Point Group D10h



D10h E 2C10 2C5 2(C10)3 2(C5)2 C2 5C'2 5C''2 i 2(S5)3 2(S10)3 2S5 2S10 σh 5σd 5σv
A1g 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A2g 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1
B1g 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
B2g 1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 1 -1 -1 1
E1g 2 2cos(π/5) 2cos(2π/5) -2cos(2π/5) -2cos(π/5) -2 0 0 2 2cos(π/5) 2cos(2π/5) -2cos(2π/5) -2cos(π/5) -2 0 0
E2g 2 2cos(2π/5) -2cos(π/5) -2cos(π/5) 2cos(2π/5) 2 0 0 2 2cos(2π/5) -2cos(π/5) -2cos(π/5) 2cos(2π/5) 2 0 0
E3g 2 -2cos(2π/5) -2cos(π/5) 2cos(π/5) 2cos(2π/5) -2 0 0 2 -2cos(2π/5) -2cos(π/5) 2cos(π/5) 2cos(2π/5) -2 0 0
E4g 2 -2cos(π/5) 2cos(2π/5) 2cos(2π/5) -2cos(π/5) 2 0 0 2 -2cos(π/5) 2cos(2π/5) 2cos(2π/5) -2cos(π/5) 2 0 0
A1u 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1
A2u 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1
B1u 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1
B2u 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1
E1u 2 2cos(π/5) 2cos(2π/5) -2cos(2π/5) -2cos(π/5) -2 0 0 -2 -2cos(π/5) -2cos(2π/5) 2cos(2π/5) 2cos(π/5) 2 0 0
E2u 2 2cos(2π/5) -2cos(π/5) -2cos(π/5) 2cos(2π/5) 2 0 0 -2 -2cos(2π/5) 2cos(π/5) 2cos(π/5) -2cos(2π/5) -2 0 0
E3u 2 -2cos(2π/5) -2cos(π/5) 2cos(π/5) 2cos(2π/5) -2 0 0 -2 2cos(2π/5) 2cos(π/5) -2cos(π/5) -2cos(2π/5) 2 0 0
E4u 2 -2cos(π/5) 2cos(2π/5) 2cos(2π/5) -2cos(π/5) 2 0 0 -2 2cos(π/5) -2cos(2π/5) -2cos(2π/5) 2cos(π/5) -2 0 0
View A
✅
View B


Additional information

Number of symmetry elements h = 40
Number of classes, irreps n = 16
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E 2C10 2C5 2(C10)3 2(C5)2 C2 5C'2 5C''2 i 2(S5)3 2(S10)3 2S5 2S10 σh 5σd 5σv



Genrate representation from irreducible representations


A1g A2g B1g B2g E1g E2g E3g E4g A1u A2u B1u B2u E1u E2u E3u E4u



Direct products of irreducible representations


Binary products
⊙ A1g A2g B1g B2g E1g E2g E3g E4g A1u A2u B1u B2u E1u E2u E3u E4u
A1g A1g
A2g A2gA1g
B1g B1gB2gA1g
B2g B2gB1gA2gA1g
E1g E1gE1gE4gE4gA1g⊕A2g⊕E2g
E2g E2gE2gE3gE3gE1g⊕E3gA1g⊕A2g⊕E4g
E3g E3gE3gE2gE2gE2g⊕E4gB1g⊕B2g⊕E1gA1g⊕A2g⊕E4g
E4g E4gE4gE1gE1gB1g⊕B2g⊕E3gE2g⊕E4gE1g⊕E3gA1g⊕A2g⊕E2g
A1u A1uA2uB1uB2uE1uE2uE3uE4uA1g
A2u A2uA1uB2uB1uE1uE2uE3uE4uA2gA1g
B1u B1uB2uA1uA2uE4uE3uE2uE1uB1gB2gA1g
B2u B2uB1uA2uA1uE4uE3uE2uE1uB2gB1gA2gA1g
E1u E1uE1uE4uE4uA1u⊕A2u⊕E2uE1u⊕E3uE2u⊕E4uB1u⊕B2u⊕E3uE1gE1gE4gE4gA1g⊕A2g⊕E2g
E2u E2uE2uE3uE3uE1u⊕E3uA1u⊕A2u⊕E4uB1u⊕B2u⊕E1uE2u⊕E4uE2gE2gE3gE3gE1g⊕E3gA1g⊕A2g⊕E4g
E3u E3uE3uE2uE2uE2u⊕E4uB1u⊕B2u⊕E1uA1u⊕A2u⊕E4uE1u⊕E3uE3gE3gE2gE2gE2g⊕E4gB1g⊕B2g⊕E1gA1g⊕A2g⊕E4g
E4u E4uE4uE1uE1uB1u⊕B2u⊕E3uE2u⊕E4uE1u⊕E3uA1u⊕A2u⊕E2uE4gE4gE1gE1gB1g⊕B2g⊕E3gE2g⊕E4gE1g⊕E3gA1g⊕A2g⊕E2g

