Point Group D12d



D12d E 2S24 2C12 2S8 2C6 2(S24)5 2C4 2(S24)7 2C3 2(S8)3 2(C12)5 2(S24)11 C2 12C'2 12σd
A1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A2 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1
B1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1
B2 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1
E1 2 2cos(π/12) 2cos(π/6) 2cos(π/4) 1 2cos(5π/12) 0 -2cos(5π/12) -1 -2cos(π/4) -2cos(π/6) -2cos(π/12) -2 0 0
E2 2 2cos(π/6) 1 0 -1 -2cos(π/6) -2 -2cos(π/6) -1 0 1 2cos(π/6) 2 0 0
E3 2 2cos(π/4) 0 -2cos(π/4) -2 -2cos(π/4) 0 2cos(π/4) 2 2cos(π/4) 0 -2cos(π/4) -2 0 0
E4 2 1 -1 -2 -1 1 2 1 -1 -2 -1 1 2 0 0
E5 2 2cos(5π/12) -2cos(π/6) -2cos(π/4) 1 2cos(π/12) 0 -2cos(π/12) -1 2cos(π/4) 2cos(π/6) -2cos(5π/12) -2 0 0
E6 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 0
E7 2 -2cos(5π/12) -2cos(π/6) 2cos(π/4) 1 -2cos(π/12) 0 2cos(π/12) -1 -2cos(π/4) 2cos(π/6) 2cos(5π/12) -2 0 0
E8 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 2 0 0
E9 2 -2cos(π/4) 0 2cos(π/4) -2 2cos(π/4) 0 -2cos(π/4) 2 -2cos(π/4) 0 2cos(π/4) -2 0 0
E10 2 -2cos(π/6) 1 0 -1 2cos(π/6) -2 2cos(π/6) -1 0 1 -2cos(π/6) 2 0 0
E11 2 -2cos(π/12) 2cos(π/6) -2cos(π/4) 1 -2cos(5π/12) 0 2cos(5π/12) -1 2cos(π/4) -2cos(π/6) 2cos(π/12) -2 0 0


Additional information

Number of symmetry elements h = 48
Number of classes, irreps n = 15
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E 2S24 2C12 2S8 2C6 2(S24)5 2C4 2(S24)7 2C3 2(S8)3 2(C12)5 2(S24)11 C2 12C'2 12σd



Genrate representation from irreducible representations


A1 A2 B1 B2 E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11




Direct products of irreducible representations


Binary products
A1 A2 B1 B2 E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11
A1 A1
A2 A2A1
B1 B1B2A1
B2 B2B1A2A1
E1 E1E1E11E11A1⊕A2⊕E2
E2 E2E2E10E10E1⊕E3A1⊕A2⊕E4
E3 E3E3E9E9E2⊕E4E1⊕E5A1⊕A2⊕E6
E4 E4E4E8E8E3⊕E5E2⊕E6E1⊕E7A1⊕A2⊕E8
E5 E5E5E7E7E4⊕E6E3⊕E7E2⊕E8E1⊕E9A1⊕A2⊕E10
E6 E6E6E6E6E5⊕E7E4⊕E8E3⊕E9E2⊕E10E1⊕E11A1⊕A2⊕B1⊕B2
E7 E7E7E5E5E6⊕E8E5⊕E9E4⊕E10E3⊕E11B1⊕B2⊕E2E1⊕E11A1⊕A2⊕E10
E8 E8E8E4E4E7⊕E9E6⊕E10E5⊕E11B1⊕B2⊕E4E3⊕E11E2⊕E10E1⊕E9A1⊕A2⊕E8
E9 E9E9E3E3E8⊕E10E7⊕E11B1⊕B2⊕E6E5⊕E11E4⊕E10E3⊕E9E2⊕E8E1⊕E7A1⊕A2⊕E6
E10 E10E10E2E2E9⊕E11B1⊕B2⊕E8E7⊕E11E6⊕E10E5⊕E9E4⊕E8E3⊕E7E2⊕E6E1⊕E5A1⊕A2⊕E4
E11 E11E11E1E1B1⊕B2⊕E10E9⊕E11E8⊕E10E7⊕E9E6⊕E8E5⊕E7E4⊕E6E3⊕E5E2⊕E4E1⊕E3A1⊕A2⊕E2

