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	18	0	6	0
Translation (x,y,z)	3	-1	1	1
Rotation (R_x,R_y,R_z)	3	-1	-1	-1
Vibration	12	2	6	0

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	Total
Cartesian 3N	6	3	6	3	18
Translation (x,y,z)	1	0	1	1	3
Rotation (R_x,R_y,R_z)	0	1	1	1	3
Vibration	5	2	4	1	12

**Molecule Parameter**
Number of Atoms (N)	6
Number of internal coordinates	12
Number of independant internal coordinates	5
Number of vibrational modes	12

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	Total
Linear (IR)	5	2	4	1	10 / 2
Quadratic (Raman)	5	2	4	1	12 / 0
IR + Raman	5	-	4	1	10 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	_v(xz)	_v(yz)
linear	1	12	2	6	0
quadratic	2	78	8	24	6
cubic	3	364	14	74	0
quartic	4	1.365	35	195	21
quintic	5	4.368	56	456	0
sextic	6	12.376	112	976	56

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂
linear	1	5	2	4	1
quadratic	2	29	14	22	13
cubic	3	113	76	106	69
quartic	4	404	296	376	289
quintic	5	1.220	992	1.192	964
sextic	6	3.380	2.864	3.296	2.836

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