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 |
252 |
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 |
246 |
0 |
0 |
2 |
0 |
0 |
0 |
4 |
Decomposition into Irreducible representations
| Motion |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
Total |
| Cartesian 3N |
11 |
11 |
31 |
10 |
10 |
32 |
105 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
| Vibration |
11 |
11 |
30 |
10 |
10 |
31 |
103 |
Molecule Parameter
| Number of Atoms (N) |
84 |
| Number of internal coordinates |
246 |
| Number of independant internal coordinates |
11 |
| Number of vibrational modes |
103 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
Total |
| Linear (IR) |
11 |
11 |
30 |
10 |
10 |
31 |
31 / 72 |
| Quadratic (Raman) |
11 |
11 |
30 |
10 |
10 |
31 |
52 / 51 |
| IR + Raman |
- |
- |
- |
10 |
10 |
- |
0* / 20 |
* 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 |
246 |
0 |
0 |
2 |
0 |
0 |
0 |
4 |
| quadratic |
2 |
30.381 |
0 |
0 |
125 |
123 |
0 |
0 |
131 |
| cubic |
3 |
2.511.496 |
82 |
82 |
248 |
0 |
0 |
0 |
504 |
| quartic |
4 |
156.340.626 |
0 |
0 |
7.874 |
7.626 |
0 |
0 |
8.626 |
| quintic |
5 |
7.817.031.300 |
0 |
0 |
15.500 |
0 |
0 |
0 |
32.000 |
| sextic |
6 |
327.012.476.050 |
3.403 |
3.403 |
333.250 |
317.750 |
41 |
41 |
380.750 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
| linear |
1 |
11 |
11 |
30 |
10 |
10 |
31 |
| quadratic |
2 |
1.303 |
1.303 |
3.781 |
1.260 |
1.260 |
3.783 |
| cubic |
3 |
104.767 |
104.726 |
313.843 |
104.641 |
104.600 |
313.969 |
| quartic |
4 |
6.516.573 |
6.516.573 |
19.541.469 |
6.513.781 |
6.513.781 |
19.541.719 |
| quintic |
5 |
325.715.575 |
325.715.575 |
977.122.975 |
325.707.575 |
325.707.575 |
977.130.975 |
| sextic |
6 |
13.625.623.473 |
13.625.621.751 |
40.876.509.975 |
13.625.501.779 |
13.625.500.098 |
40.876.525.725 |
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