Point Group S
6
S
6
E
C
3
(C
3
)
2
i
(S
6
)
5
S
6
A
g
1
1
1
1
1
1
E
g
*
2
-1
-1
2
-1
-1
A
u
1
1
1
-1
-1
-1
E
u
*
2
-1
-1
-2
1
1
Math View
Phys View
✅
Chem View
Additional information
Number of symmetry elements
h = 6
Number of classes, irreps
n = 6
Number of real-valued irreducible representations
n = 4
Abelian group
yes
Optical Isomerism (Chirality)
no
Polar
no
Parity
yes
Reduce representation to irreducible representations
E
C
3
(C
3
)
2
i
(S
6
)
5
S
6
Genrate representation from irreducible representations
A
g
E
g
*
A
u
E
u
*
Direct products of irreducible representations
Binary products
⊙
A
g
E
g
*
A
u
E
u
*
A
g
A
g
E
g
*
E
g
2A
g
⊕E
g
A
u
A
u
E
u
A
g
E
u
*
E
u
2A
u
⊕E
u
E
g
2A
g
⊕E
g
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
g
*
A
g
⊕E
g
2A
g
⊕E
g
A
g
⊕2E
g
2A
g
⊕2E
g
3A
g
⊕2E
g
More
E
u
*
A
g
⊕E
g
2A
u
⊕E
u
A
g
⊕2E
g
2A
u
⊕2E
u
3A
g
⊕2E
g
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
g
1
A
g
p (l=1)
3
Dipole
A
u
⊕E
u
3
A
u
⊕E
u
d (l=2)
5
Quadrupole
A
g
⊕2E
g
6
2A
g
⊕2E
g
f (l=3)
7
Octupole
3A
u
⊕2E
u
10
4A
u
⊕3E
u
g (l=4)
9
Hexadecapole
3A
g
⊕3E
g
15
5A
g
⊕5E
g
h (l=5)
11
Dotricontapole
3A
u
⊕4E
u
21
7A
u
⊕7E
u
i (l=6)
13
Tetrahexacontapole
5A
g
⊕4E
g
28
10A
g
⊕9E
g
j (l=7)
15
Octacosahectapole
5A
u
⊕5E
u
36
12A
u
⊕12E
u
k (l=8)
17
256-pole
5A
g
⊕6E
g
45
15A
g
⊕15E
g
l (l=9)
19
512-pole
7A
u
⊕6E
u
55
19A
u
⊕18E
u
m (l=10)
21
1024-pole
7A
g
⊕7E
g
66
22A
g
⊕22E
g
n (l=11)
23
2048-pole
7A
u
⊕8E
u
78
26A
u
⊕26E
u
o (l=12)
25
4096-pole
9A
g
⊕8E
g
91
31A
g
⊕30E
g
More
First nonvanshing multipole:
Quadrupole
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 S
6
L
2L+1
Term Splitting
S (L=0)
1
A
g
P (L=1)
3
A
g
⊕E
g
D (L=2)
5
A
g
⊕2E
g
F (L=3)
7
3A
g
⊕2E
g
G (L=4)
9
3A
g
⊕3E
g
H (L=5)
11
3A
g
⊕4E
g
I (L=6)
13
5A
g
⊕4E
g
Last update November, 13
th
2023 by A. Gelessus,
Impressum
,
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