Point Group C11h

Point Group C_11h

ε=exp(2πi/11)
C_11h	E	C₁₁	(C₁₁)²	(C₁₁)³	(C₁₁)⁴	(C₁₁)⁵	(C₁₁)⁶	(C₁₁)⁷	(C₁₁)⁸	(C₁₁)⁹	(C₁₁)¹⁰	σ_h	S₁₁	(S₁₁)¹³	(S₁₁)³	(S₁₁)¹⁵	(S₁₁)⁵	(S₁₁)¹⁷	(S₁₁)⁷	(S₁₁)¹⁹	(S₁₁)⁹	(S₁₁)²¹
A'	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
E'₁*	1 1	ε^* ε^*	ε^2* ε^2*	ε^3* ε^3*	ε^4* ε^4*	ε^5* ε^5*	ε^5* ε^5*	ε^4* ε^4*	ε^3* ε^3*	ε^2* ε^2*	ε^* ε^*	1 1	ε^* ε^*	ε^2* ε^2*	ε^3* ε^3*	ε^4* ε^4*	ε^5* ε^5*	ε^5* ε^5*	ε^4* ε^4*	ε^3* ε^3*	ε^2* ε^2*	ε^* ε^*
E'₂*	1 1	ε^2* ε^2*	ε^4* ε^4*	ε^5* ε^5*	ε^3* ε^3*	ε^* ε^*	ε^* ε^*	ε^3* ε^3*	ε^5* ε^5*	ε^4* ε^4*	ε^2* ε^2*	1 1	ε^2* ε^2*	ε^4* ε^4*	ε^5* ε^5*	ε^3* ε^3*	ε^* ε^*	ε^* ε^*	ε^3* ε^3*	ε^5* ε^5*	ε^4* ε^4*	ε^2* ε^2*
E'₃*	1 1	ε^3* ε^3*	ε^5* ε^5*	ε^2* ε^2*	ε^* ε^*	ε^4* ε^4*	ε^4* ε^4*	ε^* ε^*	ε^2* ε^2*	ε^5* ε^5*	ε^3* ε^3*	1 1	ε^3* ε^3*	ε^5* ε^5*	ε^2* ε^2*	ε^* ε^*	ε^4* ε^4*	ε^4* ε^4*	ε^* ε^*	ε^2* ε^2*	ε^5* ε^5*	ε^3* ε^3*
E'₄*	1 1	ε^4* ε^4*	ε^3* ε^3*	ε^* ε^*	ε^5* ε^5*	ε^2* ε^2*	ε^2* ε^2*	ε^5* ε^5*	ε^* ε^*	ε^3* ε^3*	ε^4* ε^4*	1 1	ε^4* ε^4*	ε^3* ε^3*	ε^* ε^*	ε^5* ε^5*	ε^2* ε^2*	ε^2* ε^2*	ε^5* ε^5*	ε^* ε^*	ε^3* ε^3*	ε^4* ε^4*
E'₅*	1 1	ε^5* ε^5*	ε^* ε^*	ε^4* ε^4*	ε^2* ε^2*	ε^3* ε^3*	ε^3* ε^3*	ε^2* ε^2*	ε^4* ε^4*	ε^* ε^*	ε^5* ε^5*	1 1	ε^5* ε^5*	ε^* ε^*	ε^4* ε^4*	ε^2* ε^2*	ε^3* ε^3*	ε^3* ε^3*	ε^2* ε^2*	ε^4* ε^4*	ε^* ε^*	ε^5* ε^5*
