Lecture 30

Point-group symmetry III

Non-Abelian groups and chemical applications of symmetry

 In this lecture, we learn non-Abelian point groups and the decomposition of a product of irreps.

 We also apply the symmetry theory to chemistry problems.

Degeneracy

 The particle in a square well (D 4h ) has

doubly degenerate

wave functions.

The D 4h character table (h = 16)

D 4h

A 1g A 2g B 1g B 2g

E g

A 1u A 2u B 1u B 2u

E u

1 1

2

E

1 1 1 1

2

1 1 2

C

4 1 1 −1 −1

0

1 1 −1 −1

0

1 1 1 1

−2

C

2 1 1 1 1

−2

1 −1 1 −1

0

2

C

2 ’ 1 −1 1 −1

0

2

C

2 ” 1 −1 −1 1

0

1 −1 −1 1

0

1

2

−1 −1 −1 −1

−2

i

1 1 1 2

S

4 1 1 −1 −1

0

−1 −1 1 1

0

σ h

1 1 1 1

−2

−1 −1 −1 −1

2

2

σ v

1 −1 1 −1

0

−1 1 −1 1

0

2

σ d

1 −1 −1 1

0

−1 1 1 −1

0

C 3v : another non-Abelian group

C 3v

, 3

m

A 1 A 2

E

E

1 1

2

2

C

3 1 1

−1

3

σ v

1 −1

0

h

= 6

z

,

z

2 ,

x

2 +

y

2 (

x

,

y

), (

xy

,

x

2 −

y

2 ), (

zx

,

yz

)

C 3v : expanded character table

C 3v

, 3

m

A 1 A 2

E

E

1 1

2

2

C

3 1 1

−1

3

σ v

1 −1

0

h

= 6

z

,

z

2 ,

x

2 +

y

2 (

x

,

y

), (

xy

,

x

2 −

y

2 ), (

zx

,

yz

)

C 3v

, 3

m

A 1 A 2

E

E

1 1

2

C

3 1 1

−1

C

3 2 1 1

−1

σ v

1 −1

0

σ v

1 −1

0

σ v

1 −1

0

z

,

z h

2 , = 6

x

2 +

y

2 (

x

,

y

), (

xy

,

x

2 −

y

2 ), (

zx

,

yz

)

Integral of degenerate orbitals

ò j 1 * j 2

d

t j 1 j 2

C 3v

, 3

m

A 1 A 2

E

E

1 1

2

C

3 1 1

−1

C

3 2 1 1

−1

σ v

1 −1

0

σ v

1 −1

0

σ v

1 −1

0

z

,

z h

2 , = 6

x

2 +

y

2 (

x

,

y

), (

xy

,

x

2 −

y

2 ), (

zx

,

yz

)

What is E

✕

E ?

C 3v

, 3

m

A 1 A 2 E

E

✕

E

E

1 1 2

4

C

3 1 1 −1

C

3 2 1 1 −1

σ v

1 −1 0

σ v

1 −1 0

σ v

1 −1 0

1 1 0 0 0

What is the irrep for this set of characters?

h

= 6

z

,

z

2 ,

x

2 +

y

2 (

x

,

y

), (

xy

,

x

2 −

y

2 ), (

zx

,

yz

) It is not a single irrep.

It is a linear combination of irreps

Superposition principle (review)

 Eigenfunctions of a Hermitian operator are

complete

.

 Eigenfunctions of a Hermitian operator are

orthogonal

.

Y =

c n c

1 F 1 + ò *

n

Y

c

2 F 2

d

t +

c

3 F 3 + …

Decomposition

 An

irrep

is a simultaneous eigenfunction of all symmetry operations.

G =

c

1 G 1 +

c

2 G 2 +

c

3 G 3 + …

c n

= 1

h

G ×G

n

Orthonormal character vectors

C 3v

, 3

m

A 1 A 2 E

E

1 1 2

C

3 1 1 −1

C

3 2 1 1 −1

σ v

1 −1 0

σ v

1 −1 0

σ v

1 −1 0

h

= 6

z

,

z

2 ,

x

2 +

y

2 (

x

,

y

), (

xy

,

x

2 −

y

2 ), (

zx

,

yz

)    The character vector of A 1 6 1 × 6 ( 1 6 1 ( × 2 × ( 1 1 1 1 1 1 The character vector of E is normalized.

1 1 0 0 0 The character vectors of A 1 1 1 1 1 1 1 × ) ) ( × × ) ( 2 ( is normalized. 1 1 1 1 1 1 2 1 1 1 0 0 0 and E are orthogonal.

