inorganic chemistry ii (ch333) i) symmetry elements · molecular symmetry and introduction to group...

Inorganic Chemistry II (CH333)

Molecular Symmetry and Introduction to Group Theory

Department of Chemistry, Faculty of Science,

Maejo University, Chiang Mai, Thailand

http://www.science.mju.ac.th/chemistry/

Weerinradah Tapala

( )

Course Outline (30% + 4%)

I) Symmetry elements

II) Symmetry operations

III) Multiplication of symmetry operations

IV) Molecular point groups

V) Determination of molecular point groups

VI) Examples of some applications

Selected references

1. P.H. Walton, Beginning Group Theory for Chemistry, Oxford University Press, 1998.

2. A.M. Lesk, Introduction to Symmetry and Group Theory for Chemists, Kluwer Academic Publishers, 2004.

3. P.W. Atkins, T.L. Overton, J.P. Rourke, M.T. Weller, F.A. Armstrong, Shriver and Atkins'

Inorganic Chemistry, 5th Edition, W. H. Freeman and Company New York, 2010.

4. G.L. Miessler, D.A. Tarr, Inorganic Chemistry, 3th Edition, Pearson Prentice Hall, 2004.

5. C.E. Housecroft, A.G. Sharpe, Inorganic Chemistry, 2nd Edition, Pearson Prentice Hall, 2005.

6. , , .

7. , ,

, 2550.

???

4

“Flag symmetry”

Symmetry in nature, art, and architecture

5

Symmetry elements & Symmetry operations

1

23

3

12

1/3

(Symmetry element):

A A’

(Symmetry operation): / /

/ /

6

Element Symbol Operation

Proper axis Cn Rotation about axis by 360/n

Symmetry plane Reflection in plane

Improper axis Sn 1. Rotation by 360/n 2. Reflection through plane

perpendicular to rotation axis

Inversion center i Inversion every point (x,y,z)

translated to (-x,-y,-z)

Identity E Doing nothing


7

(Proper rotation axis): Cn

“ (Cn) 360/n

”

Symmetry element =

Symmetry operation = 360/n

8

C2 C3 C4

(Proper rotation axis): Cn1

23

3

12A A’

C31

2

31A’’

C32

1

23A’’’

C33

A A’ A’’ (Equivalent)

A A’’’ (Identical)

1

23

1

23

C21

C22

1

32

1

23

A A’ A 4 C3 C2

2 (class)

- Class 1: C3 2 C31 C3

2

- Class 2: C2 1 3 3C21

2+3 = 5 9


C41

3

4

2

1C4

4

1

3

2C4

1

2

4

3C4

2

3

1

4C4

1

2

4

3C4

C42 C4

3 C44

C21

1

2

4

3C2

3

4

2

1C2

1

2

4

3C2

C22

C21

1

2

4

3

4

3

1

2

1

2

4

3C2

C22

C21

1

2

4

3

2

1

3

4

1

2

4

3

C2’’C2

2

10


C21

1

2

4

3

1

4

2

3

1

2

4

3C2

C22

C21

1

2

4

3

3

2

4

1

1

2

4

3

C22

1

2

4

3C4,C2

C2 C2’

C2’’

C2’’’

(Coincident) Cn n

Cnn = E (Identity)

C42 C2

1 C21

2 ( C4 1 ) + 5 (

C2 5 ) = 7

C2’

C2’’’

11


H2O

NH3

12


C6H6

? ?

?

coincident ?

? ?

