Reduction formula for point group D3d
Characters for molecular motions
Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3d |
Cartesian 3N |
39 |
0 |
-1 |
-3 |
0 |
5 |
Translation (x,y,z) |
3 |
0 |
-1 |
-3 |
0 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
3 |
0 |
-1 |
Vibration |
33 |
0 |
1 |
-3 |
0 |
5 |
Decomposition into Irreducible representations
Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
Cartesian 3N |
4 |
2 |
6 |
2 |
5 |
7 |
26 |
Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
0 |
0 |
0 |
2 |
Vibration |
4 |
1 |
5 |
2 |
4 |
6 |
22 |
Molecule Parameter
Number of Atoms (N) |
13 |
Number of internal coordinates |
33 |
Number of independant internal coordinates |
4 |
Number of vibrational modes |
22 |
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
Linear (IR) |
4 |
1 |
5 |
2 |
4 |
6 |
10 / 12 |
Quadratic (Raman) |
4 |
1 |
5 |
2 |
4 |
6 |
9 / 13 |
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 |
33 |
0 |
1 |
-3 |
0 |
5 |
quadratic |
2 |
561 |
0 |
17 |
21 |
0 |
29 |
cubic |
3 |
6.545 |
11 |
17 |
-55 |
-1 |
105 |
quartic |
4 |
58.905 |
0 |
153 |
225 |
0 |
385 |
quintic |
5 |
435.897 |
0 |
153 |
-531 |
0 |
1.141 |
sextic |
6 |
2.760.681 |
66 |
969 |
1.653 |
6 |
3.325 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field |
Tensor Order |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
linear |
1 |
4 |
1 |
5 |
2 |
4 |
6 |
quadratic |
2 |
60 |
37 |
97 |
42 |
48 |
90 |
cubic |
3 |
573 |
512 |
1.080 |
530 |
574 |
1.098 |
quartic |
4 |
5.062 |
4.793 |
9.855 |
4.832 |
4.948 |
9.780 |
quintic |
5 |
36.604 |
35.957 |
72.561 |
36.122 |
36.616 |
72.738 |
sextic |
6 |
231.280 |
229.133 |
460.377 |
229.340 |
230.518 |
459.828 |
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