Point Group C17h



C17h E 2C17 2(C17)2 2(C17)3 2(C17)4 2(C17)5 2(C17)6 2(C17)7 2(C17)8 σh 2S17 2(S17)15 2(S17)3 2(S17)13 2(S17)5 2(S17)11 2(S17)7 2(S17)9
A' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
E'1* 2 2cos(2π/17) 2cos(4π/17) 2cos(6π/17) 2cos(8π/17) 2cos(10π/17) 2cos(12π/17) 2cos(14π/17) 2cos(16π/17) 2 2cos(2π/17) 2cos(4π/17) 2cos(6π/17) 2cos(8π/17) 2cos(10π/17) 2cos(12π/17) 2cos(14π/17) 2cos(16π/17)
E'2* 2 2cos(4π/17) 2cos(8π/17) 2cos(12π/17) 2cos(16π/17) 2cos(14π/17) 2cos(10π/17) 2cos(6π/17) 2cos(2π/17) 2 2cos(4π/17) 2cos(8π/17) 2cos(12π/17) 2cos(16π/17) 2cos(14π/17) 2cos(10π/17) 2cos(6π/17) 2cos(2π/17)
E'3* 2 2cos(6π/17) 2cos(12π/17) 2cos(16π/17) 2cos(10π/17) 2cos(4π/17) 2cos(2π/17) 2cos(8π/17) 2cos(14π/17) 2 2cos(6π/17) 2cos(12π/17) 2cos(16π/17) 2cos(10π/17) 2cos(4π/17) 2cos(2π/17) 2cos(8π/17) 2cos(14π/17)
E'4* 2 2cos(8π/17) 2cos(16π/17) 2cos(10π/17) 2cos(2π/17) 2cos(6π/17) 2cos(14π/17) 2cos(12π/17) 2cos(4π/17) 2 2cos(8π/17) 2cos(16π/17) 2cos(10π/17) 2cos(2π/17) 2cos(6π/17) 2cos(14π/17) 2cos(12π/17) 2cos(4π/17)
E'5* 2 2cos(10π/17) 2cos(14π/17) 2cos(4π/17) 2cos(6π/17) 2cos(16π/17) 2cos(8π/17) 2cos(2π/17) 2cos(12π/17) 2 2cos(10π/17) 2cos(14π/17) 2cos(4π/17) 2cos(6π/17) 2cos(16π/17) 2cos(8π/17) 2cos(2π/17) 2cos(12π/17)
E'6* 2 2cos(12π/17) 2cos(10π/17) 2cos(2π/17) 2cos(14π/17) 2cos(8π/17) 2cos(4π/17) 2cos(16π/17) 2cos(6π/17) 2 2cos(12π/17) 2cos(10π/17) 2cos(2π/17) 2cos(14π/17) 2cos(8π/17) 2cos(4π/17) 2cos(16π/17) 2cos(6π/17)
E'7* 2 2cos(14π/17) 2cos(6π/17) 2cos(8π/17) 2cos(12π/17) 2cos(2π/17) 2cos(16π/17) 2cos(4π/17) 2cos(10π/17) 2 2cos(14π/17) 2cos(6π/17) 2cos(8π/17) 2cos(12π/17) 2cos(2π/17) 2cos(16π/17) 2cos(4π/17) 2cos(10π/17)
E'8* 2 2cos(16π/17) 2cos(2π/17) 2cos(14π/17) 2cos(4π/17) 2cos(12π/17) 2cos(6π/17) 2cos(10π/17) 2cos(8π/17) 2 2cos(16π/17) 2cos(2π/17) 2cos(14π/17) 2cos(4π/17) 2cos(12π/17) 2cos(6π/17) 2cos(10π/17) 2cos(8π/17)
A'' 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1
E''1* 2 2cos(2π/17) 2cos(4π/17) 2cos(6π/17) 2cos(8π/17) 2cos(10π/17) 2cos(12π/17) 2cos(14π/17) 2cos(16π/17) -2 -2cos(2π/17) -2cos(4π/17) -2cos(6π/17) -2cos(8π/17) -2cos(10π/17) -2cos(12π/17) -2cos(14π/17) -2cos(16π/17)
E''2* 2 2cos(4π/17) 2cos(8π/17) 2cos(12π/17) 2cos(16π/17) 2cos(14π/17) 2cos(10π/17) 2cos(6π/17) 2cos(2π/17) -2 -2cos(4π/17) -2cos(8π/17) -2cos(12π/17) -2cos(16π/17) -2cos(14π/17) -2cos(10π/17) -2cos(6π/17) -2cos(2π/17)
E''3* 2 2cos(6π/17) 2cos(12π/17) 2cos(16π/17) 2cos(10π/17) 2cos(4π/17) 2cos(2π/17) 2cos(8π/17) 2cos(14π/17) -2 -2cos(6π/17) -2cos(12π/17) -2cos(16π/17) -2cos(10π/17) -2cos(4π/17) -2cos(2π/17) -2cos(8π/17) -2cos(14π/17)
E''4* 2 2cos(8π/17) 2cos(16π/17) 2cos(10π/17) 2cos(2π/17) 2cos(6π/17) 2cos(14π/17) 2cos(12π/17) 2cos(4π/17) -2 -2cos(8π/17) -2cos(16π/17) -2cos(10π/17) -2cos(2π/17) -2cos(6π/17) -2cos(14π/17) -2cos(12π/17) -2cos(4π/17)
E''5* 2 2cos(10π/17) 2cos(14π/17) 2cos(4π/17) 2cos(6π/17) 2cos(16π/17) 2cos(8π/17) 2cos(2π/17) 2cos(12π/17) -2 -2cos(10π/17) -2cos(14π/17) -2cos(4π/17) -2cos(6π/17) -2cos(16π/17) -2cos(8π/17) -2cos(2π/17) -2cos(12π/17)
E''6* 2 2cos(12π/17) 2cos(10π/17) 2cos(2π/17) 2cos(14π/17) 2cos(8π/17) 2cos(4π/17) 2cos(16π/17) 2cos(6π/17) -2 -2cos(12π/17) -2cos(10π/17) -2cos(2π/17) -2cos(14π/17) -2cos(8π/17) -2cos(4π/17) -2cos(16π/17) -2cos(6π/17)
E''7* 2 2cos(14π/17) 2cos(6π/17) 2cos(8π/17) 2cos(12π/17) 2cos(2π/17) 2cos(16π/17) 2cos(4π/17) 2cos(10π/17) -2 -2cos(14π/17) -2cos(6π/17) -2cos(8π/17) -2cos(12π/17) -2cos(2π/17) -2cos(16π/17) -2cos(4π/17) -2cos(10π/17)
E''8* 2 2cos(16π/17) 2cos(2π/17) 2cos(14π/17) 2cos(4π/17) 2cos(12π/17) 2cos(6π/17) 2cos(10π/17) 2cos(8π/17) -2 -2cos(16π/17) -2cos(2π/17) -2cos(14π/17) -2cos(4π/17) -2cos(12π/17) -2cos(6π/17) -2cos(10π/17) -2cos(8π/17)
Math View
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Phys View B
Chem View A
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Chem View B


