Reduction formula for point group Oh
Characters for molecular motions
| Motion |
E |
8C3 |
6C2 |
6C4 |
3C2 =(C4)2 |
i |
6S4 |
8S6 |
3 h |
6d |
| Cartesian 3N |
21 |
0 |
-1 |
3 |
-3 |
-3 |
-1 |
0 |
5 |
3 |
| Translation (x,y,z) |
3 |
0 |
-1 |
1 |
-1 |
-3 |
-1 |
0 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
1 |
-1 |
3 |
1 |
0 |
-1 |
-1 |
| Vibration |
15 |
0 |
1 |
1 |
-1 |
-3 |
-1 |
0 |
5 |
3 |
Decomposition into Irreducible representations
| Motion |
A1g |
A2g |
Eg |
T1g |
T2g |
A1u |
A2u |
Eu |
T1u |
T2u |
Total |
| Cartesian 3N |
1 |
0 |
1 |
1 |
1 |
0 |
0 |
0 |
3 |
1 |
8 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
| Vibration |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
2 |
1 |
6 |
Molecule Parameter
| Number of Atoms (N) |
7 |
| Number of internal coordinates |
15 |
| Number of independant internal coordinates |
1 |
| Number of vibrational modes |
6 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
Eg |
T1g |
T2g |
A1u |
A2u |
Eu |
T1u |
T2u |
Total |
| Linear (IR) |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
2 |
1 |
2 / 4 |
| Quadratic (Raman) |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
2 |
1 |
3 / 3 |
| IR + Raman |
- |
0 |
- |
0 |
- |
0 |
0 |
0 |
- |
1 |
0* / 1 |
* Center of inversion: Mutual Exclusion Principle
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
8C3 |
6C2 |
6C4 |
3C2 =(C4)2 |
i |
6S4 |
8S6 |
3h |
6d |
| linear |
1 |
15 |
0 |
1 |
1 |
-1 |
-3 |
-1 |
0 |
5 |
3 |
| quadratic |
2 |
120 |
0 |
8 |
0 |
8 |
12 |
0 |
0 |
20 |
12 |
| cubic |
3 |
680 |
5 |
8 |
0 |
-8 |
-28 |
0 |
-1 |
60 |
28 |
| quartic |
4 |
3.060 |
0 |
36 |
4 |
36 |
72 |
4 |
0 |
160 |
72 |
| quintic |
5 |
11.628 |
0 |
36 |
4 |
-36 |
-144 |
-4 |
0 |
376 |
144 |
| sextic |
6 |
38.760 |
15 |
120 |
0 |
120 |
300 |
0 |
3 |
820 |
300 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A1g |
A2g |
Eg |
T1g |
T2g |
A1u |
A2u |
Eu |
T1u |
T2u |
| linear |
1 |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
2 |
1 |
| quadratic |
2 |
7 |
2 |
9 |
4 |
9 |
1 |
2 |
3 |
8 |
7 |
| cubic |
3 |
22 |
13 |
33 |
33 |
42 |
9 |
14 |
20 |
51 |
46 |
| quartic |
4 |
92 |
63 |
155 |
171 |
196 |
50 |
59 |
109 |
199 |
190 |
| quintic |
5 |
283 |
238 |
521 |
674 |
719 |
207 |
232 |
439 |
776 |
747 |
| sextic |
6 |
928 |
823 |
1.742 |
2.330 |
2.435 |
737 |
782 |
1.513 |
2.470 |
2.425 |
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