Point Group Oh

Point Group O_h

O_h	E	8C₃	6C₂	6C₄	3C₂	i	6S₄	8S₆	3σ_h	6σ_d
A_1g	1	1	1	1	1	1	1	1	1	1
A_2g	1	1	-1	-1	1	1	-1	1	1	-1
E_g	2	-1	0	0	2	2	0	-1	2	0
T_1g	3	0	-1	1	-1	3	1	0	-1	-1
T_2g	3	0	1	-1	-1	3	-1	0	-1	1
A_1u	1	1	1	1	1	-1	-1	-1	-1	-1
A_2u	1	1	-1	-1	1	-1	1	-1	-1	1
E_u	2	-1	0	0	2	-2	0	1	-2	0
T_1u	3	0	-1	1	-1	-3	-1	0	1	1
T_2u	3	0	1	-1	-1	-3	1	0	1	-1

Additional information

Number of symmetry elements	h = 48
Number of classes, irreps	n = 10
Abelian group	no
Optical Isomerism (Chirality)	no
Polar	no
Parity	yes

Reduce representation to irreducible representations

E	8C₃	6C₂	6C₄	3C₂	i	6S₄	8S₆	3σ_h	6σ_d

Genrate representation from irreducible representations

A_1g	A_2g	E_g	T_1g	T_2g	A_1u	A_2u	E_u	T_1u	T_2u

Examples

Sulfurhexafluoride	[B₆H₆]^2--Ion	Tungstenhexacarbonyl
Cubane	Octamethylsilsesquioxane

Direct products of irreducible representations

**Binary products**
⊙	A_1g	A_2g	E_g	T_1g	T_2g	A_1u	A_2u	E_u	T_1u	T_2u
A_1g	A_1g
A_2g	A_2g	A_1g
E_g	E_g	E_g	A_1g⊕A_2g⊕E_g
T_1g	T_1g	T_2g	T_1g⊕T_2g	A_1g⊕E_g⊕T_1g⊕T_2g
T_2g	T_2g	T_1g	T_1g⊕T_2g	A_2g⊕E_g⊕T_1g⊕T_2g	A_1g⊕E_g⊕T_1g⊕T_2g
A_1u	A_1u	A_2u	E_u	T_1u	T_2u	A_1g
A_2u	A_2u	A_1u	E_u	T_2u	T_1u	A_2g	A_1g
E_u	E_u	E_u	A_1u⊕A_2u⊕E_u	T_1u⊕T_2u	T_1u⊕T_2u	E_g	E_g	A_1g⊕A_2g⊕E_g
T_1u	T_1u	T_2u	T_1u⊕T_2u	A_1u⊕E_u⊕T_1u⊕T_2u	A_2u⊕E_u⊕T_1u⊕T_2u	T_1g	T_2g	T_1g⊕T_2g	A_1g⊕E_g⊕T_1g⊕T_2g
T_2u	T_2u	T_1u	T_1u⊕T_2u	A_2u⊕E_u⊕T_1u⊕T_2u	A_1u⊕E_u⊕T_1u⊕T_2u	T_2g	T_1g	T_1g⊕T_2g	A_2g⊕E_g⊕T_1g⊕T_2g	A_1g⊕E_g⊕T_1g⊕T_2g

Ternary Products
Quaternary Products

Symmetric powers [Γⁿ] of degenerate irreducible representations
Vibrational overtones

irrep	[Γ²]	[Γ³]	[Γ⁴]	[Γ⁵]	[Γ⁶]
E_g	A_1g⊕E_g	A_1g⊕A_2g⊕E_g	A_1g⊕2E_g	A_1g⊕A_2g⊕2E_g	2A_1g⊕A_2g⊕2E_g	More
T_1g	A_1g⊕E_g⊕T_2g	A_2g⊕2T_1g⊕T_2g	2A_1g⊕2E_g⊕T_1g⊕2T_2g	A_2g⊕E_g⊕4T_1g⊕2T_2g	3A_1g⊕A_2g⊕3E_g⊕2T_1g⊕4T_2g	More
T_2g	A_1g⊕E_g⊕T_2g	A_1g⊕T_1g⊕2T_2g	2A_1g⊕2E_g⊕T_1g⊕2T_2g	A_1g⊕E_g⊕2T_1g⊕4T_2g	3A_1g⊕A_2g⊕3E_g⊕2T_1g⊕4T_2g	More
E_u	A_1g⊕E_g	A_1u⊕A_2u⊕E_u	A_1g⊕2E_g	A_1u⊕A_2u⊕2E_u	2A_1g⊕A_2g⊕2E_g	More
T_1u	A_1g⊕E_g⊕T_2g	A_2u⊕2T_1u⊕T_2u	2A_1g⊕2E_g⊕T_1g⊕2T_2g	A_2u⊕E_u⊕4T_1u⊕2T_2u	3A_1g⊕A_2g⊕3E_g⊕2T_1g⊕4T_2g	More
T_2u	A_1g⊕E_g⊕T_2g	A_1u⊕T_1u⊕2T_2u	2A_1g⊕2E_g⊕T_1g⊕2T_2g	A_1u⊕E_u⊕2T_1u⊕4T_2u	3A_1g⊕A_2g⊕3E_g⊕2T_1g⊕4T_2g	More

