Point Group C4h



C4h E C4 C2 (C4)3 i (S4)3 σh S4
Ag 1 1 1 1 1 1 1 1
Bg 1 -1 1 -1 1 -1 1 -1
Eg* 1
1		 i
-i		 -1
-1		 -i
i		 1
1		 i
-i		 -1
-1		 -i
i
Au 1 1 1 1 -1 -1 -1 -1
Bu 1 -1 1 -1 -1 1 -1 1
Eu* 1
1		 i
-i		 -1
-1		 -i
i		 -1
-1		 -i
i		 1
1		 i
-i
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Additional information

Number of symmetry elements h = 8
Number of classes, irreps n = 8
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 C4 C2 (C4)3 i (S4)3 σh S4



Genrate representation from irreducible representations


Ag Bg Eg* Au Bu Eu*



Direct products of irreducible representations


Binary products
⊙ Ag Bg Eg* Au Bu Eu*
Ag Ag
Bg BgAg
Eg* EgEg2Ag⊕2Bg
Au AuBuEuAg
Bu BuAuEuBgAg
Eu* EuEu2Au⊕2BuEgEg2Ag⊕2Bg

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
Eg* Ag⊕2Bg2Eg3Ag⊕2Bg3Eg3Ag⊕4BgMore
Eu* Ag⊕2Bg2Eu3Ag⊕2Bg3Eu3Ag⊕4BgMore


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 Ag 1 Ag
p (l=1) 3 Dipole Au⊕Eu 3 Au⊕Eu
d (l=2) 5 Quadrupole Ag⊕2Bg⊕Eg 6 2Ag⊕2Bg⊕Eg
f (l=3) 7 Octupole Au⊕2Bu⊕2Eu 10 2Au⊕2Bu⊕3Eu
g (l=4) 9 Hexadecapole 3Ag⊕2Bg⊕2Eg 15 5Ag⊕4Bg⊕3Eg
h (l=5) 11 Dotricontapole 3Au⊕2Bu⊕3Eu 21 5Au⊕4Bu⊕6Eu
i (l=6) 13 Tetrahexacontapole 3Ag⊕4Bg⊕3Eg 28 8Ag⊕8Bg⊕6Eg
j (l=7) 15 Octacosahectapole 3Au⊕4Bu⊕4Eu 36 8Au⊕8Bu⊕10Eu
k (l=8) 17 256-pole 5Ag⊕4Bg⊕4Eg 45 13Ag⊕12Bg⊕10Eg
l (l=9) 19 512-pole 5Au⊕4Bu⊕5Eu 55 13Au⊕12Bu⊕15Eu
m (l=10) 21 1024-pole 5Ag⊕6Bg⊕5Eg 66 18Ag⊕18Bg⊕15Eg
n (l=11) 23 2048-pole 5Au⊕6Bu⊕6Eu 78 18Au⊕18Bu⊕21Eu
o (l=12) 25 4096-pole 7Ag⊕6Bg⊕6Eg 91 25Ag⊕24Bg⊕21Eg
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 C4h
L 2L+1 Term Splitting
S (L=0) 1 Ag
P (L=1) 3 Ag⊕Eg
D (L=2) 5 Ag⊕2Bg⊕Eg
F (L=3) 7 Ag⊕2Bg⊕2Eg
G (L=4) 9 3Ag⊕2Bg⊕2Eg
H (L=5) 11 3Ag⊕2Bg⊕3Eg
I (L=6) 13 3Ag⊕4Bg⊕3Eg

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