Reduction formula for point group D3d
Characters for molecular motions
| Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3 d |
| Cartesian 3N |
72 |
0 |
0 |
0 |
0 |
8 |
| Translation (x,y,z) |
3 |
0 |
-1 |
-3 |
0 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
3 |
0 |
-1 |
| Vibration |
66 |
0 |
2 |
0 |
0 |
8 |
Decomposition into Irreducible representations
| Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Cartesian 3N |
8 |
4 |
12 |
4 |
8 |
12 |
48 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
0 |
0 |
0 |
2 |
| Vibration |
8 |
3 |
11 |
4 |
7 |
11 |
44 |
Molecule Parameter
| Number of Atoms (N) |
24 |
| Number of internal coordinates |
66 |
| Number of independant internal coordinates |
8 |
| Number of vibrational modes |
44 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Linear (IR) |
8 |
3 |
11 |
4 |
7 |
11 |
18 / 26 |
| Quadratic (Raman) |
8 |
3 |
11 |
4 |
7 |
11 |
19 / 25 |
| IR + Raman |
- |
3 |
- |
4 |
- |
- |
0* / 7 |
* Center of inversion: Mutual Exclusion Principle
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
2C3 |
3C'2 |
i |
2S6 |
3d |
| linear |
1 |
66 |
0 |
2 |
0 |
0 |
8 |
| quadratic |
2 |
2.211 |
0 |
35 |
33 |
0 |
65 |
| cubic |
3 |
50.116 |
22 |
68 |
0 |
0 |
352 |
| quartic |
4 |
864.501 |
0 |
629 |
561 |
0 |
1.809 |
| quintic |
5 |
12.103.014 |
0 |
1.190 |
0 |
0 |
7.752 |
| sextic |
6 |
143.218.999 |
253 |
7.735 |
6.545 |
11 |
31.441 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
| linear |
1 |
8 |
3 |
11 |
4 |
7 |
11 |
| quadratic |
2 |
212 |
162 |
374 |
174 |
189 |
363 |
| cubic |
3 |
4.285 |
4.075 |
8.349 |
4.109 |
4.251 |
8.349 |
| quartic |
4 |
72.698 |
71.479 |
144.177 |
71.700 |
72.290 |
143.990 |
| quintic |
5 |
1.010.820 |
1.006.349 |
2.017.169 |
1.006.944 |
1.010.225 |
2.017.169 |
| sextic |
6 |
11.945.300 |
11.925.712 |
23.870.880 |
11.928.485 |
11.940.338 |
23.868.702 |
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