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	39	-1	9	5
Translation (x,y,z)	3	-1	1	1
Rotation (R_x,R_y,R_z)	3	-1	-1	-1
Vibration	33	1	9	5

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	Total
Cartesian 3N	13	6	11	9	39
Translation (x,y,z)	1	0	1	1	3
Rotation (R_x,R_y,R_z)	0	1	1	1	3
Vibration	12	5	9	7	33

**Molecule Parameter**
Number of Atoms (N)	13
Number of internal coordinates	33
Number of independant internal coordinates	12
Number of vibrational modes	33

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	Total
Linear (IR)	12	5	9	7	28 / 5
Quadratic (Raman)	12	5	9	7	33 / 0
IR + Raman	12	-	9	7	28 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	_v(xz)	_v(yz)
linear	1	33	1	9	5
quadratic	2	561	17	57	29
cubic	3	6.545	17	273	105
quartic	4	58.905	153	1.113	385
quintic	5	435.897	153	3.969	1.141
sextic	6	2.760.681	969	12.817	3.325

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂
linear	1	12	5	9	7
quadratic	2	166	123	143	129
cubic	3	1.735	1.546	1.674	1.590
quartic	4	15.139	14.390	14.870	14.506
quintic	5	110.290	107.735	109.643	108.229
sextic	6	694.448	686.377	692.301	687.555

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