Reduction formula for point group O

Reduction formula for point group O

**Characters for molecular motions**
Motion	E	8C₃	6C'₂	6C₄	3C₂ =(C₄)²
Cartesian 3N	36	0	0	4	-4
Translation (x,y,z)	3	0	-1	1	-1
Rotation (R_x,R_y,R_z)	3	0	-1	1	-1
Vibration	30	0	2	2	-2

**Decomposition into Irreducible representations**
Motion	A₁	A₂	E	T₁	T₂	Total
Cartesian 3N	2	0	2	6	4	14
Translation (x,y,z)	0	0	0	1	0	1
Rotation (R_x,R_y,R_z)	0	0	0	1	0	1
Vibration	2	0	2	4	4	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	A₁	A₂	E	T₁	T₂	Total
Linear (IR)	2	0	2	4	4	4 / 8
Quadratic (Raman)	2	0	2	4	4	8 / 4
IR + Raman	-	0	-	-	-	0 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	8C₃	6C'₂	6C₄	3C₂ =(C₄)²
linear	1	30	0	2	2	-2
quadratic	2	465	0	17	1	17
cubic	3	4.960	10	32	0	-32
quartic	4	40.920	0	152	8	152
quintic	5	278.256	0	272	16	-272
sextic	6	1.623.160	55	952	8	952

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	E	T₁	T₂
linear	1	2	0	2	4	4
quadratic	2	26	17	43	52	60
cubic	3	214	198	402	616	632
quartic	4	1.764	1.684	3.448	5.060	5.132
quintic	5	11.632	11.488	23.120	34.752	34.880
sextic	6	68.009	67.529	135.483	202.540	203.012

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 O

Constructor University Bremen

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