Analytical methods for calculating continuous symmetry measures and the chirality measure

We provide analytical solutions of the Continuous Symmetry Measure (CSM) equation for several symmetry point-groups, and for the associated Continuous Chirality Measure (CCM), which are quantitative estimates of the degree of a symmetry-point group or chirality in a structure, respectively. We do it by solving analytically the problem of finding the minimal distance between the original structure and the result obtained by operating on it all of the operations of a specific G symmetry point group. Specifically, we provide solutions for the symmetry measures of all of the improper rotations point group symmetries, S_n, including the mirror (S₁, C_s), inversion (S₂, C_i) as well as the higher S_ns (n > 2 is even) point group symmetries, for the rotational C₂ point group symmetry, for the higher rotational C_n symmetries (n > 2), and finally for the C_nh symmetry point group. The chirality measure is the minimal of all S_n measures.