Analytical methods for calculating continuous symmetry measures and the chirality measure

Mark Pinsky, Chaim Dryzun, David Casanova, Pere Alemany, David Avnir*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

81 Scopus citations


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.

Original languageAmerican English
Pages (from-to)2712-2721
Number of pages10
JournalJournal of Computational Chemistry
Issue number16
StatePublished - Dec 2008


  • Chirality
  • Continuous symmetry
  • Improper-rotation
  • Inversion
  • Reflection
  • Rotation
  • Symmetry


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