Point Group D5d



D5d E 2C5 2(C5)2 5C'2 i 2(S10)3 2S10 d
A1g 1 1 1 1 1 1 1 1
A2g 1 1 1 -1 1 1 1 -1
E1g 2 2cos(2π/5) 2cos(4π/5) 0 2 2cos(2π/5) 2cos(4π/5) 0
E2g 2 2cos(4π/5) 2cos(2π/5) 0 2 2cos(4π/5) 2cos(2π/5) 0
A1u 1 1 1 1 -1 -1 -1 -1
A2u 1 1 1 -1 -1 -1 -1 1
E1u 2 2cos(2π/5) 2cos(4π/5) 0 -2 -2cos(2π/5) -2cos(4π/5) 0
E2u 2 2cos(4π/5) 2cos(2π/5) 0 -2 -2cos(4π/5) -2cos(2π/5) 0


Additional information

Number of symmetry elements h = 20
Number of classes, irreps n = 8
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E 2C5 2(C5)2 5C'2 i 2(S10)3 2S10 d



Genrate representation from irreducible representations


A1g A2g E1g E2g A1u A2u E1u E2u




Examples

Ferrocene (staggered)



Direct products of irreducible representations


Binary products
A1g A2g E1g E2g A1u A2u E1u E2u
A1g A1g
A2g A2gA1g
E1g E1gE1gA1g⊕A2g⊕E2g
E2g E2gE2gE1g⊕E2gA1g⊕A2g⊕E1g
A1u A1uA2uE1uE2uA1g
A2u A2uA1uE1uE2uA2gA1g
E1u E1uE1uA1u⊕A2u⊕E2uE1u⊕E2uE1gE1gA1g⊕A2g⊕E2g
E2u E2uE2uE1u⊕E2uA1u⊕A2u⊕E1uE2gE2gE1g⊕E2gA1g⊕A2g⊕E1g

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1g A1g⊕E2gE1g⊕E2gA1g⊕E1g⊕E2gA1g⊕A2g⊕E1g⊕E2gA1g⊕2E1g⊕E2gMore
E2g A1g⊕E1gE1g⊕E2gA1g⊕E1g⊕E2gA1g⊕A2g⊕E1g⊕E2gA1g⊕E1g⊕2E2gMore
E1u A1g⊕E2gE1u⊕E2uA1g⊕E1g⊕E2gA1u⊕A2u⊕E1u⊕E2uA1g⊕2E1g⊕E2gMore
E2u A1g⊕E1gE1u⊕E2uA1g⊕E1g⊕E2gA1u⊕A2u⊕E1u⊕E2uA1g⊕E1g⊕2E2gMore



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⊕E1u 3 A2u⊕E1u
d (l=2) 5 Quadrupole A1g⊕E1g⊕E2g 6 2A1g⊕E1g⊕E2g
f (l=3) 7 Octupole A2u⊕E1u⊕2E2u 10 2A2u⊕2E1u⊕2E2u
g (l=4) 9 Hexadecapole A1g⊕2E1g⊕2E2g 15 3A1g⊕3E1g⊕3E2g
h (l=5) 11 Dotricontapole A1u⊕2A2u⊕2E1u⊕2E2u 21 A1u⊕4A2u⊕4E1u⊕4E2u
i (l=6) 13 Tetrahexacontapole 2A1g⊕A2g⊕3E1g⊕2E2g 28 5A1g⊕A2g⊕6E1g⊕5E2g
j (l=7) 15 Octacosahectapole A1u⊕2A2u⊕3E1u⊕3E2u 36 2A1u⊕6A2u⊕7E1u⊕7E2u
k (l=8) 17 256-pole 2A1g⊕A2g⊕3E1g⊕4E2g 45 7A1g⊕2A2g⊕9E1g⊕9E2g
l (l=9) 19 512-pole A1u⊕2A2u⊕4E1u⊕4E2u 55 3A1u⊕8A2u⊕11E1u⊕11E2u
m (l=10) 21 1024-pole 3A1g⊕2A2g⊕4E1g⊕4E2g 66 10A1g⊕4A2g⊕13E1g⊕13E2g
n (l=11) 23 2048-pole 2A1u⊕3A2u⊕5E1u⊕4E2u 78 5A1u⊕11A2u⊕16E1u⊕15E2u
o (l=12) 25 4096-pole 3A1g⊕2A2g⊕5E1g⊕5E2g 91 13A1g⊕6A2g⊕18E1g⊕18E2g
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 D5d
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⊕2E2g
G (L=4) 9 A1g⊕2E1g⊕2E2g
H (L=5) 11 A1g⊕2A2g⊕2E1g⊕2E2g
I (L=6) 13 2A1g⊕A2g⊕3E1g⊕2E2g


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