Reduction formula for point group D6h

Reduction formula for point group D_6h

**Characters for molecular motions**
Motion	E	2C₆ (z)	2C₃	C₂	3C'₂	3C''₂	i	2S₃	2S₆	_h (xy)	3_d	3_v
Cartesian 3N	75	2	0	-1	-1	-1	-3	-2	0	1	1	9
Translation (x,y,z)	3	2	0	-1	-1	-1	-3	-2	0	1	1	1
Rotation (R_x,R_y,R_z)	3	2	0	-1	-1	-1	3	2	0	-1	-1	-1
Vibration	69	-2	0	1	1	1	-3	-2	0	1	1	9

**Decomposition into Irreducible representations**
Motion	A_1g	A_2g	B_1g	B_2g	E_1g	E_2g	A_1u	A_2u	B_1u	B_2u	E_1u	E_2u	Total
Cartesian 3N	4	2	2	4	6	6	2	5	4	2	7	6	50
Translation (x,y,z)	0	0	0	0	0	0	0	1	0	0	1	0	2
Rotation (R_x,R_y,R_z)	0	1	0	0	1	0	0	0	0	0	0	0	2
Vibration	4	1	2	4	5	6	2	4	4	2	6	6	46

**Molecule Parameter**
Number of Atoms (N)	25
Number of internal coordinates	69
Number of independant internal coordinates	4
Number of vibrational modes	46

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A_1g	A_2g	B_1g	B_2g	E_1g	E_2g	A_1u	A_2u	B_1u	B_2u	E_1u	E_2u	Total
Linear (IR)	4	1	2	4	5	6	2	4	4	2	6	6	10 / 36
Quadratic (Raman)	4	1	2	4	5	6	2	4	4	2	6	6	15 / 31
IR + Raman	-	1	2	4	-	-	2	-	4	2	-	6	0^* / 21

* Center of inversion: Mutual Exclusion Principle

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2C₆ (z)	2C₃	C₂	3C'₂	3C''₂	i	2S₃	2S₆	_h (xy)	3_d	3_v
linear	1	69	-2	0	1	1	1	-3	-2	0	1	1	9
quadratic	2	2.415	2	0	35	35	35	39	2	0	35	35	75
cubic	3	57.155	-1	23	35	35	35	-109	-1	-1	35	35	435
quartic	4	1.028.790	0	0	630	630	630	774	0	0	630	630	2.310
quintic	5	15.020.334	0	0	630	630	630	-2.034	0	0	630	630	10.422
sextic	6	185.250.786	12	276	7.770	7.770	7.770	10.434	12	12	7.770	7.770	43.738

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A_1g	A_2g	B_1g	B_2g	E_1g	E_2g	A_1u	A_2u	B_1u	B_2u	E_1u	E_2u
linear	1	4	1	2	4	5	6	2	4	4	2	6	6
quadratic	2	128	83	94	104	199	210	94	104	104	94	198	198
cubic	3	2.449	2.314	2.326	2.426	4.746	4.758	2.338	2.438	2.438	2.338	4.770	4.770
quartic	4	43.476	42.426	42.636	43.056	85.692	85.902	42.624	43.044	43.044	42.624	85.668	85.668
quintic	5	627.354	624.276	624.486	626.934	1.251.420	1.251.630	624.708	627.156	627.156	624.708	1.251.864	1.251.864
sextic	6	7.728.272	7.711.510	7.714.096	7.723.088	15.437.118	15.439.704	7.713.874	7.722.866	7.722.866	7.713.874	15.436.674	15.436.674

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 D6h

Constructor University Bremen

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