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 |
99 |
1 |
-1 |
-1 |
-1 |
-3 |
-1 |
1 |
9 |
9 |
| 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 |
93 |
-1 |
1 |
1 |
1 |
-3 |
-1 |
1 |
9 |
9 |
Decomposition into Irreducible representations
| Motion |
A1g |
A2g |
B1g |
B2g |
Eg |
A1u |
A2u |
B1u |
B2u |
Eu |
Total |
| Cartesian 3N |
8 |
4 |
6 |
6 |
12 |
4 |
9 |
6 |
6 |
13 |
74 |
| 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 |
8 |
3 |
6 |
6 |
11 |
4 |
8 |
6 |
6 |
12 |
70 |
Molecule Parameter
| Number of Atoms (N) |
33 |
| Number of internal coordinates |
93 |
| Number of independant internal coordinates |
8 |
| Number of vibrational modes |
70 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
B1g |
B2g |
Eg |
A1u |
A2u |
B1u |
B2u |
Eu |
Total |
| Linear (IR) |
8 |
3 |
6 |
6 |
11 |
4 |
8 |
6 |
6 |
12 |
20 / 50 |
| Quadratic (Raman) |
8 |
3 |
6 |
6 |
11 |
4 |
8 |
6 |
6 |
12 |
31 / 39 |
| IR + Raman |
- |
3 |
- |
- |
- |
4 |
- |
6 |
6 |
- |
0* / 19 |
* 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 |
93 |
-1 |
1 |
1 |
1 |
-3 |
-1 |
1 |
9 |
9 |
| quadratic |
2 |
4.371 |
1 |
47 |
47 |
47 |
51 |
1 |
47 |
87 |
87 |
| cubic |
3 |
138.415 |
-1 |
47 |
47 |
47 |
-145 |
-1 |
47 |
543 |
543 |
| quartic |
4 |
3.321.960 |
24 |
1.128 |
1.128 |
1.128 |
1.320 |
24 |
1.128 |
3.288 |
3.288 |
| quintic |
5 |
64.446.024 |
-24 |
1.128 |
1.128 |
1.128 |
-3.576 |
-24 |
1.128 |
16.344 |
16.344 |
| sextic |
6 |
1.052.618.392 |
24 |
18.424 |
18.424 |
18.424 |
23.128 |
24 |
18.424 |
77.672 |
77.672 |
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 |
8 |
3 |
6 |
6 |
11 |
4 |
8 |
6 |
6 |
12 |
| quadratic |
2 |
316 |
249 |
282 |
282 |
541 |
260 |
280 |
270 |
270 |
540 |
| cubic |
3 |
8.795 |
8.500 |
8.648 |
8.648 |
17.272 |
8.536 |
8.784 |
8.660 |
8.660 |
17.320 |
| quartic |
4 |
208.956 |
206.748 |
207.840 |
207.840 |
415.128 |
207.000 |
208.080 |
207.540 |
207.540 |
415.080 |
| quintic |
5 |
4.032.156 |
4.023.420 |
4.027.800 |
4.027.800 |
8.055.024 |
4.024.296 |
4.031.904 |
4.028.100 |
4.028.100 |
8.056.200 |
| sextic |
6 |
65.816.428 |
65.768.380 |
65.792.392 |
65.792.392 |
131.575.584 |
65.772.392 |
65.802.016 |
65.787.204 |
65.787.204 |
131.574.408 |
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