Reduction formula for point group C4v
Characters for molecular motions
Motion |
E |
2C4 (z) |
C2 |
2v |
2d |
Cartesian 3N |
12 |
4 |
-4 |
4 |
4 |
Translation (x,y,z) |
3 |
1 |
-1 |
1 |
1 |
Rotation (Rx,Ry)* |
2 |
0 |
-2 |
0 |
0 |
Vibration |
7 |
3 |
-1 |
3 |
3 |
* Linear molecule
Decomposition into Irreducible representations
Motion |
A1 |
A2 |
B1 |
B2 |
E |
Total |
Cartesian 3N |
4 |
0 |
0 |
0 |
4 |
8 |
Translation (x,y,z) |
1 |
0 |
0 |
0 |
1 |
2 |
Rotation (Rx,Ry)* |
0 |
0 |
0 |
0 |
1 |
1 |
Vibration |
3 |
0 |
0 |
0 |
2 |
5 |
* Linear molecule
Molecule Parameter
Number of Atoms (N) |
4 |
Number of internal coordinates |
7 |
Number of independant internal coordinates |
3 |
Number of vibrational modes |
5 |
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1 |
A2 |
B1 |
B2 |
E |
Total |
Linear (IR) |
3 |
0 |
0 |
0 |
2 |
5 / 0 |
Quadratic (Raman) |
3 |
0 |
0 |
0 |
2 |
5 / 0 |
IR + Raman |
3 |
0 |
- |
- |
2 |
5 / 0 |
Characters of symmetric powers for vibration representation
Force field |
Tensor Order |
E |
2C4 (z) |
C2 |
2v |
2d |
linear |
1 |
7 |
3 |
-1 |
3 |
3 |
quadratic |
2 |
28 |
4 |
4 |
8 |
8 |
cubic |
3 |
84 |
4 |
-4 |
16 |
16 |
quartic |
4 |
210 |
6 |
10 |
30 |
30 |
quintic |
5 |
462 |
10 |
-10 |
50 |
50 |
sextic |
6 |
924 |
12 |
20 |
80 |
80 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field |
Tensor Order |
A1 |
A2 |
B1 |
B2 |
E |
linear |
1 |
3 |
0 |
0 |
0 |
2 |
quadratic |
2 |
9 |
1 |
3 |
3 |
6 |
cubic |
3 |
19 |
3 |
9 |
9 |
22 |
quartic |
4 |
44 |
14 |
26 |
26 |
50 |
quintic |
5 |
84 |
34 |
54 |
54 |
118 |
sextic |
6 |
161 |
81 |
115 |
115 |
226 |
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