Additional information

Number of symmetry elements h = 34
Number of classes, irreps n = 34
Number of real-valued irreducible representations n = 18
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E 2C17 2(C17)2 2(C17)3 2(C17)4 2(C17)5 2(C17)6 2(C17)7 2(C17)8 σh 2S17 2(S17)15 2(S17)3 2(S17)13 2(S17)5 2(S17)11 2(S17)7 2(S17)9



Genrate representation from irreducible representations


A' E'1* E'2* E'3* E'4* E'5* E'6* E'7* E'8* A'' E''1* E''2* E''3* E''4* E''5* E''6* E''7* E''8*



Direct products of irreducible representations


Binary products
⊙ A' E'1* E'2* E'3* E'4* E'5* E'6* E'7* E'8* A'' E''1* E''2* E''3* E''4* E''5* E''6* E''7* E''8*
A' A'
E'1* E'12A'⊕E'2
E'2* E'2E'1⊕E'32A'⊕E'4
E'3* E'3E'2⊕E'4E'1⊕E'52A'⊕E'6
E'4* E'4E'3⊕E'5E'2⊕E'6E'1⊕E'72A'⊕E'8
E'5* E'5E'4⊕E'6E'3⊕E'7E'2⊕E'8E'1⊕E'82A'⊕E'7
E'6* E'6E'5⊕E'7E'4⊕E'8E'3⊕E'8E'2⊕E'7E'1⊕E'62A'⊕E'5
E'7* E'7E'6⊕E'8E'5⊕E'8E'4⊕E'7E'3⊕E'6E'2⊕E'5E'1⊕E'42A'⊕E'3
E'8* E'8E'7⊕E'8E'6⊕E'7E'5⊕E'6E'4⊕E'5E'3⊕E'4E'2⊕E'3E'1⊕E'22A'⊕E'1
A'' A''E''1E''2E''3E''4E''5E''6E''7E''8A'
E''1* E''12A''⊕E''2E''1⊕E''3E''2⊕E''4E''3⊕E''5E''4⊕E''6E''5⊕E''7E''6⊕E''8E''7⊕E''8E'12A'⊕E'2
E''2* E''2E''1⊕E''32A''⊕E''4E''1⊕E''5E''2⊕E''6E''3⊕E''7E''4⊕E''8E''5⊕E''8E''6⊕E''7E'2E'1⊕E'32A'⊕E'4
E''3* E''3E''2⊕E''4E''1⊕E''52A''⊕E''6E''1⊕E''7E''2⊕E''8E''3⊕E''8E''4⊕E''7E''5⊕E''6E'3E'2⊕E'4E'1⊕E'52A'⊕E'6
E''4* E''4E''3⊕E''5E''2⊕E''6E''1⊕E''72A''⊕E''8E''1⊕E''8E''2⊕E''7E''3⊕E''6E''4⊕E''5E'4E'3⊕E'5E'2⊕E'6E'1⊕E'72A'⊕E'8
E''5* E''5E''4⊕E''6E''3⊕E''7E''2⊕E''8E''1⊕E''82A''⊕E''7E''1⊕E''6E''2⊕E''5E''3⊕E''4E'5E'4⊕E'6E'3⊕E'7E'2⊕E'8E'1⊕E'82A'⊕E'7
E''6* E''6E''5⊕E''7E''4⊕E''8E''3⊕E''8E''2⊕E''7E''1⊕E''62A''⊕E''5E''1⊕E''4E''2⊕E''3E'6E'5⊕E'7E'4⊕E'8E'3⊕E'8E'2⊕E'7E'1⊕E'62A'⊕E'5
E''7* E''7E''6⊕E''8E''5⊕E''8E''4⊕E''7E''3⊕E''6E''2⊕E''5E''1⊕E''42A''⊕E''3E''1⊕E''2E'7E'6⊕E'8E'5⊕E'8E'4⊕E'7E'3⊕E'6E'2⊕E'5E'1⊕E'42A'⊕E'3
E''8* E''8E''7⊕E''8E''6⊕E''7E''5⊕E''6E''4⊕E''5E''3⊕E''4E''2⊕E''3E''1⊕E''22A''⊕E''1E'8E'7⊕E'8E'6⊕E'7E'5⊕E'6E'4⊕E'5E'3⊕E'4E'2⊕E'3E'1⊕E'22A'⊕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'2E'1⊕E'3A'⊕E'2⊕E'4E'1⊕E'3⊕E'5A'⊕E'2⊕E'4⊕E'6More
E'2* A'⊕E'4E'2⊕E'6A'⊕E'4⊕E'8E'2⊕E'6⊕E'7A'⊕E'4⊕E'5⊕E'8More
E'3* A'⊕E'6E'3⊕E'8A'⊕E'5⊕E'6E'2⊕E'3⊕E'8A'⊕E'1⊕E'5⊕E'6More
E'4* A'⊕E'8E'4⊕E'5A'⊕E'1⊕E'8E'3⊕E'4⊕E'5A'⊕E'1⊕E'7⊕E'8More
E'5* A'⊕E'7E'2⊕E'5A'⊕E'3⊕E'7E'2⊕E'5⊕E'8A'⊕E'3⊕E'4⊕E'7More
E'6* A'⊕E'5E'1⊕E'6A'⊕E'5⊕E'7E'1⊕E'4⊕E'6A'⊕E'2⊕E'5⊕E'7More
E'7* A'⊕E'3E'4⊕E'7A'⊕E'3⊕E'6E'1⊕E'4⊕E'7A'⊕E'3⊕E'6⊕E'8More
E'8* A'⊕E'1E'7⊕E'8A'⊕E'1⊕E'2E'6⊕E'7⊕E'8A'⊕E'1⊕E'2⊕E'3More
E''1* A'⊕E'2E''1⊕E''3A'⊕E'2⊕E'4E''1⊕E''3⊕E''5A'⊕E'2⊕E'4⊕E'6More
E''2* A'⊕E'4E''2⊕E''6A'⊕E'4⊕E'8E''2⊕E''6⊕E''7A'⊕E'4⊕E'5⊕E'8More
E''3* A'⊕E'6E''3⊕E''8A'⊕E'5⊕E'6E''2⊕E''3⊕E''8A'⊕E'1⊕E'5⊕E'6More
E''4* A'⊕E'8E''4⊕E''5A'⊕E'1⊕E'8E''3⊕E''4⊕E''5A'⊕E'1⊕E'7⊕E'8More
E''5* A'⊕E'7E''2⊕E''5A'⊕E'3⊕E'7E''2⊕E''5⊕E''8A'⊕E'3⊕E'4⊕E'7More
E''6* A'⊕E'5E''1⊕E''6A'⊕E'5⊕E'7E''1⊕E''4⊕E''6A'⊕E'2⊕E'5⊕E'7More
E''7* A'⊕E'3E''4⊕E''7A'⊕E'3⊕E'6E''1⊕E''4⊕E''7A'⊕E'3⊕E'6⊕E'8More
E''8* A'⊕E'1E''7⊕E''8A'⊕E'1⊕E'2E''6⊕E''7⊕E''8A'⊕E'1⊕E'2⊕E'3More


