Reduction formula for point group Th
Characters for molecular motions
Motion |
E |
4C3 |
4(C3)2 |
3C2 |
i |
4(S6)5 |
4S6 |
3h |
Cartesian 3N |
48 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
Translation (x,y,z) |
3 |
0 |
0 |
-1 |
-3 |
0 |
0 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
0 |
-1 |
3 |
0 |
0 |
-1 |
Vibration |
42 |
0 |
0 |
2 |
0 |
0 |
0 |
0 |
Decomposition into Irreducible representations
Motion |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
Total |
Cartesian 3N |
2 |
2 |
6 |
2 |
2 |
6 |
20 |
Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
Rotation (Rx,Ry,Rz) |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
Vibration |
2 |
2 |
5 |
2 |
2 |
5 |
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 |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
Total |
Linear (IR) |
2 |
2 |
5 |
2 |
2 |
5 |
5 / 13 |
Quadratic (Raman) |
2 |
2 |
5 |
2 |
2 |
5 |
9 / 9 |
IR + Raman |
- |
- |
- |
2 |
2 |
- |
0* / 4 |
* Center of inversion: Mutual Exclusion Principle
Characters of symmetric powers for vibration representation
Force field |
Tensor Order |
E |
4C3 |
4(C3)2 |
3C2 |
i |
4(S6)5 |
4S6 |
3h |
linear |
1 |
42 |
0 |
0 |
2 |
0 |
0 |
0 |
0 |
quadratic |
2 |
903 |
0 |
0 |
23 |
21 |
0 |
0 |
21 |
cubic |
3 |
13.244 |
14 |
14 |
44 |
0 |
0 |
0 |
0 |
quartic |
4 |
148.995 |
0 |
0 |
275 |
231 |
0 |
0 |
231 |
quintic |
5 |
1.370.754 |
0 |
0 |
506 |
0 |
0 |
0 |
0 |
sextic |
6 |
10.737.573 |
105 |
105 |
2.277 |
1.771 |
7 |
7 |
1.771 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field |
Tensor Order |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
linear |
1 |
2 |
2 |
5 |
2 |
2 |
5 |
quadratic |
2 |
44 |
44 |
110 |
37 |
37 |
110 |
cubic |
3 |
562 |
555 |
1.650 |
562 |
555 |
1.650 |
quartic |
4 |
6.281 |
6.281 |
18.590 |
6.204 |
6.204 |
18.590 |
quintic |
5 |
57.178 |
57.178 |
171.281 |
57.178 |
57.178 |
171.281 |
sextic |
6 |
448.016 |
447.960 |
1.341.912 |
447.421 |
447.372 |
1.341.912 |
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