Reduction formula for point group D2h
Characters for molecular motions
| Motion |
E |
C2 (z) |
C2 (y) |
C2 (x) |
i |
 (xy) |
(xz) |
(yz) |
| Cartesian 3N |
72 |
0 |
-4 |
-4 |
0 |
24 |
4 |
4 |
| Translation (x,y,z) |
3 |
-1 |
-1 |
-1 |
-3 |
1 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
-1 |
-1 |
-1 |
3 |
-1 |
-1 |
-1 |
| Vibration |
66 |
2 |
-2 |
-2 |
0 |
24 |
4 |
4 |
Decomposition into Irreducible representations
| Motion |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
Total |
| Cartesian 3N |
12 |
12 |
6 |
6 |
4 |
8 |
12 |
12 |
72 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
3 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
3 |
| Vibration |
12 |
11 |
5 |
5 |
4 |
7 |
11 |
11 |
66 |
Molecule Parameter
| Number of Atoms (N) |
24 |
| Number of internal coordinates |
66 |
| Number of independant internal coordinates |
12 |
| Number of vibrational modes |
66 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
Total |
| Linear (IR) |
12 |
11 |
5 |
5 |
4 |
7 |
11 |
11 |
29 / 37 |
| Quadratic (Raman) |
12 |
11 |
5 |
5 |
4 |
7 |
11 |
11 |
33 / 33 |
| IR + Raman |
- |
- |
- |
- |
4 |
- |
- |
- |
0* / 4 |
* Center of inversion: Mutual Exclusion Principle
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
C2 (z) |
C2 (y) |
C2 (x) |
i |
(xy) |
(xz) |
(yz) |
| linear |
1 |
66 |
2 |
-2 |
-2 |
0 |
24 |
4 |
4 |
| quadratic |
2 |
2.211 |
35 |
35 |
35 |
33 |
321 |
41 |
41 |
| cubic |
3 |
50.116 |
68 |
-68 |
-68 |
0 |
3.104 |
144 |
144 |
| quartic |
4 |
864.501 |
629 |
629 |
629 |
561 |
24.081 |
841 |
841 |
| quintic |
5 |
12.103.014 |
1.190 |
-1.190 |
-1.190 |
0 |
158.424 |
2.660 |
2.660 |
| sextic |
6 |
143.218.999 |
7.735 |
7.735 |
7.735 |
6.545 |
914.641 |
11.585 |
11.585 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
| linear |
1 |
12 |
11 |
5 |
5 |
4 |
7 |
11 |
11 |
| quadratic |
2 |
344 |
306 |
236 |
236 |
235 |
238 |
308 |
308 |
| cubic |
3 |
6.680 |
6.642 |
5.868 |
5.868 |
5.832 |
5.938 |
6.644 |
6.644 |
| quartic |
4 |
111.589 |
110.854 |
105.044 |
105.044 |
105.008 |
105.114 |
110.924 |
110.924 |
| quintic |
5 |
1.533.196 |
1.532.461 |
1.492.925 |
1.492.925 |
1.492.260 |
1.494.185 |
1.532.531 |
1.532.531 |
| sextic |
6 |
18.023.320 |
18.013.660 |
17.787.896 |
17.787.896 |
17.787.231 |
17.789.156 |
18.014.920 |
18.014.920 |
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