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	78	-2	8	8
Translation (x,y,z)	3	-1	1	1
Rotation (R_x,R_y,R_z)	3	-1	-1	-1
Vibration	72	0	8	8

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	Total
Cartesian 3N	23	15	20	20	78
Translation (x,y,z)	1	0	1	1	3
Rotation (R_x,R_y,R_z)	0	1	1	1	3
Vibration	22	14	18	18	72

**Molecule Parameter**
Number of Atoms (N)	26
Number of internal coordinates	72
Number of independant internal coordinates	22
Number of vibrational modes	72

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	Total
Linear (IR)	22	14	18	18	58 / 14
Quadratic (Raman)	22	14	18	18	72 / 0
IR + Raman	22	-	18	18	58 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	_v(xz)	_v(yz)
linear	1	72	0	8	8
quadratic	2	2.628	36	68	68
cubic	3	64.824	0	376	376
quartic	4	1.215.450	666	2.010	2.010
quintic	5	18.474.840	0	8.856	8.856
sextic	6	237.093.780	8.436	37.268	37.268

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂
linear	1	22	14	18	18
quadratic	2	700	632	648	648
cubic	3	16.394	16.018	16.206	16.206
quartic	4	305.034	303.024	303.696	303.696
quintic	5	4.623.138	4.614.282	4.618.710	4.618.710
sextic	6	59.294.188	59.256.920	59.271.336	59.271.336

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