Point Group C12h



C12h E C12 C6 C4 C3 (C12)5 C2 (C12)7 (C3)2 (C4)3 (C6)5 (C12)11 i (S12)7 (S3)5 (S4)3 (S6)5 (S12)11 σh S12 S6 S4 S3 (S12)5
Ag 1 1 1 1 1 1 1 1 1 1 1 1 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 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
E1g* 2 2cos(π/6) 1 0 -1 -2cos(π/6) -2 -2cos(π/6) -1 0 1 2cos(π/6) 2 2cos(π/6) 1 0 -1 -2cos(π/6) -2 -2cos(π/6) -1 0 1 2cos(π/6)
E2g* 2 1 -1 -2 -1 1 2 1 -1 -2 -1 1 2 1 -1 -2 -1 1 2 1 -1 -2 -1 1
E3g* 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0
E4g* 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1
E5g* 2 -2cos(π/6) 1 0 -1 2cos(π/6) -2 2cos(π/6) -1 0 1 -2cos(π/6) 2 -2cos(π/6) 1 0 -1 2cos(π/6) -2 2cos(π/6) -1 0 1 -2cos(π/6)
Au 1 1 1 1 1 1 1 1 1 1 1 1 -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 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1
E1u* 2 2cos(π/6) 1 0 -1 -2cos(π/6) -2 -2cos(π/6) -1 0 1 2cos(π/6) -2 -2cos(π/6) -1 0 1 2cos(π/6) 2 2cos(π/6) 1 0 -1 -2cos(π/6)
E2u* 2 1 -1 -2 -1 1 2 1 -1 -2 -1 1 -2 -1 1 2 1 -1 -2 -1 1 2 1 -1
E3u* 2 0 -2 0 2 0 -2 0 2 0 -2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0
E4u* 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 -2 1 1 -2 1 1 -2 1 1 -2 1 1
E5u* 2 -2cos(π/6) 1 0 -1 2cos(π/6) -2 2cos(π/6) -1 0 1 -2cos(π/6) -2 2cos(π/6) -1 0 1 -2cos(π/6) 2 -2cos(π/6) 1 0 -1 2cos(π/6)


Additional information

Number of symmetry elements h = 24
Number of classes, irreps n = 24
Number of real-valued irreducible representations n = 14
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E C12 C6 C4 C3 (C12)5 C2 (C12)7 (C3)2 (C4)3 (C6)5 (C12)11 i (S12)7 (S3)5 (S4)3 (S6)5 (S12)11 σh S12 S6 S4 S3 (S12)5



Genrate representation from irreducible representations


Ag Bg E1g* E2g* E3g* E4g* E5g* Au Bu E1u* E2u* E3u* E4u* E5u*




Direct products of irreducible representations


Binary products
Ag Bg E1g* E2g* E3g* E4g* E5g* Au Bu E1u* E2u* E3u* E4u* E5u*
Ag Ag
Bg BgAg
E1g* E1gE5g2Ag⊕E2g
E2g* E2gE4gE1g⊕E3g2Ag⊕E4g
E3g* E3gE3gE2g⊕E4gE1g⊕E5g2Ag⊕2Bg
E4g* E4gE2gE3g⊕E5g2Bg⊕E2gE1g⊕E5g2Ag⊕E4g
E5g* E5gE1g2Bg⊕E4gE3g⊕E5gE2g⊕E4gE1g⊕E3g2Ag⊕E2g
Au AuBuE1uE2uE3uE4uE5uAg
Bu BuAuE5uE4uE3uE2uE1uBgAg
E1u* E1uE5u2Au⊕E2uE1u⊕E3uE2u⊕E4uE3u⊕E5u2Bu⊕E4uE1gE5g2Ag⊕E2g
E2u* E2uE4uE1u⊕E3u2Au⊕E4uE1u⊕E5u2Bu⊕E2uE3u⊕E5uE2gE4gE1g⊕E3g2Ag⊕E4g
E3u* E3uE3uE2u⊕E4uE1u⊕E5u2Au⊕2BuE1u⊕E5uE2u⊕E4uE3gE3gE2g⊕E4gE1g⊕E5g2Ag⊕2Bg
E4u* E4uE2uE3u⊕E5u2Bu⊕E2uE1u⊕E5u2Au⊕E4uE1u⊕E3uE4gE2gE3g⊕E5g2Bg⊕E2gE1g⊕E5g2Ag⊕E4g
E5u* E5uE1u2Bu⊕E4uE3u⊕E5uE2u⊕E4uE1u⊕E3u2Au⊕E2uE5gE1g2Bg⊕E4gE3g⊕E5gE2g⊕E4gE1g⊕E3g2Ag⊕E2g

