Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
Tables for Group Theory
By P. W. ATKINS, M. S. CHILD, and C. S. G. PHILLIPS
This provides the essential tables (character tables, direct products, descent in symmetry and
subgroups) required for those using group theory, together with general formulae, examples,
and other relevant information.
Character Tables:
1
2
3
4
5
6
7
8
9
10
11
12
The Groups C1, Cs, Ci
The Groups Cn (n = 2, 3, …, 8)
The Groups Dn (n = 2, 3, 4, 5, 6)
The Groups Cnv (n = 2, 3, 4, 5, 6)
The Groups Cnh (n = 2, 3, 4, 5, 6)
The Groups Dnh (n = 2, 3, 4, 5, 6)
The Groups Dnd (n = 2, 3, 4, 5, 6)
The Groups Sn (n = 4, 6, 8)
The Cubic Groups:
T, Td, Th
O, Oh
The Groups I, Ih
The Groups C∞ v and D∞ h
The Full Rotation Group (SU2 and R3)
3
4
6
7
8
10
12
14
15
17
18
19
Direct Products:
1
2
3
4
5
6
7
8
9
10
11
General Rules
C2, C3, C6, D3, D6, C2v, C3v, C6v, C2h, C3h, C6h, D3h, D6h, D3d, S6
D2, D2h
C4, D4, C4v, C4h, D4h, D2d, S4
C5, D5, C5v, C5h, D5h, D5d
D4d, S8
T, O, Th, Oh, Td
D6d
I, Ih
C∞v, D∞h
The Full Rotation Group (SU2 and R3)
The extended rotation groups (double groups):
character tables and direct product table
Descent in symmetry and subgroups
20
20
20
20
21
21
21
22
22
22
23
24
26
Notes and Illustrations:
General formulae
Worked examples
Examples of bases for some representations
Illustrative examples of point groups:
I Shapes
II Molecules
29
31
35
37
39
OXFORD
© Oxford University Press, 2010. All rights reserved.
Higher Education
1
Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
Character Tables
Notes:
(1) Schönflies symbols are given for all point groups. Hermann–Maugin symbols are given for
the 32 crystaliographic point groups.
(2) In the groups containing the operation C5 the following relations are useful:
η + = 12 (1 + 5 2 ) = 1·61803L = −2 cos144o
1
η − = 12 (1 − 5 2 ) = −0·61803L = −2 cos 72o
1
η +η + = 1 + η +
η + +η − = 1
η −η − = 1 + η −
η +η − = –1
2 cos 72o + 2 cos144o = −1
OXFORD
© Oxford University Press, 2010. All rights reserved.
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
1. The Groups C1, Cs, Ci
C1
(1)
A
E
Cs=Ch
(m)
E
σh
A′
A″
1
1
1
–1
Ci = S2
E
i
Ag
1
1
Au
1
–1
1
x, y, Rz
z, Rx, Ry
x2, y2, z2, xy
yz, xz
Rx, Ry, Rz
x2, y2, z2,
xy, xz, yz
(1)
x, y, z
OXFORD
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
2. The Groups Cn (n = 2, 3,…,8)
C2
(2)
A
B
C2
E
1
1
C3
(3)
A
E
C4
(4)
A
B
1
–1
C3
C32
1
1
1
⎧1
⎨
⎩1
ε
ε*
ε*⎫
⎬
ε⎭
z, Rz
x2 + y2, z2
(x, y)(Rx, Ry)
(x2 – y2, 2xy)(yz, xz)
C4
C2
C43
1
1
1
–1
1
1
1
–1
⎪⎧1
⎨
⎩⎪1
i
−i
−1
−1
− i ⎪⎫
⎬
i ⎭⎪
z, Rz
x2 + y2, z2
x2 – y2, 2xy
(x, y)(Rx, Ry)
(yz, xz)
C5
E
C5
C52
C53
C54
A
1
1
1
1
1
E1
⎧1
⎨
⎩1
ε
ε*
ε2
ε *2
ε *2
ε2
ε*⎫
⎬
ε⎭
E2
⎧1
⎨
⎩1
ε2
ε *2
ε*
ε
ε
ε*
ε *2 ⎫
⎬
ε2 ⎭
E
C6
C3
C2
C32
C65
1
1
1
–1
1
1
1
–1
1
1
1
–1
−ε *
−ε
−1
−ε
−ε *
ε*⎫
⎬
ε⎭
C6
(6)
A
B
E1
⎧1
⎨
⎩1
ε
ε*
E2
⎧1
⎨
⎩1
−ε *
−ε
−ε
−ε *
−1
1
1
x2, y2, z2, xy
yz, xz
ε = exp (2πi/3)
E
E
E
z, Rz
x, y, Rx, Ry
−ε *
−ε
ε = exp(2πi/5)
z, Rz
x2 + y2, z2
(x,y)(Rx, Ry)
(yz, xz)
(x2 – y2, 2xy)
ε = exp(2πi/6)
z, Rz
x2 + y2, z2
(x, y)
(Rz, Ry)
(xy, yz)
−ε ⎫
⎬
−ε * ⎭
(x2 – y2, 2xy)
OXFORD
© Oxford University Press, 2010. All rights reserved.
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
2. The Groups Cn (n = 2, 3,…,8) (cont..)
C7
E
C7
C72
C73
C74
C75
C76
A
1
1
1
1
1
1
1
E1
⎧1
⎨
⎩1
ε
ε*
ε2
ε *2
ε3
ε *3
ε *3
ε3
ε *2
ε2
ε*⎫
⎬ (x, y)
ε ⎭ (Rx, Ry)
E2
⎧1
⎨
⎩1
ε2
ε *2
ε *3
ε3
ε*
ε
ε
ε*
ε3
ε *3
ε *2 ⎫
⎬
ε2 ⎭
E3
⎧1
⎨
⎩1
ε3
ε *3
ε*
ε
ε2
ε *2
ε *2
ε2
ε
ε*
ε *3 ⎫
⎬
ε3 ⎭
ε = exp (2πi/7)
z, Rz
x2 + y2, z2
(xz, yz)
(x2 – y2, 2xy)
C8
E
C8
C4
C2
C43
C83
C85
C87
A
B
1
1
1
–1
1
1
1
1
1
1
1
–1
1
–1
1
–1
z, Rz
x2 + y2, z2
E1
⎧1
⎨
⎩1
ε
ε*
i
−i
−1
−1
−ε *
−ε
−ε
−ε
ε*⎫
⎬
ε ⎭
(x, y)
(Rx, Ry)
(xz, yz)
E2
⎧1
⎨
⎩1
i
−i
−1
−1
1
1
−1
−1
−i
i
i
−i
E3
⎧1
⎨
⎩1
−ε
−ε *
i
−i
−1
−1
−i
i
ε*
ε
ε
ε*
−i
i
*
ε = exp (2πi/8)
−i ⎫
⎬
i⎭
(x2 – y2, 2xy)
−ε * ⎫
⎬
−ε ⎭
OXFORD
© Oxford University Press, 2010. All rights reserved.
