Reduction formula for point group D3d
Characters for molecular motions
Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3d |
Cartesian 3N |
75 |
0 |
-1 |
-3 |
0 |
1 |
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 |
1 |
Decomposition into Irreducible representations
Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
Cartesian 3N |
6 |
6 |
12 |
6 |
7 |
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 |
6 |
5 |
11 |
6 |
6 |
12 |
46 |
Molecule Parameter
Number of Atoms (N) |
25 |
Number of internal coordinates |
69 |
Number of independant internal coordinates |
6 |
Number of vibrational modes |
46 |
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
Linear (IR) |
6 |
5 |
11 |
6 |
6 |
12 |
18 / 28 |
Quadratic (Raman) |
6 |
5 |
11 |
6 |
6 |
12 |
17 / 29 |
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 |
69 |
0 |
1 |
-3 |
0 |
1 |
quadratic |
2 |
2.415 |
0 |
35 |
39 |
0 |
35 |
cubic |
3 |
57.155 |
23 |
35 |
-109 |
-1 |
35 |
quartic |
4 |
1.028.790 |
0 |
630 |
774 |
0 |
630 |
quintic |
5 |
15.020.334 |
0 |
630 |
-2.034 |
0 |
630 |
sextic |
6 |
185.250.786 |
276 |
7.770 |
10.434 |
12 |
7.770 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field |
Tensor Order |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
linear |
1 |
6 |
5 |
11 |
6 |
6 |
12 |
quadratic |
2 |
222 |
187 |
409 |
198 |
198 |
396 |
cubic |
3 |
4.775 |
4.740 |
9.504 |
4.776 |
4.776 |
9.540 |
quartic |
4 |
86.112 |
85.482 |
171.594 |
85.668 |
85.668 |
171.336 |
quintic |
5 |
1.251.840 |
1.251.210 |
2.503.050 |
1.251.864 |
1.251.864 |
2.503.728 |
sextic |
6 |
15.442.368 |
15.434.598 |
30.876.822 |
15.436.740 |
15.436.740 |
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