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	33	-1	1	3
Translation (x,y,z)	3	-1	1	1
Rotation (R_x,R_y,R_z)	3	-1	-1	-1
Vibration	27	1	1	3

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	Total
Cartesian 3N	9	7	8	9	33
Translation (x,y,z)	1	0	1	1	3
Rotation (R_x,R_y,R_z)	0	1	1	1	3
Vibration	8	6	6	7	27

**Molecule Parameter**
Number of Atoms (N)	11
Number of internal coordinates	27
Number of independant internal coordinates	8
Number of vibrational modes	27

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	Total
Linear (IR)	8	6	6	7	21 / 6
Quadratic (Raman)	8	6	6	7	27 / 0
IR + Raman	8	-	6	7	21 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	_v(xz)	_v(yz)
linear	1	27	1	1	3
quadratic	2	378	14	14	18
cubic	3	3.654	14	14	46
quartic	4	27.405	105	105	165
quintic	5	169.911	105	105	375
sextic	6	906.192	560	560	1.040

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂
linear	1	8	6	6	7
quadratic	2	106	90	90	92
cubic	3	932	902	902	918
quartic	4	6.945	6.810	6.810	6.840
quintic	5	42.624	42.384	42.384	42.519
sextic	6	227.088	226.288	226.288	226.528

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