Higher Education
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
3. The Groups Dn (n = 2, 3, 4, 5, 6)
D2
(222)
A
B1
B2
B3
B
B
B
E
C2(z)
C2(y)
C2(x)
1
1
1
1
1
1
–1
–1
1
–1
1
–1
1
–1
–1
1
D3
(32)
A1
A2
E
D4
(422)
A1
A2
B1
B2
E
E
2C3
3C2
1
1
2
1
1
–1
1
–1
0
x2 + y2, z2
z, Rz
(x, y)(Rx,, Ry)
2C4
C2 (= C42 )
2C2'
2C2"
1
1
1
1
2
1
1
–1
–1
0
1
1
1
1
–2
1
–1
1
–1
0
1
–1
–1
1
0
B
D5
E
2C5
2C52
5C2
A1
A2
E1
E2
1
1
2
2
1
1
2 cos 72º
2 cos 144º
1
1
2 cos 144°
2 cos 72°
1
–1
0
0
D6
(622)
A1
A2
B1
B2
E1
E2
B
B
z, Rz
y, Ry
x, Rx
E
B
E
2C6
2C3
1
1
1
1
2
2
1
1
–1
–1
1
–1
1
1
1
1
–1
–1
C2
1
1
–1
–1
–2
2
x2, y2, z2
xy
xz
yz
3C2′
3C2′′
1
–1
1
–1
0
0
1
–1
–1
1
0
0
(x2 – y2, 2xy) (xz, yz)
x2 + y2, z2
z, Rz
(x, y)(Rx, Ry)
x2 – y2
xy
(xz, yz)
x2 + y2, z2
z, Rz
(x, y)(Rx, Ry)
(xz, yz)
(x2 – y2, 2xy)
x2 + y2, z2
z, Rz
(x, y)(Rx, Ry)
OXFORD
© Oxford University Press, 2010. All rights reserved.
(xz, yz)
(x2 – y2, 2xy)
Higher Education
6
Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
4. The Groups Cnν (n = 2, 3, 4, 5, 6)
C2ν
(2mm)
A1
A2
B1
B2
E
C2
σν(xz)
σ′v (yz)
1
1
1
1
1
1
–1
–1
1
–1
1
–1
1
–1
–1
1
C3ν
(3m)
A1
A2
E
E
2C3
3σν
1
1
2
1
1
–1
1
–1
0
C4ν
(4mm)
A1
A2
B1
B2
E
E
2C4
C2
C5ν
E
2C5
2C52
5σν
A1
A2
E1
E2
1
1
2
2
1
1
2 cos 72°
2 cos 144°
1
1
2 cos 144°
2 cos 72°
1
–1
0
0
C6ν
(6mm)
A1
A2
B1
B2
E1
E2
E
2C6
2C3
C2
3σν
3σd
1
1
1
1
2
2
1
1
–1
–1
1
–1
1
1
1
1
–1
–1
1
1
–1
–1
–2
2
1
–1
1
–1
0
0
1
–1
–1
1
0
0
B
B
1
1
1
1
2
B
B
B
B
1
1
–1
–1
0
x2 + y2, z2
z
Rz
(x, y)(Rx, Ry)
2σν
1
1
1
1
–2
1
–1
1
–1
0
x2, y2, z2
xy
xz
yz
z
Rz
x, Ry
y, Rx
(x2 – y2, 2xy)(xz, yz)
2σd
1
–1
–1
1
0
x2 + y2, z2
z
Rz
(x, y)(Rx, Ry)
z
Rz
(x, y)(Rx, Ry)
x2 – y2
xy
(xz, yz)
x2 + y2, z2
(xz, yz)
(x2 – y2, 2xy)
z
Rz
x2 + y2, z2
(x, y)(Rx, Ry)
(xz, yz)
(x2 – y2, 2xy)
OXFORD
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
5. The Groups Cnh (n = 2, 3, 4, 5, 6)
C2h
(2/m)
Ag
Bg
Au
Bu
B
B
E
C2
I
σh
1
1
1
1
1
–1
1
–1
1
1
–1
–1
1
–1
–1
1
C3h
E
C3
C32
σh
S3
S35
A'
1
1
1
1
1
1
E'
⎧⎪ 1
⎨
⎪⎩ 1
ε
ε*
ε*
ε
1
1
ε
ε*
A''
1
1
1
–1
–1
E''
⎪⎧1
⎨
⎩⎪1
ε
ε*
ε
ε
−1
−1
−ε
−ε *
( 6)
C4h
(4/m)
E
C4
C2
*
Rz
Rx, Ry
z
x, y
ε = exp (2πi/3)
Rz
x2 + y2, z2
ε * ⎫⎪
⎬
ε ⎪⎭
(x, y)
(x2 – y2, 2xy)
–1
z
− ε ⎪⎫
⎬
− ε ⎭⎪
*
(Rx, Ry)
C43 i
S43
σh
S4
1
–1
1
1
1
–1
Ag
Bg
1
1
1
–1
1
1
1
–1
Eg
⎧⎪1
⎨
⎩⎪1
i
−i
−1
−1
−i
i
1
1
i
−i
−1
−1
− i ⎫⎪
⎬
i ⎭⎪
Au
Bu
1
1
1
–1
1
1
1
–1
–1
–1
–1
1
–1
–1
–1
1
Eu
⎪⎧1
⎨
⎪⎩1
i
−i
−1
−1
−i
i
B
B
1
1
−1
−1
x2, y2, z2, xy
xz, yz
−i
i
1
1
(xz, yz)
Rz
x2 + y2, z2
(x2 – y2, 2xy)
(Rx, Ry)
(xz, yz)
z
i ⎪⎫
⎬
− i ⎪⎭
(x, y)
OXFORD
© Oxford University Press, 2010. All rights reserved.
Higher Education
8
Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
5. The Groups Cnh (n = 2, 3, 4, 5, 6) (cont…)
C5h
E
C5
C52
C53
C54
σh
S5
S57
S53
S59
A′
1
1
1
1
1
1
1
1
1
1
E1′
⎧1
⎨
⎩1
ε
ε*
ε2
ε *2
ε *2
ε2
ε*
ε
1
ε
ε*
ε2
ε *2
ε *2
ε2
ε*⎫
⎬
ε ⎭
E′2
⎧1
⎨
⎩1
ε2
ε *2
ε*
ε
ε
ε*
ε *2
ε2
1
ε2
ε *2
ε*
ε
ε
ε*
ε *2 ⎫
⎬
ε2 ⎭
A′′
1
1
1
1
E1′′
⎧1
⎨
⎩1
ε
ε*
ε
ε *2
ε
ε2
ε
ε
E′′2
⎧1
⎨
⎩1
ε2
ε *2
ε*
ε
ε
ε*
ε *2
ε2
2
1
1
*2
–1
−1
−1
–1
−ε
−ε *
−1
−1
−ε 2
−ε *2
*
C6h
(6/m)
E
C6
C3
C2
C32
C65
Ag
Bg
1
1
1
–1
1
1
1
–1
1
1
1
–1
E1g
⎧1
⎨
⎩1
ε
ε*
−1
−1
−ε
−ε *
−ε *
−ε
E2g
⎧1
⎨
⎩1
−ε
−ε
Au
Bu
1
1
1
–1
E1u
⎧1
⎨
⎩1
ε
ε*
E2u
⎧1
⎨
⎩1
−ε *
−ε
B
B
−ε
−ε
*
−ε *
1
ε
ε
i
1
1
−ε
−ε *2
−ε
−ε 2
−ε * ⎫
⎬
−ε ⎭
−ε *
−ε
−ε
−ε *
−ε *2 ⎫
⎬
−ε 2 ⎭
ε = exp(2πi/6)
1
–1
1
1
1
–1
1
1
1
–1
x2+y2, z2
(xz, yz)
−1
−1
−ε
−ε *
−ε
−ε
1
–1
–1
–1
−ε
−ε *
ε*
ε
1
1
−ε *
−ε
−ε
−ε *
−ε
−ε
–1
1
−1
−1
−1
−1
(xz, yz)
S3
−ε
−1
−1
(Rx, Ry)
S6
−ε *
−ε *
−ε
(x2 – y2, 2xy)
σh
1
1
1
1
z
S65
−ε
−ε *
1
–1
(x, y)
S35
ε
ε*
1
1
x2+y2, z2
Rz
–1
*2
1
1
−ε
−ε
−ε *
–1
2
*
1
1
*
–1
ε = exp(2πi/5)
*
*
–1
–1
1
1
−ε *
−ε
(Rx, Ry)
ε ⎫
⎬
ε ⎭
*
−ε ⎫
⎬
−ε * ⎭
–1
1
–1
–1
–1
1
ε
ε*
−ε * ⎫
⎬
−ε ⎭
−ε
−ε *
ε*
ε
1
1
ε*
ε
ε
ε*
−1
−1
ε*
ε
Z
(x, y)
ε ⎫
⎬
ε*⎭
OXFORD
© Oxford University Press, 2010. All rights reserved.
