Reduction formula for point group Oh
Characters for molecular motions
Motion |
E |
8C3 |
6C2 |
6C4 |
3C2 =(C4)2 |
i |
6S4 |
8S6 |
3h |
6d |
Cartesian 3N |
156 |
0 |
-2 |
0 |
0 |
0 |
0 |
0 |
4 |
14 |
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 |
150 |
0 |
0 |
-2 |
2 |
0 |
0 |
0 |
4 |
14 |
Decomposition into Irreducible representations
Motion |
A1g |
A2g |
Eg |
T1g |
T2g |
A1u |
A2u |
Eu |
T1u |
T2u |
Total |
Cartesian 3N |
5 |
2 |
7 |
8 |
11 |
1 |
5 |
6 |
12 |
8 |
65 |
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 |
5 |
2 |
7 |
7 |
11 |
1 |
5 |
6 |
11 |
8 |
63 |
Molecule Parameter
Number of Atoms (N) |
52 |
Number of internal coordinates |
150 |
Number of independant internal coordinates |
5 |
Number of vibrational modes |
63 |
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1g |
A2g |
Eg |
T1g |
T2g |
A1u |
A2u |
Eu |
T1u |
T2u |
Total |
Linear (IR) |
5 |
2 |
7 |
7 |
11 |
1 |
5 |
6 |
11 |
8 |
11 / 52 |
Quadratic (Raman) |
5 |
2 |
7 |
7 |
11 |
1 |
5 |
6 |
11 |
8 |
23 / 40 |
IR + Raman |
- |
2 |
- |
7 |
- |
1 |
5 |
6 |
- |
8 |
0* / 29 |
* 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 |
150 |
0 |
0 |
-2 |
2 |
0 |
0 |
0 |
4 |
14 |
quadratic |
2 |
11.325 |
0 |
75 |
3 |
77 |
75 |
1 |
0 |
83 |
173 |
cubic |
3 |
573.800 |
50 |
0 |
-4 |
152 |
0 |
0 |
0 |
312 |
1.512 |
quartic |
4 |
21.947.850 |
0 |
2.850 |
42 |
3.002 |
2.850 |
38 |
0 |
3.466 |
11.866 |
quintic |
5 |
675.993.780 |
0 |
0 |
-80 |
5.852 |
0 |
0 |
0 |
12.320 |
79.492 |
sextic |
6 |
17.463.172.650 |
1.275 |
73.150 |
118 |
79.002 |
73.150 |
38 |
25 |
97.174 |
490.042 |
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 |
5 |
2 |
7 |
7 |
11 |
1 |
5 |
6 |
11 |
8 |
quadratic |
2 |
279 |
216 |
495 |
672 |
733 |
222 |
246 |
468 |
716 |
691 |
cubic |
3 |
12.180 |
11.803 |
23.958 |
35.644 |
36.023 |
11.763 |
12.142 |
23.880 |
36.061 |
35.684 |
quartic |
4 |
459.560 |
455.861 |
915.421 |
1.369.685 |
1.373.344 |
456.032 |
458.285 |
914.317 |
1.372.719 |
1.370.464 |
quintic |
5 |
14.094.266 |
14.074.413 |
28.168.679 |
42.238.529 |
42.258.422 |
14.072.853 |
14.092.746 |
28.165.599 |
42.259.942 |
42.240.089 |
sextic |
6 |
363.899.267 |
363.758.430 |
727.657.047 |
1.091.371.472 |
1.091.512.231 |
363.761.544 |
363.865.747 |
727.626.666 |
1.091.496.976 |
1.091.392.733 |
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