chapter5 symmetry after lecture
TRANSCRIPT
1
Chapter 5. Molecular Symmetry Outline A. Introduction B. Symmetry elements and symmetry operation C. Point groups D. Applications of symmetry
References 1. Catherine E. Housecroft and Alan G. Sharpe, Inorganic Chemistry, 1st Ed., Pearson Education Ltd., 2008, Chapter 4. 2. D.F.Shriver, P.W.Atkins et al, Inorganic Chemistry, 4th Ed., Oxford University Press, 2006, Chapter 7. 3. B. Douglas, D. McDaniel and J. Alexander, Concepts and Models of Inorganic Chemistry, 3r Ed., John Wiley & Sons, Inc. 1994, Chapter 3. 4.Gary L. Miessler, Donald A. Tarr, Inorganic Chemistry, Pearson Education Ltd., 2004, Chapter 4.
2
A. Introductions
Symmetry is is all around us and is present in nature and in human culture
3
Many Inorganic compounds also have symmetry.
O C OFB
FF Cl Pt
Cl
CO
CO
FP F
F
F
F
FS FF
F
FF
FF
F
FF
I
F
F
Cl
Cl
Co
ClCl
2-
B12H122-
Ru
Will be useful to identify symmetry elements of molecules – implications for bonding and spectroscopy
4
B. Symmetry Operations and Elements
Definitions Symmetry Operation = a movement of a body such that the
appearance after the operation is indistinguishable from the original appearance (if you can tell the difference, it wasn’t a symmetry operation)
e.g. reflection, rotation, …
Symmetry Element = geometrical entity with respect to which
one or more symmetry operations can be carried out.
e.g. – could be a point, an axis or a plane
5
Types of Symmetry elements and operation
There are five important types of symmetry operations summarized on the right with their associated symmetry elements
Symmetry element Symmetry operation
Symbol
Identity* n-Fold symmetry axis Mirror plane Center of inversion n-Fold axis of improper rotation
None Rotation by 2π/n Reflection Inversion Rotation by 2π/n
followed by reflection
perpendicular to rotation axis
E Cn σ I Sn
6
1. Identity, E. The identity E. The identity E operation does nothing. No change in the object. May not seem like an operation at all, but is important when we consider a set of symmetry elements which form a group. ---Needed for mathematical completeness ---Every molecule has this symmetry element
2. Proper rotation, Cn Simple rotation about an axis passing through the molecule by an angle of 360o/n. This operation is called a proper rotation (or simply rotation) and symetry element is symbolized by Cn
e.g. Water – has a two-fold rotation axis (C2 operation)
O
H H
O
H H180o
Rotation by 180˚ leaves H2O in identical orientation.
We say water has a C2 rotation axis.
7
C3 Symmetry element and operations
e.g. NH3 has a C3 rotation axis. One can perform two C3 operations
8
Common n-fold rotation axis Cn - 360o/n rotation
Rotation symbol rotation by 180o
rotation by 120o rotation by 90o
rotation by 72o
fold rotation by 60o
Exercise. Locate the Cn axis in the following molecules.
O C OFB
FF Cl Pt
Cl
Cl
Cl
2-
9
Common n-fold rotation axis Cn - 360o/n rotation
Rotation symbol rotation by 180o C2
rotation by 120o C3 rotation by 90o C4
rotation by 72o C5
fold rotation by 60o C6
Exercise. Locate the Cn axis in the following molecules.
O C OFB
FF Cl Pt
Cl
Cl
Cl
2-
10
O C OFB
FF Cl Pt
Cl
Cl
Cl
2-
O C OFB
FF Cl Pt
Cl
Cl
Cl
2-
11
O C OFB
FF Cl Pt
Cl
Cl
Cl
2-
∞C2
C∞
O C OFB
FF Cl Pt
Cl
Cl
Cl
2-
12
O C OFB
FF Cl Pt
Cl
Cl
Cl
2-
∞C2
C∞
C2
C2
O C OFB
FF Cl Pt
Cl
Cl
Cl
2-
C2
C3
13
O C OFB
FF Cl Pt
Cl
Cl
Cl
2-
∞C2
C∞
C2
C2
C2
C2
C2
C2
O C OFB
FF Cl Pt
Cl
Cl
Cl
2-
C2
C4 C3
14
Cl Pt
Cl
Cl
Cl
2-Also has this C2 symmetry element?
C2 ??
15
Cl Pt
Cl
Cl
Cl
2-Also has this C2 symmetry element?
Cl Pt
Cl
Cl
Cl
2-
C2 ??
C4
No. One can perform a C2 operation about the axis. This axis is the C4 axis.