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
E1g A1g⊕E2gE1g⊕E3gA1g⊕E2g⊕E4gB1g⊕B2g⊕E1g⊕E3gA1g⊕E2g⊕2E4gMore
E2g A1g⊕E4gE2g⊕E4gA1g⊕E2g⊕E4gA1g⊕A2g⊕E2g⊕E4gA1g⊕2E2g⊕E4gMore
E3g A1g⊕E4gE1g⊕E3gA1g⊕E2g⊕E4gB1g⊕B2g⊕E1g⊕E3gA1g⊕2E2g⊕E4gMore
E4g A1g⊕E2gE2g⊕E4gA1g⊕E2g⊕E4gA1g⊕A2g⊕E2g⊕E4gA1g⊕E2g⊕2E4gMore
E1u A1g⊕E2gE1u⊕E3uA1g⊕E2g⊕E4gB1u⊕B2u⊕E1u⊕E3uA1g⊕E2g⊕2E4gMore
E2u A1g⊕E4gE2u⊕E4uA1g⊕E2g⊕E4gA1u⊕A2u⊕E2u⊕E4uA1g⊕2E2g⊕E4gMore
E3u A1g⊕E4gE1u⊕E3uA1g⊕E2g⊕E4gB1u⊕B2u⊕E1u⊕E3uA1g⊕2E2g⊕E4gMore
E4u A1g⊕E2gE2u⊕E4uA1g⊕E2g⊕E4gA1u⊕A2u⊕E2u⊕E4uA1g⊕E2g⊕2E4gMore


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⊕E3g⊕E4g 15 3A1g⊕2E1g⊕2E2g⊕E3g⊕E4g
h (l=5) 11 Dotricontapole A2u⊕B1u⊕B2u⊕E1u⊕E2u⊕E3u⊕E4u 21 3A2u⊕B1u⊕B2u⊕3E1u⊕2E2u⊕2E3u⊕E4u
i (l=6) 13 Tetrahexacontapole A1g⊕B1g⊕B2g⊕E1g⊕E2g⊕E3g⊕2E4g 28 4A1g⊕B1g⊕B2g⊕3E1g⊕3E2g⊕2E3g⊕3E4g
j (l=7) 15 Octacosahectapole A2u⊕B1u⊕B2u⊕E1u⊕E2u⊕2E3u⊕2E4u 36 4A2u⊕2B1u⊕2B2u⊕4E1u⊕3E2u⊕4E3u⊕3E4u
k (l=8) 17 256-pole A1g⊕B1g⊕B2g⊕E1g⊕2E2g⊕2E3g⊕2E4g 45 5A1g⊕2B1g⊕2B2g⊕4E1g⊕5E2g⊕4E3g⊕5E4g
l (l=9) 19 512-pole A2u⊕B1u⊕B2u⊕2E1u⊕2E2u⊕2E3u⊕2E4u 55 5A2u⊕3B1u⊕3B2u⊕6E1u⊕5E2u⊕6E3u⊕5E4u
m (l=10) 21 1024-pole 2A1g⊕A2g⊕B1g⊕B2g⊕2E1g⊕2E2g⊕2E3g⊕2E4g 66 7A1g⊕A2g⊕3B1g⊕3B2g⊕6E1g⊕7E2g⊕6E3g⊕7E4g
n (l=11) 23 2048-pole A1u⊕2A2u⊕B1u⊕B2u⊕3E1u⊕2E2u⊕2E3u⊕2E4u 78 A1u⊕7A2u⊕4B1u⊕4B2u⊕9E1u⊕7E2u⊕8E3u⊕7E4u
o (l=12) 25 4096-pole 2A1g⊕A2g⊕B1g⊕B2g⊕3E1g⊕3E2g⊕2E3g⊕2E4g 91 9A1g⊕2A2g⊕4B1g⊕4B2g⊕9E1g⊕10E2g⊕8E3g⊕9E4g
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 D10h
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⊕E3g⊕E4g
H (L=5) 11 A2g⊕B1g⊕B2g⊕E1g⊕E2g⊕E3g⊕E4g
I (L=6) 13 A1g⊕B1g⊕B2g⊕E1g⊕E2g⊕E3g⊕2E4g

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