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	216	0	8
Translation (x,y,z)	3	0	1
Rotation (R_x,R_y,R_z)	3	0	-1
Vibration	210	0	8

**Decomposition into Irreducible representations**
Motion	A₁	A₂	E	Total
Cartesian 3N	40	32	72	144
Translation (x,y,z)	1	0	1	2
Rotation (R_x,R_y,R_z)	0	1	1	2
Vibration	39	31	70	140

**Molecule Parameter**
Number of Atoms (N)	72
Number of internal coordinates	210
Number of independant internal coordinates	39
Number of vibrational modes	140

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	E	Total
Linear (IR)	39	31	70	109 / 31
Quadratic (Raman)	39	31	70	109 / 31
IR + Raman	39	31	70	109 / 31

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2C₃ (z)	3_v
linear	1	210	0	8
quadratic	2	22.155	0	137
cubic	3	1.565.620	70	928
quartic	4	83.369.265	0	9.117
quintic	5	3.568.204.542	0	54.120
sextic	6	127.860.662.755	2.485	397.333

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	E
linear	1	39	31	70
quadratic	2	3.761	3.624	7.385
cubic	3	261.424	260.496	521.850
quartic	4	13.899.436	13.890.319	27.789.755
quintic	5	594.727.817	594.673.697	1.189.401.514
sextic	6	21.310.309.954	21.309.912.621	42.620.220.090

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