Reduction formula for point group C3v

Reduction formula for point group C_3v

**Characters for molecular motions**
Motion	E	2C₃ (z)	3_v
Cartesian 3N	144	0	8
Translation (x,y,z)	3	0	1
Rotation (R_x,R_y,R_z)	3	0	-1
Vibration	138	0	8

**Decomposition into Irreducible representations**
Motion	A₁	A₂	E	Total
Cartesian 3N	28	20	48	96
Translation (x,y,z)	1	0	1	2
Rotation (R_x,R_y,R_z)	0	1	1	2
Vibration	27	19	46	92

**Molecule Parameter**
Number of Atoms (N)	48
Number of internal coordinates	138
Number of independant internal coordinates	27
Number of vibrational modes	92

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	E	Total
Linear (IR)	27	19	46	73 / 19
Quadratic (Raman)	27	19	46	73 / 19
IR + Raman	27	19	46	73 / 19

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2C₃ (z)	3_v
linear	1	138	0	8
quadratic	2	9.591	0	101
cubic	3	447.580	46	640
quartic	4	15.777.195	0	4.815
quintic	5	448.072.338	0	25.752
sextic	6	10.679.057.389	1.081	148.291

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	E
linear	1	27	19	46
quadratic	2	1.649	1.548	3.197
cubic	3	74.932	74.292	149.178
quartic	4	2.631.940	2.627.125	5.259.065
quintic	5	74.691.599	74.665.847	149.357.446
sextic	6	1.779.917.404	1.779.769.113	3.559.685.436

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 C3v

Constructor University Bremen

Last update November, 13^th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement