Point Group C6h



C6h E C6 C3 C2 (C3)2 (C6)5 i (S3)5 (S6)5 σh S6 S3
Ag 1 1 1 1 1 1 1 1 1 1 1 1
Bg 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
E1g* 2 1 -1 -2 -1 1 2 1 -1 -2 -1 1
E2g* 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1
Au 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1
Bu 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1
E1u* 2 1 -1 -2 -1 1 -2 -1 1 2 1 -1
E2u* 2 -1 -1 2 -1 -1 -2 1 1 -2 1 1


Additional information

Number of symmetry elements h = 12
Number of classes, irreps n = 12
Number of real-valued irreducible representations n = 8
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E C6 C3 C2 (C3)2 (C6)5 i (S3)5 (S6)5 σh S6 S3



Genrate representation from irreducible representations


Ag Bg E1g* E2g* Au Bu E1u* E2u*




Direct products of irreducible representations


Binary products
Ag Bg E1g* E2g* Au Bu E1u* E2u*
Ag Ag
Bg BgAg
E1g* E1gE2g2Ag⊕E2g
E2g* E2gE1g2Bg⊕E1g2Ag⊕E2g
Au AuBuE1uE2uAg
Bu BuAuE2uE1uBgAg
E1u* E1uE2u2Au⊕E2u2Bu⊕E1uE1gE2g2Ag⊕E2g
E2u* E2uE1u2Bu⊕E1u2Au⊕E2uE2gE1g2Bg⊕E1g2Ag⊕E2g

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1g* Ag⊕E2g2Bg⊕E1gAg⊕2E2g2Bg⊕2E1g3Ag⊕2E2gMore
E2g* Ag⊕E2g2Ag⊕E2gAg⊕2E2g2Ag⊕2E2g3Ag⊕2E2gMore
E1u* Ag⊕E2g2Bu⊕E1uAg⊕2E2g2Bu⊕2E1u3Ag⊕2E2gMore
E2u* Ag⊕E2g2Au⊕E2uAg⊕2E2g2Au⊕2E2u3Ag⊕2E2gMore



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 Ag 1 Ag
p (l=1) 3 Dipole Au⊕E1u 3 Au⊕E1u
d (l=2) 5 Quadrupole Ag⊕E1g⊕E2g 6 2Ag⊕E1g⊕E2g
f (l=3) 7 Octupole Au⊕2Bu⊕E1u⊕E2u 10 2Au⊕2Bu⊕2E1u⊕E2u
g (l=4) 9 Hexadecapole Ag⊕2Bg⊕E1g⊕2E2g 15 3Ag⊕2Bg⊕2E1g⊕3E2g
h (l=5) 11 Dotricontapole Au⊕2Bu⊕2E1u⊕2E2u 21 3Au⊕4Bu⊕4E1u⊕3E2u
i (l=6) 13 Tetrahexacontapole 3Ag⊕2Bg⊕2E1g⊕2E2g 28 6Ag⊕4Bg⊕4E1g⊕5E2g
j (l=7) 15 Octacosahectapole 3Au⊕2Bu⊕3E1u⊕2E2u 36 6Au⊕6Bu⊕7E1u⊕5E2u
k (l=8) 17 256-pole 3Ag⊕2Bg⊕3E1g⊕3E2g 45 9Ag⊕6Bg⊕7E1g⊕8E2g
l (l=9) 19 512-pole 3Au⊕4Bu⊕3E1u⊕3E2u 55 9Au⊕10Bu⊕10E1u⊕8E2u
m (l=10) 21 1024-pole 3Ag⊕4Bg⊕3E1g⊕4E2g 66 12Ag⊕10Bg⊕10E1g⊕12E2g
n (l=11) 23 2048-pole 3Au⊕4Bu⊕4E1u⊕4E2u 78 12Au⊕14Bu⊕14E1u⊕12E2u
o (l=12) 25 4096-pole 5Ag⊕4Bg⊕4E1g⊕4E2g 91 17Ag⊕14Bg⊕14E1g⊕16E2g
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 C6h
L 2L+1 Term Splitting
S (L=0) 1 Ag
P (L=1) 3 Ag⊕E1g
D (L=2) 5 Ag⊕E1g⊕E2g
F (L=3) 7 Ag⊕2Bg⊕E1g⊕E2g
G (L=4) 9 Ag⊕2Bg⊕E1g⊕2E2g
H (L=5) 11 Ag⊕2Bg⊕2E1g⊕2E2g
I (L=6) 13 3Ag⊕2Bg⊕2E1g⊕2E2g


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