Reduction formula for point group D4
Characters for molecular motions
Motion |
E |
2C4 (z) |
C2 (z) |
2C'2 |
2C''2 |
Cartesian 3N |
21 |
3 |
-3 |
-3 |
-1 |
Translation (x,y,z) |
3 |
1 |
-1 |
-1 |
-1 |
Rotation (Rx,Ry,Rz) |
3 |
1 |
-1 |
-1 |
-1 |
Vibration |
15 |
1 |
-1 |
-1 |
1 |
Decomposition into Irreducible representations
Motion |
A1 |
A2 |
B1 |
B2 |
E |
Total |
Cartesian 3N |
2 |
4 |
1 |
2 |
6 |
15 |
Translation (x,y,z) |
0 |
1 |
0 |
0 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
2 |
Vibration |
2 |
2 |
1 |
2 |
4 |
11 |
Molecule Parameter
Number of Atoms (N) |
7 |
Number of internal coordinates |
15 |
Number of independant internal coordinates |
2 |
Number of vibrational modes |
11 |
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1 |
A2 |
B1 |
B2 |
E |
Total |
Linear (IR) |
2 |
2 |
1 |
2 |
4 |
6 / 5 |
Quadratic (Raman) |
2 |
2 |
1 |
2 |
4 |
9 / 2 |
IR + Raman |
- |
- |
- |
- |
4 |
4 / 0 |
Characters of symmetric powers for vibration representation
Force field |
Tensor Order |
E |
2C4 (z) |
C2 (z) |
2C'2 |
2C''2 |
linear |
1 |
15 |
1 |
-1 |
-1 |
1 |
quadratic |
2 |
120 |
0 |
8 |
8 |
8 |
cubic |
3 |
680 |
0 |
-8 |
-8 |
8 |
quartic |
4 |
3.060 |
4 |
36 |
36 |
36 |
quintic |
5 |
11.628 |
4 |
-36 |
-36 |
36 |
sextic |
6 |
38.760 |
0 |
120 |
120 |
120 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field |
Tensor Order |
A1 |
A2 |
B1 |
B2 |
E |
linear |
1 |
2 |
2 |
1 |
2 |
4 |
quadratic |
2 |
20 |
12 |
16 |
16 |
28 |
cubic |
3 |
84 |
84 |
80 |
88 |
172 |
quartic |
4 |
406 |
370 |
386 |
386 |
756 |
quintic |
5 |
1.450 |
1.450 |
1.430 |
1.466 |
2.916 |
sextic |
6 |
4.920 |
4.800 |
4.860 |
4.860 |
9.660 |
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