A HAUSDORFF CHIRALITY MEASURE

Chirality is a property that is independent of its physical and chemical manifestations, It is therefore possible to quantify chirality without any reference to pseudoscalar observables. In this paper we propose a new measure of chirality that is based on Hausdorff's concept of distances between sets and that is a natural choice as a measure for molecular models that represent structures as sets of atomic coordinates. This Hausdorff chirality measure, a continuous and similarity-invariant function of an object's shape, is zero if and only if the object is achiral. We have applied this measure to study the chirality of tetrahedral shapes-classical models of tetracoordinate carbon atoms-and have identified the extremal (most chiral) shapes for every chiral subsymmetry of T(d) that can be realized by a tetrahedron. Our calculations show that the degree of chirality of the extremal objects increases with a decrease in symmetry, although the most symmetric chiral tetrahedron, with D2 symmetry, already possesses 87% of the maximal chirality value available for tetrahedra, and that the shape of the most chiral tetrahedron, with C1 symmetry, is very close to that of the most chiral C2 tetrahedron. The properties and the applicability of the Hausdorff chirality measure are compared with those of other measures of chirality that are based on common-volume or root-mean-square approaches.