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 |
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 |
30 |
0 |
0 |
2 |
0 |
0 |
0 |
4 |
Decomposition into Irreducible representations
| Motion |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
Total |
| Cartesian 3N |
2 |
2 |
4 |
1 |
1 |
5 |
15 |
| 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 |
1 |
1 |
4 |
13 |
Molecule Parameter
| Number of Atoms (N) |
12 |
| Number of internal coordinates |
30 |
| Number of independant internal coordinates |
2 |
| Number of vibrational modes |
13 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
Total |
| Linear (IR) |
2 |
2 |
3 |
1 |
1 |
4 |
4 / 9 |
| Quadratic (Raman) |
2 |
2 |
3 |
1 |
1 |
4 |
7 / 6 |
| IR + Raman |
- |
- |
- |
1 |
1 |
- |
0* / 2 |
* 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 |
4 |
| quadratic |
2 |
465 |
0 |
0 |
17 |
15 |
0 |
0 |
23 |
| cubic |
3 |
4.960 |
10 |
10 |
32 |
0 |
0 |
0 |
72 |
| quartic |
4 |
40.920 |
0 |
0 |
152 |
120 |
0 |
0 |
256 |
| quintic |
5 |
278.256 |
0 |
0 |
272 |
0 |
0 |
0 |
680 |
| sextic |
6 |
1.623.160 |
55 |
55 |
952 |
680 |
5 |
5 |
1.904 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
| linear |
1 |
2 |
2 |
3 |
1 |
1 |
4 |
| quadratic |
2 |
25 |
25 |
55 |
18 |
18 |
57 |
| cubic |
3 |
223 |
218 |
607 |
205 |
200 |
625 |
| quartic |
4 |
1.761 |
1.761 |
5.079 |
1.687 |
1.687 |
5.113 |
| quintic |
5 |
11.713 |
11.713 |
34.663 |
11.543 |
11.543 |
34.833 |
| sextic |
6 |
68.037 |
68.007 |
202.623 |
67.501 |
67.476 |
202.929 |
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