16
3. Reflection (Mirror), σ. Reflection of all atoms through a plane, e.g. water molecule
O
H H
O
H H
O
H HO
H H
Note that a water molecule can be reflected by two planes. Mirror plane symmetry operation is denoted as σ
Exercise. Locate σ planes in the following molecules.
O C OFB
FF Cl Pt
Cl
Cl
Cl
2-
17
O C OFB
FF Cl Pt
Cl
Cl
Cl
2-Answers
18
O C O
O C O
O C OFB
FF Cl Pt
Cl
Cl
Cl
2-Answers
∞
19
O C O
O C O
FB
FF
FB
FF
FB
FF
FB
FF
O C OFB
FF Cl Pt
Cl
Cl
Cl
2-Answers
∞
20
O C O
O C O
FB
FF
FB
FF
FB
FF
FB
FF
Cl Pt
Cl
Cl
Cl
2-
Cl Pt
Cl
Cl
Cl Cl Pt
Cl
Cl
Cl
Cl Pt
Cl
Cl
Cl Cl Pt
Cl
Cl
Cl
O C OFB
FF Cl Pt
Cl
Cl
Cl
2-Answers
∞
21
FB
FF
FB
F
F
FB
F
F
FB
F
F
C2 C3
σhσv
Type of mirrors
σh = plane perpendicular to principal axis σv = plane includes the principal axis (σd = a kind of σv plane includes the principal axis, but not the outer atoms)
O C O OH H
σh σv
σd
C∞
C2
22
4. Inversion (Center of Inversion), i = each point moves through a common central point to a position opposite and equidistant.
Reflection of all atoms through a point. Each atom has a identical counter-part on the other side of the molecular center. e.g.
FS FF
F
FF
N N
The inversion center I could be midway along a bond, e.g. as in N2 molecule. Or may be situated at an atomic center itself, e.g. as in SF6.
23
Question. Which of these molecules have a center of inversion? H2O, O2, NO2, CO2
OH H
O O NO O
O C O
24
Question. Which of these molecules have a center of inversion? H2O, O2, NO2, CO2
OH H
O O NO O
O C O
25
Question. Which of these molecules have a center of inversion? H2O, O2, NO2, CO2
OH H
O O NO O
O C O
26
4. Improper rotation Sn. also called Rotation-Reflection
The combination, in either order, of rotating the molecule about an axis passing through it by 360o/n and reflecting all atoms through a plane that is perpendicular to the axis of rotation.
Sn = Cn + σh
Question. How many S4 rotation axis does a CH4 molecule have?
27
H1
H3
H2
H4
C
H3
H4
H1
H2C
H4
H1
H3
H2
C
C4
C4
σhσh
28
H
H
H
H
C S4
S4
S4
Three
29
Exercise. Which of the following species have a Sn axis? FB
F
F Cl Pt
Cl
Cl
ClL
M LL
L
L
LM LL
L
LRu
30
Exercise. Which of the following species have Sn axis? FB
F
F Cl Pt
Cl
Cl
ClL
M LL
L
L
LM LL
L
LRu
S3
31
Exercise. Which of the following species have Sn axis? FB
F
F Cl Pt
Cl
Cl
ClL
M LL
L
L
LM LL
L
LRu
S3 S4
32
Exercise. Which of the following species have Sn axis? FB
F
F Cl Pt
Cl
Cl
ClL
M LL
L
L
LM LL
L
LRu
S3 S4 S3
33
Exercise. Which of the following species have Sn axis? FB
F
F Cl Pt
Cl
Cl
ClL
M LL
L
L
LM LL
L
LRu
S3 S4 S3
S5
34
FaB
Fb
Fc Clb Pt
Cla
Clc
CldLa
M LcLb
Ld
LeFc
B
Fa
Fb
C3
σh
C3
FcB
Fa
Fb
C4
Cla Pt
Cld
Clb
Clc
C4
σh
Cla Pt
Cld
Clb
Clc
C3
LcM Lb
La
Ld
Le
C3
LcM Lb
La
Le
Ld
σh
35
LM LL
L
LC4
LM LL
L
L
LM LL
LL
σhRu
C5
Ruσh
Ru
36
Some interesting points:
S1 operation is equivalent to the mirror reflection σ
S2 is equivalent to a center of inversion i
37
Summary of symmetry operations and symmetry elements
38
Examples:
39
40
C. Point Groups
1. Definition. One molecule may have more than one symmetry elements and one can perform various symmetry operations. E.g.
The complete set of symmetry operations than can be performed on a molecule is called the symmetry group for the molecule.
OH H
NH H
H
E, C2, 2σv
E, C3, 3σv
41
Not accurate!
