Reduction formula for point group D4
Characters for molecular motions
Motion |
E |
2C4 (z) |
C2 (z) |
2C'2 |
2C''2 |
Cartesian 3N |
15 |
1 |
-1 |
-3 |
-1 |
Translation (x,y,z) |
3 |
1 |
-1 |
-1 |
-1 |
Rotation (Rx,Ry,Rz) |
3 |
1 |
-1 |
-1 |
-1 |
Vibration |
9 |
-1 |
1 |
-1 |
1 |
Decomposition into Irreducible representations
Motion |
A1 |
A2 |
B1 |
B2 |
E |
Total |
Cartesian 3N |
1 |
3 |
1 |
2 |
4 |
11 |
Translation (x,y,z) |
0 |
1 |
0 |
0 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
2 |
Vibration |
1 |
1 |
1 |
2 |
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 |
A1 |
A2 |
B1 |
B2 |
E |
Total |
Linear (IR) |
1 |
1 |
1 |
2 |
2 |
3 / 4 |
Quadratic (Raman) |
1 |
1 |
1 |
2 |
2 |
6 / 1 |
IR + Raman |
- |
- |
- |
- |
2 |
2 / 0 |
Characters of symmetric powers for vibration representation
Force field |
Tensor Order |
E |
2C4 (z) |
C2 (z) |
2C'2 |
2C''2 |
linear |
1 |
9 |
-1 |
1 |
-1 |
1 |
quadratic |
2 |
45 |
1 |
5 |
5 |
5 |
cubic |
3 |
165 |
-1 |
5 |
-5 |
5 |
quartic |
4 |
495 |
3 |
15 |
15 |
15 |
quintic |
5 |
1.287 |
-3 |
15 |
-15 |
15 |
sextic |
6 |
3.003 |
3 |
35 |
35 |
35 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field |
Tensor Order |
A1 |
A2 |
B1 |
B2 |
E |
linear |
1 |
1 |
1 |
1 |
2 |
2 |
quadratic |
2 |
9 |
4 |
6 |
6 |
10 |
cubic |
3 |
21 |
21 |
19 |
24 |
40 |
quartic |
4 |
72 |
57 |
63 |
63 |
120 |
quintic |
5 |
162 |
162 |
156 |
171 |
318 |
sextic |
6 |
398 |
363 |
379 |
379 |
742 |
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