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

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	E	Total
Cartesian 3N	2	4	2	4	6	18
Translation (x,y,z)	0	1	0	0	1	2
Rotation (R_x,R_y,R_z)	0	1	0	0	1	2
Vibration	2	2	2	4	4	14

**Molecule Parameter**
Number of Atoms (N)	8
Number of internal coordinates	18
Number of independant internal coordinates	2
Number of vibrational modes	14

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	E	Total
Linear (IR)	2	2	2	4	4	6 / 8
Quadratic (Raman)	2	2	2	4	4	12 / 2
IR + Raman	-	-	-	-	4	4 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2C₄ (z)	C₂ (z)	2C'₂	2C''₂
linear	1	18	-2	2	-2	2
quadratic	2	171	3	11	11	11
cubic	3	1.140	-4	20	-20	20
quartic	4	5.985	9	65	65	65
quintic	5	26.334	-14	110	-110	110
sextic	6	100.947	19	275	275	275

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂	E
linear	1	2	2	2	4	4
quadratic	2	29	18	22	22	40
cubic	3	144	144	136	156	280
quartic	4	791	726	754	754	1.480
quintic	5	3.302	3.302	3.254	3.364	6.556
sextic	6	12.795	12.520	12.648	12.648	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 D4

Constructor University Bremen

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