This is now the fifth lecture of the condensed matter course.
In the first four lectures. We studied vibrations in solids and we studied electrons in metals and we learned quite a bit about these topics.
But we also developed a rather long list of puzzles, things that we couldn't understand.
I wrote a list of them at the end of Last Lecture. Some of them included the sign of the hall coefficient.
We couldn't figure out why. It sometimes says one sign, sometimes there's another sign.
We also had a rather unreasonably long, mean, free path in Sommerfeld theory.
We couldn't understand. And if you go back to earlier into by theory, there were also some puzzles there.
We didn't understand why it was we introduced an ad hoc cut-off.
We had some motivation for it, but it was certainly an ad hoc prescription.
And all of these problems stem from the same place. And the same place is that we're not taking seriously the microscopic structure of materials.
We're not really considering that they're made up of individual atoms, and the atoms are bonded together in very particular and often periodic ways.
So much of the remainder of the term is going to be devoted to understanding
how the microscopic structure of materials affects its macroscopic properties.
But before doing that,
it's probably worth backing up and understanding why it is that the atoms stick together in the first place to make the materials.
So this brings us to the domain of chemistry.
And as much as we hate to admit it to both our friends and our enemies, chemistry actually is interesting.
It's a it's an important subject, both scientifically and certainly industrially technologically.
And it's actually has some fundamental deepness to it as well.
And you could certainly spend three or four years studying chemistry or even longer.
And we're going to study it for only a day or two. And that will probably be all we need to know in order to go on further.
Now, if you had chemistry before, some of these things might be a little bit familiar.
If you haven't had chemistry before, it's maybe not so much of a problem because most of what I'm going to introduce is fairly basic.
But the main idea is to reconsider some of the chemistry from a physics perspective.
Some of the things that may have seemed strange when you even learned chemistry
the first time will maybe seem a little bit more natural this time through,
and this will give us sort of foundation for understanding much of what comes comes later.
So a good place to start is to realise that, you know, chemistry is all based on the Schrödinger equation.
The sorting equation solves everything. It tells you why the electrons stick to nuclei and why nuclei stick to each other and so forth.
It's all a shorthand equation. But that's not actually a very useful way to think about things,
because solving the Schrödinger equation for more than one or two particles becomes impossibly hard,
and you can never develop any sort of intuition of what's going on. So what we should do is we should think about very,
very simple cartoon level problems and try to figure out what the general rules are of chemistry from that.
A good place to start is with hydrogen, hydrogen or more generally, a hydrogen atom, meaning that it has a nucleus of charge.
Some genuine nucleus. Nucleus of charge.
The. And one electron.
And you'll remember from your atomic physics courses or your quantum mechanics courses
that the eigen states of this one electron are indexed by a number of quantum numbers,
the principal quantum number and equals 1 to 3 and so forth.
The angular momentum quantum number, which is takes the number, the value zero one to up to and minus one.
And these anchor momentum quantum numbers are sometimes called S, P, the F, G and so forth coming from spectroscopic notation.
Now for one chocolate bar. Does anyone know what speed and F stand for?
Something a gentleman officer had sharp his ass.
That gives you a quarter. Is this diffuse? Yes.
You're almost there. Two more. Principal.
Someone said principal. Someone who? Who got it? Who has it?
Someone said it over there. Okay, so he gets it. So start principal diffuse and find from spectroscopic notation your chocolate.
Sorry, stolen out from under you. G does anyone.
One more chocolate bar for ten. Which is for. It's not for anything.
Actually, it's. It's. It's for giraffe. They just need another letter after that.
Okay. Anyway, in addition to the angular momentum quantum numbers,
M takes the values minus l2l and then the spin quantum number can be plus or minus one half.
The most important thing to keep track of is that if you have s electrons as eigen states, maybe say, how should I says?
So in sl I guess of of eigen states can hold two.
Has two possible eigen states partial? So why is it two?
Because x means l equals zero. That means m has to be equal zero.
But the spin can be plus or minus one half. So two possibilities. With a p shell there are six because p means l equals one.
M can be minus one zero one, that's three. And then the spin can take two values that make six D gives you ten and so forth.
Now you have all these eigen states in a particular atom,
and then you suppose to yourself that you have many electrons in that atom, not just a single electron.
The atom. Also, I should I should probably give you the energies as well.
The energies of these eigen states are minus the red berg over n squared times z squared if the charge of the nucleus is more than one.
Plus small corrections. Things like fine and hyper fine fine structure.
But the main energy just comes from the principal quantum number. Okay,
so now you have a bunch of electrons in your atom and you need to figure out which
of these are eigen states you fill with electrons and which ones you leave empty.
