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	0	0	0	0	0	0	8
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	0	0	8

**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	4	6	2	4	4	2	6	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	1	2	4	5	2	3	4	2	5	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	1	2	4	5	2	3	4	2	5	8 / 24
Quadratic (Raman)	4	1	2	4	5	2	3	4	2	5	15 / 17
IR + Raman	-	1	-	-	-	2	-	4	2	-	0^* / 9

* 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	0	0	8
quadratic	2	903	3	23	23	23	21	1	21	21	53
cubic	3	13.244	-4	44	44	44	0	0	0	0	256
quartic	4	148.995	15	275	275	275	231	11	231	231	1.095
quintic	5	1.370.754	-26	506	506	506	0	0	0	0	4.056
sextic	6	10.737.573	37	2.277	2.277	2.277	1.771	11	1.771	1.771	13.803

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	4	5	2	3	4	2	5
quadratic	2	76	46	56	64	110	52	59	59	51	110
cubic	3	873	787	799	863	1.650	809	851	863	799	1.650
quartic	4	9.596	9.127	9.247	9.463	18.590	9.204	9.398	9.408	9.192	18.590
quintic	5	86.334	85.067	85.200	86.214	171.281	85.320	86.081	86.214	85.200	171.281
sextic	6	673.984	668.952	669.952	672.960	1.341.912	669.645	672.400	672.520	669.512	1.341.912

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