This is a recently published[1] (hypothetical) molecule which has such unusual properties that I cannot resist sharing it with you. It is an annulene with 144 all-cis CH groups, being a (very) much larger cousin of (also hypothetical) systems mooted in 2009[2],[3].
A 144-carbon annulene. Click for 3D.
One fascinating novel aspect of Berger’s work is that he identifies that such helical systems will exhibit a distinct anapolar ring current structure in a constant and homogeneous magnetic field, perpendicular to the main molecular plane. Such anapolar magnetism is distinctly different from the dipolar (diatropic) ring currents normally associated with aromatic molecules, and with the current interest in the magnetic properties of graphene-like objects (see also this blog post and also the helical metal wire) such molecules can only help to excite our imaginations.
I also show one of the more stable molecular orbitals for the [144]-annulene (ωB97XD/6-31G(d,p) calculation).‡ Molecular art indeed!
MO 461. Click for 3D.
If you go to the Knotplot site, there you will find a torus link of form (2,18), which displays as the below. Look familiar? Notice the chirality is opposite however!
‡Orbitals for smaller rings with such form can be found here.
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What would you expect for an odd-numbered C-chain, lets say [((CH)9)17]+?
For investigating such systems we would need a good algo for generating starting coordinates for ((CH)n)m. Ideally it would read in n, m and the average C-C length. But then there is at least one more degree of freedom describing how "stretched" the system is. This parameter would couple both radii of the "torus" (lets use this term here for simplicity). Complicated task!
For an odd-numbered chain, the topology changes from a link to a knot. There are an odd number of crossings in the electron density torus. The MOs can no longer be considered singly, but one must add pairs of (approximately degenerate) MOs. In effect, the pseudo-MO now "traverses" the cycle twice (as appropriate for any odd-twisted band).
As for generating the coordinates, yes, not easy! When we constructed smaller objects up to say C7 symmetry (ie C35H35+) a brute-force method of sketching out the molecule and letting the geometry optimiser do the initial refinement, followed by attempted symmetrisation managed the trick. But this would not work for these larger systems.
Oh, now I see :-)
I'll try to do first the currents in [144]2+.
Rob Scharein, the mathematician who produced Knotplot, clearly has worked out algorithms for doing this sort of thing. You might try to contact him?
A sideways shift! The image below is of the NCI (non-covalent-interactions) for the neutral [144]-annulene. It looks at the topology of the electron density in the non-covalent region, and colour codes it according to the second eigenvalue of the density Hessian. This reveals mildly attractive regions (green) and mildly repulsive regions (yellow). The technique attempts to identify regions were eg mildly attractive dispersion forces might be operating. As you can see, there is a LOT of dispersion in this helix; it is in some ways the ultimate stacked system!
Cool!
Btw. the structure motive itself is realized in nature in cyclic DNA or RNA strands - is it messenger RNA ? It does not need alltoomuch of speculation to assume that there is also lots of dispersion in these.
Regarding the odd-twisted bands and the possibilities for a Möbius-curent: I always get in trouble when trying to understand how a diatropic odd-twist-Möbius-current could be explained in the MO-picture. You might know Patrick Fowler's series of papers on the MO basis of mag. induced currents (like doi: 10.1351/pac200779060969).
The bottom line is that diatropic currents are the result of virtual transitions with translational symmetry and paratropic ones of virtual transitions of rotational symmetry (each weighted with the difference between the orbital energies), (which in some beautiful way explains the paratropic nature of the currents in cyclobutadiene). My problem now with pi-Orbitals and odd-twist Möbius currents is, that one would need some (set of) totally symmetric MOs from which the virtual excitation starts. And this appears to me somehow impossible since the phases of pi-orbitals cannot be positive and negative at the same time.
Or simply: How can π-conjugation take place over the whole odd-twist-Möbius band?
Regarding the last point, you can start at any node in the p-AO basis of an odd-twist Möbius system and track its overlap continuously until you reach the start. You will have made two circuits of the basis by doing this. If the system has a lot of writhe, the overlap between any two adjacent AOs will be almost optimal (since there is little twist left in the system). If you take a pair of MOs of this type, and square and add them, you get a continuous torus of electron density with no other nodes. That torus looks exactly like the orange torus knot reproduced above. I suppose we can call that conjugation. For an even twist system, you get instead two torus curves for the density, each making only one circuit of the system; this is called a torus link.
Perhaps because you need to do two circuits, that is how you can get both positive and negative overlaps at the same time?
Re: cyclic dna. Indeed yes, the linking number of such dna is often analysed in terms of the sum of the twist and the writhe. Here to express whether the duplex is over-coiled or under-coiled. It is of course the two individual chains of dna that forms the helix. With π-systems, the two nodes of each p-AO play the role of the dna chains.