Reduction formula for point group D2h
Characters for molecular motions
| Motion |
E |
C2 (z) |
C2 (y) |
C2 (x) |
i |
 (xy) |
(xz) |
(yz) |
| Cartesian 3N |
24 |
0 |
-4 |
-4 |
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 |
18 |
2 |
-2 |
-2 |
0 |
8 |
4 |
4 |
Decomposition into Irreducible representations
| Motion |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
Total |
| Cartesian 3N |
4 |
4 |
2 |
2 |
0 |
4 |
4 |
4 |
24 |
| 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 |
4 |
3 |
1 |
1 |
0 |
3 |
3 |
3 |
18 |
Molecule Parameter
| Number of Atoms (N) |
8 |
| Number of internal coordinates |
18 |
| Number of independant internal coordinates |
4 |
| Number of vibrational modes |
18 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
Total |
| Linear (IR) |
4 |
3 |
1 |
1 |
0 |
3 |
3 |
3 |
9 / 9 |
| Quadratic (Raman) |
4 |
3 |
1 |
1 |
0 |
3 |
3 |
3 |
9 / 9 |
| IR + Raman |
- |
- |
- |
- |
0 |
- |
- |
- |
0* / 0 |
* 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 |
18 |
2 |
-2 |
-2 |
0 |
8 |
4 |
4 |
| quadratic |
2 |
171 |
11 |
11 |
11 |
9 |
41 |
17 |
17 |
| cubic |
3 |
1.140 |
20 |
-20 |
-20 |
0 |
160 |
48 |
48 |
| quartic |
4 |
5.985 |
65 |
65 |
65 |
45 |
525 |
133 |
133 |
| quintic |
5 |
26.334 |
110 |
-110 |
-110 |
0 |
1.512 |
308 |
308 |
| sextic |
6 |
100.947 |
275 |
275 |
275 |
165 |
3.941 |
693 |
693 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
B1g |
B2g |
B3g |
Au |
B1u |
B2u |
B3u |
| linear |
1 |
4 |
3 |
1 |
1 |
0 |
3 |
3 |
3 |
| quadratic |
2 |
36 |
22 |
16 |
16 |
15 |
18 |
24 |
24 |
| cubic |
3 |
172 |
158 |
120 |
120 |
108 |
142 |
160 |
160 |
| quartic |
4 |
877 |
778 |
680 |
680 |
668 |
702 |
800 |
800 |
| quintic |
5 |
3.544 |
3.445 |
3.089 |
3.089 |
3.012 |
3.221 |
3.467 |
3.467 |
| sextic |
6 |
13.408 |
12.924 |
12.112 |
12.112 |
12.035 |
12.244 |
13.056 |
13.056 |
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