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	48	0	0	-4	-4	0	0	16	4	4
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	42	-2	2	-2	-2	0	0	16	4	4

**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	4	4	4	4	0	4	2	2	8	36
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	3	4	4	3	0	3	2	2	7	32

**Molecule Parameter**
Number of Atoms (N)	16
Number of internal coordinates	42
Number of independant internal coordinates	4
Number of vibrational modes	32

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	3	4	4	3	0	3	2	2	7	10 / 22
Quadratic (Raman)	4	3	4	4	3	0	3	2	2	7	15 / 17
IR + Raman	-	3	-	-	-	0	-	2	2	-	0^* / 7

* 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	42	-2	2	-2	-2	0	0	16	4	4
quadratic	2	903	3	23	23	23	21	1	149	29	29
cubic	3	13.244	-4	44	-44	-44	0	0	1.024	96	96
quartic	4	148.995	15	275	275	275	231	11	5.735	415	415
quintic	5	1.370.754	-26	506	-506	-506	0	0	27.568	1.196	1.196
sextic	6	10.737.573	37	2.277	2.277	2.277	1.771	11	117.483	3.979	3.979

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	3	4	4	3	0	3	2	2	7
quadratic	2	82	56	68	68	94	46	49	47	47	126
cubic	3	907	881	895	895	1.522	731	801	767	767	1.778
quartic	4	9.878	9.533	9.699	9.699	17.902	8.922	8.992	8.956	8.956	19.278
quintic	5	87.596	87.251	87.430	87.430	167.835	83.552	84.403	83.984	83.984	174.727
sextic	6	680.264	677.136	678.688	678.688	1.327.448	663.365	664.216	663.784	663.784	1.356.376

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