(x2 – y2, 2xy)
Higher Education
9
Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
6. The Groups Dnh (n = 2, 3, 4, 5, 6)
C2(y)
C2(x)
i
1
1
1
1
1
1
1
1
1
1
–1
–1
1
1
–1
–1
1
–1
1
–1
1
–1
1
–1
1
–1
–1
1
1
–1
–1
1
1
1
1
1
–1
–1
–1
–1
E
2C3
3C2
σh
2S3
E
D3h
(6) m2
B
B
B
B
B
B
σ(xy)
C2(z)
D2h
(mmm)
Ag
B1g
B2g
B3g
Au
B1u
B2u
B3u
1
1
–1
–1
–1
–1
1
1
σ(xz)
σ(yz)
1
–1
1
–1
–1
1
–1
1
1
–1
–1
1
–1
1
1
–1
x2 + y2, z2
1
1
1
1
1
1
A′2
1
1
–1
1
1
–1
Rz
E′
A1′′
2
1
–1
1
0
1
2
–1
–1
–1
0
–1
(x, y)
A′′2
1
1
–1
–1
–1
1
z
E′′
2
–1
0
–2
1
0
(Rx, Ry)
D4h
(4/mmm)
A1g
A2g
B1g
B2g
Eg
A1u
A2u
B1u
B2u
Eu
E
2C4
C2
2C2′
2C2′′
1
1
1
1
2
1
1
1
1
2
1
1
–1
–1
0
1
1
–1
–1
0
1
1
1
1
–2
1
1
1
1
–2
1
–1
1
–1
0
1
–1
1
–1
0
1
–1
–1
1
0
1
–1
–1
1
0
B
B
B
B
1
1
1
1
2
–1
–1
–1
–1
–2
z
y
x
3σv
A1′
i
Rz
Ry
Rx
x2, y2, z2
xy
xz
yz
(x2 – y2, 2xy)
2S4
σh
2σv
2σd
1
1
–1
–1
0
–1
–1
1
1
0
1
1
1
1
–2
–1
–1
–1
–1
2
1
–1
1
–1
0
–1
1
–1
1
0
1
–1
–1
1
0
–1
1
1
–1
0
OXFORD
© Oxford University Press, 2010. All rights reserved.
(xy, yz)
x2 + y2, z2
Rz
(Rx, Ry)
x2 – y2
xy
(xz, yz)
Z
(x, y)
Higher Education
10
Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
6. The Groups Dnh (n = 2, 3, 4, 5, 6) (cont…)
D5h
E
2C5
2C52
5C2
σh
2S5
2S53
5σv
A1′
1
1
1
1
1
1
1
1
A′2
1
1
1
–1
1
1
1
–1
Rz
E1′
2
2 cos 72°
2 cos 144°
0
2
2 cos 72°
2 cos 144°
0
(x, y)
E ′2
2
2 cos 144°
2 cos 72°
0
2
2 cos 144°
2 cos 72°
0
A1′′
1
1
1
1
–1
–1
–1
–1
A′′2
1
1
1
–1
–1
–1
–1
1
z
E1′′
2
2 cos 72°
2 cos 144°
0
–2
–2 cos 72°
–2 cos 144°
0
(Rx, Ry)
E ′′2
2
2 cos 144°
2 cos 72°
0
–2
–2 cos 144°
–2 cos 72°
0
2C6
2C3
C2
1
1
–1
–1
1
–1
1
1
–1
–1
1
–1
1
1
1
1
–1
–1
1
1
1
1
–1
–1
D6h
(6/mmm)
A1g
A2g
B1g
B2g
E1g
E2g
A1u
A2u
B1u
B2u
E1u
E2u
B
B
B
B
E
1
1
1
1
2
2
1
1
1
1
2
2
1
1
–1
–1
–2
2
1
1
–1
–1
–2
2
3C′2
1
–1
1
–1
0
0
1
–1
1
–1
0
0
3C2′′
1
–1
–1
1
0
0
1
–1
–1
1
0
0
i
1
1
1
1
2
2
–1
–1
–1
–1
–2
–2
2S3
2S6
1
1
–1
–1
1
–1
–1
–1
1
1
–1
1
1
1
1
1
–1
–1
–1
–1
–1
–1
1
1
σh
1
1
–1
–1
–2
2
–1
–1
1
1
2
–2
OXFORD
© Oxford University Press, 2010. All rights reserved.
3σd
3σv
1
–1
1
–1
0
0
–1
1
–1
1
0
0
1
–1
–1
1
0
0
–1
1
1
–1
0
0
x2 + y2, z2
(x2 – y2, 2xy)
(xy, yz)
x2 + y2, z2
Rz
(Rx – Ry)
(xz, yz)
(x2 – y2, 2xy)
z
(x, y)
Higher Education
11
Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
7. The Groups Dnd (n = 2, 3, 4, 5, 6)
D2d = Vd
42 m
2S4
E
( )
A1
A2
B1
B2
E
1
1
1
1
2
B
B
D3d
2C2′
C2
1
1
–1
–1
0
1
1
1
1
–2
2σd
1
–1
1
–1
0
2S6
1
–1
–1
1
0
x2 + y2, z2
Rz
z
(x, y)
(Rx, Ry)
E
2C3
3C2
A1g
A2g
Eg
1
1
2
1
1
–1
1
–1
0
1
1
2
1
1
–1
1
–1
0
Rz
(Rx, Ry)
A1u
A2u
Eu
1
1
2
1
1
–1
1
–1
0
–1
–1
–2
–1
–1
1
–1
1
0
z
(x, y)
i
x2 – y2
xy
(xz, yz)
3σd
(3)m
D4d
2S8
E
2C4
A1
A2
B1
B2
E1
1
1
1
1
2
1
1
–1
–1
E2
E3
2
2
0
B
B
2
– 2
1
1
1
1
0
–2
0
2S83
1
1
–1
–1
– 2
0
2
x2 + y2, z2
4C2′
4σd
1
1
1
1
–2
1
–1
1
–1
0
1
–1
–1
1
0
2
–2
0
0
0
0
C2
(x2 – y2, 2xy)
(xz, yz)
x2 + y2, z2
Rz
z
(x, y)
(Rx, Ry)
OXFORD
© Oxford University Press, 2010. All rights reserved.