42
E, 3C4, 4C3, 4S6, 3S4, i, (6+3)C2, 9σ, à Oh
E, 4C3, 3C2, 3S4, 6σd à Td
43
44
2. Characteristics of Common Point Groups
2.1. Groups of Low Symmetry
E
E, σ
E, i
45
2.2. Cn groups (n = 1, 2, …)
Contain only one Cn rotational axis
2.3. Cnv groups Contain Cn and nσv
OH H
σv
σdC2
H
H
HN
B
N
N
H
H
H
F F
Br
Sb
Cl
F F
Examples
C3
C2v
C3v
46
2.2. Cn groups (n = 1, 2, …)
Contain only one Cn rotational axis
2.3. Cnv groups Contain Cn and nσv
H
H
HN
B
N
N
H
H
H
F F
Br
Sb
Cl
F F
Examples
C3
C2v
C2
47
2.2. Cn groups (n = 1, 2, …)
Contain only one Cn rotational axis
2.3. Cnv groups Contain Cn and nσv
H
H
HN
B
N
N
H
H
H
F F
Br
Sb
Cl
F F
Examples
C3
C2v C3v C2
C3
48
2.2. Cn groups (n = 1, 2, …)
Contain only one Cn rotational axis
2.3. Cnv groups Contain Cn and nσv
H
H
HN
B
N
N
H
H
H
F F
Br
Sb
Cl
F F
Examples
C3
C2v C3v
C4v
C2 C3
C4
49
How about the following compounds?
SF4 S
F
F
F
FS
F
F
F
FS
F
F
F
F
SClF5 S
F
Cl
FFF
F
50
SF4 S
F
F
F
FS
F
F
F
FS
F
F
F
F
SClF5 S
F
Cl
FFF
FS
F
Cl
FFF
FC4v
C2v
other 3 σv are not shown
How about the following compounds?
51
2.4. Cnh groups
Cn + σh
O
H
H
B
O
O
H
Cl
H
H
Cl
Mo Mo
O O
O O
MeC
CMe
R3P X
PR3X ORe
O
Re
O
O OS
OO
SOO
OO O
SO
O
SO
O
C2h
52
2.4. Cnh groups
Cn + σh
O
H
H
B
O
O
H
Cl
H
H
Cl
Mo Mo
O O
O O
MeC
CMe
R3P X
PR3X ORe
O
Re
O
O OS
OO
SOO
OO O
SO
O
SO
O
C2h
C2
C2
53
2.4. Cnh groups
Cn + σh
O
H
H
B
O
O
H
Cl
H
H
Cl
Mo Mo
O O
O O
MeC
CMe
R3P X
PR3X ORe
O
Re
O
O OS
OO
SOO
OO O
SO
O
SO
O
C2h C3h
C2 C3
54
2.4. Cnh groups
Cn + σh
O
H
H
B
O
O
H
Cl
H
H
Cl
Mo Mo
O O
O O
MeC
CMe
R3P X
PR3X ORe
O
Re
O
O OS
OO
SOO
OO O
SO
O
SO
O
C2h C3h C3h
C2
C2 C3 C3
55
2.4. Cnh groups
Cn + σh
O
H
H
B
O
O
H
Cl
H
H
Cl
Mo Mo
O O
O O
MeC
CMe
R3P X
PR3X ORe
O
Re
O
O OS
OO
SOO
OO O
SO
O
SO
O
C2h C3h C3h
C2h
C2
C2 C3 C3
C2
56
2.4. Cnh groups
Cn + σh
O
H
H
B
O
O
H
Cl
H
H
Cl
Mo Mo
O O
O O
MeC
CMe
R3P X
PR3X ORe
O
Re
O
O OS
OO
SOO
OO O
SO
O
SO
O
C2h C3h C3h
C2h
C4h
C2
C2 C3 C3
C2
C4
57
ORe
O
Re
O
O OS
OO
SOO
OO O
SO
O
SO
OC4
ORe
O
Re
O
O OS
OO
SOO
OO O
SO
O
SO
O σh
Any other symmetry element?
58
ORe
O
Re
O
O OS
OO
SOO
OO O
SO
O
SO
O
C4
ORe
O
Re
O
O OS
OO
SOO
OO O
SO
O
SO
O σh
Any other symmetry element? E, I, S4
ORe
O
Re
O
O OS
OO
SOO
OO O
SO
O
SO
O
S4
59
2.5. D Groups
Dn groups: Cn + n C2 Dnh group: Cn + n C2 + σh
Dnd group: Cn + n C2 + nσv
A Cn principal rotational axis is accompanied by a set of n C2 axes perpendicular to it.
