Reduction formula for point group D3d
Characters for molecular motions
| Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3 d |
| Cartesian 3N |
75 |
0 |
-1 |
-3 |
0 |
9 |
| Translation (x,y,z) |
3 |
0 |
-1 |
-3 |
0 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
3 |
0 |
-1 |
| Vibration |
69 |
0 |
1 |
-3 |
0 |
9 |
Decomposition into Irreducible representations
| Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Cartesian 3N |
8 |
4 |
12 |
4 |
9 |
13 |
50 |
| 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 |
8 |
12 |
46 |
Molecule Parameter
| Number of Atoms (N) |
25 |
| Number of internal coordinates |
69 |
| Number of independant internal coordinates |
8 |
| Number of vibrational modes |
46 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Linear (IR) |
8 |
3 |
11 |
4 |
8 |
12 |
20 / 26 |
| Quadratic (Raman) |
8 |
3 |
11 |
4 |
8 |
12 |
19 / 27 |
| 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 |
69 |
0 |
1 |
-3 |
0 |
9 |
| quadratic |
2 |
2.415 |
0 |
35 |
39 |
0 |
75 |
| cubic |
3 |
57.155 |
23 |
35 |
-109 |
-1 |
435 |
| quartic |
4 |
1.028.790 |
0 |
630 |
774 |
0 |
2.310 |
| quintic |
5 |
15.020.334 |
0 |
630 |
-2.034 |
0 |
10.422 |
| sextic |
6 |
185.250.786 |
276 |
7.770 |
10.434 |
12 |
43.738 |
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 |
8 |
12 |
| quadratic |
2 |
232 |
177 |
409 |
188 |
208 |
396 |
| cubic |
3 |
4.875 |
4.640 |
9.504 |
4.676 |
4.876 |
9.540 |
| quartic |
4 |
86.532 |
85.062 |
171.594 |
85.248 |
86.088 |
171.336 |
| quintic |
5 |
1.254.288 |
1.248.762 |
2.503.050 |
1.249.416 |
1.254.312 |
2.503.728 |
| sextic |
6 |
15.451.360 |
15.425.606 |
30.876.822 |
15.427.748 |
15.445.732 |
30.873.348 |
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