Character table for point group D8d

Character table for point group D_8d

D_8d	E	2S₁₆	2C₈	2(S₁₆)³	2C₄	2(S₁₆)⁵	2(C₈)³	2(S₁₆)⁷	C₂	8C'₂	8_d	linear functions, rotations	quadratic functions	cubic functions
A₁	+1	+1	+1	+1	+1	+1	+1	+1	+1	+1	+1	-	x²+y², z²	-
A₂	+1	+1	+1	+1	+1	+1	+1	+1	+1	-1	-1	R_z	-	-
B₁	+1	-1	+1	-1	+1	-1	+1	-1	+1	+1	-1	-	-	-
B₂	+1	-1	+1	-1	+1	-1	+1	-1	+1	-1	+1	z	-	z³, z(x²+y²)
E₁	+2	+2cos(1/8)	+(2)^½	+2cos(3/8)	0	-2cos(3/8)	-(2)^½	-2cos(1/8)	-2	0	0	(x, y)	-	(xz², yz²) [x(x²+y²), y(x²+y²)]
E₂	+2	+(2)^½	0	-(2)^½	-2	-(2)^½	0	+(2)^½	+2	0	0	-	(x²-y², xy)	-
E₃	+2	+2cos(3/8)	-(2)^½	-2cos(1/8)	0	+2cos(1/8)	+(2)^½	-2cos(3/8)	-2	0	0	-	-	[x(x²-3y²), y(3x²-y²)]
E₄	+2	0	-2	0	+2	0	-2	0	+2	0	0	-	-	-
E₅	+2	-2cos(3/8)	-(2)^½	+2cos(1/8)	0	-2cos(1/8)	+(2)^½	+2cos(3/8)	-2	0	0	-	-	-
E₆	+2	-(2)^½	0	+(2)^½	-2	+(2)^½	0	-(2)^½	+2	0	0	-	-	[xyz, z(x²-y²)]
E₇	+2	-2cos(1/8)	+(2)^½	-2cos(3/8)	0	+2cos(3/8)	-(2)^½	+2cos(1/8)	-2	0	0	(R_x, R_y)	(xz, yz)	-

Additional information

Number of symmetry elements	h = 32
Number of irreducible representations	n = 11
Abelian group	no
Number of subgroups	12
Number of distinct subgroups	11
Subgroups (Number of different orientations)	C_s , C₂ (2) , C₄ , C₈ , D₂ , D₄ , D₈ , C_2v , C_4v , C_8v , S₁₆
Optical Isomerism (Chirality)	no
Polar	no

Reduction formula for point group D_8d

Examples

Uranocene (staggered)

Multipoles

dipole (p)	B₂+E₁
quadrupole (d)	A₁+E₂+E₇
octopole (f)	B₂+E₁+E₃+E₆
hexadecapole (g)	A₁+E₂+E₄+E₅+E₇
32-pole (h)	B₂+E₁+E₃+E₄+E₅+E₆
64-pole (i)	A₁+E₂+E₃+E₄+E₅+E₆+E₇
128-pole (j)	B₂+E₁+E₂+E₃+E₄+E₅+E₆+E₇
256-pole(k)	A₁+B₁+B₂+E₁+E₂+E₃+E₄+E₅+E₆+E₇
512-pole (l)	A₁+A₂+B₂+E₁+E₂+E₃+E₄+E₅+E₆+2E₇

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