13

(Plane of symmetry): “ (reflection)

(mirror image)”

Symmetry element =

Symmetry operation =

14

(Plane of symmetry): 7

12

34

5

6

12

34

5

6

1

2

3

4

5

6

’’’

’’’

1

2

34

5

6

’

1

1

6

54

3

2

’

2

1

2

34

5

6

’

1

2

3

4

5

6

’’’’ ’’’’’

’’’’’’

1

2

34

5

61

6

5

43

2

12

1

2

34

5

6’’’’ ’’’’ ’’’’

15

(Plane of symmetry): 1

2

34

5

6

’1

2

3

4

5

6’’

’’’

1

2

3

4

5

6

’’’’ ’’’’’

’’’’’’

7

2 = E

2

- Cn (vertical plane): v

v C2

v

(dihedral plane): d

- Cn (horizontal plane): h

1 h, 3 v 3 d 16

H2O

NH3

(Plane of symmetry):

17

(Improper rotation axis): Sn

“ 360/n

”

Cn + h

-

18

(Improper rotation axis): Sn

19

“ (x,y,z) (project)

(-x,-y,-z)

”

20

(Inversion centre): i

“Centrosymmetric molecule”

i2 = E 21

(Inversion centre): i (Inversion centre): i

22

(Inversion centre)

i

(Identity): E“ ”

23

C41

3

4

2

1C4

4

1

3

2C4

1

2

4

3C4

2

3

1

4C4

2

3

1

4C4

C42 C4

3 C44

Cnn = E

1

2

34

5

6

’

1

1

6

54

3

2

’

2

1

2

34

5

6

’

2 = E


24

(VSEPR)

Symmetry elements Symmetry operations

1. Proper axis (Cn) 360/n

2. Symmetry plane

3. Improper axis 360/n h

4. Inversion (x,y,z) (-x,-y,-z)

5. Identity

VSEPR model

25

VSEPR model

26

27

Symmetry elements & Symmetry operations (symmetry element)

1. H2O

2. XeF4

3. CH4

4. CH2Cl25. PCl5

28

“ A B

B A BA A B

(Group theory)

(successive symmetry operation”

A B AB B(1st) and then A(2nd)

A B C ABC C(1st), B(2nd) and then A(3rd)

29

C2 V

V C2

V’

C2 V

30

V

V

C3

V

C3

V C3

C3 V

31

v C2 C2 v v’ (commutative property)

v C3 C3 v (non-commutative property)

32

(PH3)

1. C31

v

2. v C31

(PH3)

1. C31

v v’

2. (C31

v) v’

3. C31 ( v v’) P

H(1) H(2)

H(3)

V

V’V’’

C31

v v’ (C31

v) v’ C31 ( v v’)

(associate property) 33

(Inverse operation)

A

A’

A A’

AA-1 A-1A E

A A-1

A B inverse

AB E

AB BA34

35

C31

(PH3)

P

H(1) H(2)

H(3)

V

V’V’’

C31 C3

-1 E

C3-1 C3

2

C31 C3

2

-1 E

2 E-1

36

C31 C3

-1 E

C3-1 C3

2 C31 C3

2 E

C31

P

H(1) H(2)

H(3)

V

V’V’’

Cn-1Cn

1 = E Cnn-1Cn

1 = E

Cn-1 = Cn

n-1

P

H(1) H(2)

H(3)

V

V’V’’

C31

P

H(3) H(1)

H(2)

V

V’V’’

C32

P

H(2) H(3)

H(1)

V

V’V’’

C3-1

P

H(2) H(3)

H(1)

V

V’V’’

C32

37

i2 = E

i = i-1

2 = E

= -1

Cn-1Cn

1 = E Cnn-1Cn

1 = E

Cn-1 = Cn

n-1

38

“ ”C2

v

v’E C2 v v

E EE C2E vE v E

C2 EC2 C2C2 vC2 v C2

vE v C2 v v v v v

v E v C2 v v v v v

E C2 v v

E E C2 v v

C2 C2 E v v

v v v E C2

v v v C2 E

39

trans-2-butene

E C2 i

E E C2 i

C2 C2 E i (C2) (E)

i (C2) E i

i i (E) i (E) E

E C2 i

E EE C2E E iE

C2 EC2 C2C2 C2 iC2

E C2 i

i Ei C2i i ii

40

1. 2

AB = C A B C

2. “commutative”

