Simple H\"uckel Molecular Orbital Theory for M\"obius Carbon Nanobelts

Abstract

The recently synthesized M\"obius carbon nanobelts (CNBs) have gained attention owing to their unique $\pi$-conjugation topology, which results in distinctive electronic properties with both fundamental and practical implications. Although M\"obius conjugation with phase inversion in atomic orbital (AO) basis is well-established for monocyclic systems, the extension of this understanding to double-stranded M\"obius CNBs remains uncertain. This study thoroughly examines the simple H\"uckel molecular orbital (SHMO) theory for describing the $\pi$ electronic structures of M\"obius CNBs. We demonstrate that the adjacency matrix for any M\"obius 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 M\"obius CNBs, including the experimentally synthesized one, show that the H\"uckel molecular orbitals (MOs) strikingly resemble the DFT-computed $\pi$ MOs, which were obtained using a herein proposed technique based on the localization and re-delocalization of DFT canonical MOs. Interestingly, the lower-lying $\pi$ MOs exhibit an odd number of nodal planes and are doubly quasidegenerate as a consequence of the phase inversion in M\"obius macrocycles, contrasting with macrocyclic H\"uckel systems. Coulson bond orders derived from SHMO theory correlate well with DFT-calculated Wiberg bond indices for all C-C bonds in tested M\"obius 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 M\"obius CNBs. Thus, the SHMO model not only captures the essence of $\pi$ electronic structure of M\"obius CNBs, but also provides reliable quantitative predictions comparable to DFT results.