Quantifying Chirality in Multiscale Nanostructured Systems

Chirality is an essential geometric property unifying small molecules, biological macromolecules, inorganic nanomaterials, biological microparticles, and many other chemical structures. The extent of the asymmetry of chiral nanostructures influences their physical and chemical properties. Numerous chirality measures have been developed quantify this geometric property of mirror asymmetry as a way to predict or explain these properties. Many chirality measures require prohibitively demanding computations, especially for chiral structures comprised of thousands of atoms. There is a large toolbox of chirality measures available to quantify different types of structures, from more classical chiral molecules to complex chiral nanostructures. Two of the most commonly used chirality measures are the Osipov-Pickup-Dunmur Chirality Index (OPD) and the Hausdorff Chirality Measure (HCM). Comparing the two measures eludidates the frequency of so-called chiral zeros in OPD and other pseudoscalar chirality measures. Acknowledging the fundamental problems with quantification of mirror asymmetry, including the ambiguity of sign-variable pseudoscalar chirality measures, we revisit this subject because of their unifying significance and new aspects of chirality surfaced in nanoscale materials that display chirality continuum and scale-dependent mirror asymmetry. To address some of the fundamental problems and practical limitations of existing chirality measures, we applied the concept of torsion within the framework of differential geometry to the graph theoretical representations of molecular and nanoscale helicoids. The resulting graph-theoretical chirality measure (GTC) provides a description of both the sign and magnitude of mirror asymmetry for crystalline structures.