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	33	-3	7	7
Translation (x,y,z)	3	-1	1	1
Rotation (R_x,R_y,R_z)	3	-1	-1	-1
Vibration	27	-1	7	7

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	Total
Cartesian 3N	11	4	9	9	33
Translation (x,y,z)	1	0	1	1	3
Rotation (R_x,R_y,R_z)	0	1	1	1	3
Vibration	10	3	7	7	27

**Molecule Parameter**
Number of Atoms (N)	11
Number of internal coordinates	27
Number of independant internal coordinates	10
Number of vibrational modes	27

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	Total
Linear (IR)	10	3	7	7	24 / 3
Quadratic (Raman)	10	3	7	7	27 / 0
IR + Raman	10	-	7	7	24 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	_v(xz)	_v(yz)
linear	1	27	-1	7	7
quadratic	2	378	14	38	38
cubic	3	3.654	-14	154	154
quartic	4	27.405	105	545	545
quintic	5	169.911	-105	1.687	1.687
sextic	6	906.192	560	4.784	4.784

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂
linear	1	10	3	7	7
quadratic	2	117	79	91	91
cubic	3	987	833	917	917
quartic	4	7.150	6.605	6.825	6.825
quintic	5	43.295	41.608	42.504	42.504
sextic	6	229.080	224.296	226.408	226.408

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