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	39	-1	-5	-5	5
Translation (x,y,z)	3	-1	-1	-1	1
Rotation (R_x,R_y,R_z)	3	1	-1	-1	-1
Vibration	33	-1	-3	-3	5

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	E	Total
Cartesian 3N	4	4	2	7	11	28
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	3	2	6	9	24

**Molecule Parameter**
Number of Atoms (N)	13
Number of internal coordinates	33
Number of independant internal coordinates	4
Number of vibrational modes	24

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	E	Total
Linear (IR)	4	3	2	6	9	15 / 9
Quadratic (Raman)	4	3	2	6	9	21 / 3
IR + Raman	-	3	-	6	9	15 / 3

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2S₄	C₂ (z)	2C'₂	2_d
linear	1	33	-1	-3	-3	5
quadratic	2	561	-1	21	21	29
cubic	3	6.545	1	-55	-55	105
quartic	4	58.905	9	225	225	385
quintic	5	435.897	-9	-531	-531	1.141
sextic	6	2.760.681	-9	1.653	1.653	3.325

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂	E
linear	1	4	3	2	6	9
quadratic	2	85	60	71	75	135
cubic	3	824	799	771	851	1.650
quartic	4	7.546	7.241	7.349	7.429	14.670
quintic	5	54.571	54.266	54.005	54.841	109.107
sextic	6	346.534	344.045	344.876	345.712	689.757

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