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 |
15 |
1 |
-1 |
-3 |
-1 |
-3 |
-1 |
5 |
3 |
1 |
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 |
9 |
-1 |
1 |
-1 |
1 |
-3 |
-1 |
5 |
3 |
1 |
Decomposition into Irreducible representations
Motion |
A1g |
A2g |
B1g |
B2g |
Eg |
A1u |
A2u |
B1u |
B2u |
Eu |
Total |
Cartesian 3N |
1 |
1 |
1 |
1 |
1 |
0 |
2 |
0 |
1 |
3 |
11 |
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 |
1 |
0 |
1 |
2 |
7 |
Molecule Parameter
Number of Atoms (N) |
5 |
Number of internal coordinates |
9 |
Number of independant internal coordinates |
1 |
Number of vibrational modes |
7 |
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 |
1 |
0 |
1 |
2 |
3 / 4 |
Quadratic (Raman) |
1 |
0 |
1 |
1 |
0 |
0 |
1 |
0 |
1 |
2 |
3 / 4 |
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 |
9 |
-1 |
1 |
-1 |
1 |
-3 |
-1 |
5 |
3 |
1 |
quadratic |
2 |
45 |
1 |
5 |
5 |
5 |
9 |
1 |
17 |
9 |
5 |
cubic |
3 |
165 |
-1 |
5 |
-5 |
5 |
-19 |
-1 |
45 |
19 |
5 |
quartic |
4 |
495 |
3 |
15 |
15 |
15 |
39 |
3 |
103 |
39 |
15 |
quintic |
5 |
1.287 |
-3 |
15 |
-15 |
15 |
-69 |
-3 |
211 |
69 |
15 |
sextic |
6 |
3.003 |
3 |
35 |
35 |
35 |
119 |
3 |
399 |
119 |
35 |
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 |
1 |
0 |
1 |
2 |
quadratic |
2 |
8 |
2 |
5 |
4 |
4 |
1 |
2 |
1 |
2 |
6 |
cubic |
3 |
15 |
9 |
13 |
12 |
12 |
6 |
12 |
6 |
12 |
28 |
quartic |
4 |
52 |
31 |
43 |
37 |
52 |
20 |
26 |
20 |
26 |
68 |
quintic |
5 |
100 |
79 |
94 |
88 |
124 |
62 |
83 |
62 |
83 |
194 |
sextic |
6 |
251 |
195 |
232 |
211 |
336 |
147 |
168 |
147 |
168 |
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