## Reduction formula for point group D_{2}

**Characters for molecular motions**
Motion |
E |
C_{2} (z) |
C_{2} (y) |
C_{2} (x) |

Cartesian 3N |
120 |
0 |
0 |
0 |

Translation |
3 |
-1 |
-1 |
-1 |

Rotation |
3 |
-1 |
-1 |
-1 |

Vibration |
114 |
2 |
2 |
2 |

**Decomposition into Irreducible representations**
Motion |
A |
B_{1} |
B_{2} |
B_{3} |
Total |

Cartesian 3N |
30 |
30 |
30 |
30 |
120 |

Translation |
0 |
1 |
1 |
1 |
3 |

Rotation |
0 |
1 |
1 |
1 |
3 |

Vibration |
30 |
28 |
28 |
28 |
114 |

**Molecule Parameter**
Number of Atoms (N) |
40 |

Number of internal coordinates |
114 |

Number of independant internal coordinates |
30 |

Number of vibrational modes |
114 |

###
Force field analysis

**Allowed / forbidden vibronational transitions**
Operator |
A |
B_{1} |
B_{2} |
B_{3} |
Total |

Linear (IR) |
30 |
28 |
28 |
28 |
84 / 30 |

Quadratic (Raman) |
30 |
28 |
28 |
28 |
114 / 0 |

IR + Raman |
- |
28 |
28 |
28 |
84 / 0 |

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

linear |
1 |
114 |
2 |
2 |
2 |

quadratic |
2 |
6.555 |
59 |
59 |
59 |

cubic |
3 |
253.460 |
116 |
116 |
116 |

quartic |
4 |
7.413.705 |
1.769 |
1.769 |
1.769 |

quintic |
5 |
174.963.438 |
3.422 |
3.422 |
3.422 |

sextic |
6 |
3.470.108.187 |
35.931 |
35.931 |
35.931 |

**Decomposition into Irreducible representations**
Force field |
Tensor Order |
A |
B_{1} |
B_{2} |
B_{3} |

linear |
1 |
30 |
28 |
28 |
28 |

quadratic |
2 |
1.683 |
1.624 |
1.624 |
1.624 |

cubic |
3 |
63.452 |
63.336 |
63.336 |
63.336 |

quartic |
4 |
1.854.753 |
1.852.984 |
1.852.984 |
1.852.984 |

quintic |
5 |
43.743.426 |
43.740.004 |
43.740.004 |
43.740.004 |

sextic |
6 |
867.553.995 |
867.518.064 |
867.518.064 |
867.518.064 |

### 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