EA = AE = A

3. “associative law” (AB)C =

A(BC) = ABC

4. “inverse”

GH = HG = E G = H-1 H = G-1 G H E

(Point group)

Cn n>2 2

Td ( tetrahedral)

Oh ( octahedral)

Ih ( Icosahedral) B12H122-

Point group: (Td, Oh, Ih)

41

Td IhOh Oh

42

C1 ( C1 E ) CHFClBr

Cs ( E ) phenol

Ci ( E i ) ClFHC.CHFCl (stagger)

C1 Cs Ci

Point group: (C1, Cs, Ci)

C ( Cn)

- 1 Cn

- Cn h Cnh

- Cn v Cnv

- C V

D ( dihedral nC2 Cn)

- Dn

- h Dnh

- n d Dnd

S ( - Sn n 4 )

S1 = Cs S2 = Ci Sn (n= ) = Cnh

43

Point group: Cn, Cnv, Cnh Point group

44

H2O; C2v NH3; C3v

B(OH)3; C3h

“ Cn + n v ”

“ Cn + h ”P(C6H5)3; C3

“ Cn ”

45

Cnv Point group: Cn n v ( Cn)

46

n i

Cnh Point group: Cn h ( Cn)

( Cn)

47

C2

C3

Cn Point group: Cn

Dnh point group Cn, nC2 Cn h

48

n

i

Dnd, Dnh, Dn Point group “ Cn + nC2 Cn ”

Dnd point group Cn, nC2 Cn , S2n n d

49

Dnd point group Dnd vs. Dnh

50

C2H4; D2hB2Cl4; D2d

Dnd point group Cn, S2n, nC2 Cn n d

Dnh point group Cn, nC2 Cn h

h

D5d D5h

Fe(C5H5)2

51

Dnd vs. Dnh

52

Dn point group: Cn, nC2 Cn

[Cu(en)2]2+

D2[Co(en)3]

2+

D3

C v, D h Point group “linear molecule”

53

O2; D h CO2; D h

CO; C v HCN; C v

“ i h ”

Sn n

54

Sn point group

S1 = Cs S2 = Ci

point group Sn n > 2

1.

Linear: C V, D h

Tetrahedral (Td), Octahedral (Oh), Icosahedral (Ih)

55

2. Cn Sn

E (C1), (Cs), I (Ci)Tetrahedral (Td), Octahedral (Oh), Icosahedral (Ih)

3. Sn: S4, S6,…

4. Cn nC2 Cn

(Cn) h (Cnh) n v (Cnv)

5. Cn nC2 Cn

(Dn) h (Dnh) n d (Dnd)56

57

Oh point group

Symmetry elements

E

4C3

3C2

3C4

6C2

3S4

4S6

i

3 h

3 d

START

Linear

“Decision tree”

58

1. HCN

2. H2S

3. BeF2

4. C6H6

5. BeF3

6. SiCl4

7. PCl5

8. XeF4

9. CH4

10. CH3Cl

59

11. CH2Cl2

12. SF6

13. SF5Cl

14. trans-SF4Cl2

15. 1,2-dimethylcyclopentane

16. Au(CN)2-

17. cis-PtCl2Br2

18. trans-[Co(NH3)4(H2O)2]2+

19. C2H6 (staggered)

20. C2H6 (eclipsed)

60

matrix

matrix

Identity (E)

Inversion (i)

Reflection ( )

Rotation (Cn)

Rotation-Relection (Sn)

61

(cartesian coordinate)

62

1) Identity (E)

