Reduction formula for point group D4h

Reduction formula for point group D_4h

**Characters for molecular motions**
Motion	E	2C₄ (z)	C₂	2C'₂	2C''₂	i	2S₄	_h	2_v	2_d
Cartesian 3N	156	0	0	0	-2	0	0	4	4	14
Translation (x,y,z)	3	1	-1	-1	-1	-3	-1	1	1	1
Rotation (R_x,R_y,R_z)	3	1	-1	-1	-1	3	1	-1	-1	-1
Vibration	150	-2	2	2	0	0	0	4	4	14

**Decomposition into Irreducible representations**
Motion	A_1g	A_2g	B_1g	B_2g	E_g	A_1u	A_2u	B_1u	B_2u	E_u	Total
Cartesian 3N	12	8	9	11	19	7	12	11	8	20	117
Translation (x,y,z)	0	0	0	0	0	0	1	0	0	1	2
Rotation (R_x,R_y,R_z)	0	1	0	0	1	0	0	0	0	0	2
Vibration	12	7	9	11	18	7	11	11	8	19	113

**Molecule Parameter**
Number of Atoms (N)	52
Number of internal coordinates	150
Number of independant internal coordinates	12
Number of vibrational modes	113

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A_1g	A_2g	B_1g	B_2g	E_g	A_1u	A_2u	B_1u	B_2u	E_u	Total
Linear (IR)	12	7	9	11	18	7	11	11	8	19	30 / 83
Quadratic (Raman)	12	7	9	11	18	7	11	11	8	19	50 / 63
IR + Raman	-	7	-	-	-	7	-	11	8	-	0^* / 33

* Center of inversion: Mutual Exclusion Principle

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2C₄ (z)	C₂	2C'₂	2C''₂	i	2S₄	_h	2_v	2_d
linear	1	150	-2	2	2	0	0	0	4	4	14
quadratic	2	11.325	3	77	77	75	75	1	83	83	173
cubic	3	573.800	-4	152	152	0	0	0	312	312	1.512
quartic	4	21.947.850	42	3.002	3.002	2.850	2.850	38	3.466	3.466	11.866
quintic	5	675.993.780	-80	5.852	5.852	0	0	0	12.320	12.320	79.492
sextic	6	17.463.172.650	118	79.002	79.002	73.150	73.150	38	97.174	97.174	490.042

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A_1g	A_2g	B_1g	B_2g	E_g	A_1u	A_2u	B_1u	B_2u	E_u
linear	1	12	7	9	11	18	7	11	11	8	19
quadratic	2	774	672	711	733	1.405	690	716	714	691	1.407
cubic	3	36.138	35.644	35.761	36.023	71.667	35.643	36.061	36.022	35.684	71.745
quartic	4	1.374.981	1.369.685	1.371.282	1.373.344	2.743.029	1.370.349	1.372.719	1.372.602	1.370.464	2.743.183
quintic	5	42.262.945	42.238.529	42.243.092	42.258.422	84.496.951	42.238.452	42.259.942	42.258.345	42.240.089	84.500.031
sextic	6	1.091.556.314	1.091.371.472	1.091.415.477	1.091.512.231	2.182.883.703	1.091.388.210	1.091.496.976	1.091.492.413	1.091.392.733	2.182.889.709

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 D4h

Constructor University Bremen

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