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	30	-2	4	4
Translation (x,y,z)	3	-1	1	1
Rotation (R_x,R_y,R_z)	3	-1	-1	-1
Vibration	24	0	4	4

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	Total
Cartesian 3N	9	5	8	8	30
Translation (x,y,z)	1	0	1	1	3
Rotation (R_x,R_y,R_z)	0	1	1	1	3
Vibration	8	4	6	6	24

**Molecule Parameter**
Number of Atoms (N)	10
Number of internal coordinates	24
Number of independant internal coordinates	8
Number of vibrational modes	24

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	Total
Linear (IR)	8	4	6	6	20 / 4
Quadratic (Raman)	8	4	6	6	24 / 0
IR + Raman	8	-	6	6	20 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	_v(xz)	_v(yz)
linear	1	24	0	4	4
quadratic	2	300	12	20	20
cubic	3	2.600	0	60	60
quartic	4	17.550	78	190	190
quintic	5	98.280	0	476	476
sextic	6	475.020	364	1.204	1.204

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂
linear	1	8	4	6	6
quadratic	2	88	68	72	72
cubic	3	680	620	650	650
quartic	4	4.502	4.312	4.368	4.368
quintic	5	24.808	24.332	24.570	24.570
sextic	6	119.448	118.244	118.664	118.664

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