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	-4	-4	-4	0	8	8	8
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	8	8	8

**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	0	6	6	6	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	0	5	5	5	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	0	5	5	5	15 / 15
Quadratic (Raman)	6	3	3	3	0	5	5	5	15 / 15
IR + Raman	-	-	-	-	0	-	-	-	0^* / 0

* 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	8	8	8
quadratic	2	465	17	17	17	15	47	47	47
cubic	3	4.960	-32	-32	-32	0	208	208	208
quartic	4	40.920	152	152	152	120	792	792	792
quintic	5	278.256	-272	-272	-272	0	2.640	2.640	2.640
sextic	6	1.623.160	952	952	952	680	8.008	8.008	8.008

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	0	5	5	5
quadratic	2	84	52	52	52	45	60	60	60
cubic	3	686	598	598	598	530	650	650	650
quartic	4	5.484	5.012	5.012	5.012	4.860	5.180	5.180	5.180
quintic	5	35.670	34.486	34.486	34.486	33.690	35.146	35.146	35.146
sextic	6	206.340	201.860	201.860	201.860	200.164	203.692	203.692	203.692

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