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	39	5	-5	-5	-1	-3	-1	9	9	5
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	33	3	-3	-3	1	-3	-1	9	9	5

**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	4	2	2	2	4	0	5	0	2	7	28
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	4	1	2	2	3	0	4	0	2	6	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	A_1g	A_2g	B_1g	B_2g	E_g	A_1u	A_2u	B_1u	B_2u	E_u	Total
Linear (IR)	4	1	2	2	3	0	4	0	2	6	10 / 14
Quadratic (Raman)	4	1	2	2	3	0	4	0	2	6	11 / 13
IR + Raman	-	1	-	-	-	0	-	0	2	-	0^* / 3

* 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	33	3	-3	-3	1	-3	-1	9	9	5
quadratic	2	561	3	21	21	17	21	-1	57	57	29
cubic	3	6.545	1	-55	-55	17	-55	1	273	273	105
quartic	4	58.905	9	225	225	153	225	9	1.113	1.113	385
quintic	5	435.897	27	-531	-531	153	-531	-9	3.969	3.969	1.141
sextic	6	2.760.681	27	1.653	1.653	969	1.653	-9	12.817	12.817	3.325

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	4	1	2	2	3	0	4	0	2	6
quadratic	2	57	26	45	37	63	26	38	28	34	72
cubic	3	462	377	431	407	784	340	444	362	422	866
quartic	4	4.016	3.547	3.877	3.677	7.224	3.472	3.752	3.530	3.694	7.446
quintic	5	28.019	26.836	27.691	27.155	53.991	26.314	27.686	26.552	27.430	55.116
sextic	6	175.898	171.207	174.820	172.276	343.483	170.056	173.436	170.636	172.838	346.274

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