Reduction formula for point group D4h
Characters for molecular motions
| Motion |
E |
2C4 (z) |
C2 |
2C'2 |
2C''2 |
i |
2S4 |
 h |
2v |
2d |
| Cartesian 3N |
12 |
0 |
0 |
-2 |
0 |
0 |
0 |
4 |
2 |
0 |
| Translation (x,y,z) |
3 |
1 |
-1 |
-1 |
-1 |
-3 |
-1 |
1 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
1 |
-1 |
-1 |
-1 |
3 |
1 |
-1 |
-1 |
-1 |
| Vibration |
6 |
-2 |
2 |
0 |
2 |
0 |
0 |
4 |
2 |
0 |
Decomposition into Irreducible representations
| Motion |
A1g |
A2g |
B1g |
B2g |
Eg |
A1u |
A2u |
B1u |
B2u |
Eu |
Total |
| Cartesian 3N |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
0 |
1 |
2 |
9 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
2 |
| Vibration |
1 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
1 |
1 |
5 |
Molecule Parameter
| Number of Atoms (N) |
4 |
| Number of internal coordinates |
6 |
| Number of independant internal coordinates |
1 |
| Number of vibrational modes |
5 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
B1g |
B2g |
Eg |
A1u |
A2u |
B1u |
B2u |
Eu |
Total |
| Linear (IR) |
1 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
1 |
1 |
1 / 4 |
| Quadratic (Raman) |
1 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
1 |
1 |
3 / 2 |
| IR + Raman |
- |
0 |
- |
- |
- |
0 |
- |
0 |
1 |
- |
0* / 1 |
* Center of inversion: Mutual Exclusion Principle
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
2C4 (z) |
C2 |
2C'2 |
2C''2 |
i |
2S4 |
h |
2v |
2d |
| linear |
1 |
6 |
-2 |
2 |
0 |
2 |
0 |
0 |
4 |
2 |
0 |
| quadratic |
2 |
21 |
3 |
5 |
3 |
5 |
3 |
1 |
11 |
5 |
3 |
| cubic |
3 |
56 |
-4 |
8 |
0 |
8 |
0 |
0 |
24 |
8 |
0 |
| quartic |
4 |
126 |
6 |
14 |
6 |
14 |
6 |
2 |
46 |
14 |
6 |
| quintic |
5 |
252 |
-8 |
20 |
0 |
20 |
0 |
0 |
80 |
20 |
0 |
| sextic |
6 |
462 |
10 |
30 |
10 |
30 |
10 |
2 |
130 |
30 |
10 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A1g |
A2g |
B1g |
B2g |
Eg |
A1u |
A2u |
B1u |
B2u |
Eu |
| linear |
1 |
1 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
1 |
1 |
| quadratic |
2 |
5 |
1 |
2 |
2 |
1 |
1 |
1 |
0 |
1 |
3 |
| cubic |
3 |
7 |
3 |
6 |
6 |
3 |
2 |
2 |
1 |
5 |
9 |
| quartic |
4 |
18 |
8 |
11 |
11 |
9 |
6 |
6 |
3 |
7 |
19 |
| quintic |
5 |
26 |
16 |
23 |
23 |
19 |
11 |
11 |
8 |
18 |
39 |
| sextic |
6 |
51 |
31 |
38 |
38 |
39 |
23 |
23 |
16 |
26 |
69 |
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