Reduction formula for point group D4

Reduction formula for point group D₄

**Characters for molecular motions**
Motion	E	2C₄ (z)	C₂ (z)	2C'₂	2C''₂
Cartesian 3N	39	5	-5	-5	-1
Translation (x,y,z)	3	1	-1	-1	-1
Rotation (R_x,R_y,R_z)	3	1	-1	-1	-1
Vibration	33	3	-3	-3	1

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	E	Total
Cartesian 3N	4	7	2	4	11	28
Translation (x,y,z)	0	1	0	0	1	2
Rotation (R_x,R_y,R_z)	0	1	0	0	1	2
Vibration	4	5	2	4	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	5	2	4	9	14 / 10
Quadratic (Raman)	4	5	2	4	9	19 / 5
IR + Raman	-	-	-	-	9	9 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2C₄ (z)	C₂ (z)	2C'₂	2C''₂
linear	1	33	3	-3	-3	1
quadratic	2	561	3	21	21	17
cubic	3	6.545	1	-55	-55	17
quartic	4	58.905	9	225	225	153
quintic	5	435.897	27	-531	-531	153
sextic	6	2.760.681	27	1.653	1.653	969

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂	E
linear	1	4	5	2	4	9
quadratic	2	83	64	73	71	135
cubic	3	802	821	793	829	1.650
quartic	4	7.488	7.299	7.407	7.371	14.670
quintic	5	54.333	54.522	54.243	54.585	109.107
sextic	6	345.954	344.643	345.456	345.114	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 D4

Constructor University Bremen

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