Reduction formula for point group D3

Reduction formula for point group D₃

**Characters for molecular motions**
Motion	E	2C₃ (z)	3C'₂
Cartesian 3N	240	0	0
Translation (x,y,z)	3	0	-1
Rotation (R_x,R_y,R_z)	3	0	-1
Vibration	234	0	2

**Decomposition into Irreducible representations**
Motion	A₁	A₂	E	Total
Cartesian 3N	40	40	80	160
Translation (x,y,z)	0	1	1	2
Rotation (R_x,R_y,R_z)	0	1	1	2
Vibration	40	38	78	156

**Molecule Parameter**
Number of Atoms (N)	80
Number of internal coordinates	234
Number of independant internal coordinates	40
Number of vibrational modes	156

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	E	Total
Linear (IR)	40	38	78	116 / 40
Quadratic (Raman)	40	38	78	118 / 38
IR + Raman	-	-	78	78 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2C₃ (z)	3C'₂
linear	1	234	0	2
quadratic	2	27.495	0	119
cubic	3	2.162.940	78	236
quartic	4	128.154.195	0	7.139
quintic	5	6.100.139.682	0	14.042
sextic	6	242.988.897.333	3.081	287.861

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	E
linear	1	40	38	78
quadratic	2	4.642	4.523	9.165
cubic	3	360.634	360.398	720.954
quartic	4	21.362.602	21.355.463	42.718.065
quintic	5	1.016.696.968	1.016.682.926	2.033.379.894
sextic	6	40.498.294.513	40.498.006.652	80.996.298.084

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