1 0 0 0

T

)

T

) = 1 =

T

) 0 = 1

Decomposition

E h

= 6

C 3v

, 3

m C

3

C

3 2

σ v σ v σ v

1 1 1 1 1 1

z

,

z

2 ,

x

2 +

y

2 A 1 1 1 1 −1 −1 −1 A 2 E 2 −1 −1 0 0 0 (

x

,

y

), (

xy

,

x

2 −

y

2 ), (

zx

,

yz

)

E

✕

E 4 1 1 0 0 0

  The contribution ( 6 6 1 1 × × ( 6 1 × ( 4 1 1 0 0 0 The contribution ( 4 1 1 0 0 0 Degeneracy = 2 ×

c c c

A1 A2 E ) of A ) of A × ( ( × ) × ) ) of E: 4 1 1 0 0 0 ) 2 ( 1 2 ò 1 : : j 1 + * 1 1 1 1 j 2 1

A d

2 1 t 1 1 1 1 1 1 + 1 1 0 0 0 ) ¹

T T T

) ) = 1 0

E

= 1 = 1

Chemical applications

 While the primary benefit of point-group symmetry lies in our ability to know whether some integrals are zero by symmetry, there are other chemical concepts derived from symmetry. We discuss the following three:    Woodward-Hoffmann rule Crystal field theory Jahn-Teller distortion

Woodward-Hoffmann rule

The photo and thermal pericyclic reactions yield different isomers of cyclobutene.

Reaction A CH 3 H H CH 3 Reaction B H CH 3 photochemical H CH 3 CH H 3 H CH 3 thermal

Woodward-Hoffmann rule

What are the symmetry groups to which these reactions A and B belong? Reaction A

σ

CH 3 H H CH 3 Reaction B

C

2 H CH 3 H CH 3 photochemical / disrotary / C s CH H 3 H CH 3 thermal / conrotary / C 2

Woodward-Hoffmann rule

a Reactant b c d e Product f g h higher energy occupied

h

n higher energy occupied

f f

Photochemical / C s Thermal / C 2 “Conservation of orbital symmetry”

Crystal field theory

[Ni(NH 3 ) 6 ] 2+ , [Ni(en) 3 ] 2+ , [NiCl 4 ] 2− , [Ni(H 2 O) 6 ] 2+ Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.

Crystal field theory

T d

T 2 E

d xy

,

d yz

,

d zx h

n

d z

2 ,

d x

2−

y

2 spherical

d

orbitals

O h d z

2 ,

d x

2−

y

2

h

n

d xy

,

d yz

,

d zx

E g T 2g

E

NiCl 4 2− belongs to T

d

T d E

8

C

3 3

C

2 6

S

4 6

σ d

A 1 A 2 E T 1 T 2 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

T d

spherical T 2

d xy

,

d yz

,

d zx d z

2 ,

d x

2−

y

2

d

orbitals CT transition allowed

h

= 24

x

2 +

y

2 +

z

2 (

z

2 ,

x

2 −

y

2 ) (

xy

,

yz

,

zx

) +

d z

2

Ni(OH 2 ) 6 2+ belongs to O

h

O h

A 1g …

E

1 8

C

2 1 6C 2 1 6

C

4 1 …

h

= 48

x

2 +

y

2 +

z

2 E g … 2 −1 0 0 (

z

2 ,

x

2 −

y

2 ) T 2g … 3 0 1 −1 (

xy

,

yz

,

zx

)

d z

2 spherical

O h d z

2 ,

d x

2−

y

2 +

d

orbitals

d-d

transition forbidden

d xy

,

d yz

,

d zx

E g T 2g

Jahn-Teller distortion

O h D

4

h

Jahn-Teller distortion

(3

d

) 8 Hunt’s rule

d z

2 ,

d x

2−

y

2

d xy

,

d yz

,

d zx

(3

d

) 9

d z

2 ,

d x

2−

y

2 no Hunt’s rule

d xy

,

d yz

,

d zx

Cu(OH 2 ) 6 2+ belongs to D 4h

D

4

h

A 1g …

E

1 2

C

4 1 C 2 1 2

C

2 ’ 1 …

h

= 48

x

2 +

y

2 ,

z

2 B 1g B 2g E g … 1 1 2 −1 −1 0 1 1 −2 1 −1 0

x

2 −

y

2

xy xz

,

yz D

4

h d x

2−

y

2

O h

+

d zx

B 1g A 1g B 2g

d z

2

d xy d z

2 ,

d x

2−

y

2 E g

d yz

,

d zx d xy

,

d yz

,

d zx

E g T 2g

Jahn-Teller distortion

 In Cu(OH 2 ) 6 2+ , the distortion lowers the energy of

d

electrons, but raises the energy of Cu-O bonds. The spontaneous distortion occurs.

 In Ni(OH 2 ) 6 2+ , the distortion lowers the energy of

d

electrons, but loses the spin correlation as well as raises the energy of Ni-O bonds. The distortion does not occur.

Summary

 We have learned how to apply the symmetry theory in the case of molecules with non Abelian symmetry. We have learned the decomposition of characters into irreps.

 We have discussed three chemical concepts derived from symmetry, which are Woodward-Hoffmann rule, crystal field theory, and Jahn-Teller distortion.