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8/3/2019 Lecture 23: Introduction to Valence Bond Theory
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Lecture 23: Introduction to Valence Bond
TheoryThe material in this lecture covers the following in Atkins.
14 Molecular structure
Valence-bond theory14.1 The hydrogen molecule
(a) The spatial wavefunction
(b) The role of the electron spin
Lecture on-line
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Introduction to Valence Bond Theory (PDF)Handout for this lecture
8/3/2019 Lecture 23: Introduction to Valence Bond Theory
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Valence Bond Theory Basic Theory
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 start with the :
8/3/2019 Lecture 23: Introduction to Valence Bond Theory
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Valence Bond Theory
H HBH H
H
CH
H
H
HH Cl OH
H
NH
H
H
In valence bond theory we startby writing down the Lewis structure
of our molecule
Subsequentlyas
r ri i
we write r Rthe product of electron pair
functions as
e N
i
( , )
( , )2 1 2
( , ) ( , ) ( , )
.. ( , ) ( , ).. ( , )
r Re N 1 2i
=
r r r r
r r r r r ri i j j j n n n1 2 3 4
2 1 2 2 1 2 2 1 2
Pair 1
Pair 2
Pair i Pair j
Pair n
Basic Theory
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Valence Bond Theory
H HWe shall now illustrate this simple theory for H2
We have two well separated hydrogen
atoms A and B
each with one electronA
B
1
rA1
2
rB2
We can describe hydrogen A
by 1s 1sHA HA
( ) ( ); ( ) ( )r rA A1 1
1 1 We can describe hydrogen B
by 1s 1sHB HB
( ) ( ); ( ) ( )r rB B2 2
2 2
Or in shortA(1) A(1) ( ); ( )1 1 B(2) B(2) ( ); ( )2 2
We now bring the two hydrogens together to form H2
B
2
rB2
A
1
rA1
Basic Theory
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Valence Bond Theory
A
1
rA1
B
2
rB2
The hamiltonian of theH molecule is :2
H me
ro A= h
2
12
2
12 4
1
Hamiltonian of HA
h
2
22
2
22 4
1
m
e
re o B Hamiltonian of HB
e
ro B
2
14
1
rB1
Attraction between el. 1 and H - atom B
e
ro A
2
24
1
Attraction between el. 2 and H - atom B
rA2
+ ero
2
124
1
Repulsion between el. 2 and el. 1
r12
+e
Ro
2
4
1
R
Repulsion between A and B
Basic Theory
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Valence Bond Theory
A
1
rA1
B
2
rB2rB1
rA2
r12
R
W
Hm m
e
RV
o
e can thus write the Hamiltonian as
= + +h h
2
12
2
12
2
2 2 4
1
where
Ve
r
e
r
e
r
e
r
e
ro A o A o A o B o= +
2
1
2
2
2
2
2
1
2
124
1
4
1
4
1
4
1
4
1
Subsequently
as
r ri i
we write r R
the product of electron pair
functions as
e N
i
( , )
( , )2 1 2
A(1) A(1) ( ); ( )1 1 B(2) B(2) ( ); ( )2 2
( , , ) ( , )r r1 2 1R r r= 1 2
We r rshall further write as linear
combinations of product between
functions on A and B
1 ( , )1 2
Basic Theory
8/3/2019 Lecture 23: Introduction to Valence Bond Theory
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Valence Bond Theory
A
1
rA1
B
2
rB2rB1
rA2
r12
R
A(1) A(1) ( ); ( )1 1 B(2) B(2) ( ); ( )2 2From the functions
We can construct
the products :A(1)
A(1)
A(1)
A(1)
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
1 2 2
1 2 2
1 2 2
1 2 2
B
B
B
B
Allowing for interchange of
the two affords :
A(2)
A(2)
A(2)A(2)
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( )
2 1 1
2 1 1
2 1 12 1 1
B
B
BB
We
R r r
have further that
r r1 2 1 ( , , ) ( , )= 1 2
