Reduction formula for point group D2h
Characters for molecular motions
| Motion |
E |
C2 (z) |
C2 (y) |
C2 (x) |
i |
 (xy) |
(xz) |
(yz) |
| Cartesian 3N |
15 |
-1 |
-1 |
-1 |
-3 |
5 |
1 |
1 |
| 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 |
9 |
1 |
1 |
1 |
-3 |
5 |
1 |
1 |
Decomposition into Irreducible representations
| Motion |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
Total |
| Cartesian 3N |
2 |
2 |
1 |
1 |
1 |
2 |
3 |
3 |
15 |
| 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 |
2 |
1 |
0 |
0 |
1 |
1 |
2 |
2 |
9 |
Molecule Parameter
| Number of Atoms (N) |
5 |
| Number of internal coordinates |
9 |
| Number of independant internal coordinates |
2 |
| Number of vibrational modes |
9 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
Total |
| Linear (IR) |
2 |
1 |
0 |
0 |
1 |
1 |
2 |
2 |
5 / 4 |
| Quadratic (Raman) |
2 |
1 |
0 |
0 |
1 |
1 |
2 |
2 |
3 / 6 |
| IR + Raman |
- |
- |
- |
- |
1 |
- |
- |
- |
0* / 1 |
* 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 |
9 |
1 |
1 |
1 |
-3 |
5 |
1 |
1 |
| quadratic |
2 |
45 |
5 |
5 |
5 |
9 |
17 |
5 |
5 |
| cubic |
3 |
165 |
5 |
5 |
5 |
-19 |
45 |
5 |
5 |
| quartic |
4 |
495 |
15 |
15 |
15 |
39 |
103 |
15 |
15 |
| quintic |
5 |
1.287 |
15 |
15 |
15 |
-69 |
211 |
15 |
15 |
| sextic |
6 |
3.003 |
35 |
35 |
35 |
119 |
399 |
35 |
35 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
| linear |
1 |
2 |
1 |
0 |
0 |
1 |
1 |
2 |
2 |
| quadratic |
2 |
12 |
7 |
4 |
4 |
3 |
3 |
6 |
6 |
| cubic |
3 |
27 |
22 |
12 |
12 |
18 |
18 |
28 |
28 |
| quartic |
4 |
89 |
74 |
52 |
52 |
46 |
46 |
68 |
68 |
| quintic |
5 |
188 |
173 |
124 |
124 |
145 |
145 |
194 |
194 |
| sextic |
6 |
462 |
427 |
336 |
336 |
315 |
315 |
406 |
406 |
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