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	21	0	0	-3	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	15	0	0	-3	0	0

**Decomposition into Irreducible representations**
Motion	A_g	E_g	A_u	E_u	Total
Cartesian 3N	3	3	4	4	14
Translation (x,y,z)	0	0	1	1	2
Rotation (R_x,R_y,R_z)	1	1	0	0	2
Vibration	2	2	3	3	10

**Molecule Parameter**
Number of Atoms (N)	7
Number of internal coordinates	15
Number of independant internal coordinates	2
Number of vibrational modes	10

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A_g	E_g	A_u	E_u	Total
Linear (IR)	2	2	3	3	6 / 4
Quadratic (Raman)	2	2	3	3	4 / 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	15	0	0	-3	0	0
quadratic	2	120	0	0	12	0	0
cubic	3	680	5	5	-28	-1	-1
quartic	4	3.060	0	0	72	0	0
quintic	5	11.628	0	0	-144	0	0
sextic	6	38.760	15	15	300	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	2	2	3	3
quadratic	2	22	22	18	18
cubic	3	110	108	120	117
quartic	4	522	522	498	498
quintic	5	1.914	1.914	1.962	1.962
sextic	6	6.516	6.507	6.414	6.408

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