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	60	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	54	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	9	7	7	7	6	8	8	8	60
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	9	6	6	6	6	7	7	7	54

**Molecule Parameter**
Number of Atoms (N)	20
Number of internal coordinates	54
Number of independant internal coordinates	9
Number of vibrational modes	54

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)	9	6	6	6	6	7	7	7	21 / 33
Quadratic (Raman)	9	6	6	6	6	7	7	7	27 / 27
IR + Raman	-	-	-	-	6	-	-	-	0^* / 6

* 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	54	2	2	2	0	4	4	4
quadratic	2	1.485	29	29	29	27	35	35	35
cubic	3	27.720	56	56	56	0	120	120	120
quartic	4	395.010	434	434	434	378	610	610	610
quintic	5	4.582.116	812	812	812	0	1.856	1.856	1.856
sextic	6	45.057.474	4.466	4.466	4.466	3.654	7.134	7.134	7.134

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	9	6	6	6	6	7	7	7
quadratic	2	213	181	181	181	180	183	183	183
cubic	3	3.531	3.443	3.443	3.443	3.441	3.473	3.473	3.473
quartic	4	49.815	49.293	49.293	49.293	49.263	49.351	49.351	49.351
quintic	5	573.765	572.431	572.431	572.431	572.373	572.895	572.895	572.895
sextic	6	5.636.991	5.631.191	5.631.191	5.631.191	5.630.727	5.632.061	5.632.061	5.632.061

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