We have seen the molecular orbit approach
to hydrogen molecule and now, we will look
at the Valence bond approach. We show that,
the molecular orbit approach this in describe
the dissociation limit correctly. And in Valence
bond theory the dissociation limit is actually
describe correctly, because it starts from
the dissociation limit. So, let us imagine
that, I start with the molecule such that
the distance between the two atoms is very
large.
So, let me say here is my molecule; the distance
between the two atoms is very large. And if
the distance is very large, what would I expect?
I would expect there are two separate hydrogen
atoms. So, if this is nucleolus a and if that
is nucleolus b, what you will have is one
electron sitting in this atomic orbital 1
s a and you will have other electron sitting
in this other atomic orbital which will label
1 s b. And the two systems are far apart,
let us say if the two systems are far apart
they are not really interacting with one another,
they are independent one another. And if they
are independent of one another the total wave
function for the system will be the product
of the wave function for the individual systems.
So, therefore, you would have the total wave
function given by 1 s a, let me say electron
one is in 1 s a and electron two is in 1 s
b. So, therefore, the two are not interacting
wave function, total wave function will be
the product. And you can imagine that, I will
use this has the trial function. Well may
be I will use the symbol 1 and 2 that will
make the notations easier to write. So, now,
I can say well, let me take this function
actually it is a very approximate wave function,
but what I will do is, I will use this as
may trial function, which actually means that
even at finite distance even they, when they
are very close to each other.
I will say the wave function is simply at
product of this form. Electron one is associated
with nucleus a, electron two is associated
with nucleus b and they are sitting in the
separate atomic orbitals. There are interactions
I know that, but I can try and find out how
good this is as an approximate function so,
people have done this. And if you did the
calculation what happens is you will get,
a potential energy curve what do I mean by
the potential of the curve. A curve in which
the electronic energy is plotted as a function
of inter nuclear distance, we have been doing
that though I did not use the word earlier.
So, I will get a potential energy curve for
the system which would looks something like
this. In this case, if you did use this as
an approximate function and did the calculation
you will get a potential energy curve of this
form.
Whereas the actual potential energy curve,
we have seen it has the decreases it has dissociation
energy of 4.72 electron volts. While if we
use this as the appropriate function you will
get a something like 0.4 or even less at the
bending energy or the dissociation energy
is extremely a small. And it says that, the
system is only slightly stable in contradiction
with the experiment, where the system is find
to be quite stable.
Therefore, it is obvious that, this is not
a good approximate wave function, but then
you look at it physically, I mean never you
the physically this function is not a great
function because, you are saying that the
electron one is always associated with the
nucleus a electron two is always associated
with nucleus b, but even when the nuclear
very close. you are saying electron one is
restricted to be confined to 1 s a and electron
two is restricted to be confined to 1 s b,
but when the distance are very close you cannot
say which electron is where.
So, therefore, when the distances are small,
defiantly the electron two may be associated
with nucleus a and one may be associated with
nucleus b and therefore, there is another
possible function. The other possible function
being 1 s a 2 notice, what is happening I
am inter changing the two electrons at to
the close distances definitely the two electrons
can change places.
So, I would have 1 s a 2, 1 s b 1 as another
possible function correct. Now, if you want
you can use this alone as your trial function
and you will find that again you need to exactly
the same result. And therefore, that alone
itself is not a reasoned approximation, but
if that is the way it is, what will you do?
You will say I will take the first function,
I will have a function of the form, which
is contains the first function 
and I will also have the second function right.
I take both the functions and multiply the
first one by some coefficient c 1 the second
by some coefficient c 2 and add the two up
right. Notice what is happening, here I am
saying electron one is associated with nucleus;
electron one is associated nucleus a, electron
two is associate with nucleus b, but then
at close this is a I allow the electrons to
exchange places.
And I get this function and that should also
be important. And now, if you have such a
trial function what will we do? You have to
determine c 1 and c 2 in a variation fashion,
but it is not really necessary to do the calculation,
because physically we can guess what is going
to happen, because you see these two are actually
kind of equivalent. And therefore, they should
make the same amount of contribution to the
final wave function right.
Because, I mean all that you are doing is
inter changing the two electrons right. So,
therefore, this and that should have the same
amount of important in your final wave function
and that this immediately tells you that,
you have two possibility is just cross it
the case of molecular orbitals that were discuss
earlier in the case of H 2 plus. You will
have a possibility of c 1 being equal to c
2 and the other possibility will be the negative
sign, c 1 being equal to minus c 2 right.
So, therefore, we have now two functions.
What are these two functions? Let me write
down.
My first function will be phi, which I will
denote as 1 s a, which I will write as 1 s
a 1,1 s b 2 with the positive sign, I will
add the another one 1 s a 2, 1 s b 1 this
is just similar to what happen in the case
of H 2 plus. H 2 plus, I have two atomic orbital,
they added together with a positive sign,
over they added together with a negative,
one of them having a negative sign. In a similar
fashion, I take the first function, but these
are not molecular orbital remember that, because
these are actually wave functions for two
electrons not one electron, this is the wave
function for two electron, this is the wave
function for the two electrons and I am adding
these two functions.
So, I add the two up naturally what will happen?
I will have to have a normalizing factor here.
It is possible written in the normalizing
factor, I will not write it down. I will not
calculate it, but it is easy to see, show
that it is given by 1 by square root of 2
into 1 plus S square. In the case of the molecular
orbital theory there as only an S that is
because you are adding atomic orbital, but
here you are adding two electron wave functions
and that is, why it is S square. And this
is one possibility, what is the other possibility?
The other possibility is to have 1 s a 1,
1 s b 2 minus 1 s a 2, 1 s b 1 and naturally
the normalization factor will be different
over that will happen is, you will have 1
by square root of 2 into 1 minus S square,
S is actually our overlap integral that we
had earlier.
So, there are these two functions, the first
one, I am going to call it Valence bond approximation
for the, actually this will turn out to be
approximation for the ground state and this
will be a valence bond approximation not for
the ground state, but for excited state. As
similar to what happen in the case of molecular
orbital theory, this will represent the ground
state; this will represent the possible excited
state of the system ok. So, the now, we can
do a calculation using this function. You
use this as the trial function for each distance;
you will calculate the value of script E and
then make a plot of that script E against
the distance. So, actually what happens is
you will get a curve, which looks like this.
In comparison with this the simple molecular
orbital theory curves this is valence bond
theory. This approximate function is known
as the valence bond wave function for hydrogen
molecule and in comparison with that the molecule
orbital results are something like that. So,
these things are there in the picture that
I showed you earlier we will have a look at
it once more. So, this is the experimental,
this one is the experimental curve.
That is the valence bond result and this curve
is the molecular orbital result. You can see
that valence bond theory actually is better
in two respects, better in that it describe
the dissociation limit correctly. It is not
surprising; because that is how we started
with the wave function, this is how we constructed
the wave function. So, that part is not surprising,
but you also find that the energy predicted
by the valence bond theory actually is better
than the molecular orbital theory.
Let me tell you the value of dissociation
energy, as well as the bond lengths. I will
list the values of R e and the dissociation
energy. Simple M O theory which we have discussed
earlier, it gives you R e to be 0.83 angstrom
and dissociation energy to be 2.65 electron
volts. Valence bond theory gives you 0.80
and 3.16. Comparison with this experimental
values 0.74 and 4.72, but the even though
valence bond theory is better.
You see it is, that is true correct, thank
you, it is electron volts. So, even though
the valence bond theoretical answer is better,
it is still far away from the experimental
value it does predict to that a bond is formed
and it predict the bond length to be it I
mean, not a bad value 0.80 were as the experimental
value is 0.74. So, therefore, you can see
that there is still room for improvement correct.
So, people have done. So, many kinds of calculations,
many many calculations have been done, all
this calculations where done long ago.
For example, what you can do is, I mean, I
can tell you one typical way in which the
modification will proceed. These atomic orbital
write 1 s a, 1 s b is they are taken to be
atomic orbitals, each one of them is taken
to be atomic orbitals, which feel the nuclear
charge of unity right. They are taken to be
same atomic orbital as in the case of hydrogen
atom and then the case of the hydrogen atom
each electron, not each electron, electron
feels only one nuclear charge, but here you
have two nucleus and therefore, may be what
would happen the effective nuclear charge
felt by this atomic orbitals need not be unity,
but it may be a new number, different from
unity.
