Point Group T
ε=exp(2πi/3)
T
E
4C
3
4(C
3
)
2
3C
2
A
1
1
1
1
E
*
1
1
ε
*
ε
*
ε
*
ε
*
1
1
T
3
0
0
-1
Math View
✅
Phys View
Chem View
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
4C
3
4(C
3
)
2
3C
2
Genrate representation from irreducible representations
A
E
*
T
Examples
Neopentane (rotated)
Fullerene C
60
F
36
Direct products of irreducible representations
Binary products
⊙
A
E
*
T
A
A
E
*
E
2A⊕E
T
T
2T
A⊕E⊕2T
Ternary Products
Quaternary Products
Symmetric powers [Γ
n
] of degenerate irreducible representations
Vibrational overtones
irrep
[Γ
2
]
[Γ
3
]
[Γ
4
]
[Γ
5
]
[Γ
6
]
E
*
A⊕E
2A⊕E
A⊕2E
2A⊕2E
3A⊕2E
More
T
A⊕E⊕T
A⊕3T
2A⊕2E⊕3T
A⊕E⊕6T
4A⊕3E⊕6T
More
Spherical harmonics and Multipoles
Symmetric Powers of Γ
xyz
Spherical Harmonics Y
l
/ Multipole
Symmetric Power [Γ
l
(xyz)]
l
2l+1
Multipole
Symmetry
Rank
[Γ
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, d
n
term splitting
Term symbols for electronic configurations d
n
d
n
Term Symbols
d
1
= d
9
2
D
d
2
= d
8
1
S,
1
D,
1
G,
3
P,
3
F
d
3
= d
7
2
P,
2
D (2),
2
F,
2
G,
2
H,
4
P,
4
F
d
4
= d
6
1
S (2),
1
D (2),
1
F,
1
G (2),
1
I,
3
P (2),
3
D,
3
F (2),
3
G,
3
H,
5
D
d
5
2
S,
2
P,
2
D (3),
2
F (2),
2
G (2),
2
H,
2
I,
4
P,
4
D,
4
F,
4
G,
6
S
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 November, 13
th
2023 by A. Gelessus,
Impressum
,
Datenschutzerklärung/DataPrivacyStatement