Point Group D11d



D11d E 2C11 2(C11)2 2(C11)3 2(C11)4 2(C11)5 11C'2 i 2(S22)9 2(S22)7 2(S22)5 2(S22)3 2S22 11σd
A1g 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
E1g 2 0 2 0
E2g 2 0 2 0
E3g 2 0 2 0
E4g 2 0 2 0
E5g 2 0 2 0
A1u 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
E1u 2 0 -2 0
E2u 2 0 -2 0
E3u 2 0 -2 0
E4u 2 0 -2 0
E5u 2 0 -2 0
View A
View B


Additional information

Number of symmetry elements h = 44
Number of classes, irreps n = 14
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E 2C11 2(C11)2 2(C11)3 2(C11)4 2(C11)5 11C'2 i 2(S22)9 2(S22)7 2(S22)5 2(S22)3 2S22 11σd



Genrate representation from irreducible representations


A1g A2g E1g E2g E3g E4g E5g A1u A2u E1u E2u E3u E4u E5u



Direct products of irreducible representations


Binary products
⊙ A1g A2g E1g E2g E3g E4g E5g A1u A2u E1u E2u E3u E4u E5u
A1g A1g
A2g A2gA1g
E1g E1gE1gA1g⊕A2g⊕E2g
E2g E2gE2gE1g⊕E3gA1g⊕A2g⊕E4g
E3g E3gE3gE2g⊕E4gE1g⊕E5gA1g⊕A2g⊕E5g
E4g E4gE4gE3g⊕E5gE2g⊕E5gE1g⊕E4gA1g⊕A2g⊕E3g
E5g E5gE5gE4g⊕E5gE3g⊕E4gE2g⊕E3gE1g⊕E2gA1g⊕A2g⊕E1g
A1u A1uA2uE1uE2uE3uE4uE5uA1g
A2u A2uA1uE1uE2uE3uE4uE5uA2gA1g
E1u E1uE1uA1u⊕A2u⊕E2uE1u⊕E3uE2u⊕E4uE3u⊕E5uE4u⊕E5uE1gE1gA1g⊕A2g⊕E2g
E2u E2uE2uE1u⊕E3uA1u⊕A2u⊕E4uE1u⊕E5uE2u⊕E5uE3u⊕E4uE2gE2gE1g⊕E3gA1g⊕A2g⊕E4g
E3u E3uE3uE2u⊕E4uE1u⊕E5uA1u⊕A2u⊕E5uE1u⊕E4uE2u⊕E3uE3gE3gE2g⊕E4gE1g⊕E5gA1g⊕A2g⊕E5g
E4u E4uE4uE3u⊕E5uE2u⊕E5uE1u⊕E4uA1u⊕A2u⊕E3uE1u⊕E2uE4gE4gE3g⊕E5gE2g⊕E5gE1g⊕E4gA1g⊕A2g⊕E3g
E5u E5uE5uE4u⊕E5uE3u⊕E4uE2u⊕E3uE1u⊕E2uA1u⊕A2u⊕E1uE5gE5gE4g⊕E5gE3g⊕E4gE2g⊕E3gE1g⊕E2gA1g⊕A2g⊕E1g

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
E1g A1g⊕E2gE1g⊕E3gA1g⊕E2g⊕E4gE1g⊕E3g⊕E5gA1g⊕E2g⊕E4g⊕E5gMore
E2g A1g⊕E4gE2g⊕E5gA1g⊕E3g⊕E4gE1g⊕E2g⊕E5gA1g⊕E1g⊕E3g⊕E4gMore
E3g A1g⊕E5gE2g⊕E3gA1g⊕E1g⊕E5gE2g⊕E3g⊕E4gA1g⊕E1g⊕E4g⊕E5gMore
E4g A1g⊕E3gE1g⊕E4gA1g⊕E3g⊕E5gE1g⊕E2g⊕E4gA1g⊕E2g⊕E3g⊕E5gMore
E5g A1g⊕E1gE4g⊕E5gA1g⊕E1g⊕E2gE3g⊕E4g⊕E5gA1g⊕E1g⊕E2g⊕E3gMore
E1u A1g⊕E2gE1u⊕E3uA1g⊕E2g⊕E4gE1u⊕E3u⊕E5uA1g⊕E2g⊕E4g⊕E5gMore
E2u A1g⊕E4gE2u⊕E5uA1g⊕E3g⊕E4gE1u⊕E2u⊕E5uA1g⊕E1g⊕E3g⊕E4gMore
E3u A1g⊕E5gE2u⊕E3uA1g⊕E1g⊕E5gE2u⊕E3u⊕E4uA1g⊕E1g⊕E4g⊕E5gMore
E4u A1g⊕E3gE1u⊕E4uA1g⊕E3g⊕E5gE1u⊕E2u⊕E4uA1g⊕E2g⊕E3g⊕E5gMore
E5u A1g⊕E1gE4u⊕E5uA1g⊕E1g⊕E2gE3u⊕E4u⊕E5uA1g⊕E1g⊕E2g⊕E3gMore


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 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⊕E1g⊕E2g⊕E3g⊕E4g⊕2E5g 28 4A1g⊕3E1g⊕3E2g⊕2E3g⊕2E4g⊕2E5g
j (l=7) 15 Octacosahectapole A2u⊕E1u⊕E2u⊕E3u⊕2E4u⊕2E5u 36 4A2u⊕4E1u⊕3E2u⊕3E3u⊕3E4u⊕3E5u
k (l=8) 17 256-pole A1g⊕E1g⊕E2g⊕2E3g⊕2E4g⊕2E5g 45 5A1g⊕4E1g⊕4E2g⊕4E3g⊕4E4g⊕4E5g
l (l=9) 19 512-pole A2u⊕E1u⊕2E2u⊕2E3u⊕2E4u⊕2E5u 55 5A2u⊕5E1u⊕5E2u⊕5E3u⊕5E4u⊕5E5u
m (l=10) 21 1024-pole A1g⊕2E1g⊕2E2g⊕2E3g⊕2E4g⊕2E5g 66 6A1g⊕6E1g⊕6E2g⊕6E3g⊕6E4g⊕6E5g
n (l=11) 23 2048-pole A1u⊕2A2u⊕2E1u⊕2E2u⊕2E3u⊕2E4u⊕2E5u 78 A1u⊕7A2u⊕7E1u⊕7E2u⊕7E3u⊕7E4u⊕7E5u
o (l=12) 25 4096-pole 2A1g⊕A2g⊕3E1g⊕2E2g⊕2E3g⊕2E4g⊕2E5g 91 8A1g⊕A2g⊕9E1g⊕8E2g⊕8E3g⊕8E4g⊕8E5g
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 D11d
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⊕E1g⊕E2g⊕E3g⊕E4g⊕2E5g

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