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	36	4	-4	-4	0
Translation (x,y,z)	3	1	-1	-1	-1
Rotation (R_x,R_y,R_z)	3	1	-1	-1	-1
Vibration	30	2	-2	-2	2

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	E	Total
Cartesian 3N	4	6	2	4	10	26
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	4	2	4	8	22

**Molecule Parameter**
Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	4
Number of vibrational modes	22

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	E	Total
Linear (IR)	4	4	2	4	8	12 / 10
Quadratic (Raman)	4	4	2	4	8	18 / 4
IR + Raman	-	-	-	-	8	8 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2C₄ (z)	C₂ (z)	2C'₂	2C''₂
linear	1	30	2	-2	-2	2
quadratic	2	465	1	17	17	17
cubic	3	4.960	0	-32	-32	32
quartic	4	40.920	8	152	152	152
quintic	5	278.256	16	-272	-272	272
sextic	6	1.623.160	8	952	952	952

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂	E
linear	1	4	4	2	4	8
quadratic	2	69	52	60	60	112
cubic	3	616	616	600	632	1.248
quartic	4	5.212	5.060	5.132	5.132	10.192
quintic	5	34.752	34.752	34.608	34.880	69.632
sextic	6	203.492	202.540	203.012	203.012	405.552

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