Point Group D12h



D12h E 2C12 2C6 2C4 2C3 2(C12)5 C2 6C'2 6C''2 i 2(S12)5 2S3 2S4 2S6 2S12 σh v d
A1g 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A2g 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1
B1g 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 1 -1
B2g 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 0 0 2 2cos(π/6) 1 0 -1 -2cos(π/6) -2 0 0
E2g 2 1 -1 -2 -1 1 2 0 0 2 1 -1 -2 -1 1 2 0 0
E3g 2 0 -2 0 2 0 -2 0 0 2 0 -2 0 2 0 -2 0 0
E4g 2 -1 -1 2 -1 -1 2 0 0 2 -1 -1 2 -1 -1 2 0 0
E5g 2 -2cos(π/6) 1 0 -1 2cos(π/6) -2 0 0 2 -2cos(π/6) 1 0 -1 2cos(π/6) -2 0 0
A1u 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1
A2u 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1
B1u 1 -1 1 -1 1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 -1 1
B2u 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 0 0 -2 -2cos(π/6) -1 0 1 2cos(π/6) 2 0 0
E2u 2 1 -1 -2 -1 1 2 0 0 -2 -1 1 2 1 -1 -2 0 0
E3u 2 0 -2 0 2 0 -2 0 0 -2 0 2 0 -2 0 2 0 0
E4u 2 -1 -1 2 -1 -1 2 0 0 -2 1 1 -2 1 1 -2 0 0
E5u 2 -2cos(π/6) 1 0 -1 2cos(π/6) -2 0 0 -2 2cos(π/6) -1 0 1 -2cos(π/6) 2 0 0


Additional information

Number of symmetry elements h = 48
Number of classes, irreps n = 18
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E 2C12 2C6 2C4 2C3 2(C12)5 C2 6C'2 6C''2 i 2(S12)5 2S3 2S4 2S6 2S12 σh v d



Genrate representation from irreducible representations


A1g A2g B1g B2g E1g E2g E3g E4g E5g A1u A2u B1u B2u E1u E2u E3u E4u E5u




Direct products of irreducible representations


Binary products
A1g A2g B1g B2g E1g E2g E3g E4g E5g A1u A2u B1u B2u E1u E2u E3u E4u E5u
A1g A1g
A2g A2gA1g
B1g B1gB2gA1g
B2g B2gB1gA2gA1g
E1g E1gE1gE5gE5gA1g⊕A2g⊕E2g
E2g E2gE2gE4gE4gE1g⊕E3gA1g⊕A2g⊕E4g
E3g E3gE3gE3gE3gE2g⊕E4gE1g⊕E5gA1g⊕A2g⊕B1g⊕B2g
E4g E4gE4gE2gE2gE3g⊕E5gB1g⊕B2g⊕E2gE1g⊕E5gA1g⊕A2g⊕E4g
E5g E5gE5gE1gE1gB1g⊕B2g⊕E4gE3g⊕E5gE2g⊕E4gE1g⊕E3gA1g⊕A2g⊕E2g
A1u A1uA2uB1uB2uE1uE2uE3uE4uE5uA1g
A2u A2uA1uB2uB1uE1uE2uE3uE4uE5uA2gA1g
B1u B1uB2uA1uA2uE5uE4uE3uE2uE1uB1gB2gA1g
B2u B2uB1uA2uA1uE5uE4uE3uE2uE1uB2gB1gA2gA1g
E1u E1uE1uE5uE5uA1u⊕A2u⊕E2uE1u⊕E3uE2u⊕E4uE3u⊕E5uB1u⊕B2u⊕E4uE1gE1gE5gE5gA1g⊕A2g⊕E2g
E2u E2uE2uE4uE4uE1u⊕E3uA1u⊕A2u⊕E4uE1u⊕E5uB1u⊕B2u⊕E2uE3u⊕E5uE2gE2gE4gE4gE1g⊕E3gA1g⊕A2g⊕E4g
E3u E3uE3uE3uE3uE2u⊕E4uE1u⊕E5uA1u⊕A2u⊕B1u⊕B2uE1u⊕E5uE2u⊕E4uE3gE3gE3gE3gE2g⊕E4gE1g⊕E5gA1g⊕A2g⊕B1g⊕B2g
E4u E4uE4uE2uE2uE3u⊕E5uB1u⊕B2u⊕E2uE1u⊕E5uA1u⊕A2u⊕E4uE1u⊕E3uE4gE4gE2gE2gE3g⊕E5gB1g⊕B2g⊕E2gE1g⊕E5gA1g⊕A2g⊕E4g
E5u E5uE5uE1uE1uB1u⊕B2u⊕E4uE3u⊕E5uE2u⊕E4uE1u⊕E3uA1u⊕A2u⊕E2uE5gE5gE1gE1gB1g⊕B2g⊕E4gE3g⊕E5gE2g⊕E4gE1g⊕E3gA1g⊕A2g⊕E2g

