## Reduction formula for point group C_{3}

**Characters for molecular motions**
Motion |
E |
C_{3} |
(C_{3})^{2} |

Cartesian 3N |
18 |
0 |
0 |

Translation (x,y,z) |
3 |
0 |
0 |

Rotation (R_{x},R_{y},R_{z}) |
3 |
0 |
0 |

Vibration |
12 |
0 |
0 |

**Decomposition into Irreducible representations**
Motion |
A |
E |
Total |

Cartesian 3N |
6 |
6 |
12 |

Translation (x,y,z) |
1 |
1 |
2 |

Rotation (R_{x},R_{y},R_{z}) |
1 |
1 |
2 |

Vibration |
4 |
4 |
8 |

**Molecule Parameter**
Number of Atoms (N) |
6 |

Number of internal coordinates |
12 |

Number of independant internal coordinates |
4 |

Number of vibrational modes |
8 |

###
Force field analysis

**Allowed / forbidden vibronational transitions**
Operator |
A |
E |
Total |

Linear (IR) |
4 |
4 |
8 / 0 |

Quadratic (Raman) |
4 |
4 |
8 / 0 |

IR + Raman |
4 |
4 |
8 / 0 |

**Characters of symmetric powers for vibration representation**
Force field |
Tensor Order |
E |
C_{3} |
(C_{3})^{2} |

linear |
1 |
12 |
0 |
0 |

quadratic |
2 |
78 |
0 |
0 |

cubic |
3 |
364 |
4 |
4 |

quartic |
4 |
1.365 |
0 |
0 |

quintic |
5 |
4.368 |
0 |
0 |

sextic |
6 |
12.376 |
10 |
10 |

**Decomposition into Irreducible representations**

Number of nonvanshing force constants
Force field |
Tensor Order |
A |
E |

linear |
1 |
4 |
4 |

quadratic |
2 |
26 |
26 |

cubic |
3 |
124 |
120 |

quartic |
4 |
455 |
455 |

quintic |
5 |
1.456 |
1.456 |

sextic |
6 |
4.132 |
4.122 |

### Literature

- J.K.G. Watson, J. Mol. Spec.
**41** 229 (1972)

The Numbers of Structural Parameters and Potential Constants of Molecules

- X.F. Zhou, P. Pulay. J. Comp. Chem.
**10** No. 7, 935-938 (1989)

Characters for Symmetric and Antisymmetric Higher Powers of Representations:

Application to the Number of Anharmonic Force Constants in Symmetrical Molecules

Last update Mai, 23^{rd} 2018 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement