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	-1	9
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	1	9

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	E	Total
Cartesian 3N	6	2	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	6	1	2	6	9	24

**Molecule Parameter**
Number of Atoms (N)	13
Number of internal coordinates	33
Number of independant internal coordinates	6
Number of vibrational modes	24

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	E	Total
Linear (IR)	6	1	2	6	9	15 / 9
Quadratic (Raman)	6	1	2	6	9	23 / 1
IR + Raman	-	1	-	6	9	15 / 1

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2S₄	C₂ (z)	2C'₂	2_d
linear	1	33	-1	-3	1	9
quadratic	2	561	-1	21	17	57
cubic	3	6.545	1	-55	17	273
quartic	4	58.905	9	225	153	1.113
quintic	5	435.897	-9	-531	153	3.969
sextic	6	2.760.681	-9	1.653	969	12.817

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂	E
linear	1	6	1	2	6	9
quadratic	2	91	54	63	83	135
cubic	3	884	739	747	875	1.650
quartic	4	7.710	7.077	7.149	7.629	14.670
quintic	5	55.449	53.388	53.469	55.377	109.107
sextic	6	348.736	341.843	342.332	348.256	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