title: Analytical methods for calculating Continuous Symmetry Measures and the Chirality Measure.
abstract: 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 (S(1), C(S)), inversion (S(2), C(i)) as well as the higher S(n)s (n > 2 is even) point group symmetries, for the rotational C(2) 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.
authors.affiliation: Institute of Chemistry and The Lise Meitner Minerva Center for Computational Quantum Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel.
authors.forename: Mark
authors.initials: M
authors.lastname: Pinsky
authors.forename: Chaim
authors.initials: C
authors.lastname: Dryzun
authors.forename: David
authors.initials: D
authors.lastname: Casanova
authors.forename: Pere
authors.initials: P
authors.lastname: Alemany
authors.forename: David
authors.initials: D
authors.lastname: Avnir