?  Perhaps!

So AIM can provide information about the curvature and density of the electrons in the region of a bond/interaction. But it does not provide any information about another simple question which the term bond implies. How many electrons might be involved? Ever since  G. N. Lewis coined the term two-electron bond in 1916,  we have become used to interpreting bonds in terms of simple (often integer) numbers of electrons.  A carbon-carbon single bond shares two electrons; a double bond four electrons, and so on. We use this concept all the time in the technique known  as arrow-pushing, which helps us delineate mechanisms of reactions. Might it be possible to  identify how many electrons are involved in bonds/interactions of the captured  S_N2 species above? Enter the ELF technique. It would not be appropriate to delve into the theory of this method here; suffice to say that  (approximately), the  bond-critical-point of the  AIM analysis in this case would map to a disynaptic basin for ELF. Thus a two-electron single bond will reveal a disynaptic basin (the centroid of which approximately matches the position of the  AIM BCP), which can be integrated to approximately two electrons. Shown below are the centroids of the disynaptic basins calculated for our S_N2 species:

The most striking difference with the AIM analysis is that that the central carbon is surrounded only by three, not five disynaptic basins. The BCPs found for the two di-axial interactions have no counterpart in synaptic basins. Of course, that does not mean that there are no electrons that can be integrated in that region, just that the curvature of the density in that region is not sufficiently well defined to define a bounded volume of space which can be clearly integrated. Perhaps that condition is what we might mean by a bond!

The three disynaptic basins that do surround the central carbon integrate to a total of 7.85 electrons, which is close enough to 8 for us to say that this carbon is NOT hypervalent!

So what is our final conclusion? The frozen S_N2 species is not hypervalent. It could reasonably be said to be coordinated by three bonds, and two diaxial substituents that interact with the central carbon weakly. Perhaps rather than penta-coordinate, the central carbon could be described as pentacoordinaloid!