A''	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1
E''₁*	1 1	ε^* ε^*	ε^2* ε^2*	ε^3* ε^3*	ε^4* ε^4*	ε^5* ε^5*	ε^5* ε^5*	ε^4* ε^4*	ε^3* ε^3*	ε^2* ε^2*	ε^* ε^*	-1 -1	-ε^* -ε^*	-ε^2* -ε^2*	-ε^3* -ε^3*	-ε^4* -ε^4*	-ε^5* -ε^5*	-ε^5* -ε^5*	-ε^4* -ε^4*	-ε^3* -ε^3*	-ε^2* -ε^2*	-ε^* -ε^*
E''₂*	1 1	ε^2* ε^2*	ε^4* ε^4*	ε^5* ε^5*	ε^3* ε^3*	ε^* ε^*	ε^* ε^*	ε^3* ε^3*	ε^5* ε^5*	ε^4* ε^4*	ε^2* ε^2*	-1 -1	-ε^2* -ε^2*	-ε^4* -ε^4*	-ε^5* -ε^5*	-ε^3* -ε^3*	-ε^* -ε^*	-ε^* -ε^*	-ε^3* -ε^3*	-ε^5* -ε^5*	-ε^4* -ε^4*	-ε^2* -ε^2*
E''₃*	1 1	ε^3* ε^3*	ε^5* ε^5*	ε^2* ε^2*	ε^* ε^*	ε^4* ε^4*	ε^4* ε^4*	ε^* ε^*	ε^2* ε^2*	ε^5* ε^5*	ε^3* ε^3*	-1 -1	-ε^3* -ε^3*	-ε^5* -ε^5*	-ε^2* -ε^2*	-ε^* -ε^*	-ε^4* -ε^4*	-ε^4* -ε^4*	-ε^* -ε^*	-ε^2* -ε^2*	-ε^5* -ε^5*	-ε^3* -ε^3*
E''₄*	1 1	ε^4* ε^4*	ε^3* ε^3*	ε^* ε^*	ε^5* ε^5*	ε^2* ε^2*	ε^2* ε^2*	ε^5* ε^5*	ε^* ε^*	ε^3* ε^3*	ε^4* ε^4*	-1 -1	-ε^4* -ε^4*	-ε^3* -ε^3*	-ε^* -ε^*	-ε^5* -ε^5*	-ε^2* -ε^2*	-ε^2* -ε^2*	-ε^5* -ε^5*	-ε^* -ε^*	-ε^3* -ε^3*	-ε^4* -ε^4*
E''₅*	1 1	ε^5* ε^5*	ε^* ε^*	ε^4* ε^4*	ε^2* ε^2*	ε^3* ε^3*	ε^3* ε^3*	ε^2* ε^2*	ε^4* ε^4*	ε^* ε^*	ε^5* ε^5*	-1 -1	-ε^5* -ε^5*	-ε^* -ε^*	-ε^4* -ε^4*	-ε^2* -ε^2*	-ε^3* -ε^3*	-ε^3* -ε^3*	-ε^2* -ε^2*	-ε^4* -ε^4*	-ε^* -ε^*	-ε^5* -ε^5*

