Point Group D4h



D4h E 2C4 C2 2C'2 2C''2 i 2S4 σh v d
A1g 1 1 1 1 1 1 1 1 1 1
A2g 1 1 1 -1 -1 1 1 1 -1 -1
B1g 1 -1 1 1 -1 1 -1 1 1 -1
B2g 1 -1 1 -1 1 1 -1 1 -1 1
Eg 2 0 -2 0 0 2 0 -2 0 0
A1u 1 1 1 1 1 -1 -1 -1 -1 -1
A2u 1 1 1 -1 -1 -1 -1 -1 1 1
B1u 1 -1 1 1 -1 -1 1 -1 -1 1
B2u 1 -1 1 -1 1 -1 1 -1 1 -1
Eu 2 0 -2 0 0 -2 0 2 0 0


Additional information

Number of symmetry elements h = 16
Number of classes, irreps n = 10
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E 2C4 C2 2C'2 2C''2 i 2S4 σh v d



Genrate representation from irreducible representations


A1g A2g B1g B2g Eg A1u A2u B1u B2u Eu




Examples

[Al4]2--Ion Xenontetrafluoride Squaric Acid Dianion



Direct products of irreducible representations


Binary products
A1g A2g B1g B2g Eg A1u A2u B1u B2u Eu
A1g A1g
A2g A2gA1g
B1g B1gB2gA1g
B2g B2gB1gA2gA1g
Eg EgEgEgEgA1g⊕A2g⊕B1g⊕B2g
A1u A1uA2uB1uB2uEuA1g
A2u A2uA1uB2uB1uEuA2gA1g
B1u B1uB2uA1uA2uEuB1gB2gA1g
B2u B2uB1uA2uA1uEuB2gB1gA2gA1g
Eu EuEuEuEuA1u⊕A2u⊕B1u⊕B2uEgEgEgEgA1g⊕A2g⊕B1g⊕B2g

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
Eg A1g⊕B1g⊕B2g2Eg2A1g⊕A2g⊕B1g⊕B2g3Eg2A1g⊕A2g⊕2B1g⊕2B2gMore
Eu A1g⊕B1g⊕B2g2Eu2A1g⊕A2g⊕B1g⊕B2g3Eu2A1g⊕A2g⊕2B1g⊕2B2gMore



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⊕Eu 3 A2u⊕Eu
d (l=2) 5 Quadrupole A1g⊕B1g⊕B2g⊕Eg 6 2A1g⊕B1g⊕B2g⊕Eg
f (l=3) 7 Octupole A2u⊕B1u⊕B2u⊕2Eu 10 2A2u⊕B1u⊕B2u⊕3Eu
g (l=4) 9 Hexadecapole 2A1g⊕A2g⊕B1g⊕B2g⊕2Eg 15 4A1g⊕A2g⊕2B1g⊕2B2g⊕3Eg
h (l=5) 11 Dotricontapole A1u⊕2A2u⊕B1u⊕B2u⊕3Eu 21 A1u⊕4A2u⊕2B1u⊕2B2u⊕6Eu
i (l=6) 13 Tetrahexacontapole 2A1g⊕A2g⊕2B1g⊕2B2g⊕3Eg 28 6A1g⊕2A2g⊕4B1g⊕4B2g⊕6Eg
j (l=7) 15 Octacosahectapole A1u⊕2A2u⊕2B1u⊕2B2u⊕4Eu 36 2A1u⊕6A2u⊕4B1u⊕4B2u⊕10Eu
k (l=8) 17 256-pole 3A1g⊕2A2g⊕2B1g⊕2B2g⊕4Eg 45 9A1g⊕4A2g⊕6B1g⊕6B2g⊕10Eg
l (l=9) 19 512-pole 2A1u⊕3A2u⊕2B1u⊕2B2u⊕5Eu 55 4A1u⊕9A2u⊕6B1u⊕6B2u⊕15Eu
m (l=10) 21 1024-pole 3A1g⊕2A2g⊕3B1g⊕3B2g⊕5Eg 66 12A1g⊕6A2g⊕9B1g⊕9B2g⊕15Eg
n (l=11) 23 2048-pole 2A1u⊕3A2u⊕3B1u⊕3B2u⊕6Eu 78 6A1u⊕12A2u⊕9B1u⊕9B2u⊕21Eu
o (l=12) 25 4096-pole 4A1g⊕3A2g⊕3B1g⊕3B2g⊕6Eg 91 16A1g⊕9A2g⊕12B1g⊕12B2g⊕21Eg
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 D4h
L 2L+1 Term Splitting
S (L=0) 1 A1g
P (L=1) 3 A2g⊕Eg
D (L=2) 5 A1g⊕B1g⊕B2g⊕Eg
F (L=3) 7 A2g⊕B1g⊕B2g⊕2Eg
G (L=4) 9 2A1g⊕A2g⊕B1g⊕B2g⊕2Eg
H (L=5) 11 A1g⊕2A2g⊕B1g⊕B2g⊕3Eg
I (L=6) 13 2A1g⊕A2g⊕2B1g⊕2B2g⊕3Eg


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