Reduction formula for point group O
Characters for molecular motions
| Motion |
E |
8C3 |
6C'2 |
6C4 |
3C2 =(C4)2 |
| Cartesian 3N |
48 |
0 |
0 |
0 |
0 |
| Translation (x,y,z) |
3 |
0 |
-1 |
1 |
-1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
1 |
-1 |
| Vibration |
42 |
0 |
2 |
-2 |
2 |
Decomposition into Irreducible representations
| Motion |
A1 |
A2 |
E |
T1 |
T2 |
Total |
| Cartesian 3N |
2 |
2 |
4 |
6 |
6 |
20 |
| Translation (x,y,z) |
0 |
0 |
0 |
1 |
0 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
0 |
1 |
0 |
1 |
| Vibration |
2 |
2 |
4 |
4 |
6 |
18 |
Molecule Parameter
| Number of Atoms (N) |
16 |
| Number of internal coordinates |
42 |
| Number of independant internal coordinates |
2 |
| Number of vibrational modes |
18 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
E |
T1 |
T2 |
Total |
| Linear (IR) |
2 |
2 |
4 |
4 |
6 |
4 / 14 |
| Quadratic (Raman) |
2 |
2 |
4 |
4 |
6 |
12 / 6 |
| IR + Raman |
- |
2 |
- |
- |
- |
0 / 2 |
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
8C3 |
6C'2 |
6C4 |
3C2 =(C4)2 |
| linear |
1 |
42 |
0 |
2 |
-2 |
2 |
| quadratic |
2 |
903 |
0 |
23 |
3 |
23 |
| cubic |
3 |
13.244 |
14 |
44 |
-4 |
44 |
| quartic |
4 |
148.995 |
0 |
275 |
15 |
275 |
| quintic |
5 |
1.370.754 |
0 |
506 |
-26 |
506 |
| sextic |
6 |
10.737.573 |
105 |
2.277 |
37 |
2.277 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A1 |
A2 |
E |
T1 |
T2 |
| linear |
1 |
2 |
2 |
4 |
4 |
6 |
| quadratic |
2 |
47 |
34 |
81 |
105 |
115 |
| cubic |
3 |
572 |
552 |
1.110 |
1.638 |
1.662 |
| quartic |
4 |
6.315 |
6.170 |
12.485 |
18.525 |
18.655 |
| quintic |
5 |
57.298 |
57.058 |
114.356 |
171.148 |
171.414 |
| sextic |
6 |
448.297 |
447.140 |
895.332 |
1.341.352 |
1.342.472 |
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