Point Group D7h



D7h E 2C7 2(C7)2 2(C7)3 7C'2 σh 2S7 2(S7)5 2(S7)3 v
A'1 1 1 1 1 1 1 1 1 1 1
A'2 1 1 1 1 -1 1 1 1 1 -1
E'1 2 2cos(2π/7) 2cos(4π/7) 2cos(6π/7) 0 2 2cos(2π/7) 2cos(4π/7) 2cos(6π/7) 0
E'2 2 2cos(4π/7) 2cos(6π/7) 2cos(2π/7) 0 2 2cos(4π/7) 2cos(6π/7) 2cos(2π/7) 0
E'3 2 2cos(6π/7) 2cos(2π/7) 2cos(4π/7) 0 2 2cos(6π/7) 2cos(2π/7) 2cos(4π/7) 0
A''1 1 1 1 1 1 -1 -1 -1 -1 -1
A''2 1 1 1 1 -1 -1 -1 -1 -1 1
E''1 2 2cos(2π/7) 2cos(4π/7) 2cos(6π/7) 0 -2 -2cos(2π/7) -2cos(4π/7) -2cos(6π/7) 0
E''2 2 2cos(4π/7) 2cos(6π/7) 2cos(2π/7) 0 -2 -2cos(4π/7) -2cos(6π/7) -2cos(2π/7) 0
E''3 2 2cos(6π/7) 2cos(2π/7) 2cos(4π/7) 0 -2 -2cos(6π/7) -2cos(2π/7) -2cos(4π/7) 0


Additional information

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


Reduce representation to irreducible representations


E 2C7 2(C7)2 2(C7)3 7C'2 σh 2S7 2(S7)5 2(S7)3 v



Genrate representation from irreducible representations


A'1 A'2 E'1 E'2 E'3 A''1 A''2 E''1 E''2 E''3




Examples

Tropylium Cation Cucurbit[7]uril



Direct products of irreducible representations


Binary products
A'1 A'2 E'1 E'2 E'3 A''1 A''2 E''1 E''2 E''3
A'1 A'1
A'2 A'2A'1
E'1 E'1E'1A'1⊕A'2⊕E'2
E'2 E'2E'2E'1⊕E'3A'1⊕A'2⊕E'3
E'3 E'3E'3E'2⊕E'3E'1⊕E'2A'1⊕A'2⊕E'1
A''1 A''1A''2E''1E''2E''3A'1
A''2 A''2A''1E''1E''2E''3A'2A'1
E''1 E''1E''1A''1⊕A''2⊕E''2E''1⊕E''3E''2⊕E''3E'1E'1A'1⊕A'2⊕E'2
E''2 E''2E''2E''1⊕E''3A''1⊕A''2⊕E''3E''1⊕E''2E'2E'2E'1⊕E'3A'1⊕A'2⊕E'3
E''3 E''3E''3E''2⊕E''3E''1⊕E''2A''1⊕A''2⊕E''1E'3E'3E'2⊕E'3E'1⊕E'2A'1⊕A'2⊕E'1

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E'1 A'1⊕E'2E'1⊕E'3A'1⊕E'2⊕E'3E'1⊕E'2⊕E'3A'1⊕E'1⊕E'2⊕E'3More
E'2 A'1⊕E'3E'1⊕E'2A'1⊕E'1⊕E'3E'1⊕E'2⊕E'3A'1⊕E'1⊕E'2⊕E'3More
E'3 A'1⊕E'1E'2⊕E'3A'1⊕E'1⊕E'2E'1⊕E'2⊕E'3A'1⊕E'1⊕E'2⊕E'3More
E''1 A'1⊕E'2E''1⊕E''3A'1⊕E'2⊕E'3E''1⊕E''2⊕E''3A'1⊕E'1⊕E'2⊕E'3More
E''2 A'1⊕E'3E''1⊕E''2A'1⊕E'1⊕E'3E''1⊕E''2⊕E''3A'1⊕E'1⊕E'2⊕E'3More
E''3 A'1⊕E'1E''2⊕E''3A'1⊕E'1⊕E'2E''1⊕E''2⊕E''3A'1⊕E'1⊕E'2⊕E'3More



