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 |
36 |
0 |
0 |
-4 |
0 |
0 |
0 |
8 |
| 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 |
30 |
0 |
0 |
-2 |
0 |
0 |
0 |
8 |
Decomposition into Irreducible representations
| Motion |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
Total |
| Cartesian 3N |
2 |
2 |
4 |
0 |
0 |
6 |
14 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
| Vibration |
2 |
2 |
3 |
0 |
0 |
5 |
12 |
Molecule Parameter
| Number of Atoms (N) |
12 |
| Number of internal coordinates |
30 |
| Number of independant internal coordinates |
2 |
| Number of vibrational modes |
12 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
Total |
| Linear (IR) |
2 |
2 |
3 |
0 |
0 |
5 |
5 / 7 |
| Quadratic (Raman) |
2 |
2 |
3 |
0 |
0 |
5 |
7 / 5 |
| IR + Raman |
- |
- |
- |
0 |
0 |
- |
0* / 0 |
* 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 |
30 |
0 |
0 |
-2 |
0 |
0 |
0 |
8 |
| quadratic |
2 |
465 |
0 |
0 |
17 |
15 |
0 |
0 |
47 |
| cubic |
3 |
4.960 |
10 |
10 |
-32 |
0 |
0 |
0 |
208 |
| quartic |
4 |
40.920 |
0 |
0 |
152 |
120 |
0 |
0 |
792 |
| quintic |
5 |
278.256 |
0 |
0 |
-272 |
0 |
0 |
0 |
2.640 |
| sextic |
6 |
1.623.160 |
55 |
55 |
952 |
680 |
5 |
5 |
8.008 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
| linear |
1 |
2 |
2 |
3 |
0 |
0 |
5 |
| quadratic |
2 |
28 |
28 |
52 |
15 |
15 |
60 |
| cubic |
3 |
232 |
227 |
598 |
180 |
175 |
650 |
| quartic |
4 |
1.828 |
1.828 |
5.012 |
1.620 |
1.620 |
5.180 |
| quintic |
5 |
11.890 |
11.890 |
34.486 |
11.230 |
11.230 |
35.146 |
| sextic |
6 |
68.800 |
68.770 |
201.860 |
66.738 |
66.713 |
203.692 |
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