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 |
57 |
0 |
0 |
-3 |
-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 |
51 |
0 |
0 |
-1 |
-3 |
0 |
0 |
9 |
Decomposition into Irreducible representations
| Motion |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
Total |
| Cartesian 3N |
3 |
3 |
6 |
1 |
1 |
9 |
23 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
| Vibration |
3 |
3 |
5 |
1 |
1 |
8 |
21 |
Molecule Parameter
| Number of Atoms (N) |
19 |
| Number of internal coordinates |
51 |
| Number of independant internal coordinates |
3 |
| Number of vibrational modes |
21 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
Total |
| Linear (IR) |
3 |
3 |
5 |
1 |
1 |
8 |
8 / 13 |
| Quadratic (Raman) |
3 |
3 |
5 |
1 |
1 |
8 |
11 / 10 |
| IR + Raman |
- |
- |
- |
1 |
1 |
- |
0* / 2 |
* 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 |
51 |
0 |
0 |
-1 |
-3 |
0 |
0 |
9 |
| quadratic |
2 |
1.326 |
0 |
0 |
26 |
30 |
0 |
0 |
66 |
| cubic |
3 |
23.426 |
17 |
17 |
-26 |
-82 |
-1 |
-1 |
354 |
| quartic |
4 |
316.251 |
0 |
0 |
351 |
459 |
0 |
0 |
1.671 |
| quintic |
5 |
3.478.761 |
0 |
0 |
-351 |
-1.161 |
0 |
0 |
6.831 |
| sextic |
6 |
32.468.436 |
153 |
153 |
3.276 |
4.788 |
9 |
9 |
25.564 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
Eg |
Tg |
Au |
Eu |
Tu |
| linear |
1 |
3 |
3 |
5 |
1 |
1 |
8 |
| quadratic |
2 |
68 |
68 |
158 |
49 |
49 |
167 |
| cubic |
3 |
1.019 |
1.011 |
2.877 |
938 |
929 |
2.986 |
| quartic |
4 |
13.449 |
13.449 |
39.336 |
12.993 |
12.993 |
39.639 |
| quintic |
5 |
145.710 |
145.710 |
433.890 |
144.099 |
144.099 |
435.888 |
| sextic |
6 |
1.356.710 |
1.356.629 |
4.055.548 |
1.349.914 |
1.349.842 |
4.060.742 |
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