Reduction formula for point group S4
Characters for molecular motions
| Motion |
E |
S4 |
C2 |
(S4)3 |
| Cartesian 3N |
36 |
0 |
0 |
0 |
| Translation (x,y,z) |
3 |
-1 |
-1 |
-1 |
| Rotation (Rx,Ry,Rz) |
3 |
1 |
-1 |
1 |
| Vibration |
30 |
0 |
2 |
0 |
Decomposition into Irreducible representations
| Motion |
A |
B |
E |
Total |
| Cartesian 3N |
9 |
9 |
9 |
27 |
| Translation (x,y,z) |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
1 |
0 |
1 |
2 |
| Vibration |
8 |
8 |
7 |
23 |
Molecule Parameter
| Number of Atoms (N) |
12 |
| Number of internal coordinates |
30 |
| Number of independant internal coordinates |
8 |
| Number of vibrational modes |
23 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A |
B |
E |
Total |
| Linear (IR) |
8 |
8 |
7 |
15 / 8 |
| Quadratic (Raman) |
8 |
8 |
7 |
23 / 0 |
| IR + Raman |
- |
8 |
7 |
15 / 0 |
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
S4 |
C2 |
(S4)3 |
| linear |
1 |
30 |
0 |
2 |
0 |
| quadratic |
2 |
465 |
1 |
17 |
1 |
| cubic |
3 |
4.960 |
0 |
32 |
0 |
| quartic |
4 |
40.920 |
8 |
152 |
8 |
| quintic |
5 |
278.256 |
0 |
272 |
0 |
| sextic |
6 |
1.623.160 |
8 |
952 |
8 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A |
B |
E |
| linear |
1 |
8 |
8 |
7 |
| quadratic |
2 |
121 |
120 |
112 |
| cubic |
3 |
1.248 |
1.248 |
1.232 |
| quartic |
4 |
10.272 |
10.264 |
10.192 |
| quintic |
5 |
69.632 |
69.632 |
69.496 |
| sextic |
6 |
406.032 |
406.024 |
405.552 |
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