The Geometry and Electronic Topology of Higher-Order Charged Möbius Annulenes

Higher-order aromatic charged Möbius-type annulenes have
been Lkrealized 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 (C12H122+, C12H122−, C14H14,
C18H182+, C18H182−, C21H21+, C24H242−, C28H282+, and C28H282−) 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 (ρ(r)π, ELFπ) constructed
using only the occupied π-MOs. The π-torus subdivides
into a torus knot for annulenes defined by an odd linking number (Lk = 1, 3π) and a torus link for those
with an even linking number (Lk = 2, 4π).
The torus topology is shown to map onto single canonical π-MOs
only for even values of Lk. Incomplete
and misleading descriptions of the topology of π-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.