Character table for point group C5v

Character table for point group C_5v

C_5v	E	2C₅ (z)	2(C₅)²	5_v	linear functions, rotations	quadratic functions	cubic functions
A₁	+1	+1	+1	+1	z	x²+y², z²	z³, z(x²+y²)
A₂	+1	+1	+1	-1	R_z	-	-
E₁	+2	+2cos(2/5)	+2cos(4/5)	0	(x, y) (R_x, R_y)	(xz, yz)	(xz², yz²) [x(x²+y²), y(x²+y²)]
E₂	+2	+2cos(4/5)	+2cos(2/5)	0	-	(x²-y², xy)	[xyz, z(x²-y²)] [y(3x²-y²), x(x²-3y²)]

Information for point groups with fivefold rotational axis

Additional information

Number of symmetry elements	h = 10
Number of irreducible representations	n = 4
Abelian group	no
Number of subgroups	2
Subgroups	C_s , C₅
Optical Isomerism (Chirality)	no
Polar	yes

Reduction formula for point group C_5v

Examples

Corannulene

Multipoles

dipole (p)	A₁+E₁
quadrupole (d)	A₁+E₁+E₂
octopole (f)	A₁+E₁+2E₂
hexadecapole (g)	A₁+2E₁+2E₂
32-pole (h)	2A₁+A₂+2E₁+2E₂
64-pole (i)	2A₁+A₂+3E₁+2E₂
128-pole (j)	2A₁+A₂+3E₁+3E₂
256-pole(k)	2A₁+A₂+3E₁+4E₂
512-pole (l)	2A₁+A₂+4E₁+4E₂

First nonvanishing multipole: dipole

Literature

A. Gelessus, W. Thiel and W. Weber, J. Chem. Educ. 72, 505 (1995)
Multipoles and Symmetry

Character tables for chemically important point groups

Computational Laboratory for Analysis, Modeling and Visualization

Constructor University Bremen

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