The geometry and electronic topology of higher-order charged Möbius annulenes

Higher-order aromatic charged Möbius-type annulenes have been L(k) realized computationally. These charged species are based on strips with more than one electronic half-twist, as defined by their linking numbers. The B3LYP/6-311+G(d,p) optimized structures and properties of annulene rings with such multiple half-twists (C(12)H(12)(2+), C(12)H(12)(2-), C(14)H(14), C(18)H(18)(2+), C(18)H(18)(2-), C(21)H(21)(+), C(24)H(24)(2-), C(28)H(28)(2+), and C(28)H(28)(2-)) have the nearly equal C-C bond lengths, small dihedral angles around the circuits, stabilization energies, and nucleus-independent chemical shift values associated with aromaticity. The topology and nature of Möbius annulene systems are analyzed in terms of the torus curves defined by electron density functions (rho(r)(pi), ELF(pi)) constructed using only the occupied pi-MOs. The pi-torus subdivides into a torus knot for annulenes defined by an odd linking number (L(k) = 1, 3pi) and a torus link for those with an even linking number (L(k) = 2, 4pi). The torus topology is shown to map onto single canonical pi-MOs only for even values of L(k). Incomplete and misleading descriptions of the topology of pi-electronic Möbius systems with an odd number of half twists result when only signed orbital diagrams are considered, as is often done for the iconic single half twist system.