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	39	0	-1	5	-5
Translation (x,y,z)	3	0	-1	1	-1
Rotation (R_x,R_y,R_z)	3	0	-1	1	-1
Vibration	33	0	1	3	-3

**Decomposition into Irreducible representations**
Motion	A₁	A₂	E	T₁	T₂	Total
Cartesian 3N	2	0	2	7	4	15
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	5	4	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	A₁	A₂	E	T₁	T₂	Total
Linear (IR)	2	0	2	5	4	5 / 8
Quadratic (Raman)	2	0	2	5	4	8 / 5
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	33	0	1	3	-3
quadratic	2	561	0	17	3	21
cubic	3	6.545	11	17	1	-55
quartic	4	58.905	0	153	9	225
quintic	5	435.897	0	153	27	-531
sextic	6	2.760.681	66	969	27	1.653

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	E	T₁	T₂
linear	1	2	0	2	5	4
quadratic	2	31	21	52	64	71
cubic	3	274	265	528	821	829
quartic	4	2.523	2.442	4.965	7.299	7.371
quintic	5	18.141	18.051	36.192	54.522	54.585
sextic	6	115.506	115.008	230.448	344.643	345.114

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