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Chem 104A, UC, Berkeley
Symmetry & Group Theory
MT Chap. 4Vincent: Molecular Symmetry and group theory
Chem 104A, UC, Berkeley
Symmetry:The properties of self-similarity
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Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
Re2(CO)10
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Chem 104A, UC, Berkeley
W(CO)6
Chem 104A, UC, Berkeley
C2F4 C60
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Chem 104A, UC, Berkeley
Symmetry:
•Construct bonding based on atomic orbitals
•Predict Raman & IR spectra
•Access reaction pathway
•Determine optical activity
Chem 104A, UC, Berkeley
Symmetry Operation:
Movement of an object into an equivalent or indistinguishableorientation
Symmetry Elements:
A point, line or plane about which a symmetry operation is carried out
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Chem 104A, UC, Berkeley
5 types of symmetry operations/elements
Identity: this operation does nothing, symbol: E
Element is entire object
Chem 104A, UC, Berkeley
Proper Rotation:
Rotation about an axis by an angle of 2/n
mnC
nnn
nn
CC
EC
1
Rotation 2m/n
C4
1
2
3
4
PtCl4
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Chem 104A, UC, Berkeley
C2
Chem 104A, UC, Berkeley
The highest order rotation axis is called the principle axis.
How about: NFO2?
H2ONH3
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Chem 104A, UC, Berkeley
Identity EProper Rotation Cn
Reflection: reflection through a mirror plane
NH3
Chem 104A, UC, Berkeley
H2O
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Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
a mirror plane containing a principle rotation axis is labeled
v
a mirror plane normal to a principle rotation axis is labeled
h
B
F F
F
B
F F
F
)(
)(
oddn
evennEn
n
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Chem 104A, UC, Berkeley
Inversion: iinversion center or center of symmetry(x,y,z) (-x,-y,-z)
)(
)(
oddnii
evennEin
n
Chem 104A, UC, Berkeley
Difference between inversion and 2-fold rotation
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Chem 104A, UC, Berkeley
Inversion ?
Chem 104A, UC, Berkeley
Improper rotation: Snrotation about an axis by an angle of 2/n followed by reflection through a perpendicular plane.(Cn,h symmetry are not necessary for Sn to exist)
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Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
S6
13
Chem 104A, UC, Berkeley
P
F
F
F
F
FP
F
F
F
F
F
1
2
3
4
5
1
2
34
5S3
Contain C3, h
Chem 104A, UC, Berkeley
In general: Sn with n even molecule contains Cn/2,
Snn=E
Sn with n odd molecule contains Cn + h;
Snn =h, Sn
2n =E
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Chem 104A, UC, Berkeley
P
F
F
F
F
F
1
2
3
4
5 P
F
F
F
F
F
1
2
34
5
Contain C3, h
P
F
F
F
F
F
2
1
53
4
P
F
F
F
F
F
1
2
45
3
63
53
43
33
23
3
S
S
S
S
S
S
P
F
F
F
F
F
2
1
34
5
P
F
F
F
F
F
1
2
53
4
P
F
F
F
F
F
2
1
45
3
=h
=E
Chem 104A, UC, Berkeley
S6
ES
S
CS
SiS
CS
S
66
56
23
46
236
326
6
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Chem 104A, UC, BerkeleyXeF4
42224' ,"2,'2,,,2,2,,, SCCCCiE vvh
2C
'2C "2C v
4S C4
Chem 104A, UC, BerkeleyBF3
323' ,3,,3,, SCCE vh
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Chem 104A, UC, Berkeley
Group TheoryDefinition of a Group:
A group is a collection of elements
• which is closed under a single-valued associative binary operation
• which contains a single element satisfying the identity law
• which possesses a reciprocal element for each element of the collection.
Chem 104A, UC, Berkeley
1. Closure: A, B G AB G
2. Associativity: A, B, C G A(BC)= (AB)C
3. Identity: There exists E G such that AE=EA=A for all A G
4. Inverse: A G there exists A-1 G such that AA-1 = A-1A=E
Mathematical Group
Order of a group: the number of elements it contains
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Chem 104A, UC, Berkeley
Example:1. set of all real number, under addition, order =
Closure: x + y GAssociativity: x + (y +z) =(x+y) +zIdentity: x +0 =0+x =xInverse: x +(-x) =(-x)+x =0
2. set of all integers, under addition3. {set of all real number}-{0}, under multiplication
Closure: x * y GAssociativity: x * (y *z) =(x*y) *zIdentity: x *1 =1*x =xInverse: x *(1/x) =(1/x)*x =1
4. {+1, -1}5. { 1, i}
Chem 104A, UC, Berkeley
Symmetry of an object point group (symmetry about a point){E, C2,v,v'} = point group C2vBinary operation: one operation followed by another
C2v E C2 v v’
E E C2 v v’
C2 C2 E v’ v
v v v’ E C2
v’ v’ v C2 E
Closure: Associativity: Identity: Inverse:
Multiplication Table
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Chem 104A, UC, Berkeley
C2v E C2 v v’
E E C2 v v’
C2 C2 E v’ v
v v v’ E C2
v’ v’ v C2 E
Rearrangement Theorem: each row and each column in a group multiplication table lists each of the elements once and only once.
Proof: suppose AB=AC, i.e. two column entries are identicalthen:
EB=ECB=C
ACAABA 11 A B C
A AA
B AB
C AC
Chem 104A, UC, Berkeley
C2v E C2 v v’
E E C2 v v’
C2 C2 E v’ v
v v v’ E C2
v’ v’ v C2 E
A group is Abelian if AB=BA ( the multiplication is completely commutative).
Not all groups are abelian.
vv CC 33
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Chem 104A, UC, Berkeley
Any object (or molecule) may be classified into a point group uniquely determined by its symmetry.
Groups with low symmetry:{E}=C1, Schönflies Symbol/notation{E,} =Cs{E, i} =Ci
ONCl, Cs
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
H
HCl
Cl
F
F
Ci
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Chem 104A, UC, Berkeley
Groups with a single Cn axis{E, Cn, Cn
2, Cn3….Cn
n-1} =Cn
H2O2
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
222ON },,,{ 2 iCE hhC2
Groups with a single Cn axis plus a perpendicular h plane: Cnh
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Chem 104A, UC, Berkeley
},,,{ 2 iCE hhC2
C = C
H
HCl
Cl
Chem 104A, UC, Berkeley
},,,,,{ 533
233 SSCCE hhC3
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Chem 104A, UC, BerkeleyGroups with a single Cn axis plus n vertical v planes: Cnv
vC2 OH 2
},,,{ '2 vvCE
Chem 104A, UC, Berkeley
},,,,,{ '''233 vvvCCE 3
3
NH
C v
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Chem 104A, UC, Berkeley
5
4
BrF
C v
},,,,,,{ ','3424 ddvvCCCE
d: dihedral reflection planes (bisects v or C2)
Br
F
FF
F F
Square pyramidal
Chem 104A, UC, Berkeley
n-gonal pyramidal shape: Cnv
vCHF
...},...,{ vCE
