Point Group D20h



D20h E 2C20 2C10 2(C20)3 2C5 2C4 2(C10)3 2(C20)7 2(C5)2 2(C20)9 C2 10C'2 10C''2 i 2(S20)9 2(S5)3 2(S20)7 2(S10)3 2S4 2S5 2(S20)3 2S10 2S20 σh 10σv 10σd
A1g 1 1 1 1 1 1 1 1 1 1 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 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 -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 -1 1 -1 1 -1 1 -1 1 -1 1
E1g 2 2cos(π/10) 2cos(π/5) 2cos(3π/10) 2cos(2π/5) 0 -2cos(2π/5) -2cos(3π/10) -2cos(π/5) -2cos(π/10) -2 0 0 2 2cos(π/10) 2cos(π/5) 2cos(3π/10) 2cos(2π/5) 0 -2cos(2π/5) -2cos(3π/10) -2cos(π/5) -2cos(π/10) -2 0 0
E2g 2 2cos(π/5) 2cos(2π/5) -2cos(2π/5) -2cos(π/5) -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 -2cos(π/5) -2cos(2π/5) 2cos(2π/5) 2cos(π/5) 2 0 0
E3g 2 2cos(3π/10) -2cos(2π/5) -2cos(π/10) -2cos(π/5) 0 2cos(π/5) 2cos(π/10) 2cos(2π/5) -2cos(3π/10) -2 0 0 2 2cos(3π/10) -2cos(2π/5) -2cos(π/10) -2cos(π/5) 0 2cos(π/5) 2cos(π/10) 2cos(2π/5) -2cos(3π/10) -2 0 0
E4g 2 2cos(2π/5) -2cos(π/5) -2cos(π/5) 2cos(2π/5) 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 2cos(2π/5) -2cos(π/5) -2cos(π/5) 2cos(2π/5) 2 0 0
E5g 2 0 -2 0 2 0 -2 0 2 0 -2 0 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 0
E6g 2 -2cos(2π/5) -2cos(π/5) 2cos(π/5) 2cos(2π/5) -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 2cos(2π/5) 2cos(π/5) -2cos(π/5) -2cos(2π/5) 2 0 0
E7g 2 -2cos(3π/10) -2cos(2π/5) 2cos(π/10) -2cos(π/5) 0 2cos(π/5) -2cos(π/10) 2cos(2π/5) 2cos(3π/10) -2 0 0 2 -2cos(3π/10) -2cos(2π/5) 2cos(π/10) -2cos(π/5) 0 2cos(π/5) -2cos(π/10) 2cos(2π/5) 2cos(3π/10) -2 0 0
E8g 2 -2cos(π/5) 2cos(2π/5) 2cos(2π/5) -2cos(π/5) 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 -2cos(π/5) 2cos(2π/5) 2cos(2π/5) -2cos(π/5) 2 0 0
E9g 2 -2cos(π/10) 2cos(π/5) -2cos(3π/10) 2cos(2π/5) 0 -2cos(2π/5) 2cos(3π/10) -2cos(π/5) 2cos(π/10) -2 0 0 2 -2cos(π/10) 2cos(π/5) -2cos(3π/10) 2cos(2π/5) 0 -2cos(2π/5) 2cos(3π/10) -2cos(π/5) 2cos(π/10) -2 0 0
A1u 1 1 1 1 1 1 1 1 1 1 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 -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 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 1 -1 1 -1 1 -1 1 -1 1 -1
E1u 2 2cos(π/10) 2cos(π/5) 2cos(3π/10) 2cos(2π/5) 0 -2cos(2π/5) -2cos(3π/10) -2cos(π/5) -2cos(π/10) -2 0 0 -2 -2cos(π/10) -2cos(π/5) -2cos(3π/10) -2cos(2π/5) 0 2cos(2π/5) 2cos(3π/10) 2cos(π/5) 2cos(π/10) 2 0 0
E2u 2 2cos(π/5) 2cos(2π/5) -2cos(2π/5) -2cos(π/5) -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 2cos(π/5) 2cos(2π/5) -2cos(2π/5) -2cos(π/5) -2 0 0
E3u 2 2cos(3π/10) -2cos(2π/5) -2cos(π/10) -2cos(π/5) 0 2cos(π/5) 2cos(π/10) 2cos(2π/5) -2cos(3π/10) -2 0 0 -2 -2cos(3π/10) 2cos(2π/5) 2cos(π/10) 2cos(π/5) 0 -2cos(π/5) -2cos(π/10) -2cos(2π/5) 2cos(3π/10) 2 0 0
E4u 2 2cos(2π/5) -2cos(π/5) -2cos(π/5) 2cos(2π/5) 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 -2cos(2π/5) 2cos(π/5) 2cos(π/5) -2cos(2π/5) -2 0 0
E5u 2 0 -2 0 2 0 -2 0 2 0 -2 0 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 0
E6u 2 -2cos(2π/5) -2cos(π/5) 2cos(π/5) 2cos(2π/5) -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 -2cos(2π/5) -2cos(π/5) 2cos(π/5) 2cos(2π/5) -2 0 0
E7u 2 -2cos(3π/10) -2cos(2π/5) 2cos(π/10) -2cos(π/5) 0 2cos(π/5) -2cos(π/10) 2cos(2π/5) 2cos(3π/10) -2 0 0 -2 2cos(3π/10) 2cos(2π/5) -2cos(π/10) 2cos(π/5) 0 -2cos(π/5) 2cos(π/10) -2cos(2π/5) -2cos(3π/10) 2 0 0
E8u 2 -2cos(π/5) 2cos(2π/5) 2cos(2π/5) -2cos(π/5) 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 2cos(π/5) -2cos(2π/5) -2cos(2π/5) 2cos(π/5) -2 0 0
E9u 2 -2cos(π/10) 2cos(π/5) -2cos(3π/10) 2cos(2π/5) 0 -2cos(2π/5) 2cos(3π/10) -2cos(π/5) 2cos(π/10) -2 0 0 -2 2cos(π/10) -2cos(π/5) 2cos(3π/10) -2cos(2π/5) 0 2cos(2π/5) -2cos(3π/10) 2cos(π/5) -2cos(π/10) 2 0 0