Additional information

Number of symmetry elements	h = 22
Number of classes, irreps	n = 22
Number of real-valued irreducible representations	n = 12
Abelian group	yes
Optical Isomerism (Chirality)	no
Polar	no
Parity	no

Reduce representation to irreducible representations

E	C₁₁	(C₁₁)²	(C₁₁)³	(C₁₁)⁴	(C₁₁)⁵	(C₁₁)⁶	(C₁₁)⁷	(C₁₁)⁸	(C₁₁)⁹	(C₁₁)¹⁰	σ_h	S₁₁	(S₁₁)¹³	(S₁₁)³	(S₁₁)¹⁵	(S₁₁)⁵	(S₁₁)¹⁷	(S₁₁)⁷	(S₁₁)¹⁹	(S₁₁)⁹	(S₁₁)²¹

Genrate representation from irreducible representations

A'	E'₁*	E'₂*	E'₃*	E'₄*	E'₅*	A''	E''₁*	E''₂*	E''₃*	E''₄*	E''₅*

Direct products of irreducible representations

**Binary products**
⊙	A'	E'₁*	E'₂*	E'₃*	E'₄*	E'₅*	A''	E''₁*	E''₂*	E''₃*	E''₄*	E''₅*
A'	A'
E'₁*	E'₁	2A'⊕E'₂
E'₂*	E'₂	E'₁⊕E'₃	2A'⊕E'₄
E'₃*	E'₃	E'₂⊕E'₄	E'₁⊕E'₅	2A'⊕E'₅
E'₄*	E'₄	E'₃⊕E'₅	E'₂⊕E'₅	E'₁⊕E'₄	2A'⊕E'₃
E'₅*	E'₅	E'₄⊕E'₅	E'₃⊕E'₄	E'₂⊕E'₃	E'₁⊕E'₂	2A'⊕E'₁
A''	A''	E''₁	E''₂	E''₃	E''₄	E''₅	A'
E''₁*	E''₁	2A''⊕E''₂	E''₁⊕E''₃	E''₂⊕E''₄	E''₃⊕E''₅	E''₄⊕E''₅	E'₁	2A'⊕E'₂
E''₂*	E''₂	E''₁⊕E''₃	2A''⊕E''₄	E''₁⊕E''₅	E''₂⊕E''₅	E''₃⊕E''₄	E'₂	E'₁⊕E'₃	2A'⊕E'₄
E''₃*	E''₃	E''₂⊕E''₄	E''₁⊕E''₅	2A''⊕E''₅	E''₁⊕E''₄	E''₂⊕E''₃	E'₃	E'₂⊕E'₄	E'₁⊕E'₅	2A'⊕E'₅
E''₄*	E''₄	E''₃⊕E''₅	E''₂⊕E''₅	E''₁⊕E''₄	2A''⊕E''₃	E''₁⊕E''₂	E'₄	E'₃⊕E'₅	E'₂⊕E'₅	E'₁⊕E'₄	2A'⊕E'₃
E''₅*	E''₅	E''₄⊕E''₅	E''₃⊕E''₄	E''₂⊕E''₃	E''₁⊕E''₂	2A''⊕E''₁	E'₅	E'₄⊕E'₅	E'₃⊕E'₄	E'₂⊕E'₃	E'₁⊕E'₂	2A'⊕E'₁

Ternary Products
Quaternary Products

Symmetric powers [Γⁿ] of degenerate irreducible representations
Vibrational overtones