Characteristic symmetry elements:
60
a). Dn groups
Cn + nC2 (┴)
A Cn principal rotational axis is accompanied by a set of nC2 axes perpendicular to it
C2
C2
C2
C3
D3 group
61
b). Dnh groups Characteristic symmetry elements: Cn + nC2 + σh σh: horizontal mirror plane, perpendicular to the principal axis Cn.
Also has n σv, and maybe others too
Examples
Symmetry elements:
C3
C2 σh
62
b). Dnh groups Characteristic symmetry elements: Cn + nC2 + σh σh: horizontal mirror plane, perpendicular to the principal axis Cn.
Also has n σv, and maybe others too
Examples
Symmetry elements:
E, C3, 3C2(┴), σh, 3σv, S3
C3
C2 σh
S3 σv Point group: D3h
63
XeF4
Xe F
F
F
F
Symmetry elements:
64
XeF4
Xe F
F
F
F
E, C4, 4C2(┴), σh, 4σv, + ?
Symmetry elements:
Point group: D4h
65
XeF4
Xe F
F
F
F
E, C4, 4C2(┴), σh, 4σv, S4, i
Symmetry elements:
Point group: D4h
66
How about the following species?
dualrelationship
?[PtCl4]2- a square planar geometry C6H6 BF3
67
How about the following species?
dualrelationship
?[PtCl4]2- a square planar geometry C6H6 BF3
D3h
68
How about the following species?
dualrelationship
?[PtCl4]2- a square planar geometry C6H6 BF3
D3h D6h
69
How about the following species?
dualrelationship
?[PtCl4]2- a square planar geometry C6H6 BF3
D3h D6h
D5h
70
How about the following species?
dualrelationship
?[PtCl4]2- a square planar geometry C6H6 BF3
D3h
D3h D6h
D5h
D4h
D6h
D3h
71
How about the following species?
dualrelationship
?[PtCl4]2- a square planar geometry C6H6 BF3
D3h
D3h D6h
D5h
D4h
D6h
D3h
D5h
72
73
c) Dnd group: Characteristic symmetry elements: Cn + nC2 + σv
Do not have σh mirror plane Examples:
RW R
R
W
RRR
RW R
R
W
RRR
RW R
R
W
RRR
Fe Fe FeFerrocene
R3W≡WR3
Point group: symmetry elements:
Point group: symmetry elements:
74
c) Dnd group: Characteristic symmetry elements: Cn + nC2 + σv
Do not have σh mirror plane Examples:
R3W≡WR3
symmetry elements:
Point group:
RW R
R
W
RRR
RW R
R
W
RRR
RW R
R
W
RRR
75
c) Dnd group: Characteristic symmetry elements: Cn + nC2 + σv
Do not have σh mirror plane Examples:
R3W≡WR3
symmetry elements: E, C3, 3C2(┴), 3σv, I, S6
RW R
R
W
RRR
RW R
R
W
RRR
RW R
R
W
RRR
C3
σv
C2
Point group: D3d
RW R
R
W
RRR
RW R
R
W
RRR
S6i
76
Point group:
symmetry elements:
Fe Fe Fe
77
Point group: D5d
symmetry elements: E, C5, 5C2(┴), 5σv, I, S10
Fe Fe Fe
C5
C2
σv
S10 i
Fe Fe
78
C2(┴)
79
Dnd groups Additional examples
e.g. triagular antiprism
square antiprism
C C CH
HH
HH H
H
Hallene
Steggered ethane
80
Dnd groups Additional examples
e.g. triagular antiprism (D3d)
Steggered ethane
C3
C2 C2
σv
81
Dnd groups Additional examples
e.g. triagular antiprism (D3d)
Steggered ethane
C3
C2 C2
σv σv
C2
D3d
C3
82
Point group:
C C CH
HH
H
83
C C CH
HH
H
H
H
H
H
C2
C2
C2 H
H
H
H
C2
σv
H H
H
H
C2C2
H H
H
H
σv
Point group: D2d
84
Point group:
85
C4A'
D
BC'
AB'
CD'
C2
σv
Point group: D4d
86
2.6. Sn groups (n = even only because one gets Cnh when n = odd S1 = Cs S2 = Ci
N
MeHMe H
HMeMeH
XB NR
NR BX
BX
RN BX
NR
When n = odd, Sn ≡ Cnh. Thus n can only be even
87
MeHH
Me
Me HH Me
MeH
HMe
HMeMeH
C4
C4σhσh
MeHH
Me
Me HH Me
Point group: S4
Symmetry elements?: S4
88
MeHH
Me
Me HH Me
MeH
HMe
HMeMeH
C4
C4σhσh
MeHH
Me
Me HH Me
Point group: S4
89
XB NR
BXNRXB RN
RN BX
NR BX
NRXBRN BX
XB RNσh
C4
C4 σh
XB NR
BXNRXB RN
RN BX
Point group: S4
90
2.7. High symmetry point groups a) Tetrahedral point groups
In addition there are molecules with Th point group. Th point group: T + I (rare)
Cl
SiCl Cl
Cl Any other symmetry elements?
