Atkins, Child, & Phillips: 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

Higher Education 1

Atkins, Child, & Phillips: 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

Higher Education 2

Atkins, Child, & Phillips: 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

Higher Education 3

Atkins, Child, & Phillips: 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

E

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)

C2

C43

1 1

1 –1

1 1

1 –1

⎪⎧1 ⎨ ⎪⎩1

i

−1

− i ⎪⎫

−i

−1

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)

C4

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)

Higher Education 4

Atkins, Child, & Phillips: 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

−i ⎫ ⎬ i⎭

E3

⎧1 ⎨ ⎩1

−ε

i −i

−1 −1

−i i

ε* ε

ε ε*

−ε *

−i i

ε = exp (2πi/8)

(x2 – y2, 2xy)

−ε * ⎫ ⎬ −ε ⎭

Higher Education 5

Atkins, Child, & Phillips: 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)

(xz, yz) (x2 – y2, 2xy)

Higher Education 6

Atkins, Child, & Phillips: 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)

Higher Education 7

Atkins, Child, & Phillips: 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

−1

−i

1

i

−1

− i ⎪⎫

−i

−1

i

1

−i

−1

i ⎭⎪

Au Bu

1 1

1 –1

1 1

1 –1

–1 –1

–1 1

–1 –1

Eu

⎧⎪1 ⎨ ⎩⎪1

i

−1

−i

−1

−i

1

i ⎫⎪

−i

−1

i

−1

i

1

− i ⎭⎪

B

B

1 1

x2, y2, z2, xy xz, yz

–1 1

(xz, yz)

Rz

x2 + y2, z2 (x2 – y2, 2xy)

(Rx, Ry)

(xz, yz)

z

(x, y)

Higher Education 8

Atkins, Child, & Phillips: 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

–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 *

E2g

⎧1 ⎨ ⎩1

−ε * −ε

−ε −ε *

1 1

−ε * −ε

−ε

Au Bu

1 1

1 –1

1 1

1 –1

1 1

1 –1

E1u

⎧1 ⎨ ⎩1

ε ε*

−ε * −ε

−1 −1

−ε −ε *

ε* ε

E2u

⎧1 ⎨ ⎩1

−ε * −ε

1 1

−ε * −ε

−ε −ε *

B

B

−ε −ε

−ε −ε *

ε ε

i 1 1 *

−ε *

–1

–1

(x, y)

z

(x2 – y2, 2xy)

−ε −ε 2

−ε * ⎫ ⎬ −ε ⎭

−ε * −ε

−ε −ε *

−ε *2 ⎫ ⎬ −ε 2 ⎭

*2

(Rx, Ry)

(xz, yz)

S35

S65

σh

S6

S3

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

−ε * −ε

−ε −ε *

1 1

−ε * −ε

–1 –1

–1 1

–1 –1

–1 1

–1 –1

–1 1

ε ε*

−ε * ⎫ ⎬ −ε ⎭

−1 −1

−ε −ε

(Rx, Ry)

1 1

−1 −1

x2+y2, z2

Rz

–1

−ε −ε *2 2

ε = exp(2πi/5)

−ε −ε *

ε* ε

1 1

ε* ε

ε ε*

−1 −1

ε* ε

ε ⎫ ⎬ ε ⎭ *

−ε ⎫ ⎬ −ε * ⎭ Z

(x, y)

ε ⎫ ⎬ ε*⎭

(x2 – y2, 2xy)

Higher Education 9

Atkins, Child, & Phillips: Tables for Group Theory 6. The Groups Dnh (n = 2, 3, 4, 5, 6) D2h (mmm) Ag B1g B2g B3g Au B1u B2u B3u

E

D3h (6) m2

B

B

B

B

B

B

σ(xy)

C2(z)

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

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

(xy, yz)

x2 + y2, z2 Rz

(Rx, Ry)

x2 – y2 xy (xz, yz)

Z

(x, y)

Higher Education 10

Atkins, Child, & Phillips: 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

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, Child, & Phillips: 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)

(x2 – y2, 2xy) (xz, yz)

Higher Education 12

Atkins, Child, & Phillips: 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)

