Reduction formula for point group D2

Reduction formula for point group D₂

**Characters for molecular motions**
Motion	E	C₂ (z)	C₂ (y)	C₂ (x)
Cartesian 3N	27	-1	-1	-1
Translation (x,y,z)	3	-1	-1	-1
Rotation (R_x,R_y,R_z)	3	-1	-1	-1
Vibration	21	1	1	1

**Decomposition into Irreducible representations**
Motion	A	B₁	B₂	B₃	Total
Cartesian 3N	6	7	7	7	27
Translation (x,y,z)	0	1	1	1	3
Rotation (R_x,R_y,R_z)	0	1	1	1	3
Vibration	6	5	5	5	21

**Molecule Parameter**
Number of Atoms (N)	9
Number of internal coordinates	21
Number of independant internal coordinates	6
Number of vibrational modes	21

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A	B₁	B₂	B₃	Total
Linear (IR)	6	5	5	5	15 / 6
Quadratic (Raman)	6	5	5	5	21 / 0
IR + Raman	-	5	5	5	15 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	C₂ (y)	C₂ (x)
linear	1	21	1	1	1
quadratic	2	231	11	11	11
cubic	3	1.771	11	11	11
quartic	4	10.626	66	66	66
quintic	5	53.130	66	66	66
sextic	6	230.230	286	286	286

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A	B₁	B₂	B₃
linear	1	6	5	5	5
quadratic	2	66	55	55	55
cubic	3	451	440	440	440
quartic	4	2.706	2.640	2.640	2.640
quintic	5	13.332	13.266	13.266	13.266
sextic	6	57.772	57.486	57.486	57.486

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 D2

Constructor University Bremen

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