Reduction formula for point group D3d
Characters for molecular motions
| Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3 d |
| Cartesian 3N |
24 |
0 |
0 |
0 |
0 |
4 |
| Translation (x,y,z) |
3 |
0 |
-1 |
-3 |
0 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
3 |
0 |
-1 |
| Vibration |
18 |
0 |
2 |
0 |
0 |
4 |
Decomposition into Irreducible representations
| Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Cartesian 3N |
3 |
1 |
4 |
1 |
3 |
4 |
16 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
0 |
0 |
0 |
2 |
| Vibration |
3 |
0 |
3 |
1 |
2 |
3 |
12 |
Molecule Parameter
| Number of Atoms (N) |
8 |
| Number of internal coordinates |
18 |
| Number of independant internal coordinates |
3 |
| Number of vibrational modes |
12 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Linear (IR) |
3 |
0 |
3 |
1 |
2 |
3 |
5 / 7 |
| Quadratic (Raman) |
3 |
0 |
3 |
1 |
2 |
3 |
6 / 6 |
| IR + Raman |
- |
0 |
- |
1 |
- |
- |
0* / 1 |
* 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 |
18 |
0 |
2 |
0 |
0 |
4 |
| quadratic |
2 |
171 |
0 |
11 |
9 |
0 |
17 |
| cubic |
3 |
1.140 |
6 |
20 |
0 |
0 |
48 |
| quartic |
4 |
5.985 |
0 |
65 |
45 |
0 |
133 |
| quintic |
5 |
26.334 |
0 |
110 |
0 |
0 |
308 |
| sextic |
6 |
100.947 |
21 |
275 |
165 |
3 |
693 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
| linear |
1 |
3 |
0 |
3 |
1 |
2 |
3 |
| quadratic |
2 |
22 |
8 |
30 |
12 |
15 |
27 |
| cubic |
3 |
113 |
79 |
189 |
89 |
103 |
189 |
| quartic |
4 |
552 |
453 |
1.005 |
478 |
512 |
990 |
| quintic |
5 |
2.299 |
2.090 |
4.389 |
2.145 |
2.244 |
4.389 |
| sextic |
6 |
8.672 |
8.188 |
16.848 |
8.297 |
8.506 |
16.794 |
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