Two-component Pseudovectoral Chirality Function for Tetrahedra

Abstract

Chirality, the lack of inversion symmetry, is a geometrical property critical to chemistry, biology and material sciences. In the three-dimensional Euclidean space $\mathbb{R}^3$ chriality can ususally be characterized with four-point structrual information. Various functions have therefore been proposed to quantify chirality of tetrahedra, which can be extended to other 3D objects, including molecules. However, existing functions are scalars or pseudoscalars and are unable to simultaneously possess all the desirable properties of chirality functions: detectability of chirality, inversion antisymmetry and continuity. We observe that to avoid this difficulty, any chirality function for tetrahedra must be a pseudovector with at least two components. In light of this, we propose a two-component pseudovectoral chirality function for tetrahedra that satisfies all the desirable properties. We plan to use this function to map the "chiral zeros" of existing pseudoscalar chirality functions and to design a microstructure descriptor for the chirality of many-body systems and multi-phase media in $\mathbb{R}^3$.