Point Group C9h



C9h E C9 (C9)2 C3 (C9)4 (C9)5 (C3)2 (C9)7 (C9)8 σh S9 (S9)11 S3 (S9)13 (S9)5 (S3)5 (S9)7 (S9)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/9) 2cos(4π/9) -1 2cos(8π/9) 2cos(8π/9) -1 2cos(4π/9) 2cos(2π/9) 2 2cos(2π/9) 2cos(4π/9) -1 2cos(8π/9) 2cos(8π/9) -1 2cos(4π/9) 2cos(2π/9)
E'2* 2 2cos(4π/9) 2cos(8π/9) -1 2cos(2π/9) 2cos(2π/9) -1 2cos(8π/9) 2cos(4π/9) 2 2cos(4π/9) 2cos(8π/9) -1 2cos(2π/9) 2cos(2π/9) -1 2cos(8π/9) 2cos(4π/9)
E'3* 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1
E'4* 2 2cos(8π/9) 2cos(2π/9) -1 2cos(4π/9) 2cos(4π/9) -1 2cos(2π/9) 2cos(8π/9) 2 2cos(8π/9) 2cos(2π/9) -1 2cos(4π/9) 2cos(4π/9) -1 2cos(2π/9) 2cos(8π/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/9) 2cos(4π/9) -1 2cos(8π/9) 2cos(8π/9) -1 2cos(4π/9) 2cos(2π/9) -2 -2cos(2π/9) -2cos(4π/9) 1 -2cos(8π/9) -2cos(8π/9) 1 -2cos(4π/9) -2cos(2π/9)
E''2* 2 2cos(4π/9) 2cos(8π/9) -1 2cos(2π/9) 2cos(2π/9) -1 2cos(8π/9) 2cos(4π/9) -2 -2cos(4π/9) -2cos(8π/9) 1 -2cos(2π/9) -2cos(2π/9) 1 -2cos(8π/9) -2cos(4π/9)
E''3* 2 -1 -1 2 -1 -1 2 -1 -1 -2 1 1 -2 1 1 -2 1 1
E''4* 2 2cos(8π/9) 2cos(2π/9) -1 2cos(4π/9) 2cos(4π/9) -1 2cos(2π/9) 2cos(8π/9) -2 -2cos(8π/9) -2cos(2π/9) 1 -2cos(4π/9) -2cos(4π/9) 1 -2cos(2π/9) -2cos(8π/9)


Additional information

Number of symmetry elements h = 18
Number of classes, irreps n = 18
Number of real-valued irreducible representations n = 10
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E C9 (C9)2 C3 (C9)4 (C9)5 (C3)2 (C9)7 (C9)8 σh S9 (S9)11 S3 (S9)13 (S9)5 (S3)5 (S9)7 (S9)17



Genrate representation from irreducible representations


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




Direct products of irreducible representations


Binary products
A' E'1* E'2* E'3* E'4* A'' E''1* E''2* E''3* E''4*
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'42A'⊕E'3
E'4* E'4E'3⊕E'4E'2⊕E'3E'1⊕E'22A'⊕E'1
A'' A''E''1E''2E''3E''4A'
E''1* E''12A''⊕E''2E''1⊕E''3E''2⊕E''4E''3⊕E''4E'12A'⊕E'2
E''2* E''2E''1⊕E''32A''⊕E''4E''1⊕E''4E''2⊕E''3E'2E'1⊕E'32A'⊕E'4
E''3* E''3E''2⊕E''4E''1⊕E''42A''⊕E''3E''1⊕E''2E'3E'2⊕E'4E'1⊕E'42A'⊕E'3
E''4* E''4E''3⊕E''4E''2⊕E''3E''1⊕E''22A''⊕E''1E'4E'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 2] 3] 4] 5] 6]
E'1* A'⊕E'2E'1⊕E'3A'⊕E'2⊕E'4E'1⊕E'3⊕E'4A'⊕E'2⊕E'3⊕E'4More
E'2* A'⊕E'4E'2⊕E'3A'⊕E'1⊕E'4E'1⊕E'2⊕E'3A'⊕E'1⊕E'3⊕E'4More
E'3* A'⊕E'32A'⊕E'3A'⊕2E'32A'⊕2E'33A'⊕2E'3More
E'4* A'⊕E'1E'3⊕E'4A'⊕E'1⊕E'2E'2⊕E'3⊕E'4A'⊕E'1⊕E'2⊕E'3More
E''1* A'⊕E'2E''1⊕E''3A'⊕E'2⊕E'4E''1⊕E''3⊕E''4A'⊕E'2⊕E'3⊕E'4More
E''2* A'⊕E'4E''2⊕E''3A'⊕E'1⊕E'4E''1⊕E''2⊕E''3A'⊕E'1⊕E'3⊕E'4More
E''3* A'⊕E'32A''⊕E''3A'⊕2E'32A''⊕2E''33A'⊕2E'3More
E''4* A'⊕E'1E''3⊕E''4A'⊕E'1⊕E'2E''2⊕E''3⊕E''4A'⊕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 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'4⊕A''⊕E''2⊕E''4 21 3E'1⊕2E'3⊕E'4⊕3A''⊕2E''2⊕E''4
i (l=6) 13 Tetrahexacontapole A'⊕E'2⊕E'3⊕E'4⊕E''1⊕E''3⊕E''4 28 4A'⊕3E'2⊕E'3⊕2E'4⊕3E''1⊕2E''3⊕E''4
j (l=7) 15 Octacosahectapole E'1⊕E'2⊕E'3⊕E'4⊕A''⊕E''2⊕E''3⊕E''4 36 4E'1⊕E'2⊕3E'3⊕2E'4⊕4A''⊕3E''2⊕E''3⊕2E''4
k (l=8) 17 256-pole A'⊕E'1⊕E'2⊕E'3⊕E'4⊕E''1⊕E''2⊕E''3⊕E''4 45 5A'⊕E'1⊕4E'2⊕2E'3⊕3E'4⊕4E''1⊕E''2⊕3E''3⊕2E''4
l (l=9) 19 512-pole 2A'⊕E'1⊕E'2⊕E'3⊕E'4⊕A''⊕E''1⊕E''2⊕E''3⊕E''4 55 2A'⊕5E'1⊕2E'2⊕4E'3⊕3E'4⊕5A''⊕E''1⊕4E''2⊕2E''3⊕3E''4
m (l=10) 21 1024-pole A'⊕2E'1⊕E'2⊕E'3⊕E'4⊕2A''⊕E''1⊕E''2⊕E''3⊕E''4 66 6A'⊕3E'1⊕5E'2⊕3E'3⊕4E'4⊕2A''⊕5E''1⊕2E''2⊕4E''3⊕3E''4
n (l=11) 23 2048-pole 2A'⊕E'1⊕2E'2⊕E'3⊕E'4⊕A''⊕2E''1⊕E''2⊕E''3⊕E''4 78 4A'⊕6E'1⊕4E'2⊕5E'3⊕4E'4⊕6A''⊕3E''1⊕5E''2⊕3E''3⊕4E''4
o (l=12) 25 4096-pole A'⊕2E'1⊕E'2⊕2E'3⊕E'4⊕2A''⊕E''1⊕2E''2⊕E''3⊕E''4 91 7A'⊕5E'1⊕6E'2⊕5E'3⊕5E'4⊕4A''⊕6E''1⊕4E''2⊕5E''3⊕4E''4
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 C9h
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''4
I (L=6) 13 A'⊕E'2⊕E'3⊕E'4⊕E''1⊕E''3⊕E''4


Last update August, 12th 2020 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement