Point Group D7d



D7d E 2C7 2(C7)2 2(C7)3 7C'2 i 2(S14)5 2(S14)3 2S14 7σd
A1g 1 1 1 1 1 1 1 1 1 1
A2g 1 1 1 1 -1 1 1 1 1 -1
E1g 2 1.2470 -0.4450 -1.8019 0 2 1.2470 -0.4450 -1.8019 0
E2g 2 -0.4450 -1.8019 1.2470 0 2 -0.4450 -1.8019 1.2470 0
E3g 2 -1.8019 1.2470 -0.4450 0 2 -1.8019 1.2470 -0.4450 0
A1u 1 1 1 1 1 -1 -1 -1 -1 -1
A2u 1 1 1 1 -1 -1 -1 -1 -1 1
E1u 2 1.2470 -0.4450 -1.8019 0 -2 -1.2470 0.4450 1.8019 0
E2u 2 -0.4450 -1.8019 1.2470 0 -2 0.4450 1.8019 -1.2470 0
E3u 2 -1.8019 1.2470 -0.4450 0 -2 1.8019 -1.2470 0.4450 0
View A
View B
✅


Additional information

Number of symmetry elements h = 28
Number of classes, irreps n = 10
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E 2C7 2(C7)2 2(C7)3 7C'2 i 2(S14)5 2(S14)3 2S14 7σd



Genrate representation from irreducible representations


A1g A2g E1g E2g E3g A1u A2u E1u E2u E3u



Direct products of irreducible representations


Binary products
⊙ A1g A2g E1g E2g E3g A1u A2u E1u E2u E3u
A1g A1g
A2g A2gA1g
E1g E1gE1gA1g⊕A2g⊕E2g
E2g E2gE2gE1g⊕E3gA1g⊕A2g⊕E3g
E3g E3gE3gE2g⊕E3gE1g⊕E2gA1g⊕A2g⊕E1g
A1u A1uA2uE1uE2uE3uA1g
A2u A2uA1uE1uE2uE3uA2gA1g
E1u E1uE1uA1u⊕A2u⊕E2uE1u⊕E3uE2u⊕E3uE1gE1gA1g⊕A2g⊕E2g
E2u E2uE2uE1u⊕E3uA1u⊕A2u⊕E3uE1u⊕E2uE2gE2gE1g⊕E3gA1g⊕A2g⊕E3g
E3u E3uE3uE2u⊕E3uE1u⊕E2uA1u⊕A2u⊕E1uE3gE3gE2g⊕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⊕E3gE1g⊕E2g⊕E3gA1g⊕E1g⊕E2g⊕E3gMore
E2g A1g⊕E3gE1g⊕E2gA1g⊕E1g⊕E3gE1g⊕E2g⊕E3gA1g⊕E1g⊕E2g⊕E3gMore
E3g A1g⊕E1gE2g⊕E3gA1g⊕E1g⊕E2gE1g⊕E2g⊕E3gA1g⊕E1g⊕E2g⊕E3gMore
E1u A1g⊕E2gE1u⊕E3uA1g⊕E2g⊕E3gE1u⊕E2u⊕E3uA1g⊕E1g⊕E2g⊕E3gMore
E2u A1g⊕E3gE1u⊕E2uA1g⊕E1g⊕E3gE1u⊕E2u⊕E3uA1g⊕E1g⊕E2g⊕E3gMore
E3u A1g⊕E1gE2u⊕E3uA1g⊕E1g⊕E2gE1u⊕E2u⊕E3uA1g⊕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⊕2E3g 15 3A1g⊕2E1g⊕2E2g⊕2E3g
h (l=5) 11 Dotricontapole A2u⊕E1u⊕2E2u⊕2E3u 21 3A2u⊕3E1u⊕3E2u⊕3E3u
i (l=6) 13 Tetrahexacontapole A1g⊕2E1g⊕2E2g⊕2E3g 28 4A1g⊕4E1g⊕4E2g⊕4E3g
j (l=7) 15 Octacosahectapole A1u⊕2A2u⊕2E1u⊕2E2u⊕2E3u 36 A1u⊕5A2u⊕5E1u⊕5E2u⊕5E3u
k (l=8) 17 256-pole 2A1g⊕A2g⊕3E1g⊕2E2g⊕2E3g 45 6A1g⊕A2g⊕7E1g⊕6E2g⊕6E3g
l (l=9) 19 512-pole A1u⊕2A2u⊕3E1u⊕3E2u⊕2E3u 55 2A1u⊕7A2u⊕8E1u⊕8E2u⊕7E3u
m (l=10) 21 1024-pole 2A1g⊕A2g⊕3E1g⊕3E2g⊕3E3g 66 8A1g⊕2A2g⊕10E1g⊕9E2g⊕9E3g
n (l=11) 23 2048-pole A1u⊕2A2u⊕3E1u⊕3E2u⊕4E3u 78 3A1u⊕9A2u⊕11E1u⊕11E2u⊕11E3u
o (l=12) 25 4096-pole 2A1g⊕A2g⊕3E1g⊕4E2g⊕4E3g 91 10A1g⊕3A2g⊕13E1g⊕13E2g⊕13E3g
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 D7d
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⊕2E3g
H (L=5) 11 A2g⊕E1g⊕2E2g⊕2E3g
I (L=6) 13 A1g⊕2E1g⊕2E2g⊕2E3g

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