Hidden duality and accidental degeneracy in cycloacene and M\"obius cycloacene

TL;DR

This paper explains accidental degeneracies in cycloacenes and Möbius cycloacene through segmentation, linking electronic levels to molecular substructures, and predicts degeneracy conditions using Chebyshev polynomials, impacting organic electronic transport.

Contribution

It introduces the concept of segmentation to explain degeneracies and connects electronic levels to molecular substructures, providing a new theoretical framework.

Findings

Degeneracies are linked to segmentation into simpler carbon structures.

Möbius cycloacene degeneracy is explained by segmentation.

Degeneracy conditions are derived using Chebyshev polynomials.

Abstract

The accidental degeneracy appearing in cycloacenes as triplets and quadruplets is explained with the concept of segmentation, introduced here with the aim of describing the effective disconnection of $\pi$ orbitals on these organic compounds. For periodic systems with time reversal symmetry, the emergent nodal domains are shown to divide the atomic chains into simpler carbon structures analog to benzene rings, diallyl chains, anthracene (triacene) chains and tetramethyl-naphtalene skeletal forms. The common electronic levels of these segments are identified as members of degenerate multiplets of the global system. The peculiar degeneracy of M\"obius cycloacene is also explained by segmentation. In the last part, it is shown that the multiplicity of energies for cycloacene can be foreseen by studying the continuous limit of the tight-binding model; the degeneracy conditions are put in terms of Chebyshev polynomials. The results obtained in this work have important consequences on the physics of electronic transport in organic wires, together with their artificial realizations.