Reduction formula for point group D2h
Characters for molecular motions
| Motion |
E |
C2 (z) |
C2 (y) |
C2 (x) |
i |
 (xy) |
(xz) |
(yz) |
| Cartesian 3N |
36 |
0 |
-4 |
0 |
0 |
12 |
0 |
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 |
30 |
2 |
-2 |
2 |
0 |
12 |
0 |
4 |
Decomposition into Irreducible representations
| Motion |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
Total |
| Cartesian 3N |
6 |
6 |
2 |
4 |
2 |
4 |
6 |
6 |
36 |
| 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 |
6 |
5 |
1 |
3 |
2 |
3 |
5 |
5 |
30 |
Molecule Parameter
| Number of Atoms (N) |
12 |
| Number of internal coordinates |
30 |
| Number of independant internal coordinates |
6 |
| Number of vibrational modes |
30 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
Total |
| Linear (IR) |
6 |
5 |
1 |
3 |
2 |
3 |
5 |
5 |
13 / 17 |
| Quadratic (Raman) |
6 |
5 |
1 |
3 |
2 |
3 |
5 |
5 |
15 / 15 |
| IR + Raman |
- |
- |
- |
- |
2 |
- |
- |
- |
0* / 2 |
* 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 |
30 |
2 |
-2 |
2 |
0 |
12 |
0 |
4 |
| quadratic |
2 |
465 |
17 |
17 |
17 |
15 |
87 |
15 |
23 |
| cubic |
3 |
4.960 |
32 |
-32 |
32 |
0 |
472 |
0 |
72 |
| quartic |
4 |
40.920 |
152 |
152 |
152 |
120 |
2.112 |
120 |
256 |
| quintic |
5 |
278.256 |
272 |
-272 |
272 |
0 |
8.184 |
0 |
680 |
| sextic |
6 |
1.623.160 |
952 |
952 |
952 |
680 |
28.336 |
680 |
1.904 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
| linear |
1 |
6 |
5 |
1 |
3 |
2 |
3 |
5 |
5 |
| quadratic |
2 |
82 |
64 |
46 |
48 |
47 |
48 |
66 |
64 |
| cubic |
3 |
692 |
674 |
540 |
574 |
556 |
574 |
676 |
674 |
| quartic |
4 |
5.498 |
5.328 |
4.830 |
4.864 |
4.846 |
4.864 |
5.362 |
5.328 |
| quintic |
5 |
35.924 |
35.754 |
33.572 |
33.878 |
33.708 |
33.878 |
35.788 |
35.754 |
| sextic |
6 |
207.202 |
206.080 |
199.166 |
199.472 |
199.302 |
199.472 |
206.386 |
206.080 |
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