Point Group S10

Point Group S₁₀

ε=exp(2πi/5)
S₁₀	E	C₅	(C₅)²	(C₅)³	(C₅)⁴	i	(S₁₀)⁷	(S₁₀)⁹	S₁₀	(S₁₀)³
A_g	1	1	1	1	1	1	1	1	1	1
E_1g*	1 1	ε^* ε^*	ε^2* ε^2*	ε^2* ε^2*	ε^* ε^*	1 1	ε^* ε^*	ε^2* ε^2*	ε^2* ε^2*	ε^* ε^*
E_2g*	1 1	ε^2* ε^2*	ε^* ε^*	ε^* ε^*	ε^2* ε^2*	1 1	ε^2* ε^2*	ε^* ε^*	ε^* ε^*	ε^2* ε^2*
A_u	1	1	1	1	1	-1	-1	-1	-1	-1
E_1u*	1 1	ε^* ε^*	ε^2* ε^2*	ε^2* ε^2*	ε^* ε^*	-1 -1	-ε^* -ε^*	-ε^2* -ε^2*	-ε^2* -ε^2*	-ε^* -ε^*
E_2u*	1 1	ε^2* ε^2*	ε^* ε^*	ε^* ε^*	ε^2* ε^2*	-1 -1	-ε^2* -ε^2*	-ε^* -ε^*	-ε^* -ε^*	-ε^2* -ε^2*

Additional information

Number of symmetry elements	h = 10
Number of classes, irreps	n = 10
Number of real-valued irreducible representations	n = 6
Abelian group	yes
Optical Isomerism (Chirality)	no
Polar	no
Parity	yes

Reduce representation to irreducible representations

E	C₅	(C₅)²	(C₅)³	(C₅)⁴	i	(S₁₀)⁷	(S₁₀)⁹	S₁₀	(S₁₀)³

Genrate representation from irreducible representations

A_g	E_1g*	E_2g*	A_u	E_1u*	E_2u*

Direct products of irreducible representations

**Binary products**
⊙	A_g	E_1g*	E_2g*	A_u	E_1u*	E_2u*
A_g	A_g
E_1g*	E_1g	2A_g⊕E_2g
E_2g*	E_2g	E_1g⊕E_2g	2A_g⊕E_1g
A_u	A_u	E_1u	E_2u	A_g
E_1u*	E_1u	2A_u⊕E_2u	E_1u⊕E_2u	E_1g	2A_g⊕E_2g
E_2u*	E_2u	E_1u⊕E_2u	2A_u⊕E_1u	E_2g	E_1g⊕E_2g	2A_g⊕E_1g

Ternary Products
Quaternary Products

Symmetric powers [Γⁿ] of degenerate irreducible representations
Vibrational overtones

irrep	[Γ²]	[Γ³]	[Γ⁴]	[Γ⁵]	[Γ⁶]
E_1g*	A_g⊕E_2g	E_1g⊕E_2g	A_g⊕E_1g⊕E_2g	2A_g⊕E_1g⊕E_2g	A_g⊕2E_1g⊕E_2g	More
E_2g*	A_g⊕E_1g	E_1g⊕E_2g	A_g⊕E_1g⊕E_2g	2A_g⊕E_1g⊕E_2g	A_g⊕E_1g⊕2E_2g	More
E_1u*	A_g⊕E_2g	E_1u⊕E_2u	A_g⊕E_1g⊕E_2g	2A_u⊕E_1u⊕E_2u	A_g⊕2E_1g⊕E_2g	More
E_2u*	A_g⊕E_1g	E_1u⊕E_2u	A_g⊕E_1g⊕E_2g	2A_u⊕E_1u⊕E_2u	A_g⊕E_1g⊕2E_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_g	1	A_g
p (l=1)	3	Dipole	A_u⊕E_1u	3	A_u⊕E_1u
d (l=2)	5	Quadrupole	A_g⊕E_1g⊕E_2g	6	2A_g⊕E_1g⊕E_2g
f (l=3)	7	Octupole	A_u⊕E_1u⊕2E_2u	10	2A_u⊕2E_1u⊕2E_2u
g (l=4)	9	Hexadecapole	A_g⊕2E_1g⊕2E_2g	15	3A_g⊕3E_1g⊕3E_2g
h (l=5)	11	Dotricontapole	3A_u⊕2E_1u⊕2E_2u	21	5A_u⊕4E_1u⊕4E_2u
i (l=6)	13	Tetrahexacontapole	3A_g⊕3E_1g⊕2E_2g	28	6A_g⊕6E_1g⊕5E_2g
j (l=7)	15	Octacosahectapole	3A_u⊕3E_1u⊕3E_2u	36	8A_u⊕7E_1u⊕7E_2u
k (l=8)	17	256-pole	3A_g⊕3E_1g⊕4E_2g	45	9A_g⊕9E_1g⊕9E_2g
l (l=9)	19	512-pole	3A_u⊕4E_1u⊕4E_2u	55	11A_u⊕11E_1u⊕11E_2u
m (l=10)	21	1024-pole	5A_g⊕4E_1g⊕4E_2g	66	14A_g⊕13E_1g⊕13E_2g
n (l=11)	23	2048-pole	5A_u⊕5E_1u⊕4E_2u	78	16A_u⊕16E_1u⊕15E_2u
o (l=12)	25	4096-pole	5A_g⊕5E_1g⊕5E_2g	91	19A_g⊕18E_1g⊕18E_2g
More

First nonvanshing multipole: Quadrupole

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 S₁₀**
L	2L+1	Term Splitting
S (L=0)	1	A_g
P (L=1)	3	A_g⊕E_1g
D (L=2)	5	A_g⊕E_1g⊕E_2g
F (L=3)	7	A_g⊕E_1g⊕2E_2g
G (L=4)	9	A_g⊕2E_1g⊕2E_2g
H (L=5)	11	3A_g⊕2E_1g⊕2E_2g
I (L=6)	13	3A_g⊕3E_1g⊕2E_2g

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