Reduction formula for point group S6
Characters for molecular motions
| Motion |
E |
C3(z) |
(C3)2 |
i |
(S6)5 |
S6 |
| Cartesian 3N |
57 |
0 |
0 |
-3 |
0 |
0 |
| Translation (x,y,z) |
3 |
0 |
0 |
-3 |
0 |
0 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
0 |
3 |
0 |
0 |
| Vibration |
51 |
0 |
0 |
-3 |
0 |
0 |
Decomposition into Irreducible representations
| Motion |
Ag |
Eg |
Au |
Eu |
Total |
| Cartesian 3N |
9 |
9 |
10 |
10 |
38 |
| Translation (x,y,z) |
0 |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
1 |
1 |
0 |
0 |
2 |
| Vibration |
8 |
8 |
9 |
9 |
34 |
Molecule Parameter
| Number of Atoms (N) |
19 |
| Number of internal coordinates |
51 |
| Number of independant internal coordinates |
8 |
| Number of vibrational modes |
34 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Eg |
Au |
Eu |
Total |
| Linear (IR) |
8 |
8 |
9 |
9 |
18 / 16 |
| Quadratic (Raman) |
8 |
8 |
9 |
9 |
16 / 18 |
| 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 |
51 |
0 |
0 |
-3 |
0 |
0 |
| quadratic |
2 |
1.326 |
0 |
0 |
30 |
0 |
0 |
| cubic |
3 |
23.426 |
17 |
17 |
-82 |
-1 |
-1 |
| quartic |
4 |
316.251 |
0 |
0 |
459 |
0 |
0 |
| quintic |
5 |
3.478.761 |
0 |
0 |
-1.161 |
0 |
0 |
| sextic |
6 |
32.468.436 |
153 |
153 |
4.788 |
9 |
9 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
Eg |
Au |
Eu |
| linear |
1 |
8 |
8 |
9 |
9 |
| quadratic |
2 |
226 |
226 |
216 |
216 |
| cubic |
3 |
3.896 |
3.888 |
3.924 |
3.915 |
| quartic |
4 |
52.785 |
52.785 |
52.632 |
52.632 |
| quintic |
5 |
579.600 |
579.600 |
579.987 |
579.987 |
| sextic |
6 |
5.412.258 |
5.412.177 |
5.410.656 |
5.410.584 |
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