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	60	0	4	4
Translation (x,y,z)	3	-1	1	1
Rotation (R_x,R_y,R_z)	3	-1	-1	-1
Vibration	54	2	4	4

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	Total
Cartesian 3N	17	13	15	15	60
Translation (x,y,z)	1	0	1	1	3
Rotation (R_x,R_y,R_z)	0	1	1	1	3
Vibration	16	12	13	13	54

**Molecule Parameter**
Number of Atoms (N)	20
Number of internal coordinates	54
Number of independant internal coordinates	16
Number of vibrational modes	54

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	Total
Linear (IR)	16	12	13	13	42 / 12
Quadratic (Raman)	16	12	13	13	54 / 0
IR + Raman	16	-	13	13	42 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	_v(xz)	_v(yz)
linear	1	54	2	4	4
quadratic	2	1.485	29	35	35
cubic	3	27.720	56	120	120
quartic	4	395.010	434	610	610
quintic	5	4.582.116	812	1.856	1.856
sextic	6	45.057.474	4.466	7.134	7.134

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂
linear	1	16	12	13	13
quadratic	2	396	361	364	364
cubic	3	7.004	6.884	6.916	6.916
quartic	4	99.166	98.556	98.644	98.644
quintic	5	1.146.660	1.144.804	1.145.326	1.145.326
sextic	6	11.269.052	11.261.918	11.263.252	11.263.252

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