## Linking numbers, and twist and writhe components for two extended porphyrins.

My last comment as appended to the previous post promised to analyse two so-called extended porphyrins for their topological descriptors. I start with the Cãlugãreanu/Fuller theorem  which decomposes the topology of a space curve into two components, its twist (Tw) and its writhe (Wr, this latter being the extent to which coiling of the central curve has relieved local twisting) and establishes a topological invariant called the linking number[1]

Lk = Tw + Wr

 HIYTAL. Click for 3D. SELQUW. Click for 3D.

Visual inspection of the models above (I really do encourage you to click on the images to load the 3D coordinates) reveals that HIYTAL[2] has a major coil that forms one and a half helical turns in a clockwise direction, and a loop connecting the ends of the coil which forms a half-helical turn in an anti-clockwise direction. SELQUW[3] has a major coil comprising one and half helical turns in an anti-clockwise direction and a connecting loop which also coils anti-clockwise. So the former sustains a total of one full (clockwise) helical turn and the latter two full (anti-clockwise) helical turns.

The nomenclature for helical molecules includes a chiral descriptor P (for a positive helical turn) and M (for a negative helical turn). What such a descriptor does not do is quantify the total number of helices describing the topology. So I suggest we use instead the linking number Lk. Instead of P and M, we have positive and negative integers (in units of 2π) providing this quantitative information.

The linking number analysis for these two molecules comes out as below. I have multiplied the linking number unit from 2π to 1π for a reason that I will explain shortly:

 π-electrons Lk Tw Wr Δr (meso) SELQUW 56=4n -4 -1.34 -2.66 0.048 HIYTAL 62=4n+2 +2 +0.46 +1.54 0.045

You can see that the linking numbers (and their signs) correspond exactly to the visual analysis of the helical turns above. My reason for including the factor of 2 is that it enables us to make a further link to the Hückel aromaticity rule:

1. Cyclic conjugated systems are aromatic if they contain 4n+2 π-electrons and have an even or zero linking number (in units of 1π).
2. Cyclic conjugated systems are aromatic if they contain 4n π-electrons and have an odd linking number (in units of 1π).
3. Cyclic conjugated systems are anti-aromatic if they contain 4n π-electrons and have an even or zero linking number (in units of 1π).
4. Cyclic conjugated systems are anti-aromatic if they contain 4n+2 π-electrons and have an odd linking number (in units of 1π).

By these rules, SELQUW contains (by the shortest path) 56 π-electrons, belongs to the 4n electron rule (n=14) and hence is formally anti-aromatic (rule 3 above). HIYTAL has a path of 62-electrons, belongs to the 4n+2 rule (n=15) and hence is formally aromatic (rule 1 above).

For systems with so many (correlated) electrons, it is probably tenuous to make a connection between the bond-length alternation at the meso position and the aromaticity (or lack of it). I comment only that HIYTAL converts more of the coiling into writhing of the central curve than does SELQUW, and this destroys less π-π overlap by reducing the overall degree of twisting. I might also speculate that nevertheless a modest degree of twisting may impact upon the intrinsic distortivity of π-electrons in cyclically conjugated systems (such as that in benzene[4]), as noted in this earlier post. Such effects may make the interpretation of bond-alternation in such helical systems more difficult.

‡ A program for calculating these components can be found here. For a fun-packed journey through linking numbers and the association with valentine cards, go see this post here!

## References

1. S.M. Rappaport, and H.S. Rzepa, "Intrinsically Chiral Aromaticity. Rules Incorporating Linking Number, Twist, and Writhe for Higher-Twist Möbius Annulenes", Journal of the American Chemical Society, vol. 130, pp. 7613-7619, 2008. http://dx.doi.org/10.1021/ja710438j
2. S. Shimizu, W. Cho, J. Sessler, H. Shinokubo, and A. Osuka, "meso-Aryl Substituted Rubyrin and Its Higher Homologues: Structural Characterization and Chemical Properties", Chemistry - A European Journal, vol. 14, pp. 2668-2678, 2008. http://dx.doi.org/10.1002/chem.200701909
3. S. Shimizu, N. Aratani, and A. Osuka, "meso-Trifluoromethyl-Substituted Expanded Porphyrins", Chemistry - A European Journal, vol. 12, pp. 4909-4918, 2006. http://dx.doi.org/10.1002/chem.200600158
4. S. Shaik, A. Shurki, D. Danovich, and P.C. Hiberty, " A Different Story of π-DelocalizationThe Distortivity of π-Electrons and Its Chemical Manifestations † ", Chemical Reviews, vol. 101, pp. 1501-1540, 2001. http://dx.doi.org/10.1021/cr990363l