Molecular chirality has traditionally been viewed as a binary property where a molecule is classified as either chiral or achiral, yet in recent decades, mathematical methods for quantifying chirality have been explored. Here, we use toy molecular systems to systematically compare the performance of two state-of-the-art chirality measures: (1) the Continuous Chirality Measure (CCM) and (2) the Chirality Characteristic (χ). We find that both methods exhibit qualitatively similar behavior when applied to simple molecular systems such as a four-site molecule or the polymer double-helix, but we show that the CCM may be more suitable for evaluating the chirality of arbitrary molecules or abstract structures such as normal vibrational modes. We discuss a range of considerations for applying these methods to molecular systems in general, and we provide links to user-friendly codes for both methods. We aim for this paper to serve as a concise resource for scientists attempting to familiarize themselves with these chirality measures or attempting to implement chirality measures in their own work.
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Rigorously, chirality is defined by the absence of improper symmetry in general, which includes inversion and with even n, in addition to mirror symmetry .
It has been suggested by mathematicians that there may be no universal measure of handedness and that handedness is just an agreed upon convention for labeling. The handedness assigned by the Chirality Characteristic to the structures in this work is in accord with the conventional classification of helices as either left-handed or right-handed. For certain non-helical structures, or special points along the continuous deformation of a helix into its enantiomer, this classification may be problematic.
The notion of helicity as defined here is analogous with the conventional definition of helicity in physics as the projection of a pseudovector on a vector, such as the projection of angular momentum onto linear momentum.
As noted in Ref. 14, other improper symmetry measures such as and (with n even) should be considered in addition to . However, it has been found that in the vast majority of cases (>90%) the operation yields the correct (minimum) CCM. In this work, we consider only σ for simplicity.
With the codes used, this restriction can be specified using the argument “--connectivity_file” as described in the documentation cited in the Data Availability section.
If the atoms are not identical, then one must confront the question of how such differences should be accounted for, such as weighting by mass.
Note that the untwisted wire may not represent a perfectly achiral conformation. This is because for a long and flexible wire, the conformations corresponding to local energy minima can be slightly asymmetric.