Characteristic symmetry elements: 4C3 + 3C2 (e.g. CH4, SiCl4, etc…)
Td point group
91
2.7. High symmetry point groups a) Tetrahedral point groups
In addition there are molecules with Th point group. Th point group: T + I (rare)
Cl
SiCl Cl
Cl Any other symmetry elements?
Characteristic symmetry elements: 4C3 + 3C2 (e.g. CH4, SiCl4, etc…)
Td point group
C2
C2
C2
C3
C3
C3
C3
92
2.7. High symmetry point groups a) Tetrahedral point groups
In addition there are molecules with Th point group. Th point group: T + I (rare)
Cl
SiCl Cl
Cl Any other symmetry elements?
Characteristic symmetry elements: 4C3 + 3C2 (e.g. CH4, SiCl4, etc…) Td point group
C3
C3
C3
C3
C2
C2
C2
S4
S4
S4
6σ
93
b) Octahedral point groups Oh point group: Characteristic symmetry elements 3C4 e.g. SF6
Icosahedral point groups Ih point group: Characteristic symmetry elements 6C5 + i
E, 4C3, 4S6, 3S4, i, 6C2, 9σ
E, 10C3, 6S10, 10S6, 15C2, 15σ, à Ih
Additional symmetry elements
Additional symmetry elements:
C4
C5
94
Summary
• T group: 4C3
• O group: 3C4
• I group: 6C5
95
3. Identification of point groups. To identify a molecules point group, a list of symmetry elements could be made and then compare them with list characteristic of each point group.
e.g. To what point groups do H2O and NH3 belong?
OH H
NH H
H
(a) H2O possesses the following symmetry elements: E, C2, 2σv. The set of elements (E, C2, 2σv) corresponds to the group C2v.
(b) NH3 possesses following symmetry elements: E, C3, 3σv. The set of elements (E, C3, 3σv) corresponds to the group C3v.
96
Alternatively, one can use set of rules for assigning the point group – follow a flow chart (see below).
Notes: The symmetry elements used to identify a point group (as shown in the chart) do not form a complete set of symmetry elements for the molecule under consideration
97
Exercises: Give the point group of following molecules.
(a) Mer-[Fe(CN)3Cl3]3- (b) IF7 (c) Mo(CN)8
4- (d) trans-HClC=CClH (e) Mn2(CO)10 (f) P4 (g) 1,5-dibromonaphthalene
98
Answers
Low symmetry? High symmetry (Td, Oh, Ih)? Highest Cn? Any C2 ⊥ Cn? Any σh? Any σv?
(a) C Fe
C
C
ClCl
ClN
N
N
3-
C Fe
C
C
ClCl
ClN
N
N
C Fe
C
C
ClCl
ClN
N
N
C2 C Fe
C
C
ClCl
ClN
N
N
99
Answers
Low symmetry? No High symmetry (Td, Oh, Ih)? No Highest Cn? C2 Any C2 ⊥ Cn? No Any σh? No Any σv? Yes
(a) C Fe
C
C
ClCl
ClN
N
N
3-
C Fe
C
C
ClCl
ClN
N
N
C Fe
C
C
ClCl
ClN
N
N
C2 C Fe
C
C
ClCl
ClN
N
N
C2v
100
F F
F
FF F
F
I(b)
Low symmetry? High symmetry (Td, Oh, Ih)? Highest Cn? Any C2 ⊥ Cn? Any σh?
F F
F
FF F
F
IF F
F
FF F
F
I
101
F F
F
FF F
F
I(b)
Low symmetry? No High symmetry (Td, Oh, Ih)? No Highest Cn? C5 Any C2 ⊥ Cn? Yes Any σh? Yes
F F
F
FF F
F
IF F
F
FF F
F
I
C2
σh
D5h
C5
102
Low symmetry? High symmetry (Td, Oh, Ih)? Highest Cn? Any C2 ⊥ Cn? Any σh? Any σd (or σv)?
