Point Group T



ε=exp(2πi/3)
T E 4C3 4(C3)2 3C2
A 1 1 1 1
E* 1
1
ε*
ε*
ε*
ε*
1
1
T 3 0 0 -1


Additional information

Number of symmetry elements h = 12
Number of classes, irreps n = 4
Number of real-valued irreducible representations n = 3
Abelian group no
Optical Isomerism (Chirality) yes
Polar no
Parity no


Reduce representation to irreducible representations


E 4C3 4(C3)2 3C2



Genrate representation from irreducible representations


A E* T




Examples

Neopentane (rotated) Fullerene C60F36



Direct products of irreducible representations


Binary products
A E* T
A A
E* E2A⊕E
T T2TA⊕E⊕2T

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E* A⊕E2A⊕EA⊕2E2A⊕2E3A⊕2EMore
T A⊕E⊕TA⊕3T2A⊕2E⊕3TA⊕E⊕6T4A⊕3E⊕6TMore



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

First nonvanshing multipole: Octupole

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 T
L 2L+1 Term Splitting
S (L=0) 1 A
P (L=1) 3 T
D (L=2) 5 E⊕T
F (L=3) 7 A⊕2T
G (L=4) 9 A⊕E⊕2T
H (L=5) 11 E⊕3T
I (L=6) 13 2A⊕E⊕3T


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