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	21	3	-3	-3	-1	-3	-1	5	5	3
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	15	1	-1	-1	1	-3	-1	5	5	3

**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	2	1	1	1	2	0	3	0	1	4	15
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	2	0	1	1	1	0	2	0	1	3	11

**Molecule Parameter**
Number of Atoms (N)	7
Number of internal coordinates	15
Number of independant internal coordinates	2
Number of vibrational modes	11

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)	2	0	1	1	1	0	2	0	1	3	5 / 6
Quadratic (Raman)	2	0	1	1	1	0	2	0	1	3	5 / 6
IR + Raman	-	0	-	-	-	0	-	0	1	-	0^* / 1

* 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	15	1	-1	-1	1	-3	-1	5	5	3
quadratic	2	120	0	8	8	8	12	0	20	20	12
cubic	3	680	0	-8	-8	8	-28	0	60	60	28
quartic	4	3.060	4	36	36	36	72	4	160	160	72
quintic	5	11.628	4	-36	-36	36	-144	-4	376	376	144
sextic	6	38.760	0	120	120	120	300	0	820	820	300

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	2	0	1	1	1	0	2	0	1	3
quadratic	2	16	4	11	9	13	4	8	5	7	15
cubic	3	55	33	46	42	75	29	51	34	46	97
quartic	4	247	171	218	196	367	159	199	168	190	389
quintic	5	804	674	759	719	1.393	646	776	671	747	1.523
sextic	6	2.670	2.330	2.565	2.435	4.765	2.250	2.470	2.295	2.425	4.895

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