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	12	-4	0	0
Translation (x,y,z)	3	-1	-1	-1
Rotation (R_x,R_y)^*	2	-2	0	0
Vibration	7	-1	1	1

* Linear molecule

**Decomposition into Irreducible representations**
Motion	A	B₁	B₂	B₃	Total
Cartesian 3N	2	2	4	4	12
Translation (x,y,z)	0	1	1	1	3
Rotation (R_x,R_y)^*	0	0	1	1	2
Vibration	2	1	2	2	7

* Linear molecule

**Molecule Parameter**
Number of Atoms (N)	4
Number of internal coordinates	7
Number of independant internal coordinates	2
Number of vibrational modes	7

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A	B₁	B₂	B₃	Total
Linear (IR)	2	1	2	2	5 / 2
Quadratic (Raman)	2	1	2	2	7 / 0
IR + Raman	-	1	2	2	5 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	C₂ (y)	C₂ (x)
linear	1	7	-1	1	1
quadratic	2	28	4	4	4
cubic	3	84	-4	4	4
quartic	4	210	10	10	10
quintic	5	462	-10	10	10
sextic	6	924	20	20	20

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A	B₁	B₂	B₃
linear	1	2	1	2	2
quadratic	2	10	6	6	6
cubic	3	22	18	22	22
quartic	4	60	50	50	50
quintic	5	118	108	118	118
sextic	6	246	226	226	226

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