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

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	Total
Cartesian 3N	16	14	14	16	60
Translation (x,y,z)	1	0	1	1	3
Rotation (R_x,R_y,R_z)	0	1	1	1	3
Vibration	15	13	12	14	54

**Molecule Parameter**
Number of Atoms (N)	20
Number of internal coordinates	54
Number of independant internal coordinates	15
Number of vibrational modes	54

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	Total
Linear (IR)	15	13	12	14	41 / 13
Quadratic (Raman)	15	13	12	14	54 / 0
IR + Raman	15	-	12	14	41 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	_v(xz)	_v(yz)
linear	1	54	2	0	4
quadratic	2	1.485	29	27	35
cubic	3	27.720	56	0	120
quartic	4	395.010	434	378	610
quintic	5	4.582.116	812	0	1.856
sextic	6	45.057.474	4.466	3.654	7.134

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂
linear	1	15	13	12	14
quadratic	2	394	363	362	366
cubic	3	6.974	6.914	6.886	6.946
quartic	4	99.108	98.614	98.586	98.702
quintic	5	1.146.196	1.145.268	1.144.862	1.145.790
sextic	6	11.268.182	11.262.788	11.262.382	11.264.122

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