# 1.12: Normal Modes of Vibration

- Page ID
- 204713

## Molecular vibrations

**Molecular vibrations** are one of three kinds of motion, occurs when atoms in a molecule are in periodic motion. Molecular vibrations include constant translational and rotational motion. Translational motion occurs when the whole molecule goes in the same direction while the rotational motion occurs when the molecule spins like a top. Molecule vibrations fall into two main categories of stretching and bending. **Stretching** changes in interatomic distance along bong axis, while **bending** changes in angle between two bonds in a molecule.

There are two types of stretching, symmetric stretching and asymmetric stretching as the following Figure shows:

There are four types of bend, rocking, scissoring, wagging, and twisting.

## Normal modes of vibration

Each atom in a molecule has three degree of freedom. A molecule with n atoms has 3n degree of freedom. 3n degree of freedom composes of translation, rotations and vibrations. All 3n degrees of freedom have symmetry relationships consistent with the irreducible representations of the molecule's point groups. **Non-vibration modes** (NVM) include translations and rotations. The vibrational motions of the atoms in a molecule can always be resolved into fundamental vibrational modes for the entire molecule.

### Atomic displacement coordinates

The number of normal modes of vibration:

- 3n-6 for non-linear molecules
- 3n-5 for linear molecules

To indicate the the number of normal modes of vibration:

- Locate a set of three vectors along the Cartesian coordinates at each atom, representing the 3n degrees of freedom
- Find the reducible representation
- Reduce it to irreducible representations, subtract rotations and translations
- The rest of irreducible representations will give the symmetry of the NMVs

## Samples

### vibrations for SO_{2}

C_{2v} |
E |
C_{2} |
σ_{v(xz)} |
σ’_{v(yz)} |
||
---|---|---|---|---|---|---|

A_{1} |
1 | 1 | 1 | 1 | z | x^{2}, y^{2}, z^{2} |

A_{2} |
1 | 1 | -1 | -1 | R_{z} |
xy |

B_{1} |
1 | -1 | 1 | -1 | x, R_{y} |
xz |

B_{2} |
1 | -1 | -1 | 1 | y, R_{x} |
yz |

Γ_{3n} |
9 | -1 | 1 | 3 | ||

N_{i} |
3 | 1 | 1 | 3 | ||

X_{i} |
3 | -1 | 1 | 1 |

Reduce

Γ_{3n}=3A_{1}+A_{2}+2B_{1}+3B_{2}

Γ_{trans}=A_{1}+B_{1}+B_{2}

Γ_{rotations}=A_{2}+B_{1}+B_{2}

Γ_{NMV}=Γ_{3n-6}=2A_{1}+B_{2}

**NMVs** are also identified by frequency numbers: v_{1}, v_{2}, v_{3},... The numbering is often assigned systematically in descending order of the symmetry species and among modes of the same symmetry in descending order of the vibrational frequency. Stretching modes have **higher** frequency than bending modes.