Simple Hückel Molecular Orbital Theory for Möbius Carbon Nanobelts

Abstract

The recently synthesized Möbius carbon nanobelts (CNBs) have gained attention owing to their unique $π$-conjugation topology, which results in distinctive electronic properties with both fundamental and practical implications. Although Möbius conjugation with phase inversion in atomic orbital (AO) basis is well-established for monocyclic systems, the extension of this understanding to double-stranded Möbius CNBs remains uncertain. This study thoroughly examines the simple Hückel molecular orbital (SHMO) theory for describing the $π$ electronic structures of Möbius CNBs. We demonstrate that the adjacency matrix for any Möbius CNB can preserve its eigenvalues and eigenvectors (with possibly flipped directions) under different placements of the sign inversion, ensuring identical SHMO results regardless of AO phase inversion location. Representative examples of Möbius CNBs, including the experimentally synthesized one, show that the Hückel molecular orbitals (MOs) strikingly resemble the DFT-computed $π$ MOs, which were obtained using a herein proposed technique based on the localization and re-delocalization of DFT canonical MOs. Interestingly, the lower-lying $π$ MOs exhibit an odd number of nodal planes and are doubly quasidegenerate as a consequence of the phase inversion in Möbius macrocycles, contrasting with macrocyclic Hückel systems. Coulson bond orders derived from SHMO theory correlate well with DFT-calculated Wiberg bond indices for all C-C bonds in tested Möbius CNBs. Additionally, a remarkable correlation is observed between HOMO-LUMO gaps obtained from the SHMO and GFN2-xTB calculations for a large number of topoisomers of Möbius CNBs. Thus, the SHMO model not only captures the essence of $π$ electronic structure of Möbius CNBs, but also provides reliable quantitative predictions comparable to DFT results.