Reduction formula for point group D3d
Characters for molecular motions
| Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3 d |
| Cartesian 3N |
120 |
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 |
114 |
0 |
2 |
0 |
0 |
8 |
Decomposition into Irreducible representations
| Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Cartesian 3N |
12 |
8 |
20 |
8 |
12 |
20 |
80 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
0 |
0 |
0 |
2 |
| Vibration |
12 |
7 |
19 |
8 |
11 |
19 |
76 |
Molecule Parameter
| Number of Atoms (N) |
40 |
| Number of internal coordinates |
114 |
| Number of independant internal coordinates |
12 |
| Number of vibrational modes |
76 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Linear (IR) |
12 |
7 |
19 |
8 |
11 |
19 |
30 / 46 |
| Quadratic (Raman) |
12 |
7 |
19 |
8 |
11 |
19 |
31 / 45 |
| IR + Raman |
- |
7 |
- |
8 |
- |
- |
0* / 15 |
* 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 |
114 |
0 |
2 |
0 |
0 |
8 |
| quadratic |
2 |
6.555 |
0 |
59 |
57 |
0 |
89 |
| cubic |
3 |
253.460 |
38 |
116 |
0 |
0 |
544 |
| quartic |
4 |
7.413.705 |
0 |
1.769 |
1.653 |
0 |
3.669 |
| quintic |
5 |
174.963.438 |
0 |
3.422 |
0 |
0 |
18.600 |
| sextic |
6 |
3.470.108.187 |
741 |
35.931 |
32.509 |
19 |
96.957 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
| linear |
1 |
12 |
7 |
19 |
8 |
11 |
19 |
| quadratic |
2 |
588 |
514 |
1.102 |
534 |
549 |
1.083 |
| cubic |
3 |
21.293 |
20.963 |
42.237 |
21.021 |
21.235 |
42.237 |
| quartic |
4 |
619.306 |
616.587 |
1.235.893 |
617.196 |
618.146 |
1.235.342 |
| quintic |
5 |
14.585.792 |
14.574.781 |
29.160.573 |
14.576.492 |
14.584.081 |
29.160.573 |
| sextic |
6 |
289.211.740 |
289.145.296 |
578.356.656 |
289.157.837 |
289.188.350 |
578.345.826 |
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