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 |
0 |
0 |
0 |
8 |
8 |
8 |
| 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 |
8 |
8 |
8 |
Decomposition into Irreducible representations
| Motion |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
Total |
| Cartesian 3N |
12 |
8 |
8 |
8 |
6 |
10 |
10 |
10 |
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 |
7 |
7 |
7 |
6 |
9 |
9 |
9 |
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 |
7 |
7 |
7 |
6 |
9 |
9 |
9 |
27 / 39 |
| Quadratic (Raman) |
12 |
7 |
7 |
7 |
6 |
9 |
9 |
9 |
33 / 33 |
| IR + Raman |
- |
- |
- |
- |
6 |
- |
- |
- |
0* / 6 |
* 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 |
8 |
8 |
8 |
| quadratic |
2 |
2.211 |
35 |
35 |
35 |
33 |
65 |
65 |
65 |
| cubic |
3 |
50.116 |
68 |
68 |
68 |
0 |
352 |
352 |
352 |
| quartic |
4 |
864.501 |
629 |
629 |
629 |
561 |
1.809 |
1.809 |
1.809 |
| quintic |
5 |
12.103.014 |
1.190 |
1.190 |
1.190 |
0 |
7.752 |
7.752 |
7.752 |
| sextic |
6 |
143.218.999 |
7.735 |
7.735 |
7.735 |
6.545 |
31.441 |
31.441 |
31.441 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
| linear |
1 |
12 |
7 |
7 |
7 |
6 |
9 |
9 |
9 |
| quadratic |
2 |
318 |
268 |
268 |
268 |
261 |
276 |
276 |
276 |
| cubic |
3 |
6.422 |
6.212 |
6.212 |
6.212 |
6.158 |
6.300 |
6.300 |
6.300 |
| quartic |
4 |
109.047 |
107.828 |
107.828 |
107.828 |
107.550 |
108.140 |
108.140 |
108.140 |
| quintic |
5 |
1.516.230 |
1.511.759 |
1.511.759 |
1.511.759 |
1.510.416 |
1.513.697 |
1.513.697 |
1.513.697 |
| sextic |
6 |
17.917.884 |
17.898.296 |
17.898.296 |
17.898.296 |
17.892.667 |
17.904.520 |
17.904.520 |
17.904.520 |
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