Spherical harmonics and Multipoles
Symmetric Powers of Γ_xyz

Spherical Harmonics Y^l / Multipole				Symmetric Power [Γ^l(xyz)]
l	2l+1	Multipole	Symmetry	Rank	[Γ^l(xyz)]
s (l=0)	1	Monopole	A_1g	1	A_1g
p (l=1)	3	Dipole	T_1u	3	T_1u
d (l=2)	5	Quadrupole	E_g⊕T_2g	6	A_1g⊕E_g⊕T_2g
f (l=3)	7	Octupole	A_2u⊕T_1u⊕T_2u	10	A_2u⊕2T_1u⊕T_2u
g (l=4)	9	Hexadecapole	A_1g⊕E_g⊕T_1g⊕T_2g	15	2A_1g⊕2E_g⊕T_1g⊕2T_2g
h (l=5)	11	Dotricontapole	E_u⊕2T_1u⊕T_2u	21	A_2u⊕E_u⊕4T_1u⊕2T_2u
i (l=6)	13	Tetrahexacontapole	A_1g⊕A_2g⊕E_g⊕T_1g⊕2T_2g	28	3A_1g⊕A_2g⊕3E_g⊕2T_1g⊕4T_2g
j (l=7)	15	Octacosahectapole	A_2u⊕E_u⊕2T_1u⊕2T_2u	36	2A_2u⊕2E_u⊕6T_1u⊕4T_2u
k (l=8)	17	256-pole	A_1g⊕2E_g⊕2T_1g⊕2T_2g	45	4A_1g⊕A_2g⊕5E_g⊕4T_1g⊕6T_2g
l (l=9)	19	512-pole	A_1u⊕A_2u⊕E_u⊕3T_1u⊕2T_2u	55	A_1u⊕3A_2u⊕3E_u⊕9T_1u⊕6T_2u
m (l=10)	21	1024-pole	A_1g⊕A_2g⊕2E_g⊕2T_1g⊕3T_2g	66	5A_1g⊕2A_2g⊕7E_g⊕6T_1g⊕9T_2g
n (l=11)	23	2048-pole	A_2u⊕2E_u⊕3T_1u⊕3T_2u	78	A_1u⊕4A_2u⊕5E_u⊕12T_1u⊕9T_2u
o (l=12)	25	4096-pole	2A_1g⊕A_2g⊕2E_g⊕3T_1g⊕3T_2g	91	7A_1g⊕3A_2g⊕9E_g⊕9T_1g⊕12T_2g
More

First nonvanshing multipole: Hexadecapole

Further Reading

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

Ligand Field, dⁿ term splitting

**Term symbols for electronic configurations dⁿ**
dⁿ	Term Symbols
d¹ = d⁹	²D
d² = d⁸	¹S, ¹D, ¹G, ³P, ³F
d³ = d⁷	²P, ²D (2), ²F, ²G, ²H, ⁴P, ⁴F
d⁴ = d⁶	¹S (2), ¹D (2), ¹F, ¹G (2), ¹I, ³P (2), ³D, ³F (2), ³G, ³H, ⁵D
d⁵	²S, ²P, ²D (3), ²F (2), ²G (2), ²H, ²I, ⁴P, ⁴D, ⁴F, ⁴G, ⁶S

**Term splitting in point group O_h**
L	2L+1	Term Splitting
S (L=0)	1	A_1g
P (L=1)	3	T_1g
D (L=2)	5	E_g⊕T_2g
F (L=3)	7	A_2g⊕T_1g⊕T_2g
G (L=4)	9	A_1g⊕E_g⊕T_1g⊕T_2g
H (L=5)	11	E_g⊕2T_1g⊕T_2g
I (L=6)	13	A_1g⊕A_2g⊕E_g⊕T_1g⊕2T_2g

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