Reduction formula for point group C6v

Reduction formula for point group C_6v

**Characters for molecular motions**
Motion	E	2C₆ (z)	2C₃ (z)	C₂ (z)	3_v	3_d
Cartesian 3N	15	10	0	-5	5	5
Translation (x,y,z)	3	2	0	-1	1	1
Rotation (R_x,R_y)^*	2	1	-1	-2	0	0
Vibration	10	7	1	-2	4	4

* Linear molecule

**Decomposition into Irreducible representations**
Motion	A₁	A₂	B₁	B₂	E₁	E₂	Total
Cartesian 3N	5	0	0	0	5	0	10
Translation (x,y,z)	1	0	0	0	1	0	2
Rotation (R_x,R_y)^*	0	0	0	0	1	0	1
Vibration	4	0	0	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	A₁	A₂	B₁	B₂	E₁	E₂	Total
Linear (IR)	4	0	0	0	3	0	7 / 0
Quadratic (Raman)	4	0	0	0	3	0	7 / 0
IR + Raman	4	0	0	0	3	-	7 / 0

**Characters of symmetric powers for vibration representation**
Force field	Tensor Order	E	2C₆ (z)	2C₃ (z)	C₂ (z)	3_v	3_d
linear	1	10	7	1	-2	4	4
quadratic	2	55	25	1	7	13	13
cubic	3	220	60	4	-12	32	32
quartic	4	715	108	4	27	71	71
quintic	5	2.002	156	4	-42	140	140
sextic	6	5.005	194	10	77	259	259

Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field	Tensor Order	A₁	A₂	B₁	B₂	E₁	E₂
linear	1	4	0	0	0	3	0
quadratic	2	16	3	0	0	12	6
cubic	3	44	12	10	10	48	24
quartic	4	116	45	40	40	132	105
quintic	5	260	120	145	145	366	300
sextic	6	587	328	380	380	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 C6v

Constructor University Bremen

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