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	36	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	30	0	0	0	0	0

**Decomposition into Irreducible representations**
Motion	A_g	E_g	A_u	E_u	Total
Cartesian 3N	6	6	6	6	24
Translation (x,y,z)	0	0	1	1	2
Rotation (R_x,R_y,R_z)	1	1	0	0	2
Vibration	5	5	5	5	20

**Molecule Parameter**
Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	5
Number of vibrational modes	20

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A_g	E_g	A_u	E_u	Total
Linear (IR)	5	5	5	5	10 / 10
Quadratic (Raman)	5	5	5	5	10 / 10
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	30	0	0	0	0	0
quadratic	2	465	0	0	15	0	0
cubic	3	4.960	10	10	0	0	0
quartic	4	40.920	0	0	120	0	0
quintic	5	278.256	0	0	0	0	0
sextic	6	1.623.160	55	55	680	5	5

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A_g	E_g	A_u	E_u
linear	1	5	5	5	5
quadratic	2	80	80	75	75
cubic	3	830	825	830	825
quartic	4	6.840	6.840	6.800	6.800
quintic	5	46.376	46.376	46.376	46.376
sextic	6	270.660	270.630	270.430	270.405

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