## Number and symmetry of vibrations

With any molecule, the energy can be divided into translational energy, vibrational energy and rotational energy. Translational energy can be described in terms of three vectors 90° to each other and so has three degrees of freedom. Rotational energy for most molecules can also be described in terms of three degrees of freedom. However, for a linear molecule there are only two rotations. The molecule can either rotate around the axis or about it. Thus, molecules are said to have 3 translational degrees of freedom and 3 rotational degrees of freedom with the exception of linear molecules, which have two degrees of rotational freedom. All other degrees of freedom will be vibrational degrees of freedom and each is equivalent to one vibration.

Therefore, the number of vibrations to be expected from a molecule with N atoms is 3N – 6 for all molecules except linear systems where it is 3N – 5 (Smith and Dent, 2005).

## Symmetry elements and point groups

Any molecule can be classified by its symmetry elements. It is then possible to assign the molecule to a group called a point group which has these same elements. This can then be used to predict which bands are infrared and which are Raman active.

To do this it is necessary to work out the symmetry elements in the molecule. The main symmetry elements we need to recognize are the following:

Symbol | Operation | Element |

E |
The indentify operation. | none |

C_{n} |
An axis of symmetry in which the molecule is rotated about a molecular axis. n Defines how often the molecule requires to be rotated to arrive back at the starting point. (n-fold rotation, rotation by 2π/n angle, n is an integer). |
rotation axis |

σ_{h} |
A plane of symmetry in which the plane is perpendicular to the principle axis of the molecule (horizontal reflection plane). | mirror plane |

σ_{v} |
A plane of symmetry in which the plane is parallel to the principle axis of the molecule (vertical reflection plane). | mirror plane |

i |
A centre of inversion in which every point inverted through the centre arrives at an identical point on the other side. | inversion centre |

S_{n} |
An axis which combines a rotation and an inversion. | rotation – reflection axis |

Having assigned the molecule to a point group, group theory can be used to predict whether or not a band will be Raman or infrared active. It is particularly important to note that symmetry considerations allow us to determine whether or not a band is allowed in a Raman or infrared spectrum. This does not tell us how strong it will be; this would require a calculation (Smith and Dent, 2005).

[* References: SMITH E., DENT G. (2005) – Modern Raman Spectroscopy – A Practical Approach. John Wiley and Sons, England*.]