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8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
1/27
Lecture 25: Introduction to Molecular Orbital
Theory
The material in this lecture covers the following in Atkins.
14 Molecular structure
Molecular Orbital Theory
(a) Linear combinations of atomic orbitals
(b) bonding orbitals
(c) anti-bonding orbitals
Lecture on-line
Introduction to Molecular Orbital theory (PowerPoint)Introduction to Molecular Orbital Theory (PDF)
Handout for this lecture
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
We shall now discuss ways toapproximately solve :
H r R r R E (R r Re e N e N e N e N( , ) ( , ) ) ( , ) =
and represent the many - electronwave - function r Re N( , )
Here
H = T V + V + Ve e Ne ee NN +
We shall this time use :
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
1 = [A(1) A(2)B B( ) ( )]
[ ( ) ( ) ( ) ( )]
2 1
1 2 1 2
+
BA
In valence bond theory we localizedone electron to each atom
el 1. el 2.
or BA
In molecular orbital theory each electron movesover the whole molecule
BA
el 1.el 2.
BAand
el 2.el 1.
The two theories are limiting cases (see later) of a more exact theory
Both electrons can be on the same nuclei
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
We used the orbitals of theone - electron hydrogen to build up
wavefunctions for many - electron
atoms
We shall use the orbitals of
the one - electron H molecule
to describe many electron
diatomic molecules
2+
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
The
molecule
hamiltonian for the H2+
:is given by
H =
h
22
2me e
Kinetic energy of electron
e
ro A
2
14
1
Attraction of el. 1 by A
e
ro B
2
14
1
Attraction of el. 1 by B
+e
Ro
2
4
1
rA1
A B
1
rB1
R
Repulsion between A and B
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
We R R E R Rcould now solve : H(r (r (r1 1 1
, ) , ) ( ) , ) =With
Ve
r r Ro A B
= + 2
1 14
1 1 1
[
Where
H
m
V r R
e
e
= +h
22
1
2
( , )
To this end we write (r as a linear combinationof atomic orbitals (LCAO)
1 , )R
This is possible but tedious
We shall instead find approximate soluions
The atomic orbitals are in general those centered on
the atoms of our molecule
Was first done by a Danish astronomer in 1926
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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1 1 1 3 3 2
1
s r A r ea
A A A
r
o
A ao( ) ( )
( )
/
/= =
Molecular Orbital Theory H2+
For H that is :2+
1 1 1 3 3 2
1
s r A r ea
A B B
r
o
B ao
( ) ( )( )
/
/= =
Where r and r are related by :1A 1B
r r R r RB A A1 1 2 2 12= + cos
rA1
A B
1
rB1
R
We now assume the electron to be equally
likely to be on each nuclei and write the
molecular orbitals as :+ = +( ) ( ) ( )1 1 1A B = ( ) ( ) ( )1 1 1A B
Note : we have as many linear combinations
as we have atomic orbitals
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
+
= +( ) ( ) ( )1 1 1A B
= ( ) ( ) ( )1 1 1A B
We can normalize our orbitals for easy energy
calculation and use in probability density
+ + = + + =( ) ( ) ( ( ) ( ))( ( ) ( ))1 1 1 1 1 1 12dv N A B A B dv
+ + = + + =
( ) ( ) ( ( ) ( ) ( ( ) ( )
( ( ) ( )
1 1 1 1 1 1
2 1 1 1
2 2
2
dv N A A dv N B B dv
N A B dv
A normalized B normalized overlap S > 1
+ + +
( ) ( )1 1 1 2dv N= +1 N
+2S N = 1
2
2
NS= +
12 1( )
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
+ = +( ) ( ) ( )1 1 1A B = ( ) ( ) ( )1 1 1A B
After normalization
+
=+
+( )( )
[ ( ) ( )]11
2 11 1
SA B
=
( )( )
[ ( ) ( )]11
2 11 1
SA B
The electron probability density :
+ + = + +( ) ( ) ( ) [ ( ) ( )]1 1
1
2 1 1 1
2
S A B
=+
++
++
1
2 11
1
2 11 2
1
2 112 2
( )( )
( )( )
( )( )
SA
SB
SAB
Density at A
(Reduced)
Density at B
(Reduced)
Density between A and B
(Increased)
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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(a)The amplitude of
the bonding molecularorbital in a hydrogen
molecule-ion in a plane
containing the two nuclei and
(b) a contour
representation
of the amplitude.
Molecular Orbital Theory H2+
+ =
+
+( )
( )
[ ( ) ( )]11
2 1
1 1
S
A B
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
A representation of theconstructive interference
that occurs when two H 1s
orbitals overlap and forma bonding orbital.