(x2 – y2, 2xy)
(xz, yz)
Higher Education
12
Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
7. The Groups Dnd (n = 2, 3, 4, 5, 6) (cont..)
D5d
E
2C5
2C52
5C2
i
A1g
A2g
E1g
E2g
A1u
A2u
E1u
E2u
1
1
2
2
1
1
2
2
1
1
2 cos 72°
2 cos 144°
1
1
2 cos 72°
2 cos 144°
1
1
2 cos 144°
2 cos 72°
1
1
2 cos 144°
2 cos 72°
1
–1
0
0
1
–1
0
0
1
1
2
2
–1
–1
–2
–2
D6d
E
1
1
2 cos 72°
2 cos 144°
–1
–1
–2 cos 72°
–2 cos 144°
1
1
2 cos 144°
2 cos 72°
–1
–1
–2 cos 144°
–2 cos 72°
2C6
2S4
2C3
2S125
C2
6C2′
6σd
1
1
1
1
1
1
1
–1
–1
0
1
1
1
1
–1
1
1
–1
–1
1
1
1
1
–2
1
–1
1
–1
0
1
–1
–1
1
0
–1
–2
–1
1
–2
0
2
0
–1
2
–1
–1
2
–2
2
–2
0
0
0
0
0
0
0
0
1
1
1
1
2
1
1
–1
–1
E2
E3
E4
E5
2
2
2
2
1
0
–1
B
2S10
2S12
A1
A2
B1
B2
E1
B
2S103
3
– 3
– 3
1
0
–1
3
5σd
x2 + y2, z2
Rz
(Rx, Ry)
(xy, yz)
(x2 – y2, 2xy)
z
(x, y)
x2 + y2, z2
Rz
z
(x, y)
(x2 – y2, 2xy)
(Rx, Ry)
OXFORD
© Oxford University Press, 2010. All rights reserved.
1
–1
0
0
–1
1
0
0
(xy, yz)
Higher Education
13
Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
8. The Groups Sn (n = 4, 6, 8)
S4
S4
E
C2
S 43
(4)
A
B
1
1
E
⎧1
⎨
⎩1
S6
1
–1
1
1
i
−1
−i
−1
E
C3
C32
Ag
1
1
1
Eg
⎧1
⎨
⎩1
ε
ε*
Au
1
1
1
Eu
⎧1
⎨
⎩1
ε
ε*
ε
ε
S8
E
S8
1
–1
(3)
A
B
E1
E2
E3
1
1
⎧1
⎨
⎩1
1
–1
ε
ε*
−i ⎫
⎬
i⎭
Rz
z
x2 + y2, z2
(x2 – y2, 2xy)
(x, y) (Rx, Ry)
(xz, yz)
S 65
S6
1
1
1
1
1
ε
ε*
–1
1
i
ε = exp (2πi/3)
Rz
x2 + y2, z2
ε*⎫
⎬
ε ⎭
(Rx, Ry)
(x2 – y2, 2xy) (xy, yz)
–1
–1
z
1
ε
ε*
ε ⎫
⎬
ε ⎭
(x, y)
C4
S83
C2
S85
C43
S87
1
1
1
–1
1
1
1
–1
1
1
1
–1
i
−i
−ε *
−ε
ε*
ε
*
*
−1
−1
−ε
−ε *
−i
i
⎧1
⎨
⎩1
i
−i
−1
−1
−i
i
1
1
i
−i
−1
−1
⎧1
⎨
⎩1
−ε *
−ε
−i
ε
ε*
−1
ε*
ε
i
i
−1
−i
Rz
z
x2 + y2, z2
ε*⎫
⎬ (x, y)
ε ⎭
−i ⎫
⎬
i⎭
−ε ⎫
⎬
−ε * ⎭ (Rx, Ry)
OXFORD
© Oxford University Press, 2010. All rights reserved.
ε = exp (2πi/8)
(x2 – y2, 2xy)
(xy, yz)
Higher Education
14
Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
9. The Cubic Groups
T
(23)
4C3
E
ε = exp (2πi/3)
4C32 3C2
x2 + y2 + z2
A
1
1
1
1
E
⎧1
⎨
⎩1
ε
ε*
ε*
ε
1⎫
⎬
1⎭
T
3
0
Td
0
(
–1
3
(x2 – y2)2z2 – x2 – y2)
(xy, xz, yz)
(x, y, z)
(Rx, Ry, Rz)
E
8C3
3C2
6S4
6σd
1
1
2
3
3
1
1
–1
0
0
1
1
2
–1
–1
1
–1
0
1
–1
1
–1
0
–1
1
4S6
4S62
(43m)
A1
A2
E
T1
T2
Th
(m3)
Ag
3C2
4C3
4C32
1
1
1
1
Eg
⎧1
⎨
⎩1
ε
ε*
ε*
ε
1
1
1
Tg
Au
3
1
0
1
0
1
Eu
⎧1
⎨
⎩1
ε
ε*
Tu
3
0
E
i
(Rx, Ry, Rz)
(x, y, z)
3
(x2 – y2)
(xy, xz, yz)
ε = exp (2πi/3)
3σd
1
x2 + y2 + z2
1
ε
ε*
ε*
ε
1⎫
⎬
1⎭
(2z2 – x2 –y2,
2
2
3 (x – y )
–1
1
3
–1
0
–1
0
–1
–1
–1
ε*
ε
1
1
−1
−1
−ε
−ε*
−ε *
−1⎫
⎬
−1⎭
0
–1
0
0
–3
1
3C2
6C4
6C2′
1
1
2
1
1
–1
1
1
2
1
–1
0
1
–1
0
T1
3
0
–1
1
–1
T2
3
0
–1
–1
1
E
(2z2 – x2 – y2,
1
1
8C3
O
(432)
A1
A2
E
x2 + y2 + z2
−ε
1
(Rx, Ry, Rz)
(xy, yz, xz)
(x, y, z)
x2 + y2 + z2
(2z2 – x2 – y2,
2
2
3 (x – y ))
(x, y, z)
(Rx, Ry, Rz)
(xy, xz, yz)
OXFORD
© Oxford University Press, 2010. All rights reserved.
Higher Education
15
Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
9. The Cubic Groups (cont…)
Oh
(m3m)
E
8C3
6C2
6C4
3C2
i
6S4
8S6
3σh
6σd
(= C42 )
A1g
A2g
Eg
1
1
2
1
1
–1
1
–1
0
1
–1
0
1
1
2
1
1
2
1
–1
0
1
1
–1
1
1
2
1
–1
0
T1g
T2g
A1u
A2u
Eu
T1u
T2u
3
3
1
1
2
3
3
0
0
1
1
–1
0
0
–1
1
1
–1
0
–1
1
1
–1
1
–1
0
1
–1
–1
–1
1
1
2
–1
–1
3
3
–1
–1
–2
–3
–3
1
–1
–1
1
0
–1
1
0
0
–1
–1
1
0
0
–1
–1
–1
–1
–2
1
1
–1
1
–1
1
0
1
–1
OXFORD
© Oxford University Press, 2010. All rights reserved.
x2 + y2 + z2
(2z2 – x2 –y2,
2
2
3 (x – y ))
(Rx, Ry, Rz)
(xy, xz, yz)
(x, y, z)
Higher Education
16
Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
10. The Groups I, Ih
I
12C5
E
12C52
20C3
15C2
1⎛
1
⎝
A
T1
1
3
1
η+
1
η−
1
0
1
–1
T2
G
H
3
4
5
η−
–1
0
η+
–1
0
0
1
–1
–1
0
1
⎞
η ± = ⎜1 ± 5 2 ⎟
2
2
2
⎠
2
x +y +z
(x, y, z)
(Rx, Ry, Rz)
(2z2 – x2 – y2,
3
(x2 – y2)
xy, yz, zx)
Ih
E
12C5
12C52
20C3 15C2
i
12S10
12S103
20S6 15σ
1⎛
1
⎝
Ag
T1g
T2g
Gg
Hg
1
3
3
4
5
1
η+
η−
–1
0
1
η−
η+
–1
0
1
0
0
1
–1
1
–1
–1
0
1
1
3
3
4
5
1
η−
η+
–1
0
1
η+
η−
–1
0
1
–1
0
1
–1
1
–1
–1
0
1
⎞
η ± = ⎜1 ± 5 2 ⎟
2
2
2
⎠
2
x +y +z
(Rx,Ry,Rz)
(2z2 – x2 – y2,
3
(x2 – y2))
(xy, yz, zx)
Au
T1u
T2u
Gu
Hu
1
3
3
4
5
1
η+
η−
–1
0
1
η−
η+
–1
0
1
0
0
1
–1
1
–1
–1
0
1
–1
–3
–3
–4
–5
–1
η−
η+
1
0
–1
η+
η−
1
0
–1
0
0
–1
1
–1
1
1
0
–1
OXFORD
© Oxford University Press, 2010. All rights reserved.
