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	21	-3	-1	-1
Translation (x,y,z)	3	-1	-1	-1
Rotation (R_x,R_y,R_z)	3	-1	-1	-1
Vibration	15	-1	1	1

**Decomposition into Irreducible representations**
Motion	A	B₁	B₂	B₃	Total
Cartesian 3N	4	5	6	6	21
Translation (x,y,z)	0	1	1	1	3
Rotation (R_x,R_y,R_z)	0	1	1	1	3
Vibration	4	3	4	4	15

**Molecule Parameter**
Number of Atoms (N)	7
Number of internal coordinates	15
Number of independant internal coordinates	4
Number of vibrational modes	15

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A	B₁	B₂	B₃	Total
Linear (IR)	4	3	4	4	11 / 4
Quadratic (Raman)	4	3	4	4	15 / 0
IR + Raman	-	3	4	4	11 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	C₂ (y)	C₂ (x)
linear	1	15	-1	1	1
quadratic	2	120	8	8	8
cubic	3	680	-8	8	8
quartic	4	3.060	36	36	36
quintic	5	11.628	-36	36	36
sextic	6	38.760	120	120	120

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A	B₁	B₂	B₃
linear	1	4	3	4	4
quadratic	2	36	28	28	28
cubic	3	172	164	172	172
quartic	4	792	756	756	756
quintic	5	2.916	2.880	2.916	2.916
sextic	6	9.780	9.660	9.660	9.660

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