Reduction formula for point group D2h

Reduction formula for point group D_2h

**Characters for molecular motions**
Motion	E	C₂ (z)	C₂ (y)	C₂ (x)	i	(xy)	(xz)	(yz)
Cartesian 3N	36	0	0	0	0	4	4	4
Translation (x,y,z)	3	-1	-1	-1	-3	1	1	1
Rotation (R_x,R_y,R_z)	3	-1	-1	-1	3	-1	-1	-1
Vibration	30	2	2	2	0	4	4	4

**Decomposition into Irreducible representations**
Motion	A_g	B_1g	B_2g	B_3g	A_u	B_1u	B_2u	B_3u	Total
Cartesian 3N	6	4	4	4	3	5	5	5	36
Translation (x,y,z)	0	0	0	0	0	1	1	1	3
Rotation (R_x,R_y,R_z)	0	1	1	1	0	0	0	0	3
Vibration	6	3	3	3	3	4	4	4	30

**Molecule Parameter**
Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	6
Number of vibrational modes	30

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A_g	B_1g	B_2g	B_3g	A_u	B_1u	B_2u	B_3u	Total
Linear (IR)	6	3	3	3	3	4	4	4	12 / 18
Quadratic (Raman)	6	3	3	3	3	4	4	4	15 / 15
IR + Raman	-	-	-	-	3	-	-	-	0^* / 3

* Center of inversion: Mutual Exclusion Principle

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₂ (z)	C₂ (y)	C₂ (x)	i	(xy)	(xz)	(yz)
linear	1	30	2	2	2	0	4	4	4
quadratic	2	465	17	17	17	15	23	23	23
cubic	3	4.960	32	32	32	0	72	72	72
quartic	4	40.920	152	152	152	120	256	256	256
quintic	5	278.256	272	272	272	0	680	680	680
sextic	6	1.623.160	952	952	952	680	1.904	1.904	1.904

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A_g	B_1g	B_2g	B_3g	A_u	B_1u	B_2u	B_3u
linear	1	6	3	3	3	3	4	4	4
quadratic	2	75	55	55	55	54	57	57	57
cubic	3	659	607	607	607	605	625	625	625
quartic	4	5.283	5.079	5.079	5.079	5.061	5.113	5.113	5.113
quintic	5	35.139	34.663	34.663	34.663	34.629	34.833	34.833	34.833
sextic	6	204.051	202.623	202.623	202.623	202.453	202.929	202.929	202.929

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 D2h

Constructor University Bremen

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