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1g* Ag⊕E2gE1g⊕E3gAg⊕E2g⊕E4gE1g⊕E3g⊕E5gAg⊕2Bg⊕E2g⊕E4gMore
E2g* Ag⊕E4g2Bg⊕E2gAg⊕2E4g2Bg⊕2E2g3Ag⊕2E4gMore
E3g* Ag⊕2Bg2E3g3Ag⊕2Bg3E3g3Ag⊕4BgMore
E4g* Ag⊕E4g2Ag⊕E4gAg⊕2E4g2Ag⊕2E4g3Ag⊕2E4gMore
E5g* Ag⊕E2gE3g⊕E5gAg⊕E2g⊕E4gE1g⊕E3g⊕E5gAg⊕2Bg⊕E2g⊕E4gMore
E1u* Ag⊕E2gE1u⊕E3uAg⊕E2g⊕E4gE1u⊕E3u⊕E5uAg⊕2Bg⊕E2g⊕E4gMore
E2u* Ag⊕E4g2Bu⊕E2uAg⊕2E4g2Bu⊕2E2u3Ag⊕2E4gMore
E3u* Ag⊕2Bg2E3u3Ag⊕2Bg3E3u3Ag⊕4BgMore
E4u* Ag⊕E4g2Au⊕E4uAg⊕2E4g2Au⊕2E4u3Ag⊕2E4gMore
E5u* Ag⊕E2gE3u⊕E5uAg⊕E2g⊕E4gE1u⊕E3u⊕E5uAg⊕2Bg⊕E2g⊕E4gMore



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⊕E1u⊕E2u⊕E3u 10 2Au⊕2E1u⊕E2u⊕E3u
g (l=4) 9 Hexadecapole Ag⊕E1g⊕E2g⊕E3g⊕E4g 15 3Ag⊕2E1g⊕2E2g⊕E3g⊕E4g
h (l=5) 11 Dotricontapole Au⊕E1u⊕E2u⊕E3u⊕E4u⊕E5u 21 3Au⊕3E1u⊕2E2u⊕2E3u⊕E4u⊕E5u
i (l=6) 13 Tetrahexacontapole Ag⊕2Bg⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g 28 4Ag⊕2Bg⊕3E1g⊕3E2g⊕2E3g⊕2E4g⊕E5g
j (l=7) 15 Octacosahectapole Au⊕2Bu⊕E1u⊕E2u⊕E3u⊕E4u⊕2E5u 36 4Au⊕2Bu⊕4E1u⊕3E2u⊕3E3u⊕2E4u⊕3E5u
k (l=8) 17 256-pole Ag⊕2Bg⊕E1g⊕E2g⊕E3g⊕2E4g⊕2E5g 45 5Ag⊕4Bg⊕4E1g⊕4E2g⊕3E3g⊕4E4g⊕3E5g
l (l=9) 19 512-pole Au⊕2Bu⊕E1u⊕E2u⊕2E3u⊕2E4u⊕2E5u 55 5Au⊕4Bu⊕5E1u⊕4E2u⊕5E3u⊕4E4u⊕5E5u
m (l=10) 21 1024-pole Ag⊕2Bg⊕E1g⊕2E2g⊕2E3g⊕2E4g⊕2E5g 66 6Ag⊕6Bg⊕5E1g⊕6E2g⊕5E3g⊕6E4g⊕5E5g
n (l=11) 23 2048-pole Au⊕2Bu⊕2E1u⊕2E2u⊕2E3u⊕2E4u⊕2E5u 78 6Au⊕6Bu⊕7E1u⊕6E2u⊕7E3u⊕6E4u⊕7E5u
o (l=12) 25 4096-pole 3Ag⊕2Bg⊕2E1g⊕2E2g⊕2E3g⊕2E4g⊕2E5g 91 9Ag⊕8Bg⊕7E1g⊕8E2g⊕7E3g⊕8E4g⊕7E5g
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 C12h
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⊕E1g⊕E2g⊕E3g
G (L=4) 9 Ag⊕E1g⊕E2g⊕E3g⊕E4g
H (L=5) 11 Ag⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g
I (L=6) 13 Ag⊕2Bg⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g


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