Reduction formula for point group D2h
Characters for molecular motions
Motion |
E |
C2 (z) |
C2 (y) |
C2 (x) |
i |
(xy) |
(xz) |
(yz) |
Cartesian 3N |
36 |
-4 |
0 |
0 |
0 |
8 |
4 |
4 |
Translation (x,y,z) |
3 |
-1 |
-1 |
-1 |
-3 |
1 |
1 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
-1 |
-1 |
-1 |
3 |
-1 |
-1 |
-1 |
Vibration |
30 |
-2 |
2 |
2 |
0 |
8 |
4 |
4 |
Decomposition into Irreducible representations
Motion |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
Total |
Cartesian 3N |
6 |
4 |
4 |
4 |
2 |
4 |
6 |
6 |
36 |
Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
3 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
3 |
Vibration |
6 |
3 |
3 |
3 |
2 |
3 |
5 |
5 |
30 |
Molecule Parameter
Number of Atoms (N) |
12 |
Number of internal coordinates |
30 |
Number of independant internal coordinates |
6 |
Number of vibrational modes |
30 |
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
Total |
Linear (IR) |
6 |
3 |
3 |
3 |
2 |
3 |
5 |
5 |
13 / 17 |
Quadratic (Raman) |
6 |
3 |
3 |
3 |
2 |
3 |
5 |
5 |
15 / 15 |
IR + Raman |
- |
- |
- |
- |
2 |
- |
- |
- |
0* / 2 |
* Center of inversion: Mutual Exclusion Principle
Characters of symmetric powers for vibration representation
Force field |
Tensor Order |
E |
C2 (z) |
C2 (y) |
C2 (x) |
i |
(xy) |
(xz) |
(yz) |
linear |
1 |
30 |
-2 |
2 |
2 |
0 |
8 |
4 |
4 |
quadratic |
2 |
465 |
17 |
17 |
17 |
15 |
47 |
23 |
23 |
cubic |
3 |
4.960 |
-32 |
32 |
32 |
0 |
208 |
72 |
72 |
quartic |
4 |
40.920 |
152 |
152 |
152 |
120 |
792 |
256 |
256 |
quintic |
5 |
278.256 |
-272 |
272 |
272 |
0 |
2.640 |
680 |
680 |
sextic |
6 |
1.623.160 |
952 |
952 |
952 |
680 |
8.008 |
1.904 |
1.904 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field |
Tensor Order |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
linear |
1 |
6 |
3 |
3 |
3 |
2 |
3 |
5 |
5 |
quadratic |
2 |
78 |
58 |
52 |
52 |
51 |
54 |
60 |
60 |
cubic |
3 |
668 |
616 |
598 |
598 |
580 |
600 |
650 |
650 |
quartic |
4 |
5.350 |
5.146 |
5.012 |
5.012 |
4.994 |
5.046 |
5.180 |
5.180 |
quintic |
5 |
35.316 |
34.840 |
34.486 |
34.486 |
34.316 |
34.520 |
35.146 |
35.146 |
sextic |
6 |
204.814 |
203.386 |
201.860 |
201.860 |
201.690 |
202.166 |
203.692 |
203.692 |
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