Reduction formula for point group S6
Characters for molecular motions
| Motion |
E |
C3(z) |
(C3)2 |
i |
(S6)5 |
S6 |
| Cartesian 3N |
72 |
0 |
0 |
0 |
0 |
0 |
| Translation (x,y,z) |
3 |
0 |
0 |
-3 |
0 |
0 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
0 |
3 |
0 |
0 |
| Vibration |
66 |
0 |
0 |
0 |
0 |
0 |
Decomposition into Irreducible representations
| Motion |
Ag |
Eg |
Au |
Eu |
Total |
| Cartesian 3N |
12 |
12 |
12 |
12 |
48 |
| Translation (x,y,z) |
0 |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
1 |
1 |
0 |
0 |
2 |
| Vibration |
11 |
11 |
11 |
11 |
44 |
Molecule Parameter
| Number of Atoms (N) |
24 |
| Number of internal coordinates |
66 |
| Number of independant internal coordinates |
11 |
| Number of vibrational modes |
44 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Eg |
Au |
Eu |
Total |
| Linear (IR) |
11 |
11 |
11 |
11 |
22 / 22 |
| Quadratic (Raman) |
11 |
11 |
11 |
11 |
22 / 22 |
| IR + Raman |
- |
- |
- |
- |
0* / 0 |
* Center of inversion: Mutual Exclusion Principle
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
C3(z) |
(C3)2 |
i |
(S6)5 |
S6 |
| linear |
1 |
66 |
0 |
0 |
0 |
0 |
0 |
| quadratic |
2 |
2.211 |
0 |
0 |
33 |
0 |
0 |
| cubic |
3 |
50.116 |
22 |
22 |
0 |
0 |
0 |
| quartic |
4 |
864.501 |
0 |
0 |
561 |
0 |
0 |
| quintic |
5 |
12.103.014 |
0 |
0 |
0 |
0 |
0 |
| sextic |
6 |
143.218.999 |
253 |
253 |
6.545 |
11 |
11 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
Eg |
Au |
Eu |
| linear |
1 |
11 |
11 |
11 |
11 |
| quadratic |
2 |
374 |
374 |
363 |
363 |
| cubic |
3 |
8.360 |
8.349 |
8.360 |
8.349 |
| quartic |
4 |
144.177 |
144.177 |
143.990 |
143.990 |
| quintic |
5 |
2.017.169 |
2.017.169 |
2.017.169 |
2.017.169 |
| sextic |
6 |
23.871.012 |
23.870.880 |
23.868.823 |
23.868.702 |
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