inorganic chemistry ii (ch333) i) symmetry elements · molecular symmetry and introduction to group...
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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.
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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
Symmetry elements & Symmetry operations
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
(Proper rotation axis): Cn
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
(Proper rotation axis): Cn
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
(Proper rotation axis): Cn
H2O
NH3
12
(Proper rotation axis): Cn
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
Symmetry elements & Symmetry operations
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