Point Group D7

D7 E 2C7 2(C7)2 2(C7)3 7C'2
A1 1 1 1 1 1
A2 1 1 1 1 -1
E1 2 2cos(2π/7) 2cos(4π/7) 2cos(6π/7) 0
E2 2 2cos(4π/7) 2cos(6π/7) 2cos(2π/7) 0
E3 2 2cos(6π/7) 2cos(2π/7) 2cos(4π/7) 0

Additional information

Number of symmetry elements h = 14
Number of classes, irreps n = 5
Abelian group no
Optical Isomerism (Chirality) yes
Polar no
Parity no

Reduce representation to irreducible representations

E 2C7 2(C7)2 2(C7)3 7C'2

Genrate representation from irreducible representations

A1 A2 E1 E2 E3

Direct products of irreducible representations

Binary products
A1 A2 E1 E2 E3
A1 A1
A2 A2A1
E1 E1E1A1⊕A2⊕E2
E2 E2E2E1⊕E3A1⊕A2⊕E3
E3 E3E3E2⊕E3E1⊕E2A1⊕A2⊕E1

Ternary Products
Quaternary Products

Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones

irrep 2] 3] 4] 5] 6]
E1 A1⊕E2E1⊕E3A1⊕E2⊕E3E1⊕E2⊕E3A1⊕E1⊕E2⊕E3More
E2 A1⊕E3E1⊕E2A1⊕E1⊕E3E1⊕E2⊕E3A1⊕E1⊕E2⊕E3More
E3 A1⊕E1E2⊕E3A1⊕E1⊕E2E1⊕E2⊕E3A1⊕E1⊕E2⊕E3More

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 A1 1 A1
p (l=1) 3 Dipole A2⊕E1 3 A2⊕E1
d (l=2) 5 Quadrupole A1⊕E1⊕E2 6 2A1⊕E1⊕E2
f (l=3) 7 Octupole A2⊕E1⊕E2⊕E3 10 2A2⊕2E1⊕E2⊕E3
g (l=4) 9 Hexadecapole A1⊕E1⊕E2⊕2E3 15 3A1⊕2E1⊕2E2⊕2E3
h (l=5) 11 Dotricontapole A2⊕E1⊕2E2⊕2E3 21 3A2⊕3E1⊕3E2⊕3E3
i (l=6) 13 Tetrahexacontapole A1⊕2E1⊕2E2⊕2E3 28 4A1⊕4E1⊕4E2⊕4E3
j (l=7) 15 Octacosahectapole A1⊕2A2⊕2E1⊕2E2⊕2E3 36 A1⊕5A2⊕5E1⊕5E2⊕5E3
k (l=8) 17 256-pole 2A1⊕A2⊕3E1⊕2E2⊕2E3 45 6A1⊕A2⊕7E1⊕6E2⊕6E3
l (l=9) 19 512-pole A1⊕2A2⊕3E1⊕3E2⊕2E3 55 2A1⊕7A2⊕8E1⊕8E2⊕7E3
m (l=10) 21 1024-pole 2A1⊕A2⊕3E1⊕3E2⊕3E3 66 8A1⊕2A2⊕10E1⊕9E2⊕9E3
n (l=11) 23 2048-pole A1⊕2A2⊕3E1⊕3E2⊕4E3 78 3A1⊕9A2⊕11E1⊕11E2⊕11E3
o (l=12) 25 4096-pole 2A1⊕A2⊕3E1⊕4E2⊕4E3 91 10A1⊕3A2⊕13E1⊕13E2⊕13E3

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 D7
L 2L+1 Term Splitting
S (L=0) 1 A1
P (L=1) 3 A2⊕E1
D (L=2) 5 A1⊕E1⊕E2
F (L=3) 7 A2⊕E1⊕E2⊕E3
G (L=4) 9 A1⊕E1⊕E2⊕2E3
H (L=5) 11 A2⊕E1⊕2E2⊕2E3
I (L=6) 13 A1⊕2E1⊕2E2⊕2E3

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