Quantification of the Helicality of Helical Molecular Orbitals

Abstract

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 ⇡-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 ⇡-MOs affects molecular properties.

Generation of Helical States – Breaking of Symmetries, Curie's Principle, and Excited States**

“…Fitting a helix -general case. Classical helixes are called perfect helixes in [3]. However, perfect helixes are not the rule as many examples already proven (see the next sections and [1]).…”

“…• We can use DFT (see Appendix 1) in order to determine the vector ψ(z) using the technique developed by W. Jeorgensen, M.H. Garner and G. Solomon in [3]. Such a distribution will be denoted by D DFT .…”

“…A similar idea is used by W. Bro-Jeorgensen, M.H. Garner, G. Solomon in [3], where they define the MAD index which corresponds to the mean of the absolute deviation | φ c (z j ) − f j |.…”

“…As recently reported by Garner and coll. [3] the prospective of helical states using d-orbitals has started. In particular, the metallacumulenes are potential classes of molecules that may exhibit helical MOs in the linear fragments of the molecules.…”

“…The classical picture is to draw a helix representing the twist of the π-vectors along the structure. In [3], W. Jeorgensen et al study how far a given distribution is from a perfect helix, i.e. a distribution governed by a fixed pitch.…”