chapter 3 symmetry - pennsylvania state universitychemistry.bd.psu.edu/jircitano/413c3a.pdf ·...
TRANSCRIPT
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Symmetry Element:
Symmetry Operation:
Chapter 3
Symmetry
point symmetry
a geometrical entity; a point, line or
plane; with respect to which one or more symmetry
operations may be carried out
a movement of an object such that,
after the movement has been carried out, every point on the
object is coincident with an equivalent point (or the same
point) of the object in its original orientation
Group Theory
all symmetry elements in molecule, collected together, form a:
mathematical group
2. has identity operator:
3. each element has inverse; also a group element:
AB = C
EA = A
A–1A = E
B followed by A same as C
1. product of 2 group elements = a group element:
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E
E, i
E, s
E, C2
E, 2 C3, 3 C2
E, C2, sv(xy), sv(yz)
E, 2 C3, 3 sv
E, i, C2, sh
E, C3, sh, S3
E, 2 S4, C2, 2 C2′, 2 sd
E, 2 C3, 3 C2, i, 2 S6, 3 sd
E, C2(x), C2(y), C2(z), i, sv(xy), sv(xz), sv(yz)
E, 2 C3, 3 C2, sh, 2 S3, 3 sv
E, 8 C3, 3 C2, 6 S4, 6 sd
E, 8 C3, 6 C2, 6 C4, 3 C2, i, 6 S4, 8 S6, 3 sh, 6 sd
E, 2 C, sv
E, 2 C, sv, i, 2 S , C2
E, 12 C5, 20 C3, 15 C2, i, 12 S10, 20 S6, 15 s
Point Groups
Arthur Schönflies
German
1891
Point groups represented (shorthand) by:
Schoenflies symbol
C1
Ci
Cs
C2
D3
C2v
C3v
C2h
C3h
D2d
D3d
D2h
D3h
Td
Oh
Cv
Dh
Ih
all elements not needed to
assign point group:
Flowchart for Identifying Molecular Point Groups
Br
Cl
CH
F
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Flowchart for Identifying Molecular Point Groups
H2N
Co
NH2
H2N NH2
NH2
H2N
Flowchart for Identifying Molecular Point Groups
Mn
OC
OC CO
CO
CO
Br
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Flowchart for Identifying Molecular Point Groups
S S
O
OO
O
O O
Flowchart for Identifying Molecular Point Groups
Pt
Cl
Cl Cl
Cl
Cl
Cl
Pt
Cl
Cl Cl
Cl
Cl
Cl
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Flowchart for Identifying Molecular Point Groups
Cl
Cl
AlCl
Cl
Cl
AlCl
Examples
Br
Cl
CH
F
O
CH
NCO
Fe
COOC
CO
PP
Fe
OCOC CO
OC
Me H
H Me
O O
H
HAs
Ln
H2NO
NH2O
OO
NH2
NH2
O
O
O
O
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Examples
Zr
OO
OO
OO
O
O
CH3
CH3
H3C
H3C
H3C
CH3
CH3
CH3
Co
NH2
H2N NH2
NH2
H2N
H2N
Examples
H
S
H
H
O O
HP
FF
F
O
B
O
O
H
H
H
Mn
OC
OC CO
CO
CO
Br
N
CuN
N
N
H
H
H
H
H
HH
H
H
H
H
H
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Examples
C C C
H
H
H
HCl
Cl
AlCl
Cl
Cl
AlCl
Br
Ga
Br
Br
Pt
Cl
Cl Cl
Cl
NH3
NH3
S S
O
OO
O
O O
F
F
Ta
F
F
FF
F
F
Examples
Fe
F
F
SiF
F
Pt
Cl
Cl Cl
Cl
Cl
Cl
O C O H C N
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HF
FH
FH
F H
ExamplesC4
C4 s s
HF
F H
FH
FH
HF
F H
FH
FH
Examples
BHBH
HB
BH
BH
BH
BH
BH
HB
BH
HB
BH
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Examples
Rhinovirus
sphere:
Examples
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C2v
A1
A2
B1
B2
C2 sv(xz) sv′(yz)E
1 1 1 1
1 1 –1 –1
1 –1 1 –1
1 –1 –1 1
Rz
x2, y2, z2z
xz
xy
yz
A, B character under Cn is +, –
1, 2 character under sv or C2 ┴ to Cn is +, –
′, ′′ character under sh is +, –
g, u character under i is +, – (gerade, ungerade)
E character under E is 2, (2-dimensional)
T character under E is 3, (3-dimensional)
Character Table
x
y
, Ry
, Rx
Oh Character Table
Oh
A1g
A2g
Eg
T1g
8C3 6sd8S6E
1 1 1 1
1 1 –1
2 –1 0
3 0 –1 1
x2+y2+z2
(Rx , Ry , Rz)
(xz, yz , xy)
–1
0
1 1 1 1
1 1 –1
2 2 0
–1 3 1 0
1
–1
11
2
–1 –1
–1
0
1
3 0 1 –1
1 1 1
1 1 –1
2 –1 0 0
1
–1
–1 3 –1 0
1 –1 –1
1 –1 1
2 –2 0 1
–1
–1
–1–1
–1
–2 0
–1
1
1
3 0 –1
3 0 1 –1
1 –1 –3 – 1
–1 –3 1 0
0 1
1 –1
1
T2g
A1u
A2u
Eu
T1u
T2u
6C2 6C4 3C2 i 6S4 3sh
(x , y, z)
(z2, x2 – y2)
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N # of times irreducible representation occurs
Reducing Reducible Representations
h order = total of symmetry operations
nx # of operations in class
crx, ci
x characters of reducible representation and
irreducible representation under class x
N = Σ χrx χ
ixnx
x
1
h
Optical Activity
C
Br
ClF
I
Pt
Cl
Cl NH3
NO2
NH3
NO2
NH2
H2N NH2
NH2
H2N
H2N
Co
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Dipole Moments
C
F
FF
F
P
Cl
Cl
Cl
Cl
Cl
C
H
FF
F
S
Cl
Cl
Cl
Cl
O
HH
Infrared Spectroscopy
% T
ra
nsm
itta
nce
Wavenumber, cm–1
4000 3000 2000 1500 1000 5000
50
100
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Symmetry and NMR
Bonding and Orbitals
s
A1(g) always
px py pz
dxy dxz dyz dx2–y2 dz2
Orbitals have same symmetry as character table functions.
x2 + y2,
z
(x2 – y2, xy), (xz, yz)
(x, y)2 –1 0 –2 1 0Eu
D3d
A1g
A2g
Eg
A1u
2C3E
1 1 1
1 1 –1
2 –1 0
1 1 1
1 1 1
1 1
2 –1
–1 –1 –1
–1
0
1 1 –1 –1 –1 1A2u
3C2 i 2S6 3sd
(Rx, Ry)
Rz
z2
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Bonding and Orbitals
s
A1(g) always
px py pz
dxy dxz dyz dx2–y2 dz2
orbitals have same symmetry as character table functions
x2 + y2,
z
(x2 – y2, xy), (xz, yz)
(x, y)2 –1 0 –2 1 0Eu
D3d
A1g
A2g
Eg
A1u
2C3E
1 1 1
1 1 –1
2 –1 0
1 1 1
1 1 1
1 1
2 –1
–1 –1 –1
–1
0
1 1 –1 –1 –1 1A2u
3C2 i 2S6 3sd
(Rx, Ry)
Rz
z2