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	39	0	-1	5	-5	-3	-1	0	9	5
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	33	0	1	3	-3	-3	-1	0	9	5

**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	2	0	0	0	5	2	15
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	2	0	0	0	4	2	13

**Molecule Parameter**
Number of Atoms (N)	13
Number of internal coordinates	33
Number of independant internal coordinates	2
Number of vibrational modes	13

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	2	0	0	0	4	2	4 / 9
Quadratic (Raman)	2	0	2	1	2	0	0	0	4	2	6 / 7
IR + Raman	-	0	-	1	-	0	0	0	-	2	0^* / 3

* 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	33	0	1	3	-3	-3	-1	0	9	5
quadratic	2	561	0	17	3	21	21	-1	0	57	29
cubic	3	6.545	11	17	1	-55	-55	1	-1	273	105
quartic	4	58.905	0	153	9	225	225	9	0	1.113	385
quintic	5	435.897	0	153	27	-531	-531	-9	0	3.969	1.141
sextic	6	2.760.681	66	969	27	1.653	1.653	-9	6	12.817	3.325

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	2	0	0	0	4	2
quadratic	2	23	11	34	26	37	8	10	18	38	34
cubic	3	166	135	296	377	407	108	130	232	444	422
quartic	4	1.385	1.246	2.631	3.547	3.677	1.138	1.196	2.334	3.752	3.694
quintic	5	9.449	9.121	18.570	26.836	27.155	8.692	8.930	17.622	27.686	27.430
sextic	6	59.004	57.926	116.894	171.207	172.276	56.502	57.082	113.554	173.436	172.838

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