Point Group Th

Point Group T_h

ε=exp(2πi/3)
T_h	E	4C₃	4(C₃)²	3C₂	i	4(S₆)⁵	4S₆	3σ_h
A_g	1	1	1	1	1	1	1	1
E_g*	1 1	ε^* ε^*	ε^* ε^*	1 1	1 1	ε^* ε^*	ε^* ε^*	1 1
T_g	3	0	0	-1	3	0	0	-1
A_u	1	1	1	1	-1	-1	-1	-1
E_u*	1 1	ε^* ε^*	ε^* ε^*	1 1	-1 -1	-ε^* -ε^*	-ε^* -ε^*	-1 -1
T_u	3	0	0	-1	-3	0	0	1

Additional information

Number of symmetry elements	h = 24
Number of classes, irreps	n = 8
Number of real-valued irreducible representations	n = 6
Abelian group	no
Optical Isomerism (Chirality)	no
Polar	no
Parity	yes

Reduce representation to irreducible representations

E	4C₃	4(C₃)²	3C₂	i	4(S₆)⁵	4S₆	3σ_h

Genrate representation from irreducible representations

A_g	E_g*	T_g	A_u	E_u*	T_u

Examples

[Mg(H₂O)₆]²⁺-Ion	Bromofullerene C₆₀Br₂₄

Direct products of irreducible representations

**Binary products**
⊙	A_g	E_g*	T_g	A_u	E_u*	T_u
A_g	A_g
E_g*	E_g	2A_g⊕E_g
T_g	T_g	2T_g	A_g⊕E_g⊕2T_g
A_u	A_u	E_u	T_u	A_g
E_u*	E_u	2A_u⊕E_u	2T_u	E_g	2A_g⊕E_g
T_u	T_u	2T_u	A_u⊕E_u⊕2T_u	T_g	2T_g	A_g⊕E_g⊕2T_g

Ternary Products
Quaternary Products

Symmetric powers [Γⁿ] of degenerate irreducible representations
Vibrational overtones

irrep	[Γ²]	[Γ³]	[Γ⁴]	[Γ⁵]	[Γ⁶]
E_g*	A_g⊕E_g	2A_g⊕E_g	A_g⊕2E_g	2A_g⊕2E_g	3A_g⊕2E_g	More
T_g	A_g⊕E_g⊕T_g	A_g⊕3T_g	2A_g⊕2E_g⊕3T_g	A_g⊕E_g⊕6T_g	4A_g⊕3E_g⊕6T_g	More
E_u*	A_g⊕E_g	2A_u⊕E_u	A_g⊕2E_g	2A_u⊕2E_u	3A_g⊕2E_g	More
T_u	A_g⊕E_g⊕T_g	A_u⊕3T_u	2A_g⊕2E_g⊕3T_g	A_u⊕E_u⊕6T_u	4A_g⊕3E_g⊕6T_g	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_g	1	A_g
p (l=1)	3	Dipole	T_u	3	T_u
d (l=2)	5	Quadrupole	E_g⊕T_g	6	A_g⊕E_g⊕T_g
f (l=3)	7	Octupole	A_u⊕2T_u	10	A_u⊕3T_u
g (l=4)	9	Hexadecapole	A_g⊕E_g⊕2T_g	15	2A_g⊕2E_g⊕3T_g
h (l=5)	11	Dotricontapole	E_u⊕3T_u	21	A_u⊕E_u⊕6T_u
i (l=6)	13	Tetrahexacontapole	2A_g⊕E_g⊕3T_g	28	4A_g⊕3E_g⊕6T_g
j (l=7)	15	Octacosahectapole	A_u⊕E_u⊕4T_u	36	2A_u⊕2E_u⊕10T_u
k (l=8)	17	256-pole	A_g⊕2E_g⊕4T_g	45	5A_g⊕5E_g⊕10T_g
l (l=9)	19	512-pole	2A_u⊕E_u⊕5T_u	55	4A_u⊕3E_u⊕15T_u
m (l=10)	21	1024-pole	2A_g⊕2E_g⊕5T_g	66	7A_g⊕7E_g⊕15T_g
n (l=11)	23	2048-pole	A_u⊕2E_u⊕6T_u	78	5A_u⊕5E_u⊕21T_u
o (l=12)	25	4096-pole	3A_g⊕2E_g⊕6T_g	91	10A_g⊕9E_g⊕21T_g
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 T_h**
L	2L+1	Term Splitting
S (L=0)	1	A_g
P (L=1)	3	T_g
D (L=2)	5	E_g⊕T_g
F (L=3)	7	A_g⊕2T_g
G (L=4)	9	A_g⊕E_g⊕2T_g
H (L=5)	11	E_g⊕3T_g
I (L=6)	13	2A_g⊕E_g⊕3T_g

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