Reduction formula for point group S6

Reduction formula for point group S₆

**Characters for molecular motions**
Motion	E	C₃(z)	(C₃)²	i	(S₆)⁵	S₆
Cartesian 3N	24	0	0	0	0	0
Translation (x,y,z)	3	0	0	-3	0	0
Rotation (R_x,R_y,R_z)	3	0	0	3	0	0
Vibration	18	0	0	0	0	0

**Decomposition into Irreducible representations**
Motion	A_g	E_g	A_u	E_u	Total
Cartesian 3N	4	4	4	4	16
Translation (x,y,z)	0	0	1	1	2
Rotation (R_x,R_y,R_z)	1	1	0	0	2
Vibration	3	3	3	3	12

**Molecule Parameter**
Number of Atoms (N)	8
Number of internal coordinates	18
Number of independant internal coordinates	3
Number of vibrational modes	12

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A_g	E_g	A_u	E_u	Total
Linear (IR)	3	3	3	3	6 / 6
Quadratic (Raman)	3	3	3	3	6 / 6
IR + Raman	-	-	-	-	0^* / 0

* Center of inversion: Mutual Exclusion Principle

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₃(z)	(C₃)²	i	(S₆)⁵	S₆
linear	1	18	0	0	0	0	0
quadratic	2	171	0	0	9	0	0
cubic	3	1.140	6	6	0	0	0
quartic	4	5.985	0	0	45	0	0
quintic	5	26.334	0	0	0	0	0
sextic	6	100.947	21	21	165	3	3

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A_g	E_g	A_u	E_u
linear	1	3	3	3	3
quadratic	2	30	30	27	27
cubic	3	192	189	192	189
quartic	4	1.005	1.005	990	990
quintic	5	4.389	4.389	4.389	4.389
sextic	6	16.860	16.848	16.803	16.794

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 S6

Constructor University Bremen

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