And there's several rules for figuring this out. The first of all is known as the outflow principle.
Outflow principle. Which is basically just filled by shells.
So when you add the first electron to anatomy, starting one shell and you keep filling that shell until it's filled.
When you finish filling that shell, you start filling the next challenges fill that one until it's filled.
Now, in what order do you feel the shells? Well, you fill the shells by Matt along as well.
Matt along roll, which has a nice pneumonic like this.
You write out a table one asks to as to p3s3p3d4s4p4d, four f, five s and so forth, five p after that.
And then you fill in this diagonal way. This one is first. This is second.
This is third. This is fourth.
And so forth. And this would be theft. Down that way.
Okay. So this order of filling shells is what gives you the structure of the periodic table.
So let's take a look at our friend, the periodic table. Oops. I don't see the.
Uh oh. How? Sydney.
Stop, stop, stop, stop, stop, stop, stop, stop, stop.
Okay, hold on. Boy.
Now we have a real problem. Stop.
Oh. Okay.
Well, okay. So the periodic table, let's go on and see if we can get this start up a little bit later.
So everyone kind of that you have in your head what it looks like, right?
So you start out by filling the, the, the, the top row of the periodic table,
which has hydrogen helium in it is one electron fills fills hydrogen, the next one fills helium.
Then you start the next row with with lithium and beryllium that fills up the second shell.
Then you have the next six elements boron carbon, nitrogen, oxygen, fluorine and neon.
I'm going to try this again. See if that's going to work.
Brian Nitrogen carbon, neon, fluorine, neon, and then you're filling the two piece shell and so forth and so on,
and you fill them in all and all this order. And that gives you the structure of the periodic table.
But you might wonder to yourself, Well, wait a second, why is it that we're filling the four shell before the three D shell?
Because according to that formula up there, the three shell should have lower energy than the fourth shell.
And so what gives us the right to fill up four as before?
Before we fill up three D and for that matter, why are we filling a periodic table?
Oh, thank God. So there's the periodic table, and we fill in this order.
There's the the first shell. The second shell tends to step three S, three P and there's four s filling up before the hour before the three D,
you might ask the question, well, why is it we're filling up 2 hours before two P?
Because according to that formula up there, they have more or less the same energy.
So the the answer to this question is basically because V of R is not equal to one over R, it's not proportional to the Coulomb interaction anymore.
Well, why not? Well, it's not just a single electron and a nucleus.
It's an electron interacting with all of the other electrons in the atom as well as the nucleus.
So as the electron gets closer to the nucleus, it sees more nucleus and less of the other electrons.
And it gets as far as it gets farther away, it sees more of the other electrons.
So you can think of as having some sort of effective potential,
which is the interaction between the electron and both the nucleus and the cloud of the other electron.
So you need to modify the shape of this potential somewhat.
And if you try to solve this or the radial quantum mechanics problem that you solved for the hydrogen atom,
but you change the shape of this potential, you'll all of a sudden discover that all these cells have different energies.
Now they shift around.
And that's why these the ordering of these filling the orbitals is not what you would expect from just a simple hydrogen atom, actually.
Whereas we're on the subject of the interaction with the other electrons,
you might find it surprising that we can even describe the eigen states of the hydrogen of of a general electron,
the same way we can a general atom, the same way we can describe the eigen states of a hydrogen atom,
because it's not it's just interacting with the nucleus. It's interacting with all of the other electrons as well.
But the reason more or less you can do this is because you can treat all of the other electrons besides yourself.
It's just some sort of spherical cloud. So it's still sort of a severe problem.
So it follows more or less the same rules and you can still enumerate the possible ligand states in the same way.
Okay,
so this is the general structure of the periodic table and we can now move on to asking about chemical bonds between the atoms in the periodic table.
Okay, there's filling up Madeline's rule. So there's a bunch of different types of bonding.
There's a chart that looks like this in the in the in the book, which tells you five different types of bonding,
ionic, covalent, metallic molecular hydrogen and some of their properties and what causes them.
It's probably worth familiarising yourself with this chart over the next you know,
today we're going to try to cover Ionic and covalent the other types.
We'll do a little bit later on in the and in the lecture series so that, you know, the basic, basic facts are on this.
You should probably know these, but we will just start today with the simplest type of bonding, which is ionic.
Bonding. Ionic. Bonds.
And the principle of an iconic bond is basically that an electron is transferred.
From one atom, from one atom by atom to another to another.
And then the ions attract, then ions attract.
So the kind of the kind of a sort of reaction that we we can have is a sodium atom plus a chlorine atom.