(x, y, z)
Higher Education
17
Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
11. The Groups C∞v and D∞h
C∞v
E
+
1
1
2
2
2
…
…
A1≡∑
A2≡∑–
E1≡Π
E2≡Δ
E3≡Φ
…
…
D∞h
E
+
g
1
Σ −g
1
∏g
Δg
2
2
…
…
1
Σu−
1
∏u
Δu
…
2
2
…
Σ
Σ
+
u
C2
1
1
–2
2
–2
…
…
2C∞φ
…
∞σv
1
1
2 cos φ
2 cos 2φ
2 cos 3φ
…
…
…
…
…
…
…
…
…
1
–1
0
0
0
…
…
…
∞σv
i
1
…
1
1
1
…
–1
…
…
…
1
1
2C∞φ
2 cos φ
2 cos 2φ
2 cos φ
2 cos 2φ
…
x2 + y2, z2
z
Rz
(x,y) (Rx,Ry)
(xz, yz)
(x – y2, 2xy)
2
…
∞C2
1
…
1
1
1
…
0
0
2
2
–2 cos φ
2 cos 2φ
…
…
…
…
…
1
…
–1
…
–1
…
…
…
–1
…
–1
–1
–1
…
1
…
…
…
0
–2
0
–2
… …
2 cos φ
–2 cos 2φ
…
…
…
…
2S∞φ
OXFORD
© Oxford University Press, 2010. All rights reserved.
x2 + y2, z2
–1 Rz
0 (Rx, Ry)
0
(xz, yz)
(x2 – y2, 2xy)
z
0 (x,y)
0
…
Higher Education
18
Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
12. The Full Rotation Group (SU2 and R3)
⎧ ⎛
1⎞
⎪ sin ⎜ j + 2 ⎟ φ
⎠
⎪ ⎝
φ≠0
χ( j ) (φ) = ⎨
1
φ
sin
⎪
2
⎪
φ=0
2
+1
j
⎩
Notation : Representation labelled Γ(j) with j = 0,1/2, 1, 3/2,…∞, for R3 j is confined to integral
values (and written l or L) and the labels S ≡ Γ(0), P ≡Γ(1), D ≡Γ(2), etc. are used.
OXFORD
© Oxford University Press, 2010. All rights reserved.
Higher Education
19
Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
Direct Products
1. General rules
(a) For point groups in the lists below that have representations A, B, E, T without subscripts, read
A1 = A2 = A, etc.
(b)
g
g
g
u
u
u
g
′
″
′
′
″
″
′
(c) Square brackets [ ] are used to indicate the representation spanned by the antisymmetrized
product of a degenerate representation with itself.
Examples
For D3h E′ × E′′ A1′′ + A′′2 + E
For D6h E1g × E2g = 2Bg + E1g.
2. For C2, C3, C6, D3, D6,C2v,C3v, C6v,C2h, C3h, C6h, D3h, D6h, D3d, S6
A1
A1
A1
A2
B1
B2
E1
E2
A2
A2
A1
B1
B1
B2
A1
B2
B2
B1
A2
A1
B
B
B
B
B
E1
E1
E1
E2
E2
A1 + [A2]+ E2
B
B
B
E2
E2
E2
E1
E1
B1 + B2 + E1
A1 + [A2] + E2
B
3. For D2 , D2h
A
B1
B2
B3
B
B
B
A
A
B1
B1
A
B2
B2
B3
A
B
B
B3
B3
B2
B1
A
B
B
B
OXFORD
© Oxford University Press, 2010. All rights reserved.
Higher Education
20
Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
4. For C4, D4, C4v, C4h, D4h, D2d, S4
A1
A1
A1
A2
B1
B2
E
A2
A2
A1
B1
B1
B2
A1
B2
B2
B1
A2
A1
B
B
B
B
B
E
E
E
E
E
A1 + [A2] +B1 + B2
B
B
B
5. For C5, D5, C5v, C5h, D5h, D5d
A1
A1
A1
A2
E1
E2
A2
A2
A1
E1
E1
E1
A1 + [A2] + E2
E2
E2
E2
E 1 + E2
A1 + [A2] + E1
6. For D4d, S8
A1
A2
B1
B2
E1
E2
A1
A1
A2
A2
A1
B
B1
B1
B2
A1
B
B
B
B
B2
B2
B1
A2
A1
B
B
B
E1
E1
E1
E3
E3
A1 + [A2] + E2
E2
E2
E2
E2
E2
E 1 + E2
A1 + [A2] +
B1 + B2
E3
E3
E3
E1
E1
B1 + B2 + E2
E1 + E3
B
B
E3
A1 + [A2] + E2
7. For T, O, Th, Oh, Td
A1
A2
E
T1
T2
A1
A1
A2
A2
A1
E
E
E
A1 + [A2] + E
T1
T1
T2
T 1 + T2
A1 + E + [T1] + T2
OXFORD
© Oxford University Press, 2010. All rights reserved.
T2
T2
T1
T 1 + T2
A2 + E + T1 + T2
A1 + E + [T1] +
T2
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
8. For D6d
A1
A2
B1
B2
E1
A1
A1
A2
A2
A1
B1
B1
B2
A1
B
B
B
B
B
B2
B2
B1
A2
A1
E1
E1
E1
E5
E5
A1 + [A2] +
E2
B
B
B
E2
E2
E2
E2
E4
E4
E 1 + E3
E3
E3
E3
E3
E3
E 2 + E4
E4
E4
E4
E2
E2
E 3 + E5
A1 + [A2]
+ E4
E 1 + E5
B1 + B2 +
E2
E 1 + E5
E3
A1 + [A2] +
B1 + B2
B
E5
E5
E5
E1
E1
B1 + B2 +
E4
E 3 + E5
B
E 2 + E4
B
E4
A1 + [A2]
+ E4
E5
E 1 + E3
A1 + [A2]
+ E2
9. For I, Ih
A
A
A
T1
T1
T1
A + [T1] +
H
T2
G
T2
T2
G+H
G
G
T2 + G + H
H
H
T1 +T2 + G + H
A + [T2] + H
T1 + G + H
A + [T1 +T2]
+G+H
T1 +T2 + G + H
T1 +T2 + G + 2H
H
A1 + [T1 +T2 + G] +
G + 2H
10. For C∝v, D∝h
+
Σ
Σ–
Π
Σ+
Σ–
+
–
Σ
Σ
Σ+
Π
Δ
Π
Π
+
Σ + [Σ–]
+Δ
Δ
Δ
Π+Φ
Σ+ + [Σ–] + Γ
Δ
:
Notation
Σ
Π
1
Λ=0
Λ1 × Λ2 = | Λ1 – Λ2 | + (Λ1 + Λ2)
Λ × Λ = Σ+ + [Σ–] + (2Λ).
Δ
2
Φ
3
Γ
4
OXFORD
© Oxford University Press, 2010. All rights reserved.
