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	36	0	0	4	-4	0	0	0	8	4
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	30	0	2	2	-2	0	0	0	8	4

**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	4	2	14
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	3	2	12

**Molecule Parameter**
Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	2
Number of vibrational modes	12

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	3	2	3 / 9
Quadratic (Raman)	2	0	2	1	2	0	0	0	3	2	6 / 6
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	30	0	2	2	-2	0	0	0	8	4
quadratic	2	465	0	17	1	17	15	-1	0	47	23
cubic	3	4.960	10	32	0	-32	0	0	0	208	72
quartic	4	40.920	0	152	8	152	120	8	0	792	256
quintic	5	278.256	0	272	16	-272	0	0	0	2.640	680
sextic	6	1.623.160	55	952	8	952	680	-8	5	8.008	1.904

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	3	2
quadratic	2	19	9	28	21	31	7	8	15	31	29
cubic	3	129	103	227	286	312	85	95	175	330	320
quartic	4	967	861	1.828	2.457	2.555	797	823	1.620	2.603	2.577
quintic	5	6.066	5.824	11.890	17.126	17.360	5.566	5.664	11.230	17.626	17.520
sextic	6	34.757	34.043	68.770	100.573	101.287	33.252	33.486	66.713	101.967	101.725

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