Character table for point group D8h

Character table for point group D_8h

D_8h	E	2C₈	2(C₈)³	2C₄	C₂ (z)	4C₂'	4C₂''	i	2(S₈)³	2S₈	2S₄	_h	4_v	4_d	linear functions, rotations	quadratic functions	cubic functions
A_1g	+1	+1	+1	+1	+1	+1	+1	+1	+1	+1	+1	+1	+1	+1	-	x²+y², z²	-
A_2g	+1	+1	+1	+1	+1	-1	-1	+1	+1	+1	+1	+1	-1	-1	R_z	-	-
B_1g	+1	-1	-1	+1	+1	+1	-1	+1	-1	-1	+1	+1	+1	-1	-	-	-
B_2g	+1	-1	-1	+1	+1	-1	+1	+1	-1	-1	+1	+1	-1	+1	-	-	-
E_1g	+2	+(2)^½	-(2)^½	0	-2	0	0	+2	+(2)^½	-(2)^½	0	-2	0	0	(R_x, R_y)	(xz, yz)	-
E_2g	+2	0	0	-2	+2	0	0	+2	0	0	-2	+2	0	0	-	(x²-y², xy)	-
E_3g	+2	-(2)^½	+(2)^½	0	-2	0	0	+2	-(2)^½	+(2)^½	0	-2	0	0	-	-	-
A_1u	+1	+1	+1	+1	+1	+1	+1	-1	-1	-1	-1	-1	-1	-1	-	-	-
A_2u	+1	+1	+1	+1	+1	-1	-1	-1	-1	-1	-1	-1	+1	+1	z	-	z³, z(x²+y²)
B_1u	+1	-1	-1	+1	+1	+1	-1	-1	+1	+1	-1	-1	-1	+1	-	-	-
B_2u	+1	-1	-1	+1	+1	-1	+1	-1	+1	+1	-1	-1	+1	-1	-	-	-
E_1u	+2	+(2)^½	-(2)^½	0	-2	0	0	-2	-(2)^½	+(2)^½	0	+2	0	0	(x, y)	-	(xz², yz²) [x(x²+y²), y(x²+y²)]
E_2u	+2	0	0	-2	+2	0	0	-2	0	0	+2	-2	0	0	-	-	[xyz, z(x²-y²)]
E_3u	+2	-(2)^½	+(2)^½	0	-2	0	0	-2	+(2)^½	-(2)^½	0	+2	0	0	-	-	[y(3x²-y²), x(x²-3y²)]

Additional information

Number of symmetry elements	h = 32
Number of irreducible representations	n = 14
Abelian group	no
Number of subgroups	36
Number of distinct subgroups	20
Subgroups (Number of different orientations)	C_s (3) , C_i , C₂ (3) , C₄ , C₈ , D₂ (2) , D₄ (2) , D₈ , C_2v (4) , C_4v (2) , C_8v , C_2h (3) , C_4h , C_8h , D_2h (2) , D_4h (2) , D_2d (2) , D_4d (2) , S₄ , S₈
Optical Isomerism (Chirality)	no
Polar	no

Reduction formula for point group D_8h

Examples

Cyclooctatetraene Dianion	Sulflower	Uranocene (eclipsed)

Multipoles

dipole (p)	A_2u+E_1u
quadrupole (d)	A_1g+E_1g+E_2g
octopole (f)	A_2u+E_1u+E_2u+E_3u
hexadecapole (g)	A_1g+B_1g+B_2g+E_1g+E_2g+E_3g
32-pole (h)	A_2u+B_1u+B_2u+E_1u+E_2u+2E_3u
64-pole (i)	A_1g+B_1g+B_2g+E_1g+2E_2g+2E_3g
128-pole (j)	A_2u+B_1u+B_2u+2E_1u+2E_2u+2E_3u
256-pole(k)	2A_1g+A_2g+B_1g+B_2g+2E_1g+2E_2g+2E_3g
512-pole (l)	A_1u+2A_2u+B_1u+B_2u+3E_1u+2E_2u+2E_3u

First nonvanishing multipole: quadrupole

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