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1g A1g⊕E2gE1g⊕E3gA1g⊕E2g⊕E4gE1g⊕E3g⊕E5gA1g⊕B1g⊕B2g⊕E2g⊕E4gMore
E2g A1g⊕E4gB1g⊕B2g⊕E2gA1g⊕2E4gB1g⊕B2g⊕2E2g2A1g⊕A2g⊕2E4gMore
E3g A1g⊕B1g⊕B2g2E3g2A1g⊕A2g⊕B1g⊕B2g3E3g2A1g⊕A2g⊕2B1g⊕2B2gMore
E4g A1g⊕E4gA1g⊕A2g⊕E4gA1g⊕2E4gA1g⊕A2g⊕2E4g2A1g⊕A2g⊕2E4gMore
E5g A1g⊕E2gE3g⊕E5gA1g⊕E2g⊕E4gE1g⊕E3g⊕E5gA1g⊕B1g⊕B2g⊕E2g⊕E4gMore
E1u A1g⊕E2gE1u⊕E3uA1g⊕E2g⊕E4gE1u⊕E3u⊕E5uA1g⊕B1g⊕B2g⊕E2g⊕E4gMore
E2u A1g⊕E4gB1u⊕B2u⊕E2uA1g⊕2E4gB1u⊕B2u⊕2E2u2A1g⊕A2g⊕2E4gMore
E3u A1g⊕B1g⊕B2g2E3u2A1g⊕A2g⊕B1g⊕B2g3E3u2A1g⊕A2g⊕2B1g⊕2B2gMore
E4u A1g⊕E4gA1u⊕A2u⊕E4uA1g⊕2E4gA1u⊕A2u⊕2E4u2A1g⊕A2g⊕2E4gMore
E5u A1g⊕E2gE3u⊕E5uA1g⊕E2g⊕E4gE1u⊕E3u⊕E5uA1g⊕B1g⊕B2g⊕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 A1g 1 A1g
p (l=1) 3 Dipole A2u⊕E1u 3 A2u⊕E1u
d (l=2) 5 Quadrupole A1g⊕E1g⊕E2g 6 2A1g⊕E1g⊕E2g
f (l=3) 7 Octupole A2u⊕E1u⊕E2u⊕E3u 10 2A2u⊕2E1u⊕E2u⊕E3u
g (l=4) 9 Hexadecapole A1g⊕E1g⊕E2g⊕E3g⊕E4g 15 3A1g⊕2E1g⊕2E2g⊕E3g⊕E4g
h (l=5) 11 Dotricontapole A2u⊕E1u⊕E2u⊕E3u⊕E4u⊕E5u 21 3A2u⊕3E1u⊕2E2u⊕2E3u⊕E4u⊕E5u
i (l=6) 13 Tetrahexacontapole A1g⊕B1g⊕B2g⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g 28 4A1g⊕B1g⊕B2g⊕3E1g⊕3E2g⊕2E3g⊕2E4g⊕E5g
j (l=7) 15 Octacosahectapole A2u⊕B1u⊕B2u⊕E1u⊕E2u⊕E3u⊕E4u⊕2E5u 36 4A2u⊕B1u⊕B2u⊕4E1u⊕3E2u⊕3E3u⊕2E4u⊕3E5u
k (l=8) 17 256-pole A1g⊕B1g⊕B2g⊕E1g⊕E2g⊕E3g⊕2E4g⊕2E5g 45 5A1g⊕2B1g⊕2B2g⊕4E1g⊕4E2g⊕3E3g⊕4E4g⊕3E5g
l (l=9) 19 512-pole A2u⊕B1u⊕B2u⊕E1u⊕E2u⊕2E3u⊕2E4u⊕2E5u 55 5A2u⊕2B1u⊕2B2u⊕5E1u⊕4E2u⊕5E3u⊕4E4u⊕5E5u
m (l=10) 21 1024-pole A1g⊕B1g⊕B2g⊕E1g⊕2E2g⊕2E3g⊕2E4g⊕2E5g 66 6A1g⊕3B1g⊕3B2g⊕5E1g⊕6E2g⊕5E3g⊕6E4g⊕5E5g
n (l=11) 23 2048-pole A2u⊕B1u⊕B2u⊕2E1u⊕2E2u⊕2E3u⊕2E4u⊕2E5u 78 6A2u⊕3B1u⊕3B2u⊕7E1u⊕6E2u⊕7E3u⊕6E4u⊕7E5u
o (l=12) 25 4096-pole 2A1g⊕A2g⊕B1g⊕B2g⊕2E1g⊕2E2g⊕2E3g⊕2E4g⊕2E5g 91 8A1g⊕A2g⊕4B1g⊕4B2g⊕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 D12h
L 2L+1 Term Splitting
S (L=0) 1 A1g
P (L=1) 3 A2g⊕E1g
D (L=2) 5 A1g⊕E1g⊕E2g
F (L=3) 7 A2g⊕E1g⊕E2g⊕E3g
G (L=4) 9 A1g⊕E1g⊕E2g⊕E3g⊕E4g
H (L=5) 11 A2g⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g
I (L=6) 13 A1g⊕B1g⊕B2g⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g


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