Spherical harmonics and Multipoles
Symmetric Powers of Γxyz


Spherical Harmonics Yl / Multipole Symmetric Power [Γl(xyz)]
l 2l+1 Multipole Symmetry Rank [Γl(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'6⊕E''1⊕E''3⊕E''5 28 4A'⊕3E'2⊕2E'4⊕E'6⊕3E''1⊕2E''3⊕E''5
j (l=7) 15 Octacosahectapole E'1⊕E'3⊕E'5⊕E'7⊕A''⊕E''2⊕E''4⊕E''6 36 4E'1⊕3E'3⊕2E'5⊕E'7⊕4A''⊕3E''2⊕2E''4⊕E''6
k (l=8) 17 256-pole A'⊕E'2⊕E'4⊕E'6⊕E'8⊕E''1⊕E''3⊕E''5⊕E''7 45 5A'⊕4E'2⊕3E'4⊕2E'6⊕E'8⊕4E''1⊕3E''3⊕2E''5⊕E''7
l (l=9) 19 512-pole E'1⊕E'3⊕E'5⊕E'7⊕E'8⊕A''⊕E''2⊕E''4⊕E''6⊕E''8 55 5E'1⊕4E'3⊕3E'5⊕2E'7⊕E'8⊕5A''⊕4E''2⊕3E''4⊕2E''6⊕E''8
m (l=10) 21 1024-pole A'⊕E'2⊕E'4⊕E'6⊕E'7⊕E'8⊕E''1⊕E''3⊕E''5⊕E''7⊕E''8 66 6A'⊕5E'2⊕4E'4⊕3E'6⊕E'7⊕2E'8⊕5E''1⊕4E''3⊕3E''5⊕2E''7⊕E''8
n (l=11) 23 2048-pole E'1⊕E'3⊕E'5⊕E'6⊕E'7⊕E'8⊕A''⊕E''2⊕E''4⊕E''6⊕E''7⊕E''8 78 6E'1⊕5E'3⊕4E'5⊕E'6⊕3E'7⊕2E'8⊕6A''⊕5E''2⊕4E''4⊕3E''6⊕E''7⊕2E''8
o (l=12) 25 4096-pole A'⊕E'2⊕E'4⊕E'5⊕E'6⊕E'7⊕E'8⊕E''1⊕E''3⊕E''5⊕E''6⊕E''7⊕E''8 91 7A'⊕6E'2⊕5E'4⊕E'5⊕4E'6⊕2E'7⊕3E'8⊕6E''1⊕5E''3⊕4E''5⊕E''6⊕3E''7⊕2E''8
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 C17h
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'6⊕E''1⊕E''3⊕E''5

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