Must be anti symmetric
r r r r1 2 1 2 1 1 ( , , ) ( , ) ( , , ) ( , )R r r R r r= = = 1 2 2 1
Basic Theory
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Valence Bond Theory
A(1)
A(1)
A(1)
A(1)
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
1 2 2
1 2 2
1 2 2
1 2 2
B
B
B
B
A(2)
A(2)
A(2)
A(2)
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
2 1 1
2 1 1
2 1 1
2 1 1
B
B
B
B
From the product functions
We can construct the
anti - symmetric linear
combinations : 1 = [A(1) A(1)B B( ) ( )]
[ ( ) ( ) ( ) ( )]
2 2
1 2 1 2
+
- Symmetric in space
- anti - symmetric in spin
< S singlet2 >= 0 :
2
= [A(1) A(1)B B( ) ( )]
[ ( ) ( ) ( ) ( )]
2 2
1 2 1 2
+
- Anti - symmetric in space
- symmetric in spin
< S triplet2 >= 2 2h :
3 = [A(1) A(1)B B( ) ( )] ( ) ( )2 2 1 2
4
= [A(1) A(1)B B( ) ( )] ( ) ( )2 2 1 2
ms = 0
ms = 1ms = 1
Basic Theory
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Valence Bond Theory
1 1= C [A(1) A(1)BB BB ( ) ( )]
[ ( ) ( ) ( ) ( )]2 2
1 2 1 2+
2 2= C [A(1) A(1)BB BB ( ) ( )][ ( ) ( ) ( ) ( )]2 21 2 1 2 +
3 3= C [A(1) A(1)BB BB ( ) ( )] ( ) ( )2 2 1 2 4 4= C [A(1) A(1)BB BB ( ) ( )] ( ) ( )2 2 1 2
Singlet
triplet
Electron
n n
dr dr dr dspinn
density :
(r1 ) ( , ,.. ) ( , ,.. )
..
*
= 1 2 1 2
2 3probability of finding el.
no matter where other el.are
Singlet density :
sin ( )( ) ( )
( ) ( )
gHA HB
S SA B
S
11
1
1
12 1 1
1
2 2
2
=+
++
+ +
A B BA
positive overlap density
Density build up between nuclei
A(1) B(1)
S s s dA B= 1 1 1 1( ) ( )
Overlap between
1s orbitals on A and B
Basic Theory
CCi insures normalization
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BA
Valence Bond Theory
1 1= C [A(1) A(1)BB BB ( ) ( )]
[ ( ) ( ) ( ) ( )]2 2
1 2 1 2+
2 2= C [A(1) A(1)BB BB ( ) ( )][ ( ) ( ) ( ) ( )]2 21 2 1 2 +
3 3= C [A(1) A(1)BB BB ( ) ( )] ( ) ( )2 2 1 2 4 4= C [A(1) A(1)BB BB ( ) ( )] ( ) ( )2 2 1 2
Singlet
triplet
Triplet density :
sin ( )( ) ( )
( ) ( )
gHA HB
S SA B
S
11
1
1
12 1 1
1
2 2
2
=
+
A B
negative overlap density
Density reduced between nuclei
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Valence Bond Theory
1 1= C [A(1) A(1)BB BB ( ) ( )]( ) ( ) ( ) ( )]
2 21 2 1 2
+
2 2= C [A(1) A(1)BB BB ( ) ( )]
[ ( ) ( ) ( ) ( )]2 2
1 2 1 2
+
3 3= C [A(1) A(1)BB BB ( ) ( )] ( ) ( )2 2 1 2
4 4= C [A(1) A(1)BB BB ( ) ( )] ( ) ( )2 2 1 2
Singlet :
E E J KS
eR
Ho
= + ++
+21 42
2
Triplet :E E
J K
S
e
RH
o
= +
+2
1 42
2
Energy hydrogen atom
A(1) B(1)
S s s dA B= 1 1 1 1( ) ( )
Overlap between
1s orbitals on A and B
Basic Theory
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Valence Bond Theory
Singlet :
E EJ K
S
e
RH
o
= ++
++2
1 42
2
Triplet :
E EJ K
S
e
RH
o
= +
+21 42
2
J
e
A A r dvo B=
2
114 1 1
1
( ) ( )
A
1
rA1
B
2
rB2rB1
rA2
r12
R
e
B B
r
dv
o A
2
2
2
4
2 21
( ) ( )
eB B
r
A A dv dv
o
2
12
2 1
4
2 21
1 1
( ) ( ) ( ) ( )
Int. el.1 with Nuc. B
Int. el.2 with Nuc. A
el 1 with el 2
2EH
2EH+
e2
4oR
2EH+
e2
4 oR+
J
1 S2
E
Triplet2EH
+e2
4 oR+
J
1+ S2
singlet
Basic Theory
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Valence Bond Theory
Singlet :
E EJ K
S
e
RH
o
= ++
+
+21 42
2
Triplet :
E E J KS
eR
Ho
= + +2
1 42
2
Ke
A B
r
dv
o B
= 2
1
1
4
1 11
( ) ( )
e
A B
r
dv
o A
2
2
2
4
2 21
( ) ( )
+ e
A B
r
A B dv dv
o
2
12
2 1
4
2 21
1 1
( ) ( ) ( ) ( )
Int. overlap dens. with Nuc. B
Int. overlap dens with Nuc. A
