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	324	0	8
Translation (x,y,z)	3	0	1
Rotation (R_x,R_y,R_z)	3	0	-1
Vibration	318	0	8

**Decomposition into Irreducible representations**
Motion	A₁	A₂	E	Total
Cartesian 3N	58	50	108	216
Translation (x,y,z)	1	0	1	2
Rotation (R_x,R_y,R_z)	0	1	1	2
Vibration	57	49	106	212

**Molecule Parameter**
Number of Atoms (N)	108
Number of internal coordinates	318
Number of independant internal coordinates	57
Number of vibrational modes	212

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	E	Total
Linear (IR)	57	49	106	163 / 49
Quadratic (Raman)	57	49	106	163 / 49
IR + Raman	57	49	106	163 / 49

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2C₃ (z)	3_v
linear	1	318	0	8
quadratic	2	50.721	0	191
cubic	3	5.410.240	106	1.360
quartic	4	434.171.760	0	18.000
quintic	5	27.960.661.344	0	116.112
sextic	6	1.505.215.602.352	5.671	1.120.816

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	E
linear	1	57	49	106
quadratic	2	8.549	8.358	16.907
cubic	3	902.422	901.062	1.803.378
quartic	4	72.370.960	72.352.960	144.723.920
quintic	5	4.660.168.280	4.660.052.168	9.320.220.448
sextic	6	250.869.829.357	250.868.708.541	501.738.532.227

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