Compare this illustration
with Fig.14.14.
+ =
+
+( )
( )
[ ( ) ( )]11
2 1
1 1
S
A B
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
The boundary surfaceof a orbital enclosesthe region where the
electrons that occupy
the orbital are
most likely to be found.
Note that the orbital has
cylindrical symmetry.
( ) ( ) ( ) ( ) ( ) ( ) ( )11
2 1 1
1
2 1 1 2
1
2 1 1
2 2
= + + + + +S A S B S AB
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
The electron density calculated
by forming the square of the
wavefunction used
to construct Fig.14.14.Note the accumulation
of electron density in the
internuclear region.
( )( )
( )( )
( )( )
( )11
2 11
1
2 11 2
1
2 112 2=
++
++
+SA
SB
SAB
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
Hm
er r Re
eo A B
= + h2 2 21 12 4
1 1 1
[ ] + = + +( )
( )[ ( ) ( )]1 1
2 11 1
SA B
We H dvcan now evaluate the energy : E =+ + + ( ) ( ) ( )1 1 1
E =+1
2 1
1 1 1 1 1
( )
[ ( ) ( )] ( )[ ( ) ( )]
+
+ +
S
A B H A B dv
=1
2 11
2 4
11
22
2
1( )] ( )[ ] ( )
+
SA
m
e
rA dv
ee
o A
h
+ 12 1
12 4
1 12
22
1( )] ( )[ ] ( )
+ SB
me
rB dv
ee
o Bh
+1
2 1
1
4
11
2
1( )
] ( )[ ] ( )
+
S
Ae
r
A dv
o B+
1
2 11
4
11
2
1( )] ( )[ ] ( )
+
SB
e
rB dv
o A
+ 22 1
12 4
1 1 14
1
2
2
2
1 1
2
( )] ( )[ [ ]] ( )
+ + +SA
me
r rB dv e
Ree
o A B o
h
rA1
A B
1
rB1
R
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
E =+
1
2 11 1 1 1 1
( )[ ( ) ( )] ( )[ ( ) ( )]
++
+
SA B H A B dv
=1
2 11
2 4
11
22
2
1( )] ( )[ ] ( )
+
SA
m
e
rA dv
ee
o A
h
+ 12 1
12 4
1 12
22
1( )] ( )[ ] ( )
+
SB
me
rB dv
ee
o B
h
+1
2 1
1
4
11
2
1( )
] ( )[ ] ( )
+
S
Ae
r
A dv
o B+
1
2 11
4
11
2
1( )] ( )[ ] ( )
+
SB
e
rB dv
o A
+ 22 1
12 4
1 1 12 2 2
1 1( )] ( )[ [ ]] ( )
+ +
SA
me
r rB dv
ee
o A B
h
rA1
1
rB1
+ e Ro
2
41
E
2(1+ S)1sH
E2(1+ S)
1sH
J
2(1+ S)J
2(1+ S)
K'(1+S)
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
- =J
S SA A
e
rdv
o B2 1
1
2 11 1
4
12
1( ) ( )] ( ) ( )[ ]
+ +
- J =2 1
12 1
1 14
12
1( ) ( )] ( ) ( [ ])
+ +
S SB B e
rdv
o A
A B
rB1
R
1
2(1+ S)] A(1)A(1)dv
Interaction (attraction) between nucleus B
and the charge cloud A(1)A(1)/2 1( )+ S
Interaction (attraction) between nucleus A
and the charge cloud B(1)B(1)/2 1( )+ S
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
K' =2
2 11
2 4
1 11
22
2
1 1( )] ( )[ [ ]] ( )
+ +
SA
m
e
r rB dv
ee
o A B
h
rA1
A B
1
rB1
R
=2
2 11
2 4
11
2
2 11 1
4
1
22
2
1
2
1
( )] ( )[ ]] ( )
( )] ( ) ( )[ ]
+
+
SA
m
e
rB dv
SA B
e
rdv
ee
o B
o A
h
SE SsH1 1/( )+
K
Interaction (attraction) between nucleus Aand the charge cloud A(1)B(1)/( )1+ S
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
+e
Ro
2
4
1
E+ =E
2(1+ S)
1sH +E
2(1+ S)
1sH J
2(1+ S)
J
2(1+ S)
K-SE
(1+S)
1sH
ES e
Ro+=
+ +
E E
(1+S)
J+K
(1+S)
1sH 1sH2
4
1
We
E
e
Ro
get in a similar way that
= E
J - K
(1- S)1sH +
2
4
1
Ee
Ro
+ = +EJ+K
(1+S)1sH
2
4
1
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
The calculatedand experimental molecular
potential energy curves for
a hydrogen molecule-ion.
