## 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.

Original language | American English |
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Pages (from-to) | 2712-2721 |

Number of pages | 10 |

Journal | Journal of Computational Chemistry |

Volume | 29 |

Issue number | 16 |

DOIs | |

State | Published - Dec 2008 |

## Keywords

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