int. overlap dens with itself
BA
BA
BA
K is negative for singlet
since overlap density
positive
K enters with oppositesign for triplet since
overlap density
negative
Basic Theory
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Valence Bond Theory
Singlet :
E EJ K
S
e
RH
o
= ++
+
+21 42
2
Triplet :
E EJ K
S
e
RH
o
= +
+21 42
2
2EH
2EH+ e2
4oR
2EH+ e
2
4oR+ J
1 S2
E
Triplet
2EH+ e
2
4oR+ J
1+ S2
singlet
singlet
Triplet
E = 2EH +
J + K
1+ S2 +
e2
4oR
E = 2EH +J K
1 S2+ e
2
4oR
Positive overlap density
makes singlet more
stable than triplet
and separate hydrogen
atomsorigin of chemical
bond
Basic Theory
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Valence Bond Theory Basic Theory
H H
It is very difficult to represent valence-bond
wavefunctions because they refer to twoelectrons simultaneously. However, this
illustration is an attempt. The atomic orbital
for electron 1 is represented by the blackcontours, and that of electron 2 is
represented by the green contours. The
top illustration represents A(1)B(2), and
the middle illustration represents thecontribution A(2)B(1). When the two
contributions are superimposed, there
is interference between the black
contributions and between the greencontributions, resulting in an enhanced
(two-electron) density in the internuclear
region.
8/3/2019 Lecture 23: Introduction to Valence Bond Theory
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he molecular potential energy curve for theydrogen molecule showing the variation of
he energy of the molecule as the bond length
s changed. The calculated curve refers to the
alence-bond model.
Valence Bond Theory Basic Theory
H H
8/3/2019 Lecture 23: Introduction to Valence Bond Theory
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You should know that :
In valence bond theory we start by writing down theLewis structure of our molecule Subsequently wewrite r R as the product of electron pair functions
as r Re N
i e N 1 2
i
( , )( , ) ( , ) ( , ) ( , )
.. ( , ) ( , ).. ( , )
rr rr rr rr rr rr
rr rr rr rr rr rr ii ii ii ii jj jj jj nn nn nn
2 1 2 1 2 3 4
2 1 2 2 1 2 2 1 2
=
You should know that the singlet function= C [A(1) A(1)
is more stable than the triplet (e.i. =C [A(1) A(1)
1 1
22
BB BB
BB BB
( ) ( )] [ ( ) ( ) ( ) ( )]
( ) ( )] [ ( ) ( ) ( ) ( )]
2 2 1 2 1 2
2 2 1 2 1 2
+
+
What you should learn from this lecture
You
S S
A B
SHowever
gHA HB
are not asked to derive the expressionfor the density of th singlet .
you should know that densityis increased in the bonding region and thatcontribute to the stability of the singlet
sin ( )( ) ( ) ( ) ( )
,
1 1
1
1
1
2 1 1
12 2 2=
++
++
+
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What you should learn from this lecture
You
S S
A B
SHowever
tripHA HB
are not asked to derive the expression
for the density of the triplet .
you should know that density is decreased inthe bonding region and that contribute to the higherenergy of the triplet
( )( ) ( ) ( ) ( )
,
11
1
1
1
2 1 1
12 2 2=
+
You
E EJ K
S
e
Rand
E E J KS
eR
However
g Ho
triplet Ho
will not be required to derive the energy
expression for the singlet
you should know that the (negative) exchangeintegral K is responsible for the lower energy of the singlet.
It is related to the buildup of charge in the bonding region.
the tripletsin = ++
++
= +
+
21 4
21 4
2
2
2
2