# Symmetry Tutorial - Introduction

Welcome to the world of symmetry! Symmetry plays an central role in the analysis of the structure, bonding, and spectroscopy of molecules. In this tutorial, we will explore the basic symmetry *elements* and *operations* and their use in determining the symmetry classification (point group) of different molecules.

The symmetry properties of objects (and molecules) may be described in terms of the presence of certain symmetry elements and their associated symmetry operations. The various symmetry elements and operations are described in the table below.

## Symmetry Elements and Operations

element |
operation |
symbol |

symmetry plane | reflection through plane | σ |

inversion center | inversion: every point x,y,z translated to -x,-y,-z |
i |

proper axis | rotation about axis by 360/n degrees |
C_{n} |

improper axis | 1. rotation by 360/n degrees2. reflection through plane perpendicular to rotation axis |
S_{n} |

The next few pages introduce each of these symmetry elements with example molecules having each type of symmetry. Select the checkboxes to display the individual symmetry *elements* and press the buttons to animate the corresponding symmetry *operations*.

Following this, there are several pages of molecules that show *all* of the symmetry elements present in a particular molecule. These collections of symmetry elements are what allow us to assign a particular molecule (or any object, for that matter) to a particular symmetry point group.