E
e
Ro+ = + +E
J+K
(1+S)1sH
2
4
1
Ee
Ro
+= EJ - K
(1- S)
1sH
2
4
1
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
A representation of thedestructive interference
that occurs when two H1s
orbitals overlap and form an
antibonding * orbital.Compare this illustration
with Fig.14.20.
= ( )
( )[ ( ) ( )]1 1
2 11 1
SA B
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
(a) The amplitude of the
antibonding molecular
orbital in a
hydrogen molecule-ion in
a plane containing the two
nuclei and
(b) a contour representation
of the amplitude. Note the
internuclear node.
=
( )( )
[ ( ) ( )]11
2 11 1
SA B
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
The electron density calculated
by forming the square of the
wavefunction used to constructFig.14.20. Note the elimination
of electron density from the
internuclear region.
= +
( ) ( ) ( )
( )( )
( )( )
11
2 1 1
1
2 11 2
1
2 11
2
2
S A
SB
SAB
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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Molecular Orbital Theory H2+
(b) In an antibonding
orbital, the nuclei are
attracted to an accumulation
of electron density outside
the internuclear region.
A partial explanation of the origin of bondingand antibonding effects.
(a) In a bonding orbital,
the nuclei are attracted to
the accumulation of electron
density in the internuclear
region.
+ = ++( )
( )[ ( ) ( )]1
1
2 11 1
SA B =
( )( )
[ ( ) ( )]11
2 11 1
SA B
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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A molecular orbital energylevel diagram for orbitals
constructed from the overlap
of H1s orbitals; the separation
of the levels corresponds tothat found at the equilibrium
bond length. The ground
electronic configuration ofH2 is obtained by
accommodating the two
electrons in the lowest
available orbital(the bonding orbital).
Molecular Orbital Theory H2+
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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rA1
A B
1
rB1
R
What you should learn from this lecture
Be
Hm
e
r
e
r
e
Ree
o A o B o
able to construct the Hamiltonian
= +h
22
2
1
2
1
2
2 4
1
4
1
4
1
Understand the justification for constructingthe two molecular orbitals as linear combinationsof atomic orbitals
and
be able to evaluate the normalizationconstants from the overlap integral
S = A(1)B(1)dv
+ + = + =
( ) ( ( ) ( )) ( ) ( ( ) ( ))1 1 1 1 1 1N A B N A B
and
You
E eR
or Ee
R
o
o
are not asked to derive the energy expressions
for E J + K(1+S)
EJ - K
(1-S)
+ 1sH
- 1sH
:
:
+
= +
= +
2
2
41
4
1
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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What you should learn from this lecture
However you should understand the
meening of
J = and
K = and undertand how K
stabilize and destabalizes .
A Ae
rdv
A Be
rdv
o B
o A
( ) ( )[ ]
( ) ( )[ ]
1 14
1
1 14
1
2
1
2
1
+
rA1
A B
1
rB1
R
It
SA
SB
SA B
You
A B A B
S
A B
is expected that you can derive the expressionfor the density due to as :
shoule further understand the meening of
and Also you should understand
how influence the energy E as well
as the features of Fig 14.16
+
+
( )( )
( )( )
( )( )
( ) ( )
( ) , ( ) , ( ) ( ).
( )
( ) ( )
11
2 11
1
2 11 2
1
2 11 1
1 1 1 1
21
2 1
1 1
2 2
2 2
=+
++
++
+
+
8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory
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What you should learn from this lecture
It
SA
SB
SA B
You
A B A B
SA B
is expected that you can derive the expression
for the density due to as :
shoule further understand the meening of
and Also you should understand
how influences the energy E as well
as the features of Fig 14.21
-
-
( )
( )( )
( )( )
( )( ) ( )
( ) , ( ) , ( ) ( ).
( )( ) ( )
11
2 11
1
2 11 2
1
2 11 1
1 1 1 1
21
2 11 1
2 2
2 2
=
+
Unders dfor N A B N A B
respectively
tan( ) ( ( ) ( )) ( ) ( ( ) ( )),
.
the features of the plots in Fig. 14.17 and 14.19and + + = + = 1 1 1 1 1 1
Make note of the potential energy curve in Fig 14.18 andobserve that E is lowered less in energy (compared to
E E is increased (see also Fig 14.24+
1s -than