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 |
24 |
0 |
0 |
-4 |
0 |
0 |
0 |
8 |
4 |
0 |
| 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 |
18 |
-2 |
2 |
-2 |
2 |
0 |
0 |
8 |
4 |
0 |
Decomposition into Irreducible representations
| Motion |
A1g |
A2g |
B1g |
B2g |
Eg |
A1u |
A2u |
B1u |
B2u |
Eu |
Total |
| Cartesian 3N |
2 |
2 |
2 |
2 |
2 |
0 |
2 |
0 |
2 |
4 |
18 |
| 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 |
2 |
1 |
2 |
2 |
1 |
0 |
1 |
0 |
2 |
3 |
14 |
Molecule Parameter
| Number of Atoms (N) |
8 |
| Number of internal coordinates |
18 |
| Number of independant internal coordinates |
2 |
| Number of vibrational modes |
14 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
B1g |
B2g |
Eg |
A1u |
A2u |
B1u |
B2u |
Eu |
Total |
| Linear (IR) |
2 |
1 |
2 |
2 |
1 |
0 |
1 |
0 |
2 |
3 |
4 / 10 |
| Quadratic (Raman) |
2 |
1 |
2 |
2 |
1 |
0 |
1 |
0 |
2 |
3 |
7 / 7 |
| IR + Raman |
- |
1 |
- |
- |
- |
0 |
- |
0 |
2 |
- |
0* / 3 |
* 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 |
18 |
-2 |
2 |
-2 |
2 |
0 |
0 |
8 |
4 |
0 |
| quadratic |
2 |
171 |
3 |
11 |
11 |
11 |
9 |
1 |
41 |
17 |
9 |
| cubic |
3 |
1.140 |
-4 |
20 |
-20 |
20 |
0 |
0 |
160 |
48 |
0 |
| quartic |
4 |
5.985 |
9 |
65 |
65 |
65 |
45 |
5 |
525 |
133 |
45 |
| quintic |
5 |
26.334 |
-14 |
110 |
-110 |
110 |
0 |
0 |
1.512 |
308 |
0 |
| sextic |
6 |
100.947 |
19 |
275 |
275 |
275 |
165 |
5 |
3.941 |
693 |
165 |
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 |
2 |
1 |
2 |
2 |
1 |
0 |
1 |
0 |
2 |
3 |
| quadratic |
2 |
21 |
9 |
15 |
13 |
16 |
8 |
9 |
7 |
9 |
24 |
| cubic |
3 |
88 |
76 |
84 |
82 |
120 |
56 |
68 |
52 |
74 |
160 |
| quartic |
4 |
454 |
377 |
423 |
401 |
680 |
337 |
349 |
331 |
353 |
800 |
| quintic |
5 |
1.784 |
1.707 |
1.760 |
1.738 |
3.089 |
1.518 |
1.595 |
1.494 |
1.626 |
3.467 |
| sextic |
6 |
6.762 |
6.410 |
6.646 |
6.514 |
12.112 |
6.033 |
6.110 |
6.002 |
6.134 |
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