Reduction formula for point group Th
Characters for molecular motions
| Motion |
E |
4C3 |
4(C3)2 |
3C2 |
i |
4(S6)5 |
4S6 |
3 h |
| Cartesian 3N |
39 |
0 |
0 |
-5 |
-3 |
0 |
0 |
9 |
| 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 |
33 |
0 |
0 |
-3 |
-3 |
0 |
0 |
9 |
Decomposition into Irreducible representations
| Motion |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
Total |
| Cartesian 3N |
2 |
2 |
4 |
0 |
0 |
7 |
15 |
| 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 |
3 |
0 |
0 |
6 |
13 |
Molecule Parameter
| Number of Atoms (N) |
13 |
| Number of internal coordinates |
33 |
| Number of independant internal coordinates |
2 |
| Number of vibrational modes |
13 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
Total |
| Linear (IR) |
2 |
2 |
3 |
0 |
0 |
6 |
6 / 7 |
| Quadratic (Raman) |
2 |
2 |
3 |
0 |
0 |
6 |
7 / 6 |
| IR + Raman |
- |
- |
- |
0 |
0 |
- |
0* / 0 |
* 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 |
33 |
0 |
0 |
-3 |
-3 |
0 |
0 |
9 |
| quadratic |
2 |
561 |
0 |
0 |
21 |
21 |
0 |
0 |
57 |
| cubic |
3 |
6.545 |
11 |
11 |
-55 |
-55 |
-1 |
-1 |
273 |
| quartic |
4 |
58.905 |
0 |
0 |
225 |
225 |
0 |
0 |
1.113 |
| quintic |
5 |
435.897 |
0 |
0 |
-531 |
-531 |
0 |
0 |
3.969 |
| sextic |
6 |
2.760.681 |
66 |
66 |
1.653 |
1.653 |
6 |
6 |
12.817 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
| linear |
1 |
2 |
2 |
3 |
0 |
0 |
6 |
| quadratic |
2 |
34 |
34 |
63 |
18 |
18 |
72 |
| cubic |
3 |
301 |
296 |
784 |
238 |
232 |
866 |
| quartic |
4 |
2.631 |
2.631 |
7.224 |
2.334 |
2.334 |
7.446 |
| quintic |
5 |
18.570 |
18.570 |
53.991 |
17.622 |
17.622 |
55.116 |
| sextic |
6 |
116.930 |
116.894 |
343.483 |
113.584 |
113.554 |
346.274 |
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