EX

Z

Y

X

Z

Y

1 0 0

0 1 0

0 0 1

X

Y

Z

X

Y

Z

matrix ( ) = 3

63

2) Inversion (i)

iX

Z

Y

-X

-Z

-Y

-1 0 0

0 -1 0

0 0 -1

X

Y

Z

-X

-Y

-Z

(i) = -3 64

3) Reflection ( )

xy

X

Z

Y

1 0 0

0 1 0

0 0 -1

X

Y

Z

X

Y

-Z

( xy) = 1

-Z

X

Y

65

1 0 0

0 1 0

0 0 -1

( ) = 1

Matrix xy

1 0 0

0 -1 0

0 0 1

Matrix xz

-1 0 0

0 1 0

0 0 1

Matrix yz

66

4) Rotation (Cn)

C21

X

Z

Y

-1 0 0

0 -1 0

0 0 1

X

Y

Z

- X

- Y

Z

(C21) = -1

-Y

Z-X

67

C21

X

Z

Y

1 0 0

0 -1 0

0 0 -1

X

Y

Z

X

- Y

- Z

(C21) = -1

X

Z

Y

68

matrix Cn

Cn

X

Z

Y

Y X

69

Y X

Y sin

Y cos

X cos

X sin

X = X cos - Y sin

Y = X sin + Y cosZ = Z

cos -sin 0

sin cos 00 0 1

X

Y

Z

X cos - Y sin

X sin + Y cosZ

(Cn) = 1 + 2cos

= 180

cos180 -sin180 0

sin180 cos180 0

0 0 1

-1 0 0

0 -1 0

0 0 1

70

5) Rotation-Reflection (Sn)

C21

X

Z

Y

-1 0 0

0 -1 0

0 0 1

-Y

Z-X

S2 Z

xy-Y

-Z

-X

1 0 0

0 1 0

0 0 -1

-1 0 0

0 -1 0

0 0 -1

71

matrix ( )

(E) = 3

(i) = -3

( ) = 1

(Cn) = 1 + 2cos

(Sn) = -1 + 2cos

= 360/n

72

matrix

z2

x2

y2

z1

x1

y1z3

x3

y3

-1 0 0 0 0 0 0 0 0

0 -1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 -1 0 0

0 0 0 0 0 0 0 -1 0

0 0 0 0 0 0 0 0 1

0 0 0 -1 0 0 0 0 0

0 0 0 0 -1 0 0 0 0

0 0 0 0 0 1 0 0 0

C21 (z) :

= -1

73

matrix

z2

x2

y2

z1

x1

y1z3

x3

y3

1 0 0 0 0 0 0 0 0

0 -1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 -1 0

0 0 0 0 0 0 0 0 1

0 0 0 1 0 0 0 0 0

0 0 0 0 -1 0 0 0 0

0 0 0 0 0 1 0 0 0

xz :

= 174

matrix

z2

x2

y2

z1

x1

y1z3

x3

y3

-1 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 -1 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 -1 0 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1

yz :

= 3

75

( H2O)

C2v E C2(z) xz yz

No. of unshifted atom 3 1 1 3

Type of unshifted atom O O O O, 2H

Contribution/atom ( ) 3 -1 1 1

H2O 3 39

1 -1-1

1 11

3 13 (Reducible representation)

(Character table)

76

Point group Class of symmetry operations Order of group

Mulliken

symbols

(Irreducible representationsbase

f = f =

h = order

R =

I = ( )N =

77

C2v E C2(z) xz yz

H2O 9 -1 1 3

f(A1) =(1/4) (9 1 1)+(-1 1 1)+(1 1 1)+(3 1 1) = 1/4(9-1+1+3) = 3

f(A2) =(1/4) (9 1 1)+(-1 1 1)+(1 -1 1)+(3 -1 1) = 1/4(9-1-1-3) = 1

f(B1) =(1/4) (9 1 1)+(-1 -1 1)+(1 1 1)+(3 -1 1) = 1/4(9+1+1-3) = 2

f(B2) =(1/4) (9 1 1)+(-1 -1 1)+(1 -1 1)+(3 1 1) = 1/4(9+1-1+3) = 3

H2O = 3A1 + A2 + 2B1 + 3B2

H2O (Reducible representation)

