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	15	-1	-3	-3
Translation (x,y,z)	3	-1	-1	-1
Rotation (R_x,R_y,R_z)	3	-1	-1	-1
Vibration	9	1	-1	-1

**Decomposition into Irreducible representations**
Motion	A	B₁	B₂	B₃	Total
Cartesian 3N	2	5	4	4	15
Translation (x,y,z)	0	1	1	1	3
Rotation (R_x,R_y,R_z)	0	1	1	1	3
Vibration	2	3	2	2	9

**Molecule Parameter**
Number of Atoms (N)	5
Number of internal coordinates	9
Number of independant internal coordinates	2
Number of vibrational modes	9

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A	B₁	B₂	B₃	Total
Linear (IR)	2	3	2	2	7 / 2
Quadratic (Raman)	2	3	2	2	9 / 0
IR + Raman	-	3	2	2	7 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	C₂ (y)	C₂ (x)
linear	1	9	1	-1	-1
quadratic	2	45	5	5	5
cubic	3	165	5	-5	-5
quartic	4	495	15	15	15
quintic	5	1.287	15	-15	-15
sextic	6	3.003	35	35	35

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A	B₁	B₂	B₃
linear	1	2	3	2	2
quadratic	2	15	10	10	10
cubic	3	40	45	40	40
quartic	4	135	120	120	120
quintic	5	318	333	318	318
sextic	6	777	742	742	742

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