Reduction formula for point group D2h
Characters for molecular motions
| Motion |
E |
C2 (z) |
C2 (y) |
C2 (x) |
i |
 (xy) |
(xz) |
(yz) |
| Cartesian 3N |
57 |
-3 |
-3 |
-3 |
-3 |
9 |
9 |
9 |
| 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 |
51 |
-1 |
-1 |
-1 |
-3 |
9 |
9 |
9 |
Decomposition into Irreducible representations
| Motion |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
Total |
| Cartesian 3N |
9 |
6 |
6 |
6 |
3 |
9 |
9 |
9 |
57 |
| 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 |
9 |
5 |
5 |
5 |
3 |
8 |
8 |
8 |
51 |
Molecule Parameter
| Number of Atoms (N) |
19 |
| Number of internal coordinates |
51 |
| Number of independant internal coordinates |
9 |
| Number of vibrational modes |
51 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
Total |
| Linear (IR) |
9 |
5 |
5 |
5 |
3 |
8 |
8 |
8 |
24 / 27 |
| Quadratic (Raman) |
9 |
5 |
5 |
5 |
3 |
8 |
8 |
8 |
24 / 27 |
| IR + Raman |
- |
- |
- |
- |
3 |
- |
- |
- |
0* / 3 |
* 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 |
51 |
-1 |
-1 |
-1 |
-3 |
9 |
9 |
9 |
| quadratic |
2 |
1.326 |
26 |
26 |
26 |
30 |
66 |
66 |
66 |
| cubic |
3 |
23.426 |
-26 |
-26 |
-26 |
-82 |
354 |
354 |
354 |
| quartic |
4 |
316.251 |
351 |
351 |
351 |
459 |
1.671 |
1.671 |
1.671 |
| quintic |
5 |
3.478.761 |
-351 |
-351 |
-351 |
-1.161 |
6.831 |
6.831 |
6.831 |
| sextic |
6 |
32.468.436 |
3.276 |
3.276 |
3.276 |
4.788 |
25.564 |
25.564 |
25.564 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
| linear |
1 |
9 |
5 |
5 |
5 |
3 |
8 |
8 |
8 |
| quadratic |
2 |
204 |
158 |
158 |
158 |
147 |
167 |
167 |
167 |
| cubic |
3 |
3.041 |
2.877 |
2.877 |
2.877 |
2.796 |
2.986 |
2.986 |
2.986 |
| quartic |
4 |
40.347 |
39.336 |
39.336 |
39.336 |
38.979 |
39.639 |
39.639 |
39.639 |
| quintic |
5 |
437.130 |
433.890 |
433.890 |
433.890 |
432.297 |
435.888 |
435.888 |
435.888 |
| sextic |
6 |
4.069.968 |
4.055.548 |
4.055.548 |
4.055.548 |
4.049.598 |
4.060.742 |
4.060.742 |
4.060.742 |
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