78

O H ( O H)

C2v E C2(z) xz yz

No. of unshifted bond 2 0 0 2

Contribution/atom 1 1 1 1

O H 2 12

0 10

0 10

2 12

r1 r2

f(A1) = (1/4) (2 1 1)+(0 1 1)+(0 1 1)+(2 1 1) = 1/4(2+0+0+2) = 1

f(A2) = (1/4) (2 1 1)+(0 1 1)+(0 -1 1)+(2 1 -1) = 1/4(2+0+0-2) = 0

f(B1) = (1/4) (2 1 1)+(0 -1 1)+(0 1 1)+(2 -1 1) = 1/4(2+0+0-2) = 0

f(B2) = (1/4) (2 1 1)+(0 -1 1)+(0 -1 1)+(2 1 1) = 1/4(2+0+0+2) = 1

O H = A1 + B2

C2v E C2(z) xz yz

O H 2 0 0 2

79 80

C3v E 2C3 3 v

m 5 2 1

f(A1) = (1/6) (5 1 1)+(2 1 2)+(1 1 3) = 1/6(5+4+3) = 2

f(A2) = (1/6) (5 1 1)+(2 1 2)+(1 -1 3) = 1/6(5+4-3) = 1

f(E) = (1/6) (5 2 1)+(2 -1 2)+(1 0 3) = 1/6(10-4+0) = 1

m = 2A1 + A2 + E

81

(molecular displacement)

(vibration) (translation) (rotation)

N Degree of freedom = 3N

Degree of freedom = 3N – 6 (non-linear molecule)

= 3N – 5 (linear molecule)

82

3N = vibration + translation + rotation

vibration = 3N - translation - rotation

3N = 3A1 + A2 + 2B1 + 3B2 (degree of freedom = 9)

translation = A1 + B1 + B2 (degree of freedom = 3)

rotation = A2 + B1 + B2 (degree of freedom = 3)

vibration = [3A1 + A2 + 2B1 + 3B2] – [A1 + B1 + B2] – [A2 + B1 + B2]

= 2A1 + B2 (degree of freedom = 3)

83

vibration

NH3 Point group: C3v

C3v E 2C3 3 v

No. of unshifted atom 4 1 2

Type of unshifted atom N,3H N N,H

Contribution/atom ( ) 3 0 1

3N12 0 2

84

C3v E 2C3 3 v

3N 12 0 2

f(A1) = (1/6) (12 1 1)+(0 1 2)+(2 1 3) = 1/6(12+0+6) = 3

f(A2) = (1/6) (12 1 1)+(0 1 2)+(2 -1 3) = 1/6(12+0-6) = 1

f(E) = (1/6) (12 2 1)+(0 -1 2)+(2 0 3) = 1/6(24+0+0) = 4

3N = 3A1 + A2 + 4E (degree of freedom = 12)

character table: translation = A1 + E (degree of freedom = 3)

rotation = A2 + E (degree of freedom = 3)

vibration = 2A1 + 2E (degree of freedom = 6)

85

vibration XeF4

86

(mode of molecular vibration)

(stretching) (bending)

vibration = stretching + bending

87

C2v E C2(z) xz yz

No. of unshifted bond 2 0 0 2

Contribution/atom 1 1 1 1

O H 2 12

0 10

0 10

2 12

O H stretching

stretching = A1 + B2

vibration = 2A1 + B2 bending = [2A1 + B2] - [A1 + B2] = A1 88

C3v E 2C3 3 v

No. of unshifted atom 3 0 1

Contribution/atom ( ) 1 1 1

N-H3 0 1

bending ( vibration = 2A1 + 2E)

r2r1 r3

stretching = A1 + E

vibration = 2A1 + 2E bending = [2A1 + 2E] - [A1 + E] = A1 + E

inorganic chemistry ii (ch333) i) symmetry elements · molecular symmetry and introduction to group...

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