2.3. Lemniscular or Double-twist Möbius Rings

In 2005, the Möbius approach was extended by Rzepa to rings created by imparting two half-twists to a cyclic array of 2p-AOs. They are equivalent to molecules with two antarafacial components, but components which do not cancel each other. They retain the C2 axis axis of symmetry (in fact they can have three such axes). Such systems, like the single half-twist topology, are also chiral (disymmetric). The molecular orbitals (and the electron density) take the form of 212 torus links having linking numbers of two, and these often exhibit figure-8 (lemniscular) shapes (See DOI: 10.1039/b510508k and 10.1021/ol0518333).

It was recently recognized that this figure-eight is stabilized by a topological property known as writhe (or in molecular biology, supercoiling), which makes such geometries far more likely and accessible than they otherwise would be (DOI: 10.1021/ja710438j). Systems with up to six half twists have now been identified (DOI: 10.1021/ic800987f) and there is also a fascinating link to biological molecules known as cyclic or supercoiled DNA (DOI: 10.1016/j.cub.2006.02.029). Another bit of coiled helical fun!