Reduction formula for point group S4
Characters for molecular motions
| Motion |
E |
S4 |
C2 |
(S4)3 |
| Cartesian 3N |
15 |
-1 |
-1 |
-1 |
| Translation (x,y,z) |
3 |
-1 |
-1 |
-1 |
| Rotation (Rx,Ry,Rz) |
3 |
1 |
-1 |
1 |
| Vibration |
9 |
-1 |
1 |
-1 |
Decomposition into Irreducible representations
| Motion |
A |
B |
E |
Total |
| Cartesian 3N |
3 |
4 |
4 |
11 |
| Translation (x,y,z) |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
1 |
0 |
1 |
2 |
| Vibration |
2 |
3 |
2 |
7 |
Molecule Parameter
| Number of Atoms (N) |
5 |
| Number of internal coordinates |
9 |
| Number of independant internal coordinates |
2 |
| Number of vibrational modes |
7 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A |
B |
E |
Total |
| Linear (IR) |
2 |
3 |
2 |
5 / 2 |
| Quadratic (Raman) |
2 |
3 |
2 |
7 / 0 |
| IR + Raman |
- |
3 |
2 |
5 / 0 |
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
S4 |
C2 |
(S4)3 |
| linear |
1 |
9 |
-1 |
1 |
-1 |
| quadratic |
2 |
45 |
1 |
5 |
1 |
| cubic |
3 |
165 |
-1 |
5 |
-1 |
| quartic |
4 |
495 |
3 |
15 |
3 |
| quintic |
5 |
1.287 |
-3 |
15 |
-3 |
| sextic |
6 |
3.003 |
3 |
35 |
3 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A |
B |
E |
| linear |
1 |
2 |
3 |
2 |
| quadratic |
2 |
13 |
12 |
10 |
| cubic |
3 |
42 |
43 |
40 |
| quartic |
4 |
129 |
126 |
120 |
| quintic |
5 |
324 |
327 |
318 |
| sextic |
6 |
761 |
758 |
742 |
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