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	15	-1	3	3
Translation (x,y,z)	3	-1	1	1
Rotation (R_x,R_y,R_z)	3	-1	-1	-1
Vibration	9	1	3	3

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	Total
Cartesian 3N	5	2	4	4	15
Translation (x,y,z)	1	0	1	1	3
Rotation (R_x,R_y,R_z)	0	1	1	1	3
Vibration	4	1	2	2	9

**Molecule Parameter**
Number of Atoms (N)	5
Number of internal coordinates	9
Number of independant internal coordinates	4
Number of vibrational modes	9

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	Total
Linear (IR)	4	1	2	2	8 / 1
Quadratic (Raman)	4	1	2	2	9 / 0
IR + Raman	4	-	2	2	8 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	_v(xz)	_v(yz)
linear	1	9	1	3	3
quadratic	2	45	5	9	9
cubic	3	165	5	19	19
quartic	4	495	15	39	39
quintic	5	1.287	15	69	69
sextic	6	3.003	35	119	119

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂
linear	1	4	1	2	2
quadratic	2	17	8	10	10
cubic	3	52	33	40	40
quartic	4	147	108	120	120
quintic	5	360	291	318	318
sextic	6	819	700	742	742

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