Reduction formula for point group D3d
Characters for molecular motions
| Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3 d |
| Cartesian 3N |
48 |
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 |
42 |
0 |
2 |
0 |
0 |
8 |
Decomposition into Irreducible representations
| Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Cartesian 3N |
6 |
2 |
8 |
2 |
6 |
8 |
32 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
0 |
0 |
0 |
2 |
| Vibration |
6 |
1 |
7 |
2 |
5 |
7 |
28 |
Molecule Parameter
| Number of Atoms (N) |
16 |
| Number of internal coordinates |
42 |
| Number of independant internal coordinates |
6 |
| Number of vibrational modes |
28 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Linear (IR) |
6 |
1 |
7 |
2 |
5 |
7 |
12 / 16 |
| Quadratic (Raman) |
6 |
1 |
7 |
2 |
5 |
7 |
13 / 15 |
| IR + Raman |
- |
1 |
- |
2 |
- |
- |
0* / 3 |
* 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 |
42 |
0 |
2 |
0 |
0 |
8 |
| quadratic |
2 |
903 |
0 |
23 |
21 |
0 |
53 |
| cubic |
3 |
13.244 |
14 |
44 |
0 |
0 |
256 |
| quartic |
4 |
148.995 |
0 |
275 |
231 |
0 |
1.095 |
| quintic |
5 |
1.370.754 |
0 |
506 |
0 |
0 |
4.056 |
| sextic |
6 |
10.737.573 |
105 |
2.277 |
1.771 |
7 |
13.803 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
| linear |
1 |
6 |
1 |
7 |
2 |
5 |
7 |
| quadratic |
2 |
96 |
58 |
154 |
66 |
81 |
147 |
| cubic |
3 |
1.181 |
1.031 |
2.205 |
1.053 |
1.159 |
2.205 |
| quartic |
4 |
12.778 |
12.093 |
24.871 |
12.192 |
12.602 |
24.794 |
| quintic |
5 |
115.370 |
113.089 |
228.459 |
113.342 |
115.117 |
228.459 |
| sextic |
6 |
898.984 |
890.944 |
1.789.872 |
891.785 |
897.548 |
1.789.284 |
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