Tight-binding analysis of helical states in carbyne

Carbyne is a linear allotrope of carbon that is composed of a chain of sp-hybridized carbon atoms. Through appropriate engineering of the chain termination, carbyne can harbor helical states where the π-electron delocalization twists along the axis of the chain. Herein, we present a comprehensive analysis of these helical states at the tight-binding level. We demonstrate that, in general, the molecular orbital coefficients of the helical states trace out an ellipse, in analogy to elliptically polarized light. Helical states can be realized in a model, inspired by the structure of cumulene, which considers a chain terminated by sp2-hybridized atoms oriented at a nontrivial dihedral angle. We provide a complete analytic solution for this model. Additionally, we present a variation of the model that yields perfect helical states that trace out a circle as opposed to an ellipse. Our results provide a deeper understanding of helical states and lay a foundation for more advanced levels of theory.