Reduction formula for point group C6

Reduction formula for point group C₆

**Characters for molecular motions**
Motion	E	C₆	C₃	C₂	(C₃)²	(C₆)⁵
Cartesian 3N	15	10	0	-5	0	10
Translation (x,y,z)	3	2	0	-1	0	2
Rotation (R_x,R_y)^*	2	1	-1	-2	-1	1
Vibration	10	7	1	-2	1	7

* Linear molecule

**Decomposition into Irreducible representations**
Motion	A	B	E₁	E₂	Total
Cartesian 3N	5	0	5	0	10
Translation (x,y,z)	1	0	1	0	2
Rotation (R_x,R_y)^*	0	0	1	0	1
Vibration	4	0	3	0	7

* Linear molecule

**Molecule Parameter**
Number of Atoms (N)	5
Number of internal coordinates	10
Number of independant internal coordinates	4
Number of vibrational modes	7

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A	B	E₁	E₂	Total
Linear (IR)	4	0	3	0	7 / 0
Quadratic (Raman)	4	0	3	0	7 / 0
IR + Raman	4	0	3	-	7 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	C₆	C₃	C₂	(C₃)²	(C₆)⁵
linear	1	10	7	1	-2	1	7
quadratic	2	55	25	1	7	1	25
cubic	3	220	60	4	-12	4	60
quartic	4	715	108	4	27	4	108
quintic	5	2.002	156	4	-42	4	156
sextic	6	5.005	194	10	77	10	194

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A	B	E₁	E₂
linear	1	4	0	3	0
quadratic	2	19	0	12	6
cubic	3	56	20	48	24
quartic	4	161	80	132	105
quintic	5	380	290	366	300
sextic	6	915	760	852	813

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 C6

Constructor University Bremen

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