…
…
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
11. The Full Rotation Group (SU2 and R3)
Γ(j) × Γ(j′) = Γ(j + j′) + Γ(j + j′–1) + … + Γ(|j–j′|)
Γ(j) × Γ(j) = Γ(2j) + Γ(2j – 2) + … + Γ(0) + [Γ(2j – 1) + … + Γ(1)]
OXFORD
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
Extended rotation groups (double groups):
Character tables and direct product tables
D2*
E
R
2C2(z)
2C2(y)
2C2(x)
E1/2
2
–2
0
0
0
D3*
E
R
2C3
2C3R
3C2
3C2R
E1/2
E3/2
2
–2
1
–1
0
0
−1
−1
−1
−1
1
1
i
−i
−i ⎫
⎬
i⎭
⎧1
⎨
⎩1
4C2′
4C2′′
0
0
0
0
0
0
D4
E
R
2C4
2C4R
2C2
E1/2
2
–2
2
– 2
E3/2
2
–2
– 2
2
6C2′
6C2′′
0
0
0
1
0
0
0
2
0
0
0
D6*
E
R
2C6
2C6R
2C3
2C3R
2C2
E1/2
2
–2
3
– 3
1
–1
E3/2
2
–2
– 3
3
–1
E5/2
2
–2
0
0
–2
12σ d
Td*
E
R
8C3
8C3R
6C2
6S4
6S4R
*
E
R
8C3
8C3R
6C2
6C4
6S4R
12C2′
E1/2
2
–2
1
–1
0
2
– 2
0
E5/2
2
–2
1
–1
0
– 2
2
0
G3/2
4
–4
–1
1
0
0
0
0
O
E1/2 × E1/2 = [A] +B1 +B2 + B3
E1/2
E3/2
E1/2
E3/2
E1/2
[A1] + A2 + E
E1/2
[A1] + A2 + E
E3/2
2E
[A1] + A1 + 2A2
E3/2
B1 + B2 + E
[A1] + A2 + E
B
OXFORD
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
E1/2
E3/2
E5/2
E1/2
E5/2
G3/2
E3/2
B1 + B2+ E2
[A1] +A2 + E1
E1/2
[A1] +A2 + E1
E1/2
[A1] + T1
E5/2
E 1 + E2
E 1 + E2
[A1] + A2 +B1 + B2
B
E5/2
A2 + T2
[A1] + T1
E3/2
E + T 1 + T2
E + T 1 + T2
[A1 + E + T2] + A2 + 2T1 + T2]
Direct products of ordinary and extended representations for Td* and O*
E1/2
E5/2
G3/2
A1
E1/2
E5/2
G3/2
A2
E5/2
E1/2
G3/2
E
G3/2
G3/2
E1/2 + E5/2+ G3/2
T1
E1/2 + G3/2
E5/2 + G3/2
E1/2 + E5/2+ 2G3/2
OXFORD
© Oxford University Press, 2010. All rights reserved.
T2
E5/2 + G3/2
E1/2 + G3/2
E1/2 + E5/2+ 2G3/2
Higher Education
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
Descent in symmetry and subgroups
The following tables show the correlation between the irreducible representations of a group
and those of some of its subgroups. In a number of cases more than one correlation exists
between groups. In Cs the σ of the heading indicates which of the planes in the parent group
becomes the sole plane of Cs; in C2v it becomes must be set by a convention); where there are
various possibilities for the correlation of C2 axes and σ planes in D4h and D6h with their
subgroups, the column is headed by the symmetry operation of the parent group that is
preserved in the descent.
C2v
C2
Cs
σ(zx)
Cs
σ(yz)
A1
A2
B1
B2
A
A
B
B
A′
A"
A′
A′
A"
A′
A"
A"
C3v
A1
A2
E
C3
A
A
E
Cs
A′
A"
A′ + A"
C4v
C2v
C2v
A1
A2
B1
B2
E
A1
A2
A1
A2
A1
A2
A2
A1
B1 + B2
B1 + B2
B
B
σv
B
B
σd
[Other subgroups: C4, C2, C6]
Cs
Cs
A1
C2ν
σh→σν
A1
A′
A′
A′
A2
B2
A′
A"
E'
A1′′
E'
A"
E
A2
A1 + B2
A2
2A'
A"
A' + A"
A"
A′′2
A"
A1
B1
A"
A′
E"
E"
E
A2 + B1
2A"
A' + A"
D3h
C3h
C3v
A1′
A′
A′2
B
B
σh
σν
[Other subgroups: D3, C3, C2]
OXFORD
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
D4h
A1g
A2g
B1g
B2g
Eg
A1u
A2u
B1u
B2u
Eu
B
B
B
B
D2d
D2d
D2h
D2h
D2
D2
C2′ (→ C2′ )
C2′′ (→ C2′ )
C2′
C2′′
C2′
C2′′
A1
A2
B1
B2
E
B1
B2
A1
A2
E
A1
A2
B2
B1
E
B1
B2
A2
A1
E
Ag
B1g
Ag
B1g
B2g + B3g
Au
B1u
Au
B1u
B2u + B3u
Ag
B1g
B1g
Ag
B2g + B3g
Au
B1u
B1u
Au
B2u + B3u
A
B1
A
B1
B2 + B3
A
B1
A
B1
B2 + B3
A
B1
B1
A
B2 + B3
A
B1
B1
A
B2 + B3
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
C4h
C4v
C2v
C2, σv
C2v
C2, σd
Ag
Ag
Bg
Bg
Eg
Au
Au
Bu
Bu
Eu
A1
A2
B1
B2
E
A2
A1
B2
B1
E
A1
A2
A1
A2
B1 + B2
A2
A1
A2
A1
B1 + B2
A1
A2
A2
A1
B1 + B2
A2
A1
A1
A2
B1 + B2
B
B
B
B
B
B
B
B
B
B
B
B
B
Other subgroups:D4, C4, S4, 3C2h, 3Cs,3C2,Ci, (2C2v)
D6
D3d C2′′
D3d C2′
A1g
A2g
B1g
B2g
E1g
E2g
A1u
A2u
B1u
B2u
E1u
E2u
A1g
A2g
A2g
A1g
Eg
Eg
A1u
A2u
A2u
A1u
Eu
Eu
A1g
A2g
A1g
A2g
Eg
Eg
A1g
A2g
A1u
A2u
Eu
Eu
B
B
B
B
D2h
σh → σ(xy)
σv → σ(yz)
Ag
B1g
B2g
B3g
B2g + B3g
Ag + B1g
Au
B1u
B2u
B3u
B2u + B3u
Au + B1u
B
B
B
B
B
B
B
B
C6v
C3v
σv
C2v
C2v
C2′′
C2h
C2
C2h
C2h
A1
A2
B2
B1
E1
E2
A2
A1
B1
B2
E1
E2
A1
A2
A2
A1
E
E
A2
A1
A1
A2
E
E
A1
B1
A2
B2
A2 + B2
A1 + B1
A2
B2
B1
A1
A1 + B1
A2 + B2
A1
B1
B2
A2
A2 + B2
A1 + B1
A2
B2
B1
A1
A1 + B1
A2 + B2
Ag
Ag
Bg
Bg
2Bg
2Ag
Au
Au
Bu
Bu
2Bu
2Au
Ag
Bg
Ag
Bg
Ag + Bg
Ag + Bg
Au
Bu
Au
Bu
Au + Bu
Au + Bu
Ag
Bg
Bg
Ag
Ag + Bg
Ag + Bg
Au
Bu
Bu
Au
Au + Bu
Au + Bu
B
B
B
B
C2′
B
B
B
B
B
B
B
B
B
B
B
B
C2′
C2′′
B
B
B
B
B
B
B
B
Other subgroups: D6, 2D3h, C6h, C6, C3h, 2D3, S6, D2, C3, 3C2, 3Cg, Ci
Td
A1
A2
E
T1
T2
T
A
A
E
T
T
D2d
A1
B1
A1 + B1
A2 + E
B2 + E
B
C3v
A1
A2
E
A2 + E
A1 + E
C2v
A1
A2
A1 +A2
A2 + B1 + B2
A1 + B2 + B1
Other subgroups: S4, D2, C3, C2, Cs.
