Reduction formula for point group D2d

Reduction formula for point group D_2d

**Characters for molecular motions**
Motion	E	2S₄	C₂ (z)	2C'₂	2_d
Cartesian 3N	27	-1	-1	-1	5
Translation (x,y,z)	3	-1	-1	-1	1
Rotation (R_x,R_y,R_z)	3	1	-1	-1	-1
Vibration	21	-1	1	1	5

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	E	Total
Cartesian 3N	4	2	2	5	7	20
Translation (x,y,z)	0	0	0	1	1	2
Rotation (R_x,R_y,R_z)	0	1	0	0	1	2
Vibration	4	1	2	4	5	16

**Molecule Parameter**
Number of Atoms (N)	9
Number of internal coordinates	21
Number of independant internal coordinates	4
Number of vibrational modes	16

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	E	Total
Linear (IR)	4	1	2	4	5	9 / 7
Quadratic (Raman)	4	1	2	4	5	15 / 1
IR + Raman	-	1	-	4	5	9 / 1

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2S₄	C₂ (z)	2C'₂	2_d
linear	1	21	-1	1	1	5
quadratic	2	231	1	11	11	23
cubic	3	1.771	-1	11	11	75
quartic	4	10.626	6	66	66	226
quintic	5	53.130	-6	66	66	586
sextic	6	230.230	6	286	286	1.430

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂	E
linear	1	4	1	2	4	5
quadratic	2	39	22	27	33	55
cubic	3	244	201	207	239	440
quartic	4	1.411	1.265	1.295	1.375	2.640
quintic	5	6.811	6.485	6.521	6.781	13.266
sextic	6	29.245	28.387	28.527	29.099	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 D2d

Constructor University Bremen

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