Point Group C
4
C
4
E
C
4
C
2
(C
4
)
3
A
1
1
1
1
B
1
-1
1
-1
E
*
1
1
i
-i
-1
-1
-i
i
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Additional information
Number of symmetry elements
h = 4
Number of classes, irreps
n = 4
Number of real-valued irreducible representations
n = 3
Abelian group
yes
Optical Isomerism (Chirality)
yes
Polar
yes
Parity
no
Reduce representation to irreducible representations
E
C
4
C
2
(C
4
)
3
Genrate representation from irreducible representations
A
B
E
*
Direct products of irreducible representations
Binary products
⊙
A
B
E
*
A
A
B
B
A
E
*
E
E
2A⊕2B
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⊕2B
2E
3A⊕2B
3E
3A⊕4B
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
A⊕E
3
A⊕E
d (l=2)
5
Quadrupole
A⊕2B⊕E
6
2A⊕2B⊕E
f (l=3)
7
Octupole
A⊕2B⊕2E
10
2A⊕2B⊕3E
g (l=4)
9
Hexadecapole
3A⊕2B⊕2E
15
5A⊕4B⊕3E
h (l=5)
11
Dotricontapole
3A⊕2B⊕3E
21
5A⊕4B⊕6E
i (l=6)
13
Tetrahexacontapole
3A⊕4B⊕3E
28
8A⊕8B⊕6E
j (l=7)
15
Octacosahectapole
3A⊕4B⊕4E
36
8A⊕8B⊕10E
k (l=8)
17
256-pole
5A⊕4B⊕4E
45
13A⊕12B⊕10E
l (l=9)
19
512-pole
5A⊕4B⊕5E
55
13A⊕12B⊕15E
m (l=10)
21
1024-pole
5A⊕6B⊕5E
66
18A⊕18B⊕15E
n (l=11)
23
2048-pole
5A⊕6B⊕6E
78
18A⊕18B⊕21E
o (l=12)
25
4096-pole
7A⊕6B⊕6E
91
25A⊕24B⊕21E
More
First nonvanshing multipole:
Dipole
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 C
4
L
2L+1
Term Splitting
S (L=0)
1
A
P (L=1)
3
A⊕E
D (L=2)
5
A⊕2B⊕E
F (L=3)
7
A⊕2B⊕2E
G (L=4)
9
3A⊕2B⊕2E
H (L=5)
11
3A⊕2B⊕3E
I (L=6)
13
3A⊕4B⊕3E
Last update November, 13
th
2023 by A. Gelessus,
Impressum
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