So, you can introduce an effective nuclear
charge, but once you introduce an effective
nuclear charge, you will have to do a calculation
for different effective nuclear charge and
find out which effective nuclear charge is
the best. And this will leave to improvement
in the energy of the system and people have
done this kind of calculation. I will tell
you details of one such calculation, not in
details, but I will give essentially I will
give answer for that.
But before I do that, we have to understand
why is that the molecular orbital method.
Gives a totally wrong dissociation limit,
we will try to understand that. So, we will
look at the molecular orbital wave function,
oh before I go into that description I should
also tell you what happens? If you did a variation
calculation, using this as the trial function.
In that case, you will find that there is
no bond formation, but this system is acutely
repulsive. Repulsive meaning there is no formation
of a molecule, there is no bond formed. If
you say that is the wave function for this
system.
And so, it is some excited state of the system.
Let us forget this, we are not interested.
This is the case where is a bond formed. Now,
we look at phi M O once again, what is phi
M O? It is actually sigma g 1 s with the first
electron and sigma g 1 s with the second electron
right. And what is sigma g 1 s? Let me just
remind you, it is just given by 1 by square
root of 2 into 1 plus S into 1 s a 1, 1 s
sorry, plus 1 s b 1 that is sigma g 1 s. And
sigma g 2 s has the similar appearance; it
will also contain this factor 1 by square
root of 2 into 1 plus S. So effectively there
are two such factors, I would have the square
of that and then I would have 1 s a 2 plus
1 s b 2 correct. Now, this is just a normalization
factor that is worry about what is in the
wave function. So, it is actually going to
be if I squared this, I will get 1 by 2 into
1 plus S multiplied by, see there are two
terms here, there are two terms there, so
if you multiplied out and added them together,
you will get four terms.
I am going to write those four terms, the
way I am going to write those four terms is
like this. I am going to multiply this with
that, that is acutely 1 s a 1, 1 s a 2 right.
And then I will multiply this with that so,
this plus 1 s b 1, 1 s b 2. And then, I will
have what are repeated cross term, this multiplying
that, that is acutely 1 s a 1, 1 s b 2 plus
1 s a 2, 1 s b 1. So, this is exactly what
my molecular orbital wave function is for
the hydrogen molecule, is it ok, you have
made a mistake; I do not think I have made
the mistake right. So, then if you look at
this function, you will notice something very,
very interesting. See if you looked at this
part of the function 
and compared it with the valence bond wave
function. We will find that, this and that
are the same.
So, contained inside your molecular orbital
wave function is acutely the valence bond
wave function. And what does the violence
bond wave function say? The violence bond
wave function says that ok. There is one electron
associated with nucleus a right and that is
the way it is the other electron is associated
with nucleus b. And if you look at this one,
what will happen? The electrons simply have
inter change their positions. So, therefore,
what is acutely happening is that ok, there
is one electron with nucleus a, one electron
with nucleus b right. And further I think
this is the best point for me, to introduce
the spin part of the wave function.
Let me just introduce a spin part for this
function, how would this spin part look like.
Well here, I have written only the spatial
part of my wave function. And if you looked
at the spatial part if suppose you interchange
the electrons 1 and 2, suppose you interchange
the electrons 1 and 2 what will happen? This
function if you interchange the electron 1
and 2 you will find that, this function does
not change sign. And therefore, strictly speaking
you see the wave function as it starts does
not satisfy the Pauli Exclusion Principle,
but then you can make it satisfy the Pauli
Exclusion Principle by adjusting the spin
part.
So, how was the spin part look like? It would
be 1 by square root of 2 into alpha 1 beta
2 minus beta 1 alpha 2, remember we have seen
this kind of spin function earlier. And this
function has the nice feature that, if you
interchange the two electrons, the wave function
will change signs. So, therefore, with along
with the spin part if it took this function
what is the result? The result is that it
is anti symmetric. Similarly for this function
also you can use exactly the same spin part
for the molecular orbit function also you
can use the exactly the same spin part without
any difficulty because when you interchange
the 2 electrons you see this function does
not change sign.
