Character table for point group C8v

Character table for point group C_8v

C_8v	E	2C₈	2C₄	2(C₈)³	C₂	4_v	4_d	linear functions, rotations	quadratic functions	cubic functions
A₁	+1	+1	+1	+1	+1	+1	+1	z	x²+y², z²	z³, z(x²+y²)
A₂	+1	+1	+1	+1	+1	-1	-1	R_z	-	-
B₁	+1	-1	+1	-1	+1	+1	-1	-	-	-
B₂	+1	-1	+1	-1	+1	-1	+1	-	-	-
E₁	+2	+(2)^½	0	-(2)^½	-2	0	0	(x, y) (R_x, R_y)	(xz, yz)	(xz², yz²) [x(x²+y²), y(x²+y²)]
E₂	+2	0	-2	0	+2	0	0	-	(x²-y², xy)	[xyz, z(x²-y²)]
E₃	+2	-(2)^½	0	+(2)^½	-2	0	0	-	-	[y(3x²-y²), x(x²-3y²)]

Additional information

Number of symmetry elements	h = 16
Number of irreducible representations	n = 7
Abelian group	no
Number of subgroups	9
Number of distinct subgroups	6
Subgroups (Number of different orientations)	C_s (2) , C₂ , C₄ , C₈ , C_2v (2) , C_4v (2)
Optical Isomerism (Chirality)	no
Polar	yes

Reduction formula for point group C_8v

Multipoles

dipole (p)	A₁+E₁
quadrupole (d)	A₁+E₁+E₂
octopole (f)	A₁+E₁+E₂+E₃
hexadecapole (g)	A₁+B₁+B₂+E₁+E₂+E₃
32-pole (h)	A₁+B₁+B₂+E₁+E₂+2E₃
64-pole (i)	A₁+B₁+B₂+E₁+2E₂+2E₃
128-pole (j)	A₁+B₁+B₂+2E₁+2E₂+2E₃
256-pole(k)	2A₁+A₂+B₁+B₂+2E₁+2E₂+2E₃
512-pole (l)	2A₁+A₂+B₁+B₂+3E₁+2E₂+2E₃

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