## Reduction formula for point group D_{6h}

Your representation

E |
2C_{6} (z) |
2C_{3} |
C_{2} |
3C'_{2} |
3C''_{2} |
i |
2S_{3} |
2S_{6} |
_{h} (xy) |
3_{d} |
3_{v} |

324 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
24 |
12 |
8 |

can be reduced to

A_{1g} |
A_{2g} |
B_{1g} |
B_{2g} |
E_{1g} |
E_{2g} |
A_{1u} |
A_{2u} |
B_{1u} |
B_{2u} |
E_{1u} |
E_{2u} |

17 |
12 |
13 |
12 |
25 |
29 |
10 |
15 |
14 |
15 |
29 |
25 |

Substraction of translational and rotational contributions results in

A_{1g} |
A_{2g} |
B_{1g} |
B_{2g} |
E_{1g} |
E_{2g} |
A_{1u} |
A_{2u} |
B_{1u} |
B_{2u} |
E_{1u} |
E_{2u} |

17 |
11 |
13 |
12 |
24 |
29 |
10 |
14 |
14 |
15 |
28 |
25 |

###
Force field analysis

Infrared and Raman activities (harmonic approximation)
Method |
A_{1g} |
A_{2g} |
B_{1g} |
B_{2g} |
E_{1g} |
E_{2g} |
A_{1u} |
A_{2u} |
B_{1u} |
B_{2u} |
E_{1u} |
E_{2u} |

Infrared |
- |
- |
- |
- |
- |
- |
- |
+ |
- |
- |
+ |
- |

Raman |
+ |
- |
- |
- |
+ |
+ |
- |
- |
- |
- |
- |
- |

Number of force field parameters
Force field |
No. 1 |
No. 2 |
No. 3 |

linear |
318 |
318 |
17 |

quadratic |
101124 |
50721 |
2239 |

cubic |
32157432 |
5410240 |
226229 |

quartic |
XXX^{*} | XXX^{*} | XXX^{*} |

No. 1 : Total number of force constants .

No. 2 : Total number of force constants k.

No. 3 : Total number of nonvanishing force constants k.

Symmetry properties of force field parameters
Force field |
A_{1g} |
A_{2g} |
B_{1g} |
B_{2g} |
E_{1g} |
E_{2g} |
A_{1u} |
A_{2u} |
B_{1u} |
B_{2u} |
E_{1u} |
E_{2u} |

linear |
17 |
11 |
13 |
12 |
24 |
29 |
10 |
14 |
14 |
15 |
28 |
25 |

quadratic |
2239 |
2053 |
2099 |
2089 |
4190 |
4290 |
2083 |
2108 |
2113 |
2123 |
4238 |
4189 |

cubic |
226229 |
225179 |
225272 |
225062 |
450307 |
451382 |
224827 |
225557 |
225574 |
225784 |
451327 |
450362 |

quartic |
XXX^{*} | XXX^{*} | XXX^{*} | XXX^{*} | XXX^{*} | XXX^{*} | XXX^{*} | XXX^{*} | XXX^{*} | XXX^{*} | XXX^{*} | XXX^{*} |

^{*}Not calculated for systems with more than 75 atoms

Number of independant internal coordinates |
17 |

### Literature

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

The Numbers of Structural Parameters and Potential Constants of Molecules