Reduction formula for point group C2v

Reduction formula for point group C_2v

**Characters for molecular motions**
Motion	E	C₂ (z)	_v(xz)	_v(yz)
Cartesian 3N	27	-1	5	5
Translation (x,y,z)	3	-1	1	1
Rotation (R_x,R_y,R_z)	3	-1	-1	-1
Vibration	21	1	5	5

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	Total
Cartesian 3N	9	4	7	7	27
Translation (x,y,z)	1	0	1	1	3
Rotation (R_x,R_y,R_z)	0	1	1	1	3
Vibration	8	3	5	5	21

**Molecule Parameter**
Number of Atoms (N)	9
Number of internal coordinates	21
Number of independant internal coordinates	8
Number of vibrational modes	21

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	Total
Linear (IR)	8	3	5	5	18 / 3
Quadratic (Raman)	8	3	5	5	21 / 0
IR + Raman	8	-	5	5	18 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	_v(xz)	_v(yz)
linear	1	21	1	5	5
quadratic	2	231	11	23	23
cubic	3	1.771	11	75	75
quartic	4	10.626	66	226	226
quintic	5	53.130	66	586	586
sextic	6	230.230	286	1.430	1.430

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂
linear	1	8	3	5	5
quadratic	2	72	49	55	55
cubic	3	483	408	440	440
quartic	4	2.786	2.560	2.640	2.640
quintic	5	13.592	13.006	13.266	13.266
sextic	6	58.344	56.914	57.486	57.486

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 C2v

Constructor University Bremen

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