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	48	-4	16	4
Translation (x,y,z)	3	-1	1	1
Rotation (R_x,R_y,R_z)	3	-1	-1	-1
Vibration	42	-2	16	4

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	Total
Cartesian 3N	16	6	16	10	48
Translation (x,y,z)	1	0	1	1	3
Rotation (R_x,R_y,R_z)	0	1	1	1	3
Vibration	15	5	14	8	42

**Molecule Parameter**
Number of Atoms (N)	16
Number of internal coordinates	42
Number of independant internal coordinates	15
Number of vibrational modes	42

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	Total
Linear (IR)	15	5	14	8	37 / 5
Quadratic (Raman)	15	5	14	8	42 / 0
IR + Raman	15	-	14	8	37 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	_v(xz)	_v(yz)
linear	1	42	-2	16	4
quadratic	2	903	23	149	29
cubic	3	13.244	-44	1.024	96
quartic	4	148.995	275	5.735	415
quintic	5	1.370.754	-506	27.568	1.196
sextic	6	10.737.573	2.277	117.483	3.979

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂
linear	1	15	5	14	8
quadratic	2	276	187	250	190
cubic	3	3.580	3.020	3.554	3.090
quartic	4	38.855	35.780	38.510	35.850
quintic	5	349.753	335.371	349.408	336.222
sextic	6	2.715.328	2.654.597	2.712.200	2.655.448

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