Reduction formula for point group D2h
Characters for molecular motions
| Motion |
E |
C2 (z) |
C2 (y) |
C2 (x) |
i |
 (xy) |
(xz) |
(yz) |
| Cartesian 3N |
21 |
-3 |
-3 |
-3 |
-3 |
5 |
5 |
5 |
| 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 |
15 |
-1 |
-1 |
-1 |
-3 |
5 |
5 |
5 |
Decomposition into Irreducible representations
| Motion |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
Total |
| Cartesian 3N |
3 |
2 |
2 |
2 |
0 |
4 |
4 |
4 |
21 |
| 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 |
3 |
1 |
1 |
1 |
0 |
3 |
3 |
3 |
15 |
Molecule Parameter
| Number of Atoms (N) |
7 |
| Number of internal coordinates |
15 |
| Number of independant internal coordinates |
3 |
| Number of vibrational modes |
15 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
Total |
| Linear (IR) |
3 |
1 |
1 |
1 |
0 |
3 |
3 |
3 |
9 / 6 |
| Quadratic (Raman) |
3 |
1 |
1 |
1 |
0 |
3 |
3 |
3 |
6 / 9 |
| IR + Raman |
- |
- |
- |
- |
0 |
- |
- |
- |
0* / 0 |
* 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 |
15 |
-1 |
-1 |
-1 |
-3 |
5 |
5 |
5 |
| quadratic |
2 |
120 |
8 |
8 |
8 |
12 |
20 |
20 |
20 |
| cubic |
3 |
680 |
-8 |
-8 |
-8 |
-28 |
60 |
60 |
60 |
| quartic |
4 |
3.060 |
36 |
36 |
36 |
72 |
160 |
160 |
160 |
| quintic |
5 |
11.628 |
-36 |
-36 |
-36 |
-144 |
376 |
376 |
376 |
| sextic |
6 |
38.760 |
120 |
120 |
120 |
300 |
820 |
820 |
820 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
| linear |
1 |
3 |
1 |
1 |
1 |
0 |
3 |
3 |
3 |
| quadratic |
2 |
27 |
13 |
13 |
13 |
9 |
15 |
15 |
15 |
| cubic |
3 |
101 |
75 |
75 |
75 |
63 |
97 |
97 |
97 |
| quartic |
4 |
465 |
367 |
367 |
367 |
327 |
389 |
389 |
389 |
| quintic |
5 |
1.563 |
1.393 |
1.393 |
1.393 |
1.317 |
1.523 |
1.523 |
1.523 |
| sextic |
6 |
5.235 |
4.765 |
4.765 |
4.765 |
4.545 |
4.895 |
4.895 |
4.895 |
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