irrep	[Γ²]	[Γ³]	[Γ⁴]	[Γ⁵]	[Γ⁶]
E'₁*	A'⊕E'₂	E'₁⊕E'₃	A'⊕E'₂⊕E'₄	E'₁⊕E'₃⊕E'₅	A'⊕E'₂⊕E'₄⊕E'₅	More
E'₂*	A'⊕E'₄	E'₂⊕E'₅	A'⊕E'₃⊕E'₄	E'₁⊕E'₂⊕E'₅	A'⊕E'₁⊕E'₃⊕E'₄	More
E'₃*	A'⊕E'₅	E'₂⊕E'₃	A'⊕E'₁⊕E'₅	E'₂⊕E'₃⊕E'₄	A'⊕E'₁⊕E'₄⊕E'₅	More
E'₄*	A'⊕E'₃	E'₁⊕E'₄	A'⊕E'₃⊕E'₅	E'₁⊕E'₂⊕E'₄	A'⊕E'₂⊕E'₃⊕E'₅	More
E'₅*	A'⊕E'₁	E'₄⊕E'₅	A'⊕E'₁⊕E'₂	E'₃⊕E'₄⊕E'₅	A'⊕E'₁⊕E'₂⊕E'₃	More
E''₁*	A'⊕E'₂	E''₁⊕E''₃	A'⊕E'₂⊕E'₄	E''₁⊕E''₃⊕E''₅	A'⊕E'₂⊕E'₄⊕E'₅	More
E''₂*	A'⊕E'₄	E''₂⊕E''₅	A'⊕E'₃⊕E'₄	E''₁⊕E''₂⊕E''₅	A'⊕E'₁⊕E'₃⊕E'₄	More
E''₃*	A'⊕E'₅	E''₂⊕E''₃	A'⊕E'₁⊕E'₅	E''₂⊕E''₃⊕E''₄	A'⊕E'₁⊕E'₄⊕E'₅	More
E''₄*	A'⊕E'₃	E''₁⊕E''₄	A'⊕E'₃⊕E'₅	E''₁⊕E''₂⊕E''₄	A'⊕E'₂⊕E'₃⊕E'₅	More
E''₅*	A'⊕E'₁	E''₄⊕E''₅	A'⊕E'₁⊕E'₂	E''₃⊕E''₄⊕E''₅	A'⊕E'₁⊕E'₂⊕E'₃	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'	1	A'
p (l=1)	3	Dipole	E'₁⊕A''	3	E'₁⊕A''
d (l=2)	5	Quadrupole	A'⊕E'₂⊕E''₁	6	2A'⊕E'₂⊕E''₁
f (l=3)	7	Octupole	E'₁⊕E'₃⊕A''⊕E''₂	10	2E'₁⊕E'₃⊕2A''⊕E''₂
g (l=4)	9	Hexadecapole	A'⊕E'₂⊕E'₄⊕E''₁⊕E''₃	15	3A'⊕2E'₂⊕E'₄⊕2E''₁⊕E''₃
h (l=5)	11	Dotricontapole	E'₁⊕E'₃⊕E'₅⊕A''⊕E''₂⊕E''₄	21	3E'₁⊕2E'₃⊕E'₅⊕3A''⊕2E''₂⊕E''₄
i (l=6)	13	Tetrahexacontapole	A'⊕E'₂⊕E'₄⊕E'₅⊕E''₁⊕E''₃⊕E''₅	28	4A'⊕3E'₂⊕2E'₄⊕E'₅⊕3E''₁⊕2E''₃⊕E''₅
j (l=7)	15	Octacosahectapole	E'₁⊕E'₃⊕E'₄⊕E'₅⊕A''⊕E''₂⊕E''₄⊕E''₅	36	4E'₁⊕3E'₃⊕E'₄⊕2E'₅⊕4A''⊕3E''₂⊕2E''₄⊕E''₅
k (l=8)	17	256-pole	A'⊕E'₂⊕E'₃⊕E'₄⊕E'₅⊕E''₁⊕E''₃⊕E''₄⊕E''₅	45	5A'⊕4E'₂⊕E'₃⊕3E'₄⊕2E'₅⊕4E''₁⊕3E''₃⊕E''₄⊕2E''₅
l (l=9)	19	512-pole	E'₁⊕E'₂⊕E'₃⊕E'₄⊕E'₅⊕A''⊕E''₂⊕E''₃⊕E''₄⊕E''₅	55	5E'₁⊕E'₂⊕4E'₃⊕2E'₄⊕3E'₅⊕5A''⊕4E''₂⊕E''₃⊕3E''₄⊕2E''₅
m (l=10)	21	1024-pole	A'⊕E'₁⊕E'₂⊕E'₃⊕E'₄⊕E'₅⊕E''₁⊕E''₂⊕E''₃⊕E''₄⊕E''₅	66	6A'⊕E'₁⊕5E'₂⊕2E'₃⊕4E'₄⊕3E'₅⊕5E''₁⊕E''₂⊕4E''₃⊕2E''₄⊕3E''₅
n (l=11)	23	2048-pole	2A'⊕E'₁⊕E'₂⊕E'₃⊕E'₄⊕E'₅⊕A''⊕E''₁⊕E''₂⊕E''₃⊕E''₄⊕E''₅	78	2A'⊕6E'₁⊕2E'₂⊕5E'₃⊕3E'₄⊕4E'₅⊕6A''⊕E''₁⊕5E''₂⊕2E''₃⊕4E''₄⊕3E''₅
o (l=12)	25	4096-pole	A'⊕2E'₁⊕E'₂⊕E'₃⊕E'₄⊕E'₅⊕2A''⊕E''₁⊕E''₂⊕E''₃⊕E''₄⊕E''₅	91	7A'⊕3E'₁⊕6E'₂⊕3E'₃⊕5E'₄⊕4E'₅⊕2A''⊕6E''₁⊕2E''₂⊕5E''₃⊕3E''₄⊕4E''₅
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 C_11h**
L	2L+1	Term Splitting
S (L=0)	1	A'
P (L=1)	3	A'⊕E''₁
D (L=2)	5	A'⊕E'₂⊕E''₁
F (L=3)	7	A'⊕E'₂⊕E''₁⊕E''₃
G (L=4)	9	A'⊕E'₂⊕E'₄⊕E''₁⊕E''₃
H (L=5)	11	A'⊕E'₂⊕E'₄⊕E''₁⊕E''₃⊕E''₅
I (L=6)	13	A'⊕E'₂⊕E'₄⊕E'₅⊕E''₁⊕E''₃⊕E''₅

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