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1 A1⊕E2E1⊕E3A1⊕E2⊕E4E1⊕E3⊕E5A1⊕E2⊕E4⊕E6More
E2 A1⊕E4E2⊕E6A1⊕E4⊕E8E2⊕E6⊕E10A1⊕B1⊕B2⊕E4⊕E8More
E3 A1⊕E6E3⊕E9A1⊕B1⊕B2⊕E6E3⊕2E9A1⊕B1⊕B2⊕2E6More
E4 A1⊕E8B1⊕B2⊕E4A1⊕2E8B1⊕B2⊕2E42A1⊕A2⊕2E8More
E5 A1⊕E10E5⊕E9A1⊕E4⊕E10E1⊕E5⊕E9A1⊕E4⊕E6⊕E10More
E6 A1⊕B1⊕B22E62A1⊕A2⊕B1⊕B23E62A1⊕A2⊕2B1⊕2B2More
E7 A1⊕E10E3⊕E7A1⊕E4⊕E10E3⊕E7⊕E11A1⊕E4⊕E6⊕E10More
E8 A1⊕E8A1⊕A2⊕E8A1⊕2E8A1⊕A2⊕2E82A1⊕A2⊕2E8More
E9 A1⊕E6E3⊕E9A1⊕B1⊕B2⊕E62E3⊕E9A1⊕B1⊕B2⊕2E6More
E10 A1⊕E4E6⊕E10A1⊕E4⊕E8E2⊕E6⊕E10A1⊕B1⊕B2⊕E4⊕E8More
E11 A1⊕E2E9⊕E11A1⊕E2⊕E4E7⊕E9⊕E11A1⊕E2⊕E4⊕E6More



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 A1 1 A1
p (l=1) 3 Dipole B2⊕E1 3 B2⊕E1
d (l=2) 5 Quadrupole A1⊕E2⊕E11 6 2A1⊕E2⊕E11
f (l=3) 7 Octupole B2⊕E1⊕E3⊕E10 10 2B2⊕2E1⊕E3⊕E10
g (l=4) 9 Hexadecapole A1⊕E2⊕E4⊕E9⊕E11 15 3A1⊕2E2⊕E4⊕E9⊕2E11
h (l=5) 11 Dotricontapole B2⊕E1⊕E3⊕E5⊕E8⊕E10 21 3B2⊕3E1⊕2E3⊕E5⊕E8⊕2E10
i (l=6) 13 Tetrahexacontapole A1⊕E2⊕E4⊕E6⊕E7⊕E9⊕E11 28 4A1⊕3E2⊕2E4⊕E6⊕E7⊕2E9⊕3E11
j (l=7) 15 Octacosahectapole B2⊕E1⊕E3⊕E5⊕E6⊕E7⊕E8⊕E10 36 4B2⊕4E1⊕3E3⊕2E5⊕E6⊕E7⊕2E8⊕3E10
k (l=8) 17 256-pole A1⊕E2⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E11 45 5A1⊕4E2⊕3E4⊕E5⊕2E6⊕2E7⊕E8⊕3E9⊕4E11
l (l=9) 19 512-pole B2⊕E1⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10 55 5B2⊕5E1⊕4E3⊕E4⊕3E5⊕2E6⊕2E7⊕3E8⊕E9⊕4E10
m (l=10) 21 1024-pole A1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10⊕E11 66 6A1⊕5E2⊕E3⊕4E4⊕2E5⊕3E6⊕3E7⊕2E8⊕4E9⊕E10⊕5E11
n (l=11) 23 2048-pole B2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10⊕E11 78 6B2⊕6E1⊕E2⊕5E3⊕2E4⊕4E5⊕3E6⊕3E7⊕4E8⊕2E9⊕5E10⊕E11
o (l=12) 25 4096-pole A1⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10⊕E11 91 7A1⊕B1⊕B2⊕E1⊕6E2⊕2E3⊕5E4⊕3E5⊕4E6⊕4E7⊕3E8⊕5E9⊕2E10⊕6E11
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 D12d
L 2L+1 Term Splitting
S (L=0) 1 A1
P (L=1) 3 A2⊕E11
D (L=2) 5 A1⊕E2⊕E11
F (L=3) 7 A2⊕E2⊕E9⊕E11
G (L=4) 9 A1⊕E2⊕E4⊕E9⊕E11
H (L=5) 11 A2⊕E2⊕E4⊕E7⊕E9⊕E11
I (L=6) 13 A1⊕E2⊕E4⊕E6⊕E7⊕E9⊕E11


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