Molecular torus group

Generally speaking, the highest symmetry of M?bius cyclacene molecule possesses the C₂ symmetry based on the theory of point group according to the previous works. However, based on the topology principle, the fundamental group of M?buis strip is an infinite continuous cyclic group and its border is made up of twice of the generator. Of course, the M?bius strip-like molecule is associated with a finite discrete cyclic symmetry group. For the cyclacene isomers constructed by several (n) benzene rings, these isomers include: the common cylinder Hückel cyclacene (HC-[n]) molecules, the M?bius cyclacene (MC-[n]) molecules by twisting the linear precursor one time (180°), and the multi-twisted M?bius strip-like cyclacene (M ^m C?[n]) molecules by twisting the linear precursor m times (m × 180°). The relevant results suggest that in addition to the point symmetry, there is a new kind of symmetry called the torus screw rotation (denoted as TSR). In this article, we take the M ^m C?[n] molecules as examples to discuss their TSR group and point group symmetry, and the relative symmetry adaptive atom sets as well as their atomic orbital (AO) sets. Here, the Cartesian coordinates is not quite fit for the investigation of these AOs, so it is replaced by either the torus orthogonal curvilinear coordinates (for M ^m C?[n] molecule) or the cylinder orthogonal curvilinear coordinates (for HC-[n] molecule). According to the features of cyclic group, the character table of the irreducible representation of the TSR group could be constructed easily. Some other relevant point-group symmetries maybe also belong to the molecule, so its symmetry maybe called as the torus group symmetry. For multi-twisted M?bius strip-like molecule, there are some correlations in topology characteristics.