Chiral Hausdorff metrics and structural spectroscopy in a complex system

The surface Coulomb problem consists of finding the locally stable equilibrium configurations of N equal charges confined to the surface of a sphere. In the range 2≤N ≤112, computer searches have located 2054 equilibrium states including 1824 that occur in chiral pairs. The structural difference between the 'left' and the 'right' partners of each pair, i.e. the degree of chiral symmetry breaking, can be described in quantitative terms by computing the Hausdorff distance between the two enantiomorphic configurations. The results show that there is a general, albeit not universal, correlation between increasing complexity, decreasing symmetry, and increasing Hausdorff distance. The distribution of distances generated by the relative rotations of the pairs yield an intrinsic characterization of their shapes. The orientations with minimum distances, or optimum overlap, lead to reptation transforms that provide upper bounds for the saddle point heights separating the chiral minima on the energy surfaces. Since the Hausdorff metric constructions are generally asymmetric, the 'left' → 'right' and 'right' → left' distances may differ. These distinctions are related to classical parity violation and Hilbert's thirteenth problem. (author)