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	24	0	0	0	4
Translation (x,y,z)	3	-1	-1	-1	1
Rotation (R_x,R_y,R_z)	3	1	-1	-1	-1
Vibration	18	0	2	2	4

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	E	Total
Cartesian 3N	4	2	2	4	6	18
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	3	4	14

**Molecule Parameter**
Number of Atoms (N)	8
Number of internal coordinates	18
Number of independant internal coordinates	4
Number of vibrational modes	14

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	E	Total
Linear (IR)	4	1	2	3	4	7 / 7
Quadratic (Raman)	4	1	2	3	4	13 / 1
IR + Raman	-	1	-	3	4	7 / 1

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2S₄	C₂ (z)	2C'₂	2_d
linear	1	18	0	2	2	4
quadratic	2	171	1	11	11	17
cubic	3	1.140	0	20	20	48
quartic	4	5.985	5	65	65	133
quintic	5	26.334	0	110	110	308
sextic	6	100.947	5	275	275	693

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂	E
linear	1	4	1	2	3	4
quadratic	2	30	16	21	24	40
cubic	3	162	128	138	152	280
quartic	4	807	708	738	772	1.480
quintic	5	3.410	3.201	3.256	3.355	6.556
sextic	6	12.896	12.412	12.547	12.756	25.168

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