Reduction formula for point group S6
Characters for molecular motions
| Motion |
E |
C3(z) |
(C3)2 |
i |
(S6)5 |
S6 |
| Cartesian 3N |
216 |
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 |
210 |
0 |
0 |
0 |
0 |
0 |
Decomposition into Irreducible representations
| Motion |
Ag |
Eg |
Au |
Eu |
Total |
| Cartesian 3N |
36 |
36 |
36 |
36 |
144 |
| Translation (x,y,z) |
0 |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
1 |
1 |
0 |
0 |
2 |
| Vibration |
35 |
35 |
35 |
35 |
140 |
Molecule Parameter
| Number of Atoms (N) |
72 |
| Number of internal coordinates |
210 |
| Number of independant internal coordinates |
35 |
| Number of vibrational modes |
140 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Eg |
Au |
Eu |
Total |
| Linear (IR) |
35 |
35 |
35 |
35 |
70 / 70 |
| Quadratic (Raman) |
35 |
35 |
35 |
35 |
70 / 70 |
| 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 |
210 |
0 |
0 |
0 |
0 |
0 |
| quadratic |
2 |
22.155 |
0 |
0 |
105 |
0 |
0 |
| cubic |
3 |
1.565.620 |
70 |
70 |
0 |
0 |
0 |
| quartic |
4 |
83.369.265 |
0 |
0 |
5.565 |
0 |
0 |
| quintic |
5 |
3.568.204.542 |
0 |
0 |
0 |
0 |
0 |
| sextic |
6 |
127.860.662.755 |
2.485 |
2.485 |
198.485 |
35 |
35 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
Eg |
Au |
Eu |
| linear |
1 |
35 |
35 |
35 |
35 |
| quadratic |
2 |
3.710 |
3.710 |
3.675 |
3.675 |
| cubic |
3 |
260.960 |
260.925 |
260.960 |
260.925 |
| quartic |
4 |
13.895.805 |
13.895.805 |
13.893.950 |
13.893.950 |
| quintic |
5 |
594.700.757 |
594.700.757 |
594.700.757 |
594.700.757 |
| sextic |
6 |
21.310.144.380 |
21.310.143.120 |
21.310.078.195 |
21.310.076.970 |
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