Quantification of the Helicality of Helical Molecular Orbitals

The frontier molecular orbital (MO) topology of linear carbon molecules, such as polyynes, can be visually identified as helices. However, there is no clear way to quantify the helical curvature of these π-MOs, and it is thus challenging to quantify correlations between the helical curvature and molecular properties. In this paper, we develop a method that enables us to compute the helical curvature of MOs based on their nodal planes. Using this method, we define a robust way of quantifying the helical nature of MOs (helicality) by their deviation from a perfect helix. We explore several limiting cases, including polyynes, metallacumulenes, cyclic allenes, and spiroconjugated systems, where the change in helical curvature is subtle yet clearly highlighted with this method. For example, we show that strain only has a minor effect on the helicality of the frontier orbitals of cycloallenes and that the MOs of spiroconjugated systems are close to perfect helices around the spiro-carbon. Our work provides a well-defined method for assessing orbital helicality beyond visual inspection of MO isosurfaces, thus paving the way for future studies of how the helicality of π-MOs affects molecular properties.