Character table for point group D∞h

Character table for point group D_∞h

D_∞h	E	2C_∞	...	∞σ_v	i	2S_∞	...	∞C'₂	linear functions, rotations	quadratic functions	cubic functions
A_1g=Σ⁺_g	+1	+1	...	+1	+1	+1	...	+1	-	x²+y², z²	-
A_2g=Σ^-_g	+1	+1	...	-1	+1	+1	...	-1	R_z	-	-
E_1g=Π_g	+2	+2cos(φ)	...	0	+2	-2cos(φ)	...	0	(R_x, R_y)	(xz, yz)	-
E_2g=Δ_g	+2	+2cos(2φ)	...	0	+2	+2cos(2φ)	...	0	-	(x²-y², xy)	-
E_3g=Φ_g	+2	+2cos(3φ)	...	0	+2	-2cos(3φ)	...	0	-	-	-
E_ng	+2	+2cos(nφ)	...	0	+2	(-1)ⁿ2cos(nφ)	...	0	-	-	-
...	...	...	...	...	...	...	...	...	-	-	-
A_1u=Σ⁺_u	+1	+1	...	+1	-1	-1	...	-1	z	-	z³, z(x²+y²)
A_2u=Σ^-_u	+1	+1	...	-1	-1	-1	...	+1	-	-	-
E_1u=Π_u	+2	+2cos(φ)	...	0	-2	+2cos(φ)	...	0	(x, y)	-	(xz², yz²) [x(x²+y²), y(x²+y²)]
E_2u=Δ_u	+2	+2cos(2φ)	...	0	-2	-2cos(2φ)	...	0	-	-	[xyz, z(x²-y²)]
E_3u=Φ_u	+2	+2cos(3φ)	...	0	-2	2cos(3φ)	...	0	-	-	[y(3x²-y²), x(x²-3y²)]
E_nu	+2	+2cos(nφ)	...	0	-2	(-1)ⁿ⁺¹2cos(nφ)	...	0	-	-	-
...	...	...	...	...	...	...	...	...	-	-	-

Additional information

Number of symmetry elements	h = ∞
Number of irreducible representations	n = ∞
Abelian group	no
Number of subgroups	∞
Number of distinct subgroups	∞
Subgroups	C_s C_i C₂,C₃,C₄,C₅,C₆,…,C_∞ D₂,D₃,D₄,D₅,D₆,…,D_∞ C_2v,C_3v,C_4v,C_5v,C_6v,…,C_∞v, C_2h,C_3h,C_4h,C_5h,C_6h,…,C_∞h D_2h,D_3h,D_4h,D_5h,D_6h,… D_2d,D_3d,D_4d,D_5d,D_6d,…,D_∞d S₄,S₆,S₈,S₁₀,S₁₂,…,S_∞
Optical Isomerism (Chirality)	no
Polar	no

Force field analysis for point group D_∞h

Examples

Hydrogen	Carbondioxide	Acetylene
Carbonsuboxide	Diacetylene

Multipoles

dipole (p)	Σ⁺_u+Π_u
quadrupole (d)	Σ⁺_g+Π_g+Δ_g
octopole (f)	Σ⁺_u+Π_u+Δ_u+Φ_u
hexadecapole (g)	Σ⁺_g+Π_g+Δ_g+Φ_g+Γ_g
32-pole (h)	Σ⁺_u+Π_u+Δ_u+Φ_u+Γ_u+H_u
64-pole (i)	Σ⁺_g+Π_g+Δ_g+Φ_g+Γ_g+H_g+I_g
128-pole (j)	Σ⁺_u+Π_u+Δ_u+Φ_u+Γ_u+H_u+I_u+J_u
256-pole (k)	Σ⁺_g+Π_g+Δ_g+Φ_g+Γ_g+H_g+I_g+J_g+K_g
512-pole (l)	Σ⁺_u+Π_u+Δ_u+Φ_u+Γ_u+H_u+I_u+J_u+K_u+L_u

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