1 –1 0 0 –1 1 0 0

(xy, yz)

Higher Education 13

Atkins, Child, & Phillips: 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

ε* ε *

−ε * −ε

i −i

−1

*

−i

−1

−ε −ε *

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)

ε = exp (2πi/8)

(x2 – y2, 2xy)

(xy, yz)

Higher Education 14

Atkins, Child, & Phillips: 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)

(x, y, z) (Rx, Ry, Rz)

(xy, xz, yz)

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

4S 62

(43m)

A1 A2 E T1 T2

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

Th (m3) Ag

E

3C2

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)

Higher Education 15

Atkins, Child, & Phillips: 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

x2 + y2 + z2 (2z2 – x2 –y2, 2 2 3 (x – y )) (Rx, Ry, Rz) (xy, xz, yz)

(x, y, z)

Higher Education 16

Atkins, Child, & Phillips: 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

(x, y, z)

Higher Education 17

Atkins, Child, & Phillips: 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

x2 + y2, z2

–1 Rz 0 (Rx, Ry) 0

(xz, yz) (x2 – y2, 2xy)

z

0 (x,y) 0 …

Higher Education 18

Atkins, Child, & Phillips: 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 j +1

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.

Higher Education 19

Atkins, Child, & Phillips: 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 u

g

u

g

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

E1 E1 E1 E2 E2 A1 + [A2]+ E2

B

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

Higher Education 20

Atkins, Child, & Phillips: 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

T2 T2 T1 T 1 + T2 A2 + E + T1 + T2 A1 + E + [T1] + T2

Higher Education 21

Atkins, Child, & Phillips: 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 Π Π + Σ + [Σ–] +Δ

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

… …

Higher Education 22

Atkins, Child, & Phillips: 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)]

Higher Education 23

Atkins, Child, & Phillips: 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

O*

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

E1/2 × E1/2 = [A] +B1 +B2 + B3

E1/2 E3/2

E1/2 E3/2

E1/2 [A1] + A2 + E

E3/2 2E [A1] + A1 + 2A2

E1/2 [A1] + A2 + E

E3/2 B1 + B2 + E [A1] + A2 + E B

Higher Education 24

Atkins, Child, & Phillips: Tables for Group Theory

E1/2 E3/2 E5/2

E1/2 E5/2 G3/2

E1/2 [A1] +A2 + E1

E1/2 [A1] + T1

E3/2 B1 + B2+ E2 [A1] +A2 + E1

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

T2 E5/2 + G3/2 E1/2 + G3/2 E1/2 + E5/2+ 2G3/2

Higher Education 25

Atkins, Child, & Phillips: 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

A1′

A′

A′2

C3v

B

B

σh

σν

[Other subgroups: D3, C3, C2]

Higher Education 26

Atkins, Child, & Phillips: Tables for Group Theory D4h A1g A2g B1g B2g Eg A1u A2u B1u B2u Eu B

B

B

B

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

D2d

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.

Higher Education 27

Atkins, Child, & Phillips: 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

A1 A2 + E A1 + 2E A1 + 2A2 + 2E 2A1+ A2 + 3E A1 + 2A2 + 4E

Higher Education 28

Atkins, Child, & Phillips: 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

Higher Education 29

Atkins, Child, & Phillips: 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 Ωˆ (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 )

Higher Education 30

Atkins, Child, & Phillips: 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 ⎜ C (h1 , h2 , h3 ) = (h2 , h3 , h1 ) = (h1 , h2 , h3 ) ⎜ 1 ⎜⎜ ⎝0 + 3

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

Higher Education 31

Atkins, Child, & Phillips: 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

Higher Education 32

Atkins, Child, & Phillips: 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.

Higher Education 33

Atkins, Child, & Phillips: 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

Higher Education 34

Atkins, Child, & Phillips: 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

Higher Education 35

Atkins, Child, & Phillips: 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

Higher Education 36

Atkins, Child, & Phillips: 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, Child, & Phillips: Tables for Group Theory

Cs

Ci

. 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.

Higher Education 38

Atkins, Child, & Phillips: 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