Reduction formula for point group D3d
Characters for molecular motions
| Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3 d |
| Cartesian 3N |
90 |
0 |
0 |
0 |
0 |
6 |
| Translation (x,y,z) |
3 |
0 |
-1 |
-3 |
0 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
3 |
0 |
-1 |
| Vibration |
84 |
0 |
2 |
0 |
0 |
6 |
Decomposition into Irreducible representations
| Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Cartesian 3N |
9 |
6 |
15 |
6 |
9 |
15 |
60 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
0 |
0 |
0 |
2 |
| Vibration |
9 |
5 |
14 |
6 |
8 |
14 |
56 |
Molecule Parameter
| Number of Atoms (N) |
30 |
| Number of internal coordinates |
84 |
| Number of independant internal coordinates |
9 |
| Number of vibrational modes |
56 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Linear (IR) |
9 |
5 |
14 |
6 |
8 |
14 |
22 / 34 |
| Quadratic (Raman) |
9 |
5 |
14 |
6 |
8 |
14 |
23 / 33 |
| IR + Raman |
- |
5 |
- |
6 |
- |
- |
0* / 11 |
* 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 |
84 |
0 |
2 |
0 |
0 |
6 |
| quadratic |
2 |
3.570 |
0 |
44 |
42 |
0 |
60 |
| cubic |
3 |
102.340 |
28 |
86 |
0 |
0 |
290 |
| quartic |
4 |
2.225.895 |
0 |
989 |
903 |
0 |
1.725 |
| quintic |
5 |
39.175.752 |
0 |
1.892 |
0 |
0 |
7.116 |
| sextic |
6 |
581.106.988 |
406 |
15.136 |
13.244 |
14 |
32.416 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
| linear |
1 |
9 |
5 |
14 |
6 |
8 |
14 |
| quadratic |
2 |
327 |
275 |
602 |
290 |
298 |
588 |
| cubic |
3 |
8.627 |
8.439 |
17.052 |
8.482 |
8.584 |
17.052 |
| quartic |
4 |
186.245 |
184.888 |
371.133 |
185.232 |
185.600 |
370.832 |
| quintic |
5 |
3.266.898 |
3.262.394 |
6.529.292 |
3.263.340 |
3.265.952 |
6.529.292 |
| sextic |
6 |
48.438.644 |
48.414.868 |
96.853.302 |
48.420.224 |
48.428.864 |
96.848.892 |
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