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	-5	-1	-1
Translation (x,y,z)	3	-1	-1	-1
Rotation (R_x,R_y)^*	2	-2	0	0
Vibration	10	-2	0	0

* Linear molecule

**Decomposition into Irreducible representations**
Motion	A	B₁	B₂	B₃	Total
Cartesian 3N	2	3	5	5	15
Translation (x,y,z)	0	1	1	1	3
Rotation (R_x,R_y)^*	0	0	1	1	2
Vibration	2	2	3	3	10

* Linear molecule

**Molecule Parameter**
Number of Atoms (N)	5
Number of internal coordinates	10
Number of independant internal coordinates	2
Number of vibrational modes	10

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A	B₁	B₂	B₃	Total
Linear (IR)	2	2	3	3	8 / 2
Quadratic (Raman)	2	2	3	3	10 / 0
IR + Raman	-	2	3	3	8 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	C₂ (y)	C₂ (x)
linear	1	10	-2	0	0
quadratic	2	55	7	5	5
cubic	3	220	-12	0	0
quartic	4	715	27	15	15
quintic	5	2.002	-42	0	0
sextic	6	5.005	77	35	35

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A	B₁	B₂	B₃
linear	1	2	2	3	3
quadratic	2	18	13	12	12
cubic	3	52	52	58	58
quartic	4	193	178	172	172
quintic	5	490	490	511	511
sextic	6	1.288	1.253	1.232	1.232

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