OXFORD
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
Oh
A1g
A2g
Eg
T1g
T2g
A1u
A2u
Eu
T1u
T2u
Td
A1
A2
E
T1
T2
A2
A1
E
T2
T1
O
A1
A2
E
T1
T2
A1
A2
E
T1
T2
Th
Ag
Ag
Eg
Tg
Tg
Au
Au
Eu
Tu
Tu
D4h
A1g
B1g
A1g + B1g
A2g + Eg
B2g + Eg
A1u
B1u
A1u + B1u
A2u + Eu
B2u + Eu
B
B
B
B
D3d
A1g
A2g
Eg
A2g + Eg
A1g + Eg
A1u
B1u
Eu
A2u + Eu
A1u + Eu
B
Other subgroups: T, D4, D2d, C4h, C4v, 2D2h, D3, C3v, S6, C4, S4, 3C2v, 2D2, 2C2h, C3, 2C2, S2, Cs
R3
S
P
D
F
G
H
O
A1
T1
E + T2
A2 + T1 +T2
A1 + E + T1 + T2
E + 2T1 + T2
D4
D3
A1
A2 + E
A1 + B1 +B2 + E
A2+ B1 +B2 + 2E
2A1 + A2 +B1 + B2 + 2E
A1 + 2A2 + B1 + B2 + 3E
OXFORD
© Oxford University Press, 2010. All rights reserved.
A1
A2 + E
A1 + 2E
A1 + 2A2 + 2E
2A1+ A2 + 3E
A1 + 2A2 + 4E
Higher Education
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
Notes and Illustrations
General Formulae
(a) Notation
h
the order (the number of elements) of the group.
Γ(i)
labels the irreducible representation.
Χ(i)(R)
the character of the operation R in Γ(i).
(i )
Dμν
( R)
l
i
the μv element of the representative matrix of the operation R in the irreducible
representation Γ(i).
the dimension of Γ(i).(the number of rows or columns in the matrices D(i))
(b) Formulae
(i) Number of irreducible representations of a group = number of classes.
(ii)
(iii)
2
∑ li = h
i
(i )
χ (i ) ( R) = ∑ Dμμ
( R)
μ
(iv) Orthogonality of representations:
(i )
( R)* Dμ(i''ν)' ( R) = (h / li ) δ ii' δ μμ' δνν'
∑ Dμν
(δij=1 if i = j and δij = 0 if i ≠ j
(v) Orthogonality of characters:
∑ χ (i ) ( R)* χ (i ) ( R) = h δ ii'
R
(vi) Decomposition of a direct product, reduction of a representation: If
Γ = ∑ ai Γ (i )
i
and the character of the operation R in the reducible representation is χ(R), then the coefficients at
are given by
ai = (l / h)∑ χ (i ) ( R)* χ ( R).
R
OXFORD
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
(vii) Projection operators:
The projection operator
(i )
= (li / h)∑ χ (i ) ( R)* R
R
when applied to a function f, generates a sum of functions that constitute a component of a
basis for the representation Γ(i); in order to generate the complete basis P (i) must be applied
to li distinct functions f. The resulting functions may be made mutually orthogonal. When li
= 1 the function generated is a basis for Γ(i) without ambiguity:
P
(i )
f = f (i )
(viii) Selection rules:
If f(i) is a member of the basis set for the irreducible representation Γ(i), f{k) a member of that
for Γ(k), and Ω̂ (j) an operator that is a basis for Γ(j), then the integral
( i )* ( j ) ( k )
∫ dτ f Ωˆ f
is zero unless Γ(i) occurs in the decomposition of the direct product Γ(j) × Γ(k)
s
(i )
(i )
(ix) The symmetrized direct product is written Γ ×Γ , and its characters are given by
s
χ (i ) ( R) × χ (i ) ( R) = 12 χ (i ) ( R)2 + 12 χ (i ) ( R 2 )
a
(i )
(i )
The antisymmetrized direct product is written Γ ×Γ and its characters are given by
a
χ (i ) ( R) × χ (i ) ( R) = 12 χ (i ) ( R)2 + 12 χ (i ) ( R 2 )
OXFORD
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
Worked examples
1. To show that the representation Γ based on the hydrogen 1s-orbitals in NH3 (C3v) contains A1
and E, and to generate appropriate symmetry adapted combinations.
A table in which symmetry elements in the same class are distinguished will be employed:
C3v
A1
A2
E
D(R)
x(R)
Rh1
Rh2
E
1
1
2
C3+
C3–
σ1
σ2
σ3
1
1
–1
1
1
–1
1
–1
0
1
–1
0
1
–1
0
⎛1 0 0⎞
⎜
⎟
⎜0 1 0⎟
⎜0 0 1⎟
⎝
⎠
⎛0 0 1⎞
⎜
⎟
⎜1 0 0⎟
⎜0 1 0⎟
⎝
⎠
⎛0 1 0⎞
⎜
⎟
⎜0 0 1⎟
⎜1 0 0⎟
⎝
⎠
⎛1 0 0⎞
⎜
⎟
⎜0 0 1⎟
⎜0 1 0⎟
⎝
⎠
⎛0 0 1⎞
⎜
⎟
⎜0 1 0⎟
⎜1 0 0⎟
⎝
⎠
⎛0 1 0⎞
⎜
⎟
⎜1 0 0⎟
⎜0 0 1⎟
⎝
⎠
3
h1
h2
0
h2
h3
0
h3
h1
1
h1
h3
1
h3
h2
1
h2
h1
The representative matrices are derived from the effect of the operation R on the basis (h1, h2,
h3); see the figure below. For example
⎛0
⎜
C3+ (h1 , h2 , h3 ) = (h2 , h3 , h1 ) = (h1 , h2 , h3 ) ⎜ 1
⎜⎜
⎝0
0 1⎞
⎟
0 0⎟
⎟
1 0 ⎟⎠
According to the general formula (b)(iii) the character χ(R) is the sum of the diagonal elements
of D(R). For example, χ(σ2) = 0 + 1 + 0 = 1. The decomposition of Γ follows from the formula
(b)(vi):
Γ = a1A1 + a2A2 + aEE
where
a1 = 16 {1× 3 + 2 ×1× 0 + 3×1×1} = 1
a2 = 16 {1× 3 + 2 ×1× 0 + 3×1× (–1)} = 0
aE = 16 {2 × 3 + 2 × (–1) × 0 + 3× 0 ×1} = 1
OXFORD
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
Therefore
Γ = A1 + E
Symmetry adapted combinations are generated by the projection operator in (b)(vii). Using the
last two rows of the table,
φ ( A1 ) =
℘
( A1 )
h1 =
(1 × h 1 + 1 × h 2 + 1 × h 3 + 1 × h
1
6
+ 1 × h3 + 1 × h2 ) =
⎧ φ (
⎪
⎪
⎪
⎨
⎪
⎪ φ ′(
⎪
⎩
E ) = ℘
(E )
h1 =
(E )
h2 =
1
( h1 + h 2 + h 3 )
( 2 × h1 – 1 × h 2 – 1 × h 3 + 0 × h1
2
6
+ 0 × h3 + 0 × h2 ) =
E ) = ℘
1
3
2
6
1
3
( 2 h1 – h 2 – h 3 )
( 2 × h 2 – 1 × h 3 – 1 × h1 + 0 × h 3
+ 0 × h 2 + 0 × h1 ) =
1
3
( – h1 + 2 h 2 – h 3 )
φ(E) and φ'(E) provide a valid basis for the E representation, but the orthogonal combinations
1
1
φa ( E ) = (1/ 6) 2 (2h1 – h2 – h3 ) = (3/ 2) 2 φ ( E )
1
1
φb ( E ) = (1/ 2) 2 (h2 – h3 ) = (1/ 2) 2 {φ ( E ) + 2φ' ( E )}
would be a more useful basis in most applications.