Additional information

Number of symmetry elements h = 80
Number of classes, irreps n = 26
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E 2C20 2C10 2(C20)3 2C5 2C4 2(C10)3 2(C20)7 2(C5)2 2(C20)9 C2 10C'2 10C''2 i 2(S20)9 2(S5)3 2(S20)7 2(S10)3 2S4 2S5 2(S20)3 2S10 2S20 σh 10σv 10σd



Genrate representation from irreducible representations


A1g A2g B1g B2g E1g E2g E3g E4g E5g E6g E7g E8g E9g A1u A2u B1u B2u E1u E2u E3u E4u E5u E6u E7u E8u E9u




Direct products of irreducible representations


Binary products
A1g A2g B1g B2g E1g E2g E3g E4g E5g E6g E7g E8g E9g A1u A2u B1u B2u E1u E2u E3u E4u E5u E6u E7u E8u E9u
A1g A1g
A2g A2gA1g
B1g B1gB2gA1g
B2g B2gB1gA2gA1g
E1g E1gE1gE9gE9gA1g⊕A2g⊕E2g
E2g E2gE2gE8gE8gE1g⊕E3gA1g⊕A2g⊕E4g
E3g E3gE3gE7gE7gE2g⊕E4gE1g⊕E5gA1g⊕A2g⊕E6g
E4g E4gE4gE6gE6gE3g⊕E5gE2g⊕E6gE1g⊕E7gA1g⊕A2g⊕E8g
E5g E5gE5gE5gE5gE4g⊕E6gE3g⊕E7gE2g⊕E8gE1g⊕E9gA1g⊕A2g⊕B1g⊕B2g
E6g E6gE6gE4gE4gE5g⊕E7gE4g⊕E8gE3g⊕E9gB1g⊕B2g⊕E2gE1g⊕E9gA1g⊕A2g⊕E8g
E7g E7gE7gE3gE3gE6g⊕E8gE5g⊕E9gB1g⊕B2g⊕E4gE3g⊕E9gE2g⊕E8gE1g⊕E7gA1g⊕A2g⊕E6g
E8g E8gE8gE2gE2gE7g⊕E9gB1g⊕B2g⊕E6gE5g⊕E9gE4g⊕E8gE3g⊕E7gE2g⊕E6gE1g⊕E5gA1g⊕A2g⊕E4g
E9g E9gE9gE1gE1gB1g⊕B2g⊕E8gE7g⊕E9gE6g⊕E8gE5g⊕E7gE4g⊕E6gE3g⊕E5gE2g⊕E4gE1g⊕E3gA1g⊕A2g⊕E2g
A1u A1uA2uB1uB2uE1uE2uE3uE4uE5uE6uE7uE8uE9uA1g
A2u A2uA1uB2uB1uE1uE2uE3uE4uE5uE6uE7uE8uE9uA2gA1g
B1u B1uB2uA1uA2uE9uE8uE7uE6uE5uE4uE3uE2uE1uB1gB2gA1g