And if I use the exactly the same spin part
everything is fine the wave function satisfies
the anti symmetric principle. Now, if you
look at this function, it says that one electron
is associated with nucleolus a, the other
is associated with nucleolus b. And further
it says that one of them if it is pointing
up, the other is pointing down. Now this is
actually very nice for the chemist, because
you see if you remember what Lewis said what
the chemist were thinking of the formation
of a bound in hydrogen molecule. They would
say that you have one electron from hydrogen
one and the hydrogen a and you have the other
electron from the other hydrogen and these
2 electrons are paired. And the bond means
the pair of electrons, which are acutely paired
in such a fashion that there spins are anti
parallel.
That is the Lewis way of thinking about the
bond electrons form a paired and so, you have
the bond. And you will see that the valence
bond description acutely is very much resembling
that because you are saying the two electrons
are paired, there spins are anti parallel
and when one electron is on associated with
nucleus a other electron is associated with
nucleus b and these two electrons are getting
pared. And you will also realize that, the
same thing as contained inside the M O function,
the same thing is contained, but M O function
has other possible contribution also from,
a case like 1 s a 1, 1 s a 2.
So, how will I represent that well, we can
say this represent a situation where the electrons
are shade equally by the two nuclei. This
part represent the case where the electrons
are shade equally by the nuclei, but if you
look at this 1 s a 1, 1 s a 2, this means
that both the electrons are associated with
nucleus a, while that represent the situation
were both the electrons are associated with
the nucleolus b. And therefore, if I wanted,
I can say that this one represent a chemical
structure which I will denote as H minus H
plus. While this part represents a chemical
structure according to the chemist it would
represent the structure which would be written
as H plus and H minus right. This is acutely
H minus and H plus and this will be H plus
and H minus. Yeah there is the plus there
right. Everything is been added it up ok.
So, therefore, if I looked at the molecular
orbital descriptions, what happens is that
contain in the molecular orbital description
are the valence bond wave function which is
this. It also has contribution from what may
be referred to as ionic structure, these are
the ionic structures right. And if you looked
at the contribution ionic structures inside
the molecular orbital wave function you have
the ionic structure, but what happen is that
ionic structures and this structure, which
is acutely referred has the covalent structure
this one, because you see it is representing
the covalent bond. So, this is covalent structure.
So, the covalent structures and ionic structures
have the same amount of importance in this
description.
So, therefore, if you think of dissociate
the molecule, what will happen? This is the
system the molecule may dissociate to give
you two hydrogen atoms. That is what will
happen? If you had only this part of the function,
but this part of the function implies that
this system may also dissociate into the ionic
structures, which are H plus H minus. So,
if you dissociate the molecule, the molecular
orbital theory will say that, it is possible
for you to dissociate in such a fashion that,
you will get H plus and H minus.
And that is the reason why the plot of energy
this is molecular orbital energy, energy according
to molecular orbital calculation, it does
not give you the correct dissociation limit.
It does not dissociate to hydrogen atoms,
but instead it disassociate sometimes. I mean,
if when you dissociates, you can get two hydrogen
atoms, because it contains this part, but
you cannot also get positive ion on one side
and the negative ion on other side, that is
what the molecular orbital theory says and
that definitely cannot be true. And this is
the reason for M O theory gives you totally
incorrect disassociate limits fine. But then
you look at the Valence bond theory, Valence
bond theory actually has only the neutral
structure.
Now, if I wanted to improve the valence bond
theory, what can I do? Well these ionic structures
are acutely possible that is the way in which
valence bond theory would think. It would
say that, this structures must be possible,
but they should not have the same amount of
importance has the covalent structure. So,
therefore, an improved valence bond theory
wave function can be written, how will I write
the function.
And improved valence bond theory function
will have phi improved M O sorry, improved
valence bond would be, you will have this
neutral structure right. So, let me just say
that, it is written as some phi H H, I do
not want to be writing all this things in
detail. But when I write phi H H, may be I
will put two dots between, what that is it
represents? It just represents this part of
my function. And then I would have phi H plus
H minus, what would that be, it would be this
one. It would represent 1 s b 1 into 1 s b
2 and then I would have what is the other
one phi H minus H plus.
So, these are three possible functions, right
and what does the molecular orbital method
say, it simply add all of them right, with
the same amount of importance and does the
calculation and that gives you wrong answers.