An electron is transferred from the sodium to the chlorine.
So we get sodium plus plus chlorine minus.
And then now we have to charge species and they attract each other and form sodium chloride.
Sounds easy enough. Probably the easiest way to keep track of this kind of reaction or to evaluate this kind of
analyse this type of reaction is to think about the energies associated with each step.
So one piece of the energy is the ionisation energy, the ionisation,
which is the energy to pull the electron off the sodium atom, to make a sodium ion and an electron.
So you start with the sodium atom neutral.
You pull the electron apart from it and put it out infinitely and the sodium ion infinitely far away at infinity.
And this will be the difference in energies, what's required to pull the electron off the atom.
There's also what's known as the electron affinity,
which is the energy that you get back when you put the electron on the chlorine atom to make a chlorine ion.
And then the last piece is known as the cohesive energy or the bonding energy,
which is the energy you get out when you put together the sodium ion and the chlorine ion to get sodium chloride.
So the total energy budget for this reaction is the ion, the ionisation energy you get back,
the affinity energy, the electron affinity, and you get back the cohesive energy.
And if the total delta e is less than zero, if Delta is less than zero, then you get a reaction and then it reacts.
The reaction goes forwards. Now there's a couple of caveats about about this kind of argument.
The first caveat is that if you're a chemist, you are inevitably talking about gives free energies rather than just energies.
And the reason it gives free energies is because you're doing your experiment at a finite temperature,
probably room temperature and finite pressure, your room pressure.
We're going to think just as if we're doing everything at zero temperature, zero pressure.
So we can just think about energies, but it's more or less the same thing.
The second thing one should be warned about if you're reading books,
is that the word cohesive energy or bonding energy is used in several different senses in different books.
Everyone agrees that a cohesive energy or bonding energy is the energy to put two things together.
But not everyone agrees what those two things should be.
So they'll talk about that cohesive energy for sodium chloride in some books will tell you that the whole delta is a cohesive energy,
and some books will tell you that the cohesive energy is just the energy for putting together the sodium ion and the chlorine ion.
So you get two different results depending on whether you're counting the whole Delta E and calling
that the bonding energy or if you're just talking about the energy to put together the two ions.
So that's another warning.
The last warning is that the results that you get for bonding energy will be extremely different depending if you're talking
about taking one ion of sodium and one ion and chlorine and putting them together to make a sodium chloride molecule.
Or if you're talking about taking lots of them and putting them together to make a whole big chunk of sodium chloride.
And the reason for this is because, you know, you're sodium atom.
It's either bonded to one chlorine or its bonded to a lot of different chlorine.
It will be a totally different energy depending on how many things you're bonded to.
So that's another thing to keep in mind, to be careful about.
Okay. So in this set of energies that we try to keep track of, probably the easiest thing to understand is the bonding energy of that cohesive energy.
Also known as the mad along energy or the Coulomb energy, because this Coulomb energy is basically just right out.
It's basically just the Coulomb energy between the between the relevant ions in your, in your system.
So I'm not equal to j q i QJ over four pi.
Epsilon not distance between I and J.
So Q Why is the charge? Am I on this charge of iron?
J So if I have a bunch of if I have two ions.
J I just take its usual the usual coulomb if I have two ions, one positive charge, one a negative charge.
I'm just summing up the usual Coulomb energy between those ions to get that cohesive energy.
Now,
this is a perfectly good way to express the the bonding energy or the energy you get by taking these two charge species and bringing them together.
But it obviously and actually it's fairly accurate as well for most ionic solids because
you a very good estimate of the amount of cohesive energy you get by this definition.
And so all you need to know is how far apart those ions are and you know, how much bonding energy you got.
Of course, there's a little bit of a problem with this expression, and that is it goes to minus infinity.
If you bring the sodium ion in the chlorine, I am extremely close and that's obviously doesn't make any sense.
So what's really happening here? Let's see it like this.
This is a r i j for between the two atoms and here's the energy.
So this is the one over r curve that you expect for a Coulomb interaction and the actual physical energy you get.
Looks like one over ah, it looks like one of our and then at some point it shoots off way up to infinity.
So this is actual. So if you know the distance between the two atoms, you have a just by taking this energy here, which is the Coulomb prediction,
it's extremely close to the actual bonding energy there because you because the shoot off is extremely rapid.
So what's causing this shoot off to infinity?
Well, when you start pushing two atoms really close, really close together, you realise they're not just point charges anymore.
They're going to start to overlap, their orbitals are going to start to overlap.
