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	-5	5	5
Translation (x,y,z)	3	-1	1	1
Rotation (R_x,R_y)^*	2	-2	0	0
Vibration	10	-2	4	4

* Linear molecule

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	Total
Cartesian 3N	5	0	5	5	15
Translation (x,y,z)	1	0	1	1	3
Rotation (R_x,R_y)^*	0	0	1	1	2
Vibration	4	0	3	3	10

* Linear molecule

**Molecule Parameter**
Number of Atoms (N)	5
Number of internal coordinates	10
Number of independant internal coordinates	4
Number of vibrational modes	10

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	Total
Linear (IR)	4	0	3	3	10 / 0
Quadratic (Raman)	4	0	3	3	10 / 0
IR + Raman	4	-	3	3	10 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	_v(xz)	_v(yz)
linear	1	10	-2	4	4
quadratic	2	55	7	13	13
cubic	3	220	-12	32	32
quartic	4	715	27	71	71
quintic	5	2.002	-42	140	140
sextic	6	5.005	77	259	259

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂
linear	1	4	0	3	3
quadratic	2	22	9	12	12
cubic	3	68	36	58	58
quartic	4	221	150	172	172
quintic	5	560	420	511	511
sextic	6	1.400	1.141	1.232	1.232

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