Reduction formula for point group Oh

Reduction formula for point group O_h

**Characters for molecular motions**
Motion	E	8C₃	6C₂	6C₄	3C₂ =(C₄)²	i	6S₄	8S₆	3_h	6_d
Cartesian 3N	48	0	0	0	0	0	0	0	0	8
Translation (x,y,z)	3	0	-1	1	-1	-3	-1	0	1	1
Rotation (R_x,R_y,R_z)	3	0	-1	1	-1	3	1	0	-1	-1
Vibration	42	0	2	-2	2	0	0	0	0	8

**Decomposition into Irreducible representations**
Motion	A_1g	A_2g	E_g	T_1g	T_2g	A_1u	A_2u	E_u	T_1u	T_2u	Total
Cartesian 3N	2	0	2	2	4	0	2	2	4	2	20
Translation (x,y,z)	0	0	0	0	0	0	0	0	1	0	1
Rotation (R_x,R_y,R_z)	0	0	0	1	0	0	0	0	0	0	1
Vibration	2	0	2	1	4	0	2	2	3	2	18

**Molecule Parameter**
Number of Atoms (N)	16
Number of internal coordinates	42
Number of independant internal coordinates	2
Number of vibrational modes	18

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A_1g	A_2g	E_g	T_1g	T_2g	A_1u	A_2u	E_u	T_1u	T_2u	Total
Linear (IR)	2	0	2	1	4	0	2	2	3	2	3 / 15
Quadratic (Raman)	2	0	2	1	4	0	2	2	3	2	8 / 10
IR + Raman	-	0	-	1	-	0	2	2	-	2	0^* / 7

* 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	3_h	6_d
linear	1	42	0	2	-2	2	0	0	0	0	8
quadratic	2	903	0	23	3	23	21	1	0	21	53
cubic	3	13.244	14	44	-4	44	0	0	0	0	256
quartic	4	148.995	0	275	15	275	231	11	0	231	1.095
quintic	5	1.370.754	0	506	-26	506	0	0	0	0	4.056
sextic	6	10.737.573	105	2.277	37	2.277	1.771	11	7	1.771	13.803

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A_1g	A_2g	E_g	T_1g	T_2g	A_1u	A_2u	E_u	T_1u	T_2u
linear	1	2	0	2	1	4	0	2	2	3	2
quadratic	2	32	12	44	46	64	15	22	37	59	51
cubic	3	318	244	555	787	863	254	308	555	851	799
quartic	4	3.315	2.966	6.281	9.127	9.463	3.000	3.204	6.204	9.398	9.192
quintic	5	29.156	28.022	57.178	85.067	86.214	28.142	29.036	57.178	86.081	85.200
sextic	6	226.024	221.992	447.960	668.952	672.960	222.273	225.148	447.372	672.400	669.512

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

Character tables for chemically important point groups

Character table for point group Oh

Constructor University Bremen

Last update November, 13^th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement