Point Group O
O
E
8C
3
6C
2
6C
4
3C
2
A
1
1
1
1
1
1
A
2
1
1
-1
-1
1
E
2
-1
0
0
2
T
1
3
0
-1
1
-1
T
2
3
0
1
-1
-1
Additional information
Number of symmetry elements
h = 24
Number of classes, irreps
n = 5
Abelian group
no
Optical Isomerism (Chirality)
yes
Polar
no
Parity
no
Reduce representation to irreducible representations
E
8C
3
6C
2
6C
4
3C
2
Genrate representation from irreducible representations
A
1
A
2
E
T
1
T
2
Examples
Octamethylsilsesquioxane (rotated)
Dodecaethyleneoctamine
Direct products of irreducible representations
Binary products
⊙
A
1
A
2
E
T
1
T
2
A
1
A
1
A
2
A
2
A
1
E
E
E
A
1
⊕A
2
⊕E
T
1
T
1
T
2
T
1
⊕T
2
A
1
⊕E⊕T
1
⊕T
2
T
2
T
2
T
1
T
1
⊕T
2
A
2
⊕E⊕T
1
⊕T
2
A
1
⊕E⊕T
1
⊕T
2
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
1
⊕E
A
1
⊕A
2
⊕E
A
1
⊕2E
A
1
⊕A
2
⊕2E
2A
1
⊕A
2
⊕2E
More
T
1
A
1
⊕E⊕T
2
A
2
⊕2T
1
⊕T
2
2A
1
⊕2E⊕T
1
⊕2T
2
A
2
⊕E⊕4T
1
⊕2T
2
3A
1
⊕A
2
⊕3E⊕2T
1
⊕4T
2
More
T
2
A
1
⊕E⊕T
2
A
1
⊕T
1
⊕2T
2
2A
1
⊕2E⊕T
1
⊕2T
2
A
1
⊕E⊕2T
1
⊕4T
2
3A
1
⊕A
2
⊕3E⊕2T
1
⊕4T
2
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
1
A
1
p (l=1)
3
Dipole
T
1
3
T
1
d (l=2)
5
Quadrupole
E⊕T
2
6
A
1
⊕E⊕T
2
f (l=3)
7
Octupole
A
2
⊕T
1
⊕T
2
10
A
2
⊕2T
1
⊕T
2
g (l=4)
9
Hexadecapole
A
1
⊕E⊕T
1
⊕T
2
15
2A
1
⊕2E⊕T
1
⊕2T
2
h (l=5)
11
Dotricontapole
E⊕2T
1
⊕T
2
21
A
2
⊕E⊕4T
1
⊕2T
2
i (l=6)
13
Tetrahexacontapole
A
1
⊕A
2
⊕E⊕T
1
⊕2T
2
28
3A
1
⊕A
2
⊕3E⊕2T
1
⊕4T
2
j (l=7)
15
Octacosahectapole
A
2
⊕E⊕2T
1
⊕2T
2
36
2A
2
⊕2E⊕6T
1
⊕4T
2
k (l=8)
17
256-pole
A
1
⊕2E⊕2T
1
⊕2T
2
45
4A
1
⊕A
2
⊕5E⊕4T
1
⊕6T
2
l (l=9)
19
512-pole
A
1
⊕A
2
⊕E⊕3T
1
⊕2T
2
55
A
1
⊕3A
2
⊕3E⊕9T
1
⊕6T
2
m (l=10)
21
1024-pole
A
1
⊕A
2
⊕2E⊕2T
1
⊕3T
2
66
5A
1
⊕2A
2
⊕7E⊕6T
1
⊕9T
2
n (l=11)
23
2048-pole
A
2
⊕2E⊕3T
1
⊕3T
2
78
A
1
⊕4A
2
⊕5E⊕12T
1
⊕9T
2
o (l=12)
25
4096-pole
2A
1
⊕A
2
⊕2E⊕3T
1
⊕3T
2
91
7A
1
⊕3A
2
⊕9E⊕9T
1
⊕12T
2
More
First nonvanshing multipole:
Hexadecapole
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 O
L
2L+1
Term Splitting
S (L=0)
1
A
1
P (L=1)
3
T
1
D (L=2)
5
E⊕T
2
F (L=3)
7
A
2
⊕T
1
⊕T
2
G (L=4)
9
A
1
⊕E⊕T
1
⊕T
2
H (L=5)
11
E⊕2T
1
⊕T
2
I (L=6)
13
A
1
⊕A
2
⊕E⊕T
1
⊕2T
2
Last update November, 13
th
2023 by A. Gelessus,
Impressum
,
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