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Analytical methods for calculating Continuous Symmetry Measures and the Chirality Measure

Authors

  • Mark Pinsky,

    1. Institute of Chemistry and The Lise Meitner Minerva Center for Computational Quantum Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
    2. Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
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  • Chaim Dryzun,

    1. Institute of Chemistry and The Lise Meitner Minerva Center for Computational Quantum Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
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  • David Casanova,

    1. Departament de Química Inorgànica and Institut de Química Teòrica i Computacional, Universitat de Barcelona, Martí i Franqués 1-11, 08028 Barcelona, Spain
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  • Pere Alemany,

    1. Departament de Química Inorgànica and Institut de Química Teòrica i Computacional, Universitat de Barcelona, Martí i Franqués 1-11, 08028 Barcelona, Spain
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  • David Avnir

    Corresponding author
    1. Institute of Chemistry and The Lise Meitner Minerva Center for Computational Quantum Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
    • Institute of Chemistry and The Lise Meitner Minerva Center for Computational Quantum Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
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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, Sn, including the mirror (S1, CS), inversion (S2, Ci) as well as the higher Sns (n > 2 is even) point group symmetries, for the rotational C2 point group symmetry, for the higher rotational Cn symmetries (n > 2), and finally for the Cnh symmetry point group. The chirality measure is the minimal of all Sn measures. © 2008 Wiley Periodicals, Inc. J Comput Chem 2008

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