## Reduction formula for point group C_{2v}

**Characters of input representation**
E |
C_{2} (z) |
_{v}(xz) |
_{v}(yz) |

9 |
1 |
3 |
3 |

**Decomposition into Irreducible representations**
A_{1} |
A_{2} |
B_{1} |
B_{2} |
Total |

4 |
1 |
2 |
2 |
9 |

###
Force field analysis

**Allowed / forbidden vibronational transitions**
Operator |
A_{1} |
A_{2} |
B_{1} |
B_{2} |
Total |

Linear (IR) |
4 |
1 |
2 |
2 |
8 / 1 |

Quadratic (Raman) |
4 |
1 |
2 |
2 |
9 / 0 |

IR + Raman |
4 |
- |
2 |
2 |
8 / 0 |

**Characters of symmetric powers for vibration representation**
Force field |
Tensor Order |
E |
C_{2} (z) |
_{v}(xz) |
_{v}(yz) |

linear |
1 |
9 |
1 |
3 |
3 |

quadratic |
2 |
45 |
5 |
9 |
9 |

cubic |
3 |
165 |
5 |
19 |
19 |

quartic |
4 |
495 |
15 |
39 |
39 |

quintic |
5 |
1.287 |
15 |
69 |
69 |

sextic |
6 |
3.003 |
35 |
119 |
119 |

**Decomposition into Irreducible representations**

Number of nonvanshing force constants
Force field |
Tensor Order |
A_{1} |
A_{2} |
B_{1} |
B_{2} |

linear |
1 |
4 |
1 |
2 |
2 |

quadratic |
2 |
17 |
8 |
10 |
10 |

cubic |
3 |
52 |
33 |
40 |
40 |

quartic |
4 |
147 |
108 |
120 |
120 |

quintic |
5 |
360 |
291 |
318 |
318 |

sextic |
6 |
819 |
700 |
742 |
742 |

### 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