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 A'1 1 A'1
p (l=1) 3 Dipole E'1⊕A''2 3 E'1⊕A''2
d (l=2) 5 Quadrupole A'1⊕E'2⊕E''1 6 2A'1⊕E'2⊕E''1
f (l=3) 7 Octupole E'1⊕E'3⊕A''2⊕E''2 10 2E'1⊕E'3⊕2A''2⊕E''2
g (l=4) 9 Hexadecapole A'1⊕E'2⊕E'3⊕E''1⊕E''3 15 3A'1⊕2E'2⊕E'3⊕2E''1⊕E''3
h (l=5) 11 Dotricontapole E'1⊕E'2⊕E'3⊕A''2⊕E''2⊕E''3 21 3E'1⊕E'2⊕2E'3⊕3A''2⊕2E''2⊕E''3
i (l=6) 13 Tetrahexacontapole A'1⊕E'1⊕E'2⊕E'3⊕E''1⊕E''2⊕E''3 28 4A'1⊕E'1⊕3E'2⊕2E'3⊕3E''1⊕E''2⊕2E''3
j (l=7) 15 Octacosahectapole A'1⊕A'2⊕E'1⊕E'2⊕E'3⊕A''2⊕E''1⊕E''2⊕E''3 36 A'1⊕A'2⊕4E'1⊕2E'2⊕3E'3⊕4A''2⊕E''1⊕3E''2⊕2E''3
k (l=8) 17 256-pole A'1⊕2E'1⊕E'2⊕E'3⊕A''1⊕A''2⊕E''1⊕E''2⊕E''3 45 5A'1⊕3E'1⊕4E'2⊕3E'3⊕A''1⊕A''2⊕4E''1⊕2E''2⊕3E''3
l (l=9) 19 512-pole A'1⊕A'2⊕E'1⊕2E'2⊕E'3⊕A''2⊕2E''1⊕E''2⊕E''3 55 2A'1⊕2A'2⊕5E'1⊕4E'2⊕4E'3⊕5A''2⊕3E''1⊕4E''2⊕3E''3
m (l=10) 21 1024-pole A'1⊕2E'1⊕E'2⊕2E'3⊕A''1⊕A''2⊕E''1⊕2E''2⊕E''3 66 6A'1⊕5E'1⊕5E'2⊕5E'3⊕2A''1⊕2A''2⊕5E''1⊕4E''2⊕4E''3
n (l=11) 23 2048-pole A'1⊕A'2⊕E'1⊕2E'2⊕2E'3⊕A''2⊕2E''1⊕E''2⊕2E''3 78 3A'1⊕3A'2⊕6E'1⊕6E'2⊕6E'3⊕6A''2⊕5E''1⊕5E''2⊕5E''3
o (l=12) 25 4096-pole A'1⊕2E'1⊕2E'2⊕2E'3⊕A''1⊕A''2⊕E''1⊕2E''2⊕2E''3 91 7A'1⊕7E'1⊕7E'2⊕7E'3⊕3A''1⊕3A''2⊕6E''1⊕6E''2⊕6E''3
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 D7h
L 2L+1 Term Splitting
S (L=0) 1 A'1
P (L=1) 3 A'2⊕E''1
D (L=2) 5 A'1⊕E'2⊕E''1
F (L=3) 7 A'2⊕E'2⊕E''1⊕E''3
G (L=4) 9 A'1⊕E'2⊕E'3⊕E''1⊕E''3
H (L=5) 11 A'2⊕E'2⊕E'3⊕E''1⊕E''2⊕E''3
I (L=6) 13 A'1⊕E'1⊕E'2⊕E'3⊕E''1⊕E''2⊕E''3


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