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 |
72 |
0 |
0 |
0 |
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 |
66 |
0 |
0 |
2 |
0 |
0 |
0 |
8 |
Decomposition into Irreducible representations
| Motion |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
Total |
| Cartesian 3N |
4 |
4 |
8 |
2 |
2 |
10 |
30 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
| Vibration |
4 |
4 |
7 |
2 |
2 |
9 |
28 |
Molecule Parameter
| Number of Atoms (N) |
24 |
| Number of internal coordinates |
66 |
| Number of independant internal coordinates |
4 |
| Number of vibrational modes |
28 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
Total |
| Linear (IR) |
4 |
4 |
7 |
2 |
2 |
9 |
9 / 19 |
| Quadratic (Raman) |
4 |
4 |
7 |
2 |
2 |
9 |
15 / 13 |
| 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 |
66 |
0 |
0 |
2 |
0 |
0 |
0 |
8 |
| quadratic |
2 |
2.211 |
0 |
0 |
35 |
33 |
0 |
0 |
65 |
| cubic |
3 |
50.116 |
22 |
22 |
68 |
0 |
0 |
0 |
352 |
| quartic |
4 |
864.501 |
0 |
0 |
629 |
561 |
0 |
0 |
1.809 |
| quintic |
5 |
12.103.014 |
0 |
0 |
1.190 |
0 |
0 |
0 |
7.752 |
| sextic |
6 |
143.218.999 |
253 |
253 |
7.735 |
6.545 |
11 |
11 |
31.441 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
| linear |
1 |
4 |
4 |
7 |
2 |
2 |
9 |
| quadratic |
2 |
106 |
106 |
268 |
87 |
87 |
276 |
| cubic |
3 |
2.148 |
2.137 |
6.212 |
2.060 |
2.049 |
6.300 |
| quartic |
4 |
36.349 |
36.349 |
107.828 |
35.850 |
35.850 |
108.140 |
| quintic |
5 |
505.410 |
505.410 |
1.511.759 |
503.472 |
503.472 |
1.513.697 |
| sextic |
6 |
5.972.716 |
5.972.584 |
17.898.296 |
5.964.303 |
5.964.182 |
17.904.520 |
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