2. To determine the symmetries of the states arising from the electronic configurations e2 and e1t21
for a tetrahedral complex (Td ), and to determine the group theoretical selection rules for electric
dipole transitions between them.
The spatial symmetries of the required states are given by the direct products in Table 7.
E × E = A1 + [A2] + E
E × T 2 = T1 + T2
Combination of the electron spins yields both singlet and triplet states, but for the e2
configuration some possibilities are excluded. Since the total (spin and orbital) state must be
antisymmetric under electron interchange, the antisymmetrized spatial combination [A2] must be
a triplet, and the symmetrized combinations A1 and E are singlets. For the e1t21 configuration
there are no exclusions. The required terms are therefore
e2 → 1A1 + 3A2 +1E
e1t21 → 1T1 + 1T2 +3T1 + 3T2
The selection rules are obtained from formula (b)(viii). For electric dipole transitions the operator
Ω(j) has the symmetry of a vector (x, y, z), which from the character table for Td transforms as T2.
From the table of direct products, Table 7,
A1 × T2 = T2
A2 × T1 = T2
E × T 2 = E × T 1 = T1 + T2
Assuming the spin selection rule ΔS = 0, the allowed transitions are
e2 1A1 ↔ e1t21 1T2
e2 3A2 ↔ e1t21 3T1
e2 1E ↔ e1t21 1T1,1T2
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
3. To determine the symmetries of the vibrations of a tetrahedral molecule AB4, and to predict the
appearance of its infrared and Raman spectra.
The molecule is depicted in the figure below and the character table for the point group Td is
given on page 15.
The representations spanned by the vibrational coordinates are based on the 5 × 3 cartesian
displacements less the representations T1 and T2, which are accounted for by the rotations (Rx, Ry,
Rz) and the translations (x, y, z). The stretching vibrations are the subset based on the 4 bonds of
the molecule. For a particular symmetry operation, only atoms (or bonds) that remain invariant
can contribute to the character of the cartesian displacement representation, Γ (all) (or the
stretching representation, Γ(stretch)).
C3:
Two atoms invariant, x, y, z, interchanged
One bond invariant
χ(all)(C3) = 0
χ(stretch)(C3) = 1
C2(z): Central atom invariant; x, y, sign reversed, z invariant
χ(all)(C3) = 0
No bonds invariant
χ(stretch) (C2) = 0
S4(z): Central atom invariant; x, y, interchanged, z sign reversed x(all)(S4) = – 1
No bonds invariant
χ(stretch)(S4) = 0
σd(z): Three atoms invariant; x, y, interchanged, z invariant
x (all)(σd) = 3
Two bonds invariant
χ(stretch)(σd) = 2
The characters of the representations Γ(all) and Γ(stretch) are therefore
(all)
Γ
Γ(stretch)
E
15
4
8C3
0
1
3C2
–1
0
6S4
–1
0
6σd
3
2
= A1 + E + T1 + 3T2
= A1 + T2
Γ (alI) and Γ(stretch) have been decomposed with the help of formula (b)(vi) (compare Example 1).
Allowing for the rotations and translations contained in Γ(all) there are therefore four fundamental
vibrations, conventionally labelled ν1 (A1), ν2(E), ν3(T2), and ν4(T2). ν1 and v2 are stretching and
bending vibrations respectively, ν3 and ν4 involve both stretching and bending motions.
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
The selection rule (b)(viii) gives the spectroscopic properties of the vibrations. Infrared activity
is induced by the dipole moment (a vector with symmetry T2, according to the character table for
( j)
( j)
Td) as the operator Ωˆ In the case of the Raman effect, Ωˆ is the component of the
polarizability tensor (A1 + E + T2). f(i) is the ground vibrational state (A1), and f(k) is the excited
state (with the same symmetry as the vibration in the case of the fundamental; as the direct
product of the appropriate representations in the case of an overtone or a combination band).
v1(A1)and v2(E) are therefore Raman active and ν3(T2) and ν4(T2) are infrared and Raman active.
The following overtone and combination bands are allowed in the infrared spectrum:
ν1 + ν3, ν1 + ν4, ν2 + ν3 , ν2 + ν4, 2ν3 , ν3 + ν4, 2ν4
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
Examples of bases for some representations
The customary bases—polar vector (e.g. translation x), axial vector (e.g. rotation Rx), and tensor
(e.g. xy)—are given in the character tables.
It may be of some assistance to consider other types of bases and a few examples are given here.
Base
Irreducible Representation
1
A2 in Td
2
x(1)y(2) – x(2)y(1)
3
The normal vibration of an octahedral molecule
represented by
A2 in C4v
Alg in Oh
4
The three equivalent normal vibrations of an
octahedral molecule, one of which is represented by
T2u in Oh
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
5
The π-orbital of the benzene molecule represented by
A2u in D6h
6
The π-orbital of the benzene molecule represented by
B2g in D6h
B
7
The π-orbital of the naphthalene molecule
represented by
Au in D2h
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
Illustrative Examples of Point Groups
I Shapes
The character tables for (a), Cn, are on page 4; for (b), Dn, on page 6; for (c), Cnv, on page 7; for
(d), Cnh, on page 8; for (e), Dnh, on page 10; for (f), Dnd, on page 12; and for (g), S2n, on page 14.
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
Ci
Cs
.
Oh
Td
tetrahedron
Oh
cube
Ih
octahedron
dodecahedron
R3
Ih
icosahedron
sphere
The character table for Cs is on page 3, for Ci on page 3, for Td on page 15, for Oh on page 16, for
Ih on page 17, and for R3 on page 19.
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Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory
II
Molecules
Point
group
Example
C1
Cs
Ci
C2
C2v
C3v
C4v
C2h
C3h
CHFClBr
BFClBr (planar), quinoline
meso-tartaric acid
H2O2, S2C12 (skew)
H2O, HCHO, C6H5C1
NH3 (pyramidal), POC13
SF5Cl, XeOF4
trans-dichloroethylene
H
Page number
for character
table
3
3
3
4
7
7
7
8
8
H
O
B
O
O
H
D2h
D3h
D4h
D5h
D6h
D2d
D4d
D5d
S4
Td
Oh
Ih
C∞v
D∞h
R3
(in planar configuration)
trans-PtX2Y2, C2H4
BF3 (planar), PC15 (trigonal bipyramid), 1:3: 5–trichlorobenzene
AuCl4– (square plane)
ruthenocene (pentagonal prism), IF7 (pentagonal bipyramid)
benzene
CH2=C=CH2
S8 (puckered ring)
ferrocene (pentagonal antiprism)
tetraphenylmethane
CCl4
SF6, FeF63–
B12H122–
HCN, COS
CO2, C2H2
any atom (sphere)
B
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10
10
10
11
11
12
12
13
14
15
16
17
18
18
19
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