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 |
72 |
0 |
0 |
-4 |
-4 |
0 |
0 |
24 |
4 |
4 |
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 |
66 |
-2 |
2 |
-2 |
-2 |
0 |
0 |
24 |
4 |
4 |
Decomposition into Irreducible representations
Motion |
A1g |
A2g |
B1g |
B2g |
Eg |
A1u |
A2u |
B1u |
B2u |
Eu |
Total |
Cartesian 3N |
6 |
6 |
6 |
6 |
6 |
1 |
5 |
3 |
3 |
12 |
54 |
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 |
6 |
5 |
6 |
6 |
5 |
1 |
4 |
3 |
3 |
11 |
50 |
Molecule Parameter
Number of Atoms (N) |
24 |
Number of internal coordinates |
66 |
Number of independant internal coordinates |
6 |
Number of vibrational modes |
50 |
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1g |
A2g |
B1g |
B2g |
Eg |
A1u |
A2u |
B1u |
B2u |
Eu |
Total |
Linear (IR) |
6 |
5 |
6 |
6 |
5 |
1 |
4 |
3 |
3 |
11 |
15 / 35 |
Quadratic (Raman) |
6 |
5 |
6 |
6 |
5 |
1 |
4 |
3 |
3 |
11 |
23 / 27 |
IR + Raman |
- |
5 |
- |
- |
- |
1 |
- |
3 |
3 |
- |
0* / 12 |
* 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 |
66 |
-2 |
2 |
-2 |
-2 |
0 |
0 |
24 |
4 |
4 |
quadratic |
2 |
2.211 |
3 |
35 |
35 |
35 |
33 |
1 |
321 |
41 |
41 |
cubic |
3 |
50.116 |
-4 |
68 |
-68 |
-68 |
0 |
0 |
3.104 |
144 |
144 |
quartic |
4 |
864.501 |
21 |
629 |
629 |
629 |
561 |
17 |
24.081 |
841 |
841 |
quintic |
5 |
12.103.014 |
-38 |
1.190 |
-1.190 |
-1.190 |
0 |
0 |
158.424 |
2.660 |
2.660 |
sextic |
6 |
143.218.999 |
55 |
7.735 |
7.735 |
7.735 |
6.545 |
17 |
914.641 |
11.585 |
11.585 |
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 |
6 |
5 |
6 |
6 |
5 |
1 |
4 |
3 |
3 |
11 |
quadratic |
2 |
182 |
144 |
162 |
162 |
236 |
117 |
120 |
118 |
118 |
308 |
cubic |
3 |
3.349 |
3.311 |
3.331 |
3.331 |
5.868 |
2.889 |
2.995 |
2.943 |
2.943 |
6.644 |
quartic |
4 |
55.983 |
55.248 |
55.606 |
55.606 |
105.044 |
52.478 |
52.584 |
52.530 |
52.530 |
110.924 |
quintic |
5 |
766.777 |
766.042 |
766.419 |
766.419 |
1.492.925 |
745.644 |
747.569 |
746.616 |
746.616 |
1.532.531 |
sextic |
6 |
9.014.084 |
9.004.424 |
9.009.236 |
9.009.236 |
17.787.896 |
8.893.139 |
8.895.064 |
8.894.092 |
8.894.092 |
18.014.920 |
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