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 |
120 |
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 |
114 |
0 |
0 |
2 |
0 |
0 |
0 |
8 |
Decomposition into Irreducible representations
| Motion |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
Total |
| Cartesian 3N |
6 |
6 |
14 |
4 |
4 |
16 |
50 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
| Vibration |
6 |
6 |
13 |
4 |
4 |
15 |
48 |
Molecule Parameter
| Number of Atoms (N) |
40 |
| Number of internal coordinates |
114 |
| Number of independant internal coordinates |
6 |
| Number of vibrational modes |
48 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
Total |
| Linear (IR) |
6 |
6 |
13 |
4 |
4 |
15 |
15 / 33 |
| Quadratic (Raman) |
6 |
6 |
13 |
4 |
4 |
15 |
25 / 23 |
| IR + Raman |
- |
- |
- |
4 |
4 |
- |
0* / 8 |
* 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 |
114 |
0 |
0 |
2 |
0 |
0 |
0 |
8 |
| quadratic |
2 |
6.555 |
0 |
0 |
59 |
57 |
0 |
0 |
89 |
| cubic |
3 |
253.460 |
38 |
38 |
116 |
0 |
0 |
0 |
544 |
| quartic |
4 |
7.413.705 |
0 |
0 |
1.769 |
1.653 |
0 |
0 |
3.669 |
| quintic |
5 |
174.963.438 |
0 |
0 |
3.422 |
0 |
0 |
0 |
18.600 |
| sextic |
6 |
3.470.108.187 |
741 |
741 |
35.931 |
32.509 |
19 |
19 |
96.957 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
| linear |
1 |
6 |
6 |
13 |
4 |
4 |
15 |
| quadratic |
2 |
294 |
294 |
808 |
267 |
267 |
816 |
| cubic |
3 |
10.656 |
10.637 |
31.600 |
10.520 |
10.501 |
31.736 |
| quartic |
4 |
309.653 |
309.653 |
926.240 |
308.598 |
308.598 |
926.744 |
| quintic |
5 |
7.292.896 |
7.292.896 |
21.867.677 |
7.288.246 |
7.288.246 |
21.872.327 |
| sextic |
6 |
144.606.060 |
144.605.680 |
433.750.976 |
144.579.099 |
144.578.738 |
433.767.088 |
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