Point Group D6h



D6h E 2C6 2C3 C2 3C'2 3C''2 i 2S3 2S6 σh 3σd 3σv
A1g 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
B1g 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
E1g 2 1 -1 -2 0 0 2 1 -1 -2 0 0
E2g 2 -1 -1 2 0 0 2 -1 -1 2 0 0
A1u 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
B1u 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
E1u 2 1 -1 -2 0 0 -2 -1 1 2 0 0
E2u 2 -1 -1 2 0 0 -2 1 1 -2 0 0


Additional information

Number of symmetry elements h = 24
Number of classes, irreps n = 12
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E 2C6 2C3 C2 3C'2 3C''2 i 2S3 2S6 σh 3σd 3σv



Genrate representation from irreducible representations


A1g A2g B1g B2g E1g E2g A1u A2u B1u B2u E1u E2u



Examples

Benzene Bis(benzene)chromium (eclipsed) [18]Annulene
Coronene Superphane Kekulene
Cucurbit[6]uril


Direct products of irreducible representations


Binary products
⊙ A1g A2g B1g B2g E1g E2g A1u A2u B1u B2u E1u E2u
A1g A1g
A2g A2gA1g
B1g B1gB2gA1g
B2g B2gB1gA2gA1g
E1g E1gE1gE2gE2gA1g⊕A2g⊕E2g
E2g E2gE2gE1gE1gB1g⊕B2g⊕E1gA1g⊕A2g⊕E2g
A1u A1uA2uB1uB2uE1uE2uA1g
A2u A2uA1uB2uB1uE1uE2uA2gA1g
B1u B1uB2uA1uA2uE2uE1uB1gB2gA1g
B2u B2uB1uA2uA1uE2uE1uB2gB1gA2gA1g
E1u E1uE1uE2uE2uA1u⊕A2u⊕E2uB1u⊕B2u⊕E1uE1gE1gE2gE2gA1g⊕A2g⊕E2g
E2u E2uE2uE1uE1uB1u⊕B2u⊕E1uA1u⊕A2u⊕E2uE2gE2gE1gE1gB1g⊕B2g⊕E1gA1g⊕A2g⊕E2g

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
E1g A1g⊕E2gB1g⊕B2g⊕E1gA1g⊕2E2gB1g⊕B2g⊕2E1g2A1g⊕A2g⊕2E2gMore
E2g A1g⊕E2gA1g⊕A2g⊕E2gA1g⊕2E2gA1g⊕A2g⊕2E2g2A1g⊕A2g⊕2E2gMore
E1u A1g⊕E2gB1u⊕B2u⊕E1uA1g⊕2E2gB1u⊕B2u⊕2E1u2A1g⊕A2g⊕2E2gMore
E2u A1g⊕E2gA1u⊕A2u⊕E2uA1g⊕2E2gA1u⊕A2u⊕2E2u2A1g⊕A2g⊕2E2gMore


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⊕B1u⊕B2u⊕E1u⊕E2u 10 2A2u⊕B1u⊕B2u⊕2E1u⊕E2u
g (l=4) 9 Hexadecapole A1g⊕B1g⊕B2g⊕E1g⊕2E2g 15 3A1g⊕B1g⊕B2g⊕2E1g⊕3E2g
h (l=5) 11 Dotricontapole A2u⊕B1u⊕B2u⊕2E1u⊕2E2u 21 3A2u⊕2B1u⊕2B2u⊕4E1u⊕3E2u
i (l=6) 13 Tetrahexacontapole 2A1g⊕A2g⊕B1g⊕B2g⊕2E1g⊕2E2g 28 5A1g⊕A2g⊕2B1g⊕2B2g⊕4E1g⊕5E2g
j (l=7) 15 Octacosahectapole A1u⊕2A2u⊕B1u⊕B2u⊕3E1u⊕2E2u 36 A1u⊕5A2u⊕3B1u⊕3B2u⊕7E1u⊕5E2u
k (l=8) 17 256-pole 2A1g⊕A2g⊕B1g⊕B2g⊕3E1g⊕3E2g 45 7A1g⊕2A2g⊕3B1g⊕3B2g⊕7E1g⊕8E2g
l (l=9) 19 512-pole A1u⊕2A2u⊕2B1u⊕2B2u⊕3E1u⊕3E2u 55 2A1u⊕7A2u⊕5B1u⊕5B2u⊕10E1u⊕8E2u
m (l=10) 21 1024-pole 2A1g⊕A2g⊕2B1g⊕2B2g⊕3E1g⊕4E2g 66 9A1g⊕3A2g⊕5B1g⊕5B2g⊕10E1g⊕12E2g
n (l=11) 23 2048-pole A1u⊕2A2u⊕2B1u⊕2B2u⊕4E1u⊕4E2u 78 3A1u⊕9A2u⊕7B1u⊕7B2u⊕14E1u⊕12E2u
o (l=12) 25 4096-pole 3A1g⊕2A2g⊕2B1g⊕2B2g⊕4E1g⊕4E2g 91 12A1g⊕5A2g⊕7B1g⊕7B2g⊕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 D6h
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⊕B1g⊕B2g⊕E1g⊕E2g
G (L=4) 9 A1g⊕B1g⊕B2g⊕E1g⊕2E2g
H (L=5) 11 A2g⊕B1g⊕B2g⊕2E1g⊕2E2g
I (L=6) 13 2A1g⊕A2g⊕B1g⊕B2g⊕2E1g⊕2E2g

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