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	102	0	14
Translation (x,y,z)	3	0	1
Rotation (R_x,R_y,R_z)	3	0	-1
Vibration	96	0	14

**Decomposition into Irreducible representations**
Motion	A₁	A₂	E	Total
Cartesian 3N	24	10	34	68
Translation (x,y,z)	1	0	1	2
Rotation (R_x,R_y,R_z)	0	1	1	2
Vibration	23	9	32	64

**Molecule Parameter**
Number of Atoms (N)	34
Number of internal coordinates	96
Number of independant internal coordinates	23
Number of vibrational modes	64

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	E	Total
Linear (IR)	23	9	32	55 / 9
Quadratic (Raman)	23	9	32	55 / 9
IR + Raman	23	9	32	55 / 9

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2C₃ (z)	3_v
linear	1	96	0	14
quadratic	2	4.656	0	146
cubic	3	152.096	32	1.134
quartic	4	3.764.376	0	7.546
quintic	5	75.287.520	0	43.582
sextic	6	1.267.339.920	528	227.458

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	E
linear	1	23	9	32
quadratic	2	849	703	1.552
cubic	3	25.927	24.793	50.688
quartic	4	631.169	623.623	1.254.792
quintic	5	12.569.711	12.526.129	25.095.840
sextic	6	211.337.225	211.109.767	422.446.464

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