B2u B2uB1uA2uA1uE9uE8uE7uE6uE5uE4uE3uE2uE1uB2gB1gA2gA1g
E1u E1uE1uE9uE9uA1u⊕A2u⊕E2uE1u⊕E3uE2u⊕E4uE3u⊕E5uE4u⊕E6uE5u⊕E7uE6u⊕E8uE7u⊕E9uB1u⊕B2u⊕E8uE1gE1gE9gE9gA1g⊕A2g⊕E2g
E2u E2uE2uE8uE8uE1u⊕E3uA1u⊕A2u⊕E4uE1u⊕E5uE2u⊕E6uE3u⊕E7uE4u⊕E8uE5u⊕E9uB1u⊕B2u⊕E6uE7u⊕E9uE2gE2gE8gE8gE1g⊕E3gA1g⊕A2g⊕E4g
E3u E3uE3uE7uE7uE2u⊕E4uE1u⊕E5uA1u⊕A2u⊕E6uE1u⊕E7uE2u⊕E8uE3u⊕E9uB1u⊕B2u⊕E4uE5u⊕E9uE6u⊕E8uE3gE3gE7gE7gE2g⊕E4gE1g⊕E5gA1g⊕A2g⊕E6g
E4u E4uE4uE6uE6uE3u⊕E5uE2u⊕E6uE1u⊕E7uA1u⊕A2u⊕E8uE1u⊕E9uB1u⊕B2u⊕E2uE3u⊕E9uE4u⊕E8uE5u⊕E7uE4gE4gE6gE6gE3g⊕E5gE2g⊕E6gE1g⊕E7gA1g⊕A2g⊕E8g
E5u E5uE5uE5uE5uE4u⊕E6uE3u⊕E7uE2u⊕E8uE1u⊕E9uA1u⊕A2u⊕B1u⊕B2uE1u⊕E9uE2u⊕E8uE3u⊕E7uE4u⊕E6uE5gE5gE5gE5gE4g⊕E6gE3g⊕E7gE2g⊕E8gE1g⊕E9gA1g⊕A2g⊕B1g⊕B2g
E6u E6uE6uE4uE4uE5u⊕E7uE4u⊕E8uE3u⊕E9uB1u⊕B2u⊕E2uE1u⊕E9uA1u⊕A2u⊕E8uE1u⊕E7uE2u⊕E6uE3u⊕E5uE6gE6gE4gE4gE5g⊕E7gE4g⊕E8gE3g⊕E9gB1g⊕B2g⊕E2gE1g⊕E9gA1g⊕A2g⊕E8g
E7u E7uE7uE3uE3uE6u⊕E8uE5u⊕E9uB1u⊕B2u⊕E4uE3u⊕E9uE2u⊕E8uE1u⊕E7uA1u⊕A2u⊕E6uE1u⊕E5uE2u⊕E4uE7gE7gE3gE3gE6g⊕E8gE5g⊕E9gB1g⊕B2g⊕E4gE3g⊕E9gE2g⊕E8gE1g⊕E7gA1g⊕A2g⊕E6g
E8u E8uE8uE2uE2uE7u⊕E9uB1u⊕B2u⊕E6uE5u⊕E9uE4u⊕E8uE3u⊕E7uE2u⊕E6uE1u⊕E5uA1u⊕A2u⊕E4uE1u⊕E3uE8gE8gE2gE2gE7g⊕E9gB1g⊕B2g⊕E6gE5g⊕E9gE4g⊕E8gE3g⊕E7gE2g⊕E6gE1g⊕E5gA1g⊕A2g⊕E4g
E9u E9uE9uE1uE1uB1u⊕B2u⊕E8uE7u⊕E9uE6u⊕E8uE5u⊕E7uE4u⊕E6uE3u⊕E5uE2u⊕E4uE1u⊕E3uA1u⊕A2u⊕E2uE9gE9gE1gE1gB1g⊕B2g⊕E8gE7g⊕E9gE6g⊕E8gE5g⊕E7gE4g⊕E6gE3g⊕E5gE2g⊕E4gE1g⊕E3gA1g⊕A2g⊕E2g

