An extensive discussion developed regarding my post on a fascinating helical [144]-annulene. Topics included the nature of the ring current sustained by the π-electrons and in particular the bond-length alternation around the periphery and whether this should alter if the electron count were to be changed to that of a 4n+2 system (i.e. a dication). Whilst the [144]-annulene itself is hypothetical, it emerged that some compounds known as expanded porphyrins have very similar (albeit smaller scale) helical structures. X-ray structures for two such provide useful reality checks on the calculations. Here‡ I include the (3D) coordinates of these two systems so that you can explore for yourself their helicity.
SELQUW. Click for 3D X-ray structure
HIYTAL. Click for 3D X-ray structure
I include below Δrmeso, being the mean unsigned difference in bond length (Å) at the meso positions of the porphrin ring, the calculations being at the 6-311G(d,p) level using the DFT procedure indicated below. The linking number analysis[1] for such systems will be reported elsewhere.[2]
| Method | SELQUW | HIYTAL |
| X-ray | 0.048 | 0.045 |
| B97D/6-311G(d,p) | 0.025 | 0.015 |
| B3LYP/6-311G(d,p) | 0.047 | 0.017 |
‡The WordPress system operated here does not enable 3D coordinates to be inserted into the comment section of a post, only the main body.
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I just bulit a paper model of a treefoil knot with my kids and we figured out that it is not a pi/-pi linkage but a 0pi one. I am afraid there is no elegant way of circumventing the "localization" of the Möbius twist.
A molecular trefoil knot, as shown in this post has a linking number of 6π and this is decomposed into Tw = -0.8π and Wr = +6.8π. So this indeed is NOT a system with an odd linking number. This molecule is unusual because the twist (Tw) and the writhe (Wr) are actually of opposite signs!
Perhaps systems with half-helical turns (as required) can only be constructed if the restriction of all cis-alkenes is supplemented with some trans-motifs?
Some examples of (4n π-electron) molecules with odd linking numbers can be found in this article, but these contain trans as well as cis alkene bonds. This table also includes the NICS(0) value, which indicates that they are all diamagnetic (and hence what we refer to as aromatic). By the way, the systems of D5 and D7 symmetry were not stable minima; the latter for example was a transition state for bond shifting. But this might just be because it is so small, and a larger ring may behave differently.
Re "Dealing all the time with Hamitonians we forget about the other big discovery of the congenial Irish mathmatician, the quaternions!" A lovely connection Raphael. I know that Bob Hanson has played with quaternion coordinates for large molecules, but I did not make the connection with using them for helical systems until you pointed it out above!
We, as chemists, do not use quaternions enough!