Pauli Exclusion principle is going to get into the game because you're trying to put two, two, you know,
too many electrons in one place and the energy just shoots right off very, very, very rapidly.
But right here, by the minimum, the Coulomb estimate is still a pretty good estimate of the cohesive energy.
So that's pretty much everything we need to know about the cohesive energy.
Now, what about the ionisation energy and electron affinity energies?
Well, the ionisation energy or electron affinity energy has to do with the energy levels in the atoms.
And in order to figure that out, we really need to solve the Schrödinger equation of those atoms in an electron,
in some effective potential, which has to do with the nucleus and also all of the other electrons.
And it's kind of a hard problem, but even though we're not going to actually solve the shorter equation,
there are some trends which are pretty easy to keep track of.
So this, you know, if you've done chemistry, you probably know this already. But that this plot,
this picture of the periodic table shows you where the large electron affinities and the large ionisation energies of small ionisation energy are.
It's also down here. So the small, ionisation energies are sort of down in this corner.
This are the size of the splotches. How big B on his energy, the smallest one being caesium or France.
I'm down here. And the large ionisation energies way up here in the noble gases in helium.
Electron affinities go the other way, with the exception of the noble gases, which have essentially no electron affinity.
The highest electron affinities are way over here in this corner, up by fluorine and chlorine and bromine.
Over on the right hand side. So if you want a good ionic solid,
what you need in an ionic reaction is you need to be able to easily pull the electron off
of one species and then you want to get out a lot of energy for putting it on the other.
So you want a small ionisation energy and a large electron. Metheny So good, so good.
Ionic solids are things like, you know, fluorine, sodium fluoride, rubidium chloride,
things from way over here bound together with things from way over there.
Now, chemists have an expression for this.
They're a shell of electrons. An atomic shell wants to be filled.
So if you have a shell filled plus one electron like you would have in this first column, sodium has one electron in the three shell.
Potassium has one electron in the four shell. That shell is easily pulled off because the shell is most happy when it's exactly filled.
Similarly, fluorine, it's one short of being filled, so it has a high electron affinity, it wants to be filled.
So can we understand a little bit, you know, mostly filled shells on the on the right and they have high electron affinity and high ionisation energy.
And on the left it's small ionisation energy and small electron affinities.
So can we understand in some physical way why this is?
Actually, it's not too hard to figure out why that is, at least roughly, we can make a cartoon picture of it.
So let's try to figure it out from looking at sodium.
So sodium it has if you look at this table, a periodic table, it has 11 electrons, it's the 11th element.
So what is it? What does it look like? It has a nucleus of charge plus 11.
Plus 11 in the nucleus. And then. Then we have what we have ten electrons in the one and two s shells.
So we'll draw a picture of one S and two s sorry,
and two P and in the 1s2 as two p shells we have a let a ten electrons with a charge of minus minus ten.
Then we have this one electron in the three s shell out here. So this is one electron out here in the three s shell.
And to that one electron, if you imagine yourself being that one electron, how much nuclear charge do you actually see?
Well, the total nuclear charge you would see this electron sees Z equals one.
Why? Well, because these ten electrons inside of it are more or less screening that positive charge in the nucleus.
So it sees a charge of only plus one.
So if we wanted to use our hydrogen formula to figure out the binding energy,
we would say that the binding energy is a red burg times one squared over the seasons is three s divided by three squared and squared downstairs.
Now this binding energy, ionisation energy is a little bit of an underestimate of the actual binding, binding energy of the three electron.
And the reason for this is because sometimes that three s electron gets closer to the nucleus and realises that in fact it's not a charge of plus one.
It's actually a charge bigger than plus one. So it's bound a little bit tighter than this estimate would would tell us.
Okay. But basically, at some level approximation, it's like one electron bound to just one charge.
Does that make sense? Yeah. Okay. Now, let's look at fluorine instead.
Fluorine is a little more complicated. Fluorine.
It's the ninth element right over here, almost filled to Shell.
So what is what is fluorine? C Well, okay, so we have the nucleus with a charge,
a plus nine and then we have the one s shell down here, one s shell, which has a charge of minus two.
And then in the two shell, we have seven electrons out here.
One, two, three, four, five, six, seven, seven electrons.
Now, suppose you're and they're more or less at the same radius because they're all in the two cell.
So suppose you're this electron. What do you see? You see a charge of Z equals.
Well, maybe you see plus seven. Because these electrons here aren't screening the nuclear.
So you only see two plus nine minus two. Now, plus seven is a little bit of an overestimate.
You would you would then get a binding energy of a red berg seven squared over.