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1g A1g⊕E2gE1g⊕E3gA1g⊕E2g⊕E4gE1g⊕E3g⊕E5gA1g⊕E2g⊕E4g⊕E6gMore
E2g A1g⊕E4gE2g⊕E6gA1g⊕E4g⊕E8gB1g⊕B2g⊕E2g⊕E6gA1g⊕E4g⊕2E8gMore
E3g A1g⊕E6gE3g⊕E9gA1g⊕E6g⊕E8gE3g⊕E5g⊕E9gA1g⊕E2g⊕E6g⊕E8gMore
E4g A1g⊕E8gE4g⊕E8gA1g⊕E4g⊕E8gA1g⊕A2g⊕E4g⊕E8gA1g⊕2E4g⊕E8gMore
E5g A1g⊕B1g⊕B2g2E5g2A1g⊕A2g⊕B1g⊕B2g3E5g2A1g⊕A2g⊕2B1g⊕2B2gMore
E6g A1g⊕E8gE2g⊕E6gA1g⊕E4g⊕E8gB1g⊕B2g⊕E2g⊕E6gA1g⊕2E4g⊕E8gMore
E7g A1g⊕E6gE1g⊕E7gA1g⊕E6g⊕E8gE1g⊕E5g⊕E7gA1g⊕E2g⊕E6g⊕E8gMore
E8g A1g⊕E4gE4g⊕E8gA1g⊕E4g⊕E8gA1g⊕A2g⊕E4g⊕E8gA1g⊕E4g⊕2E8gMore
E9g A1g⊕E2gE7g⊕E9gA1g⊕E2g⊕E4gE5g⊕E7g⊕E9gA1g⊕E2g⊕E4g⊕E6gMore
E1u A1g⊕E2gE1u⊕E3uA1g⊕E2g⊕E4gE1u⊕E3u⊕E5uA1g⊕E2g⊕E4g⊕E6gMore
E2u A1g⊕E4gE2u⊕E6uA1g⊕E4g⊕E8gB1u⊕B2u⊕E2u⊕E6uA1g⊕E4g⊕2E8gMore
E3u A1g⊕E6gE3u⊕E9uA1g⊕E6g⊕E8gE3u⊕E5u⊕E9uA1g⊕E2g⊕E6g⊕E8gMore
E4u A1g⊕E8gE4u⊕E8uA1g⊕E4g⊕E8gA1u⊕A2u⊕E4u⊕E8uA1g⊕2E4g⊕E8gMore
E5u A1g⊕B1g⊕B2g2E5u2A1g⊕A2g⊕B1g⊕B2g3E5u2A1g⊕A2g⊕2B1g⊕2B2gMore
E6u A1g⊕E8gE2u⊕E6uA1g⊕E4g⊕E8gB1u⊕B2u⊕E2u⊕E6uA1g⊕2E4g⊕E8gMore
E7u A1g⊕E6gE1u⊕E7uA1g⊕E6g⊕E8gE1u⊕E5u⊕E7uA1g⊕E2g⊕E6g⊕E8gMore
E8u A1g⊕E4gE4u⊕E8uA1g⊕E4g⊕E8gA1u⊕A2u⊕E4u⊕E8uA1g⊕E4g⊕2E8gMore
E9u A1g⊕E2gE7u⊕E9uA1g⊕E2g⊕E4gE5u⊕E7u⊕E9uA1g⊕E2g⊕E4g⊕E6gMore



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⊕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⊕E1u⊕E2u⊕E3u⊕E4u⊕E5u 21 3A2u⊕3E1u⊕2E2u⊕2E3u⊕E4u⊕E5u
i (l=6) 13 Tetrahexacontapole A1g⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g⊕E6g 28 4A1g⊕3E1g⊕3E2g⊕2E3g⊕2E4g⊕E5g⊕E6g
j (l=7) 15 Octacosahectapole A2u⊕E1u⊕E2u⊕E3u⊕E4u⊕E5u⊕E6u⊕E7u 36 4A2u⊕4E1u⊕3E2u⊕3E3u⊕2E4u⊕2E5u⊕E6u⊕E7u
k (l=8) 17 256-pole A1g⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g⊕E6g⊕E7g⊕E8g 45 5A1g⊕4E1g⊕4E2g⊕3E3g⊕3E4g⊕2E5g⊕2E6g⊕E7g⊕E8g
l (l=9) 19 512-pole A2u⊕E1u⊕E2u⊕E3u⊕E4u⊕E5u⊕E6u⊕E7u⊕E8u⊕E9u 55 5A2u⊕5E1u⊕4E2u⊕4E3u⊕3E4u⊕3E5u⊕2E6u⊕2E7u⊕E8u⊕E9u
m (l=10) 21 1024-pole A1g⊕B1g⊕B2g⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g⊕E6g⊕E7g⊕E8g⊕E9g 66 6A1g⊕B1g⊕B2g⊕5E1g⊕5E2g⊕4E3g⊕4E4g⊕3E5g⊕3E6g⊕2E7g⊕2E8g⊕E9g
n (l=11) 23 2048-pole A2u⊕B1u⊕B2u⊕E1u⊕E2u⊕E3u⊕E4u⊕E5u⊕E6u⊕E7u⊕E8u⊕2E9u 78 6A2u⊕B1u⊕B2u⊕6E1u⊕5E2u⊕5E3u⊕4E4u⊕4E5u⊕3E6u⊕3E7u⊕2E8u⊕3E9u
o (l=12) 25 4096-pole A1g⊕B1g⊕B2g⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g⊕E6g⊕E7g⊕2E8g⊕2E9g 91 7A1g⊕2B1g⊕2B2g⊕6E1g⊕6E2g⊕5E3g⊕5E4g⊕4E5g⊕4E6g⊕3E7g⊕4E8g⊕3E9g
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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 D20h
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⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g
I (L=6) 13 A1g⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g⊕E6g


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