## Reduction formula for point group Oh

Characters of input representation
E 8C3 6C2 6C4 3C2 =(C4)2 i 6S4 8S6 3h 6d
15 0 1 1 -1 -3 -1 0 5 3

Decomposition into Irreducible representations
A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u Total
1 0 1 0 1 0 0 0 2 1 6

### Force field analysis

Allowed / forbidden vibronational transitions
Operator A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u Total
Linear (IR) 1 0 1 0 1 0 0 0 2 1 2 / 4
Quadratic (Raman) 1 0 1 0 1 0 0 0 2 1 3 / 3
IR + Raman - 0 - 0 - 0 0 0 - 1 0* / 1
* Center of inversion: Mutual Exclusion Principle

Characters of symmetric powers for vibration representation
Force field Tensor
Order
E 8C3 6C2 6C4 3C2 =(C4)2 i 6S4 8S6 3h 6d
linear 1 15 0 1 1 -1 -3 -1 0 5 3
quadratic 2 120 0 8 0 8 12 0 0 20 12
cubic 3 680 5 8 0 -8 -28 0 -1 60 28
quartic 4 3.060 0 36 4 36 72 4 0 160 72
quintic 5 11.628 0 36 4 -36 -144 -4 0 376 144
sextic 6 38.760 15 120 0 120 300 0 3 820 300

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field Tensor
Order
A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u
linear 1 1 0 1 0 1 0 0 0 2 1
quadratic 2 7 2 9 4 9 1 2 3 8 7
cubic 3 22 13 33 33 42 9 14 20 51 46
quartic 4 92 63 155 171 196 50 59 109 199 190
quintic 5 283 238 521 674 719 207 232 439 776 747
sextic 6 928 823 1.742 2.330 2.435 737 782 1.513 2.470 2.425

### 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 Mai, 23rd 2018 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement