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	24	-2	4	2
Translation (x,y,z)	3	-1	1	1
Rotation (R_x,R_y,R_z)	3	-1	-1	-1
Vibration	18	0	4	2

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	Total
Cartesian 3N	7	4	7	6	24
Translation (x,y,z)	1	0	1	1	3
Rotation (R_x,R_y,R_z)	0	1	1	1	3
Vibration	6	3	5	4	18

**Molecule Parameter**
Number of Atoms (N)	8
Number of internal coordinates	18
Number of independant internal coordinates	6
Number of vibrational modes	18

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	Total
Linear (IR)	6	3	5	4	15 / 3
Quadratic (Raman)	6	3	5	4	18 / 0
IR + Raman	6	-	5	4	15 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	_v(xz)	_v(yz)
linear	1	18	0	4	2
quadratic	2	171	9	17	11
cubic	3	1.140	0	48	20
quartic	4	5.985	45	133	65
quintic	5	26.334	0	308	110
sextic	6	100.947	165	693	275

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂
linear	1	6	3	5	4
quadratic	2	52	38	42	39
cubic	3	302	268	292	278
quartic	4	1.557	1.458	1.502	1.468
quintic	5	6.688	6.479	6.633	6.534
sextic	6	25.520	25.036	25.300	25.091

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