Now simple valence bond theory what does it
do? It does not include these or that, but
it just says, the function is just this. So,
a better calculation will be, to say that
I will take combinations of these things after
multiplying them by suitable Constance.
So, what I will do is, I will multiply this
by may be constant, which I will referred
to as c 1 and these two, I would accept hopefully
would have the same amount of importance right.
Because you see a structure in which, you
have H plus on one side and H minus on the
other side over structure in which H minus
is one this side and H plus on the other side.
They would make the same amount of contribution.
So, therefore, these two I would accept would
have the same amount of importance.
So, what I will do is? I will simply add them
together multiply them with a coefficient
which maybe, I should call this c 1 and that
c 2. Now, if you did that, what will be your
next step? You will take c 1 and c 2 to be
variation parameters and find their value
in such a fashion that 50 as the least value.
And because of the wave, the wave function
is constructed it is guaranteed that dissociate
correctly and it is also guaranteed that the
energy will be better than the valence bond
theatrical calculation. And such calculations
have been done. There are too many curve in
this picture, but any way, let me draw this
one also I can find chalk of different color,
I will use that.
So, this one will actually lead to an improvement
in the energy and the one length so, this
is improved valence bond theory. Now I will
write with the answer of one such calculation,
the 
bond length is actually 0.77 angstroms and
the dissociation energy is 4.00 electron volts
and I should also tell you that, this particular
calculation was due to represent to Weinbaum.
And in addition to taking this as the function,
what it did was introduce this idea of effective
nuclear charge and varied the nuclear charge
in such a fashion that you got the best energy.
And you can see that the answer that you got
for the energy was fairly decent dissociation
was 4 and the bond length was to be found
0.77 angstroms.
Now, you may ask it is possible to improve
molecular orbital theory and the answer is
that, yes it is possible to improve the molecular
orbital theory also. In a similar fashion
and in fact, if you try to improve the molecular
orbital theory what will happen is that? You
will get exactly the same kind of function
now, I mean when you look at it in detail
you will find that the function is of this
form. And such procedures are known as configuration
interaction procedures.
Now, even for a system like hydrogen molecule
this calculations, which are require to configuration
interaction calculation are very difficult
they are quite involved and quite difficult
to perform. But I should also tell you that,
in the case of hydrogen molecule, there was
a calculation done long ago by two scientists.
Let me remove everything else. I will just
say James and Coolidge these were the name
of the scientists who did the calculation.
And effectively I mean they had large number
of variation parameters, but you should also
remember that the more variational parameter
you have, the more difficult is the calculation.
Because, each variational parameter you have
to find it is value in such a fashion that
script E has the least value. And these were
the days in which computers were not there
at all you had only hand, you had only mechanize,
I mean mechanical calculators using this mechanical
calculators. They did the lot of work to get
the best possible value is for all the parameter
and then essentially you got the experimental
values back right. They did such a calculation
that the answers were in agreement, perfect
agreement with the experimental results are
available and that time, but that involve
lot of calculations.
We have seen that our improved wave function
according to valence bond theory is given
by phi improved is equal to some constant
c 1 phi which corresponds to H plus and H
minus plus phi which corresponds to H minus
and H plus, plus another constant c 2 into
phi which corresponds to natural sector which
we may write as H H just remind you, this
means that, both the electrons are associated
with the hydrogen atom on the right hand side.
While this means that both the electrons are
associated with the molecule are both the
electrons are the associated with the left
hand side electron. While this wave function
means that, the two electrons are shared equally.
And the acutely wave function is a linear
combination of these three functions. And
this has been used essentially in chemistry
particular by the organic chemist.
And instead of writing such a wave function,
what they will do is? They will say that the
actual structure is a mixture of all this
things. And hence they would say that the
acutely structure is that resonance hybrid
of this structure which is H plus H minus,
the second structure which is H minus H plus
and the third structure which is H H. And
to tell you that it is a resonance hybrid,
what they would do is? They would put double
ended arrows in this fashion. So, this as
to be very clearly understood if you put this
double ended arrows what it means, is that
actually structure the wave function for the
actually structure is a linear combination
of the wave function corresponding to each
one of these structures. Having seen this,
we will now look at diatomic molecules. Thank
you for listening.
