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!