Reduction formula for point group D3

Reduction formula for point group D₃

**Characters for molecular motions**
Motion	E	2C₃ (z)	3C'₂
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	-6

**Decomposition into Irreducible representations**
Motion	A₁	A₂	E	Total
Cartesian 3N	32	40	72	144
Translation (x,y,z)	0	1	1	2
Rotation (R_x,R_y,R_z)	0	1	1	2
Vibration	32	38	70	140

**Molecule Parameter**
Number of Atoms (N)	72
Number of internal coordinates	210
Number of independant internal coordinates	32
Number of vibrational modes	140

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	E	Total
Linear (IR)	32	38	70	108 / 32
Quadratic (Raman)	32	38	70	102 / 38
IR + Raman	-	-	70	70 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2C₃ (z)	3C'₂
linear	1	210	0	-6
quadratic	2	22.155	0	123
cubic	3	1.565.620	70	-668
quartic	4	83.369.265	0	7.521
quintic	5	3.568.204.542	0	-37.482
sextic	6	127.860.662.755	2.485	305.731

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	E
linear	1	32	38	70
quadratic	2	3.754	3.631	7.385
cubic	3	260.626	261.294	521.850
quartic	4	13.898.638	13.891.117	27.789.755
quintic	5	594.682.016	594.719.498	1.189.401.514
sextic	6	21.310.264.153	21.309.958.422	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 D3

Constructor University Bremen

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