NC CNNC
CN CN
CN
NC
NC
Mo
NC CNNC
CN CN
CN
NC
NC
Mo
NC CNNC
CN CN
CN
NC
NC
Mo
103
Low symmetry? No High symmetry (Td, Oh, Ih)? No Highest Cn? C4 Any C2 ⊥ Cn? Yes Any σh? No Any σd (or σv)? Yes
NC CNNC
CN CN
CN
NC
NC
Mo
NC CNNC
CN CN
CN
NC
NC
Mo
NC CNNC
CN CN
CN
NC
NC
Mo
D4d
C4A'
D
BC'
AB'
CD'
C2
σv
C4
104
Cl
H
H
Cl
(d)
Mn MnOC
CO CO
OC
CO
CO
CO
CO
COOC(e)
P
PP
P(f)
Cl
Cl
(g)
105
Cl
H
H
Cl
(d)
Mn MnOC
CO CO
OC
CO
CO
CO
CO
COOC(e)
P
PP
P(f)
Cl
Cl
(g)
C2h
106
Cl
H
H
Cl
(d)
Mn MnOC
CO CO
OC
CO
CO
CO
CO
COOC(e)
P
PP
P(f)
Cl
Cl
(g)
C2h
D4h
107
Cl
H
H
Cl
(d)
Mn MnOC
CO CO
OC
CO
CO
CO
CO
COOC(e)
P
PP
P(f)
Cl
Cl
(g)
D4h
C2h
Td
108
Cl
H
H
Cl
(d)
Mn MnOC
CO CO
OC
CO
CO
CO
CO
COOC(e)
P
PP
P(f)
Cl
Cl
(g)
D4h
C2h
C2h
Td
109
Additional examples C60 dodecahedron
110
111
C60
6 C5
Th
112
dodecahedron
113
dodecahedron
6 C5
Th
114
115
Point group?
116
Point group?
Oh
117
E, 3C4, 4C3, 4S6, 3S4, i, 6C2, 9σ, à Oh
C4
C4
C4
Oh Oh
118
E, 3C4, 4C3, 4S6, 3S4, i, 6C2, 9σ, à Oh
C4
C4
C4
Oh Oh Oh
119
Dual polyhedra
Every polyhedron has a dual polyhedron with faces and vertices interchanged.
The tetrahedron is self-dual (i.e. its dual is another tetrahedron). The cube and the octahedron form a dual pair. The dodecahedron and the icosahedron form a dual pair.
Oh Ih
Ih
Ih
120
Point group?
121
Oh Oh Td Ih
Ih
Point group?
122
Summary: Some typical shapes of point groups
123
124
D. Applications of symmetry 1. Molecular polarity A polar molecule is a molecule with a permanent electric dipole moment.
The presence of a mirror plane or a C2 axis rules out a dipole in the direction shown.
+-F H+-
Symmetry consideration: a molecule (1) can not have a permanent dipole if it has an inversion center. (2) cannot have a permanent dipole perpendicular to any mirror plane. (3) cannot have a permanent dipole perpendicular to any axis of symmetry.
+
σ
+
C2
+
+
Xe F
F
F
F
125
B FF
F
C3
Xe FF
F
F
C4
xy
z
Cannot have a dipole in the xy plane
126
Conclusion Conclusion: Molecules having both a Cn axis and a perpendicular C2 axis or σh cannot have a dipole in any direction.
-Molecules belonging to any D, T, O or I groups cannot have permanent dipole moment.
Molecules can not have a permanent dipole if it has an inversion center
- some Cnh, Sn groups
Exercises: Which of the following molecules are polar?
Fe S
F
Cl
FFF
FS
F
F
F
F
127
Conclusion Conclusion: Molecules having both a Cn axis and a perpendicular C2 axis or σh cannot have a dipole in any direction.
-Molecules belonging to any D, T, O or I groups cannot have permanent dipole moment.
Molecules can not have a permanent dipole if it has an inversion center
- some Cnh, Sn groups
Exercises: Which of the following molecules are polar?
Fe S
F
Cl
FFF
FS
F
F
F
F
D5d, no C4v, yes C2v, yes
128
2. Molecular chirality (分子手性)
A chiral molecule (手性分子) is a molecule that is distinguished from its mirror image in the same way that left and right hands are distinguishable
Symmetry consideration: A molecule that has no axis of improper rotation (Sn) is chiral.
Remember, Sn including S1 = σ and S2 = i Conclusion: a molecule lack of Sn (including σ, i ) are chiral.
129
Exercises: Which of the following molecule is chiral?
N
MeHMe H
HMeMeH
S
F
F
F
FS
F
Cl
FFF
F
N CoNN
N
N
N
(a) (b)
(c)
(d)(e) The skew form of H2O2
130
Exercises: Which of the following molecule is chiral?
N
MeHMe H
HMeMeH
S
F
F
F
FS
F
Cl
FFF
F
N CoNN
N
N
N
(a) (b)
(c)
(d)(e) The skew form of H2O2
Has σ, not chiral
131
Exercises: Which of the following molecule is chiral?
