Reduction formula for point group Th
Characters for molecular motions
| Motion |
E |
4C3 |
4(C3)2 |
3C2 |
i |
4(S6)5 |
4S6 |
3 h |
| Cartesian 3N |
60 |
0 |
0 |
0 |
0 |
0 |
0 |
4 |
| Translation (x,y,z) |
3 |
0 |
0 |
-1 |
-3 |
0 |
0 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
0 |
-1 |
3 |
0 |
0 |
-1 |
| Vibration |
54 |
0 |
0 |
2 |
0 |
0 |
0 |
4 |
Decomposition into Irreducible representations
| Motion |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
Total |
| Cartesian 3N |
3 |
3 |
7 |
2 |
2 |
8 |
25 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
| Vibration |
3 |
3 |
6 |
2 |
2 |
7 |
23 |
Molecule Parameter
| Number of Atoms (N) |
20 |
| Number of internal coordinates |
54 |
| Number of independant internal coordinates |
3 |
| Number of vibrational modes |
23 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
Total |
| Linear (IR) |
3 |
3 |
6 |
2 |
2 |
7 |
7 / 16 |
| Quadratic (Raman) |
3 |
3 |
6 |
2 |
2 |
7 |
12 / 11 |
| IR + Raman |
- |
- |
- |
2 |
2 |
- |
0* / 4 |
* Center of inversion: Mutual Exclusion Principle
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
4C3 |
4(C3)2 |
3C2 |
i |
4(S6)5 |
4S6 |
3h |
| linear |
1 |
54 |
0 |
0 |
2 |
0 |
0 |
0 |
4 |
| quadratic |
2 |
1.485 |
0 |
0 |
29 |
27 |
0 |
0 |
35 |
| cubic |
3 |
27.720 |
18 |
18 |
56 |
0 |
0 |
0 |
120 |
| quartic |
4 |
395.010 |
0 |
0 |
434 |
378 |
0 |
0 |
610 |
| quintic |
5 |
4.582.116 |
0 |
0 |
812 |
0 |
0 |
0 |
1.856 |
| sextic |
6 |
45.057.474 |
171 |
171 |
4.466 |
3.654 |
9 |
9 |
7.134 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
| linear |
1 |
3 |
3 |
6 |
2 |
2 |
7 |
| quadratic |
2 |
71 |
71 |
181 |
60 |
60 |
183 |
| cubic |
3 |
1.183 |
1.174 |
3.443 |
1.153 |
1.144 |
3.473 |
| quartic |
4 |
16.605 |
16.605 |
49.293 |
16.421 |
16.421 |
49.351 |
| quintic |
5 |
191.255 |
191.255 |
572.431 |
190.791 |
190.791 |
572.895 |
| sextic |
6 |
1.879.057 |
1.878.967 |
5.631.191 |
1.876.963 |
1.876.882 |
5.632.061 |
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 November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement