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	15	-1	-1	-1	3
Translation (x,y,z)	3	-1	-1	-1	1
Rotation (R_x,R_y,R_z)	3	1	-1	-1	-1
Vibration	9	-1	1	1	3

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	E	Total
Cartesian 3N	2	1	1	3	4	11
Translation (x,y,z)	0	0	0	1	1	2
Rotation (R_x,R_y,R_z)	0	1	0	0	1	2
Vibration	2	0	1	2	2	7

**Molecule Parameter**
Number of Atoms (N)	5
Number of internal coordinates	9
Number of independant internal coordinates	2
Number of vibrational modes	7

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	E	Total
Linear (IR)	2	0	1	2	2	4 / 3
Quadratic (Raman)	2	0	1	2	2	7 / 0
IR + Raman	-	0	-	2	2	4 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2S₄	C₂ (z)	2C'₂	2_d
linear	1	9	-1	1	1	3
quadratic	2	45	1	5	5	9
cubic	3	165	-1	5	5	19
quartic	4	495	3	15	15	39
quintic	5	1.287	-3	15	15	69
sextic	6	3.003	3	35	35	119

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂	E
linear	1	2	0	1	2	2
quadratic	2	10	3	5	7	10
cubic	3	27	15	18	25	40
quartic	4	78	51	57	69	120
quintic	5	183	141	150	177	318
sextic	6	419	342	358	400	742

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