N
MeHMe H
HMeMeH
S
F
F
F
FS
F
Cl
FFF
F
N CoNN
N
N
N
(a) (b)
(c)
(d)(e) The skew form of H2O2
Has σ, not chiral
Has σ, not chiral
132
Exercises: Which of the following molecule is chiral?
N
MeHMe H
HMeMeH
S
F
F
F
FS
F
Cl
FFF
F
N CoNN
N
N
N
(a) (b)
(c)
(d)(e) The skew form of H2O2
Has σ, not chiral
Has σ, not chiral
Νο σ, I and Sn =>chiral
133
Exercises: Which of the following molecule is chiral?
N
MeHMe H
HMeMeH
S
F
F
F
FS
F
Cl
FFF
F
N CoNN
N
N
N
(a) (b)
(c)
(d)(e) The skew form of H2O2
Has σ, not chiral
Has σ, not chiral
Νο σ, I and Sn =>chiral
Has S4 not chiral
134
Exercises: Which of the following molecule is chiral?
N
MeHMe H
HMeMeH
S
F
F
F
FS
F
Cl
FFF
F
N CoNN
N
N
N
(a) (b)
(c)
(d)(e) The skew form of H2O2
Has σ, not chiral
Has σ, not chiral
Νο σ, I and Sn =>chiral
Has S4 not chiral
Νο σ, I and Sn =>chiral
135
3. Symmetries of Orbitals
Diatomic species
O2 and F2
σ2s
σ2s*
σ2p
σ2p*
π2p
π2p*
Based on symmetry in relation to rotation.
Other labels
136
Orbital labels for polyatomic species
2pz 2py 2px
2s
OH H
H HO
xy
z
1a1
1b22a1
1b1
2b2
3a1
2pz 2py 2px
2s
OH H
H HO
xy
z
1σ
2σ3σ
4σ
5σ
6σ
137
The more elaborated labels are based on the behavior of orbitals under all operations, which can be assigned by referring to character table.
2py 2px 2pz
2s
x
y
z
1a1'
1e'
2e'
1a2"
2a1'
a1'
e' a2"
2px2py 2pz
2s
1σ
2σ
3σ
4σ
5σ
MO of BH3
138
Character, χ Character: a number “indicating” (relating) how an object is affected by a symmetry operation.
+1 no change. -1 change sign. 0 move to a new place.
OH H
OH H
C2
χ(C2) = -1
OH H
C2
OH H
χ(C2) = 1
OH H
OH H
C2
χ(C2) = 0
139
Orbital Symmetry, Consider pz in C2v
z E χ(E) = +1 C2(z) y
χ (C2(z)) = +1 x σv(xz)
χ (σv(xz)) = +1 σv(yz)
χ (σv(xz)) = +1
O H H
Under C2v, we have E, C2, 2σv
140
Orbital Symmetry, Consider px in C2v
+ -
+ -
- +
- +
+ -
z E
y
x
χ(E) = +1
C2(z)
χ(C2(z)) = -1 σv(yz)
χ(σv(yz)) = -1
χ(σv(xz)) = +1
σv(xz)
141
Orbital Symmetry, Consider px in C2v
- +
- +
+ -
- +
+ -
z
y
x
E χ(E) = +1
C2(z)
χ(C2(z)) = -1 σv(yz)
σv(xz) χ(σ(yz)) = +1
χ(σv(xz)) = -1
142
C2 (px) = - (px)
σxz (px) = (px)
σyz (px) = - (px)
C2 (py) = - (py)
σxz (py) = - (py)
σyz (py) = (py)
C2 (pz) = (pz)
σxz (pz) = (pz)
σyz (pz) = (pz)
C2 (dxz) = - (dxz)
σxz (dxz) = (dxz)
σyz (dxz) = - (dxz)
C2 (dyz) = - (dyz)
σxz (dyz) = - (dyz)
σyz (dyz) = (dyz)
Summary χ χ
143
C2 (px) = - (px)
σxz (px) = (px)
σyz (px) = - (px)
C2 (py) = - (py)
σxz (py) = - (py)
σyz (py) = (py)
C2 (pz) = (pz)
σxz (pz) = (pz)
σyz (pz) = (pz)
C2 (dxz) = - (dxz)
σxz (dxz) = (dxz)
σyz (dxz) = - (dxz)
C2 (dyz) = - (dyz)
σxz (dyz) = - (dyz)
σyz (dyz) = (dyz)
Summary χ χ
-1
-1
-1
-1
-1
-1
-1
-1
1
1 1
1
1
1
1
144
Each point group has a character table associated with it.
Character table: summary of characters
145
Descriptions of Character table
p orbitals
d orbitals
146
An Example Name of point group
labels of symmetry species
Character
Symmetry operation Function or basis
Symmetry species and atomic orbital s orbital: the first one px, py,pz =>x,y,z dxy,dxz,dyz => xy,xz,yz dx2-y2=>x2-y2
dz2 => z2, 2z2-x2-y2
147
C2 (py) = - (py)
σxz (py) = - (py)
σyz (py) = (py)
C2 ( ) = - ( )
σxz ( ) = - ( )
σyz ( ) = ( )
Molecular orbitals may have the same symetry properties as atomic orbitals, e.g.
148
b2,
149
More complicated cases
H
BH H
x
yH
BH H
y
x
C3
H
BH H
C3
HB
H
H
C3H
BH H H
BH
H
150
x
yx'
y'z (z')
C3 (x,y):-1/2 3/2
3/2 -1/2
C3(z):[1]
E (x, y):1
1 0
0E(z): [1]
x'
y'
z'
xy
z
C3
151
x'
y'
z'
x
y
z
a
dg
b
e
h
c
f
i
χ = a+ e+ i
x' = [1]x χ = 1
x' = [-1]x χ = -1
Character (χ) and matrix
152
C3 (x,y):-1/2 3/2
3/2 -1/2
C3(z):[1]
E (x, y):1
1 0
0
E(z): [1]
χ(C3) = -1 χ(C3) = 2
χ(C3) = 1 χ(C3) = 1
153
Labels for symmetry species
154
1: symmetric with regard to mirror (σv)
2: antisymmetric with regard to mirror (σv)
155
156
157
z
y πu πg
σg
σu
A AReal examples
158
xy
z
H2O
2pz 2py 2px
2s
OH H
H HO
xy
z
1a1
1b22a1
1b1
2b2
3a1
a1
a1 b2 b1
Symmetry labels of the LGOs?
159
xy
z
H2O
2pz 2py 2px
2s
OH H
H HO
xy
z
1a1
1b22a1
1b1
2b2
3a1
a1
a1 b2 b1
Symmetry labels of the LGOs?
a1
b2
160
2py 2px 2pz
2s
x
y
z
1a1'
1e'
2e'
1a2"
2a1'
a1'
e' a2"
BH3
161
BH3
2py 2px 2pz
2s
x
y
z
1a1'
1e'
2e'
1a2"
2a1'
a1'
e' a2"
162
H
N
H H
NH3
163
NH3
x
y
z
x
y
z
s
H HH N
1a1
2a1
3a1
1e
2e
a1
a1
e
Symmetry labels of the LGOs?
164
NH3
x
y
z
x
y
z
s
H HH N
1a1
2a1
3a1
1e
2e
a1
a1
e
Symmetry labels of the LGOs?
a1
e
165
M LL
LL
Exercise. Give symmetry labels of molecular orbitals of Td ML4 and Oh ML6
s
p
d
L
ML L
L
166
M LL
LL
Exercise. Give symmetry labels of molecular orbitals of Td ML4 and Oh ML6
s
p
d
L
ML L
L
symmetry labels for AOs?
167
M LL
LL
Exercise. Give symmetry labels of molecular orbitals of Td ML4 and Oh ML6
s
p
d
L
ML L
L
symmetry labels for AOs?
e
t2
t2
symmetry labels for MOs?
a1
168
M LL
LL
Exercise. Give symmetry labels of molecular orbitals of Td ML4 and Oh ML6
s
p
d
L
ML L
L
symmetry labels for AOs?
e
t2
t2
symmetry labels for MOs?
a1
1a1
2a1
1t2
2t2
3t2
1e
169
M LL
LL
Exercise. Give symmetry labels of molecular orbitals of Td ML4 and Oh ML6
s
p
d
L
ML L
L
symmetry labels for AOs?
e
t2
t2
symmetry labels for MOs?
a1
1a1
2a1
1t2
2t2
3t2
1e
a1
t2
170
M L
L
L
L
L
L
symmetry labels for AOs and LGOs?
p
s
d
171
M L
L
L
L
L
L
symmetry labels for AOs and LGO?
t1u
a1g
t2g
eg
t1u
a1g
eg
symmetry labels for MOs?
p
s
d
172
M L
L
L
L
L
L
symmetry labels for AOs and LGOs? t1u
a1g
t2g
eg
t1u
eg
1a1g
1t2g
1t1u
2t1u
1eg
2eg
2a1g p
s
d
a1g symmetry labels for MOs?
173
Summary of Chapter 5
• Types of Symmetry elements and operation – E, Cn, σ, I, Sn
• Characteristics of Common Point Groups – C1, Cs, Ci, Cn, Cnv, Cnh, Dn, Dnh, Dnd, Td, Oh, Ih
• Applications
– Polarity, chirality, symmetry labels of orbitals