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	36	0	4	4
Translation (x,y,z)	3	-1	1	1
Rotation (R_x,R_y,R_z)	3	-1	-1	-1
Vibration	30	2	4	4

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	Total
Cartesian 3N	11	7	9	9	36
Translation (x,y,z)	1	0	1	1	3
Rotation (R_x,R_y,R_z)	0	1	1	1	3
Vibration	10	6	7	7	30

**Molecule Parameter**
Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	10
Number of vibrational modes	30

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	Total
Linear (IR)	10	6	7	7	24 / 6
Quadratic (Raman)	10	6	7	7	30 / 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	30	2	4	4
quadratic	2	465	17	23	23
cubic	3	4.960	32	72	72
quartic	4	40.920	152	256	256
quintic	5	278.256	272	680	680
sextic	6	1.623.160	952	1.904	1.904

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂
linear	1	10	6	7	7
quadratic	2	132	109	112	112
cubic	3	1.284	1.212	1.232	1.232
quartic	4	10.396	10.140	10.192	10.192
quintic	5	69.972	69.292	69.496	69.496
sextic	6	406.980	405.076	405.552	405.552

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