Reduction formula for point group D3

Reduction formula for point group D₃

**Characters for molecular motions**
Motion	E	2C₃ (z)	3C'₂
Cartesian 3N	120	0	0
Translation (x,y,z)	3	0	-1
Rotation (R_x,R_y,R_z)	3	0	-1
Vibration	114	0	2

**Decomposition into Irreducible representations**
Motion	A₁	A₂	E	Total
Cartesian 3N	20	20	40	80
Translation (x,y,z)	0	1	1	2
Rotation (R_x,R_y,R_z)	0	1	1	2
Vibration	20	18	38	76

**Molecule Parameter**
Number of Atoms (N)	40
Number of internal coordinates	114
Number of independant internal coordinates	20
Number of vibrational modes	76

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	E	Total
Linear (IR)	20	18	38	56 / 20
Quadratic (Raman)	20	18	38	58 / 18
IR + Raman	-	-	38	38 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2C₃ (z)	3C'₂
linear	1	114	0	2
quadratic	2	6.555	0	59
cubic	3	253.460	38	116
quartic	4	7.413.705	0	1.769
quintic	5	174.963.438	0	3.422
sextic	6	3.470.108.187	741	35.931

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	E
linear	1	20	18	38
quadratic	2	1.122	1.063	2.185
cubic	3	42.314	42.198	84.474
quartic	4	1.236.502	1.234.733	2.471.235
quintic	5	29.162.284	29.158.862	58.321.146
sextic	6	578.369.577	578.333.646	1.156.702.482

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