Let's see. So this is a two electron out here. So it's over two squared, the principle quantum number squared of this electron.
This is a huge overestimate of the actual binding energy in fluorine,
because sometimes these electrons do get inside of this electron and screen the nucleus better than we would estimate here.
But still, you see that there's a very good reason why it is an almost filled shell sees
a big nuclear charge and an almost empty shell sees a small nuclear charge.
So this is behind the origin of why it is that we have this this cartoon phrase that an
almost filled shell wants to be filled and an almost empty shell wants to be empty.
Does that make some sense? Yes. Yes. Yeah. Okay.
All right. So it's basically an issue of how well screened the nucleus is.
So this gives us a pretty good picture of what's going on within ionic bonds.
There are differences in how much the electrons are, how much the atoms want to have an extra electron or want to give up another electron.
And so if you have an atom, if you have an atom like potassium in an atom like fluorine, the potassium is happy to give up its electron.
The fluorine wants to get that extra electron. You get two ions, the ions then attract.
So it's a great picture of ionic bonding.
However, you also have bonding between two electrons that want between two atoms that want the electrons just as much as each other.
For example, O2, two oxygen atoms will bond with each other and of course they want the electrons just as much as each other.
So this is give brings us to the picture of what they call covalent bonding, covalent bonds.
Which is okay. There is a chemists have this picture of covalent bonds.
Which are they? They like to say that the atoms will share an electron and because they're sharing electrons, that binds the atoms together.
Now, can we understand a little bit better from physics what it is that we mean by this, that sharing electrons binds atoms together?
So there's a nice picture of this, which basically comes from just particle in a box physics.
All right, put it over here. Particle in a box. In box, we're going to analyse hydrogen.
Plus hydrogen goes to hydrogen too, just based on what we know about particle in a box.
Now, what am I talking about when I'm talking about a hydrogen atom being a particle box?
Let's draw a little box here. That box is supposed to represent the attraction to the nucleus.
So a hydrogen atom is basically its electron bound to the nucleus.
So, okay, it's not exactly in a square box, but it's still some binding to the nucleus.
So we'll put our electron in our box, give it a boxer size L and the ground state energy of the hydrogen of the
electron in that box of size L is h bar squared over 2 a.m. pi over l squared.
Now we have another hydrogen atom here, so we'll give him a box size l and he has the same energy h per square with two m pi over l squared.
Now over here where we have h two, we get a bigger box, maybe a size two, l.
For the hydrogen molecule. And what are the. So we have a ground state wave function for the bigger box here.
And the ground state wave function is an energy bar squared over two m pi over two L squared,
which is lower because we have to put two electrons in this bouncing wave function now.
But we can do that because there are two spins and we discover that this side of the equation is lower than the sum of these two equations.
So what's going on here? What's going on here is that we're allowing the electron wave function to spread
out from being confined here and confined here to being less confined here.
And by letting the wave function spread out, we're lowering its energy. This is just, you know, Heisenberg uncertainty.
You make Delta X bigger, Delta P gets smaller with smaller delta P, you can have a lower kinetic energy.
So by spreading out the wave function, you lower the kinetic energy.
Now we can consider the same thing with helium and we'll see that actually something is very different about helium.
So helium you have a box, you have a ground state wave function and you have two electrons and that's here again, a box.
Ground, same wave function, but two electrons in it. And then the question is, does it go to helium two?
Okay, now here we have a bigger box. We get a ground state wave function, we can put two electrons in it.
Then we have to start filling up an excited wave function as well because we fill the lowest the ground state wave functions.
We have to fill this guy as well and now we're not getting so much matter if I'm not
getting anything at all because we had to as well as filling the lowest energy state,
we have to fill a higher energy state as well. The lower energy state is known as the bonding orbital.
Bonding orbital. And the higher energy state is known as the anti bonding orbital.
So generally bonding orbitals are ones whose energy is lower than the original
wave functions and antibodies are higher or equal to the original wave functions.
So this is why helium, very roughly why it is of helium, doesn't bond, but hydrogen does.
Now, this is obviously very cartoonish, but we can do a better job by using what's known as molecular orbital theory.
Molecular or atomic orbital theory. Atomic.
Orbital theory. The theory also known as type binding theory, also known as linear LCA o LCA which equals linear combination.
Of atomic orbitals, of atomic orbitals.
Now. I think if you took the atomic course last term, you may have even done this calculation.
And we're going to do it again, but we're going to do it simpler than you did it last term.
And the reason I'm simplifying it is so it's easier to see the physics underlying.
Okay. And we're going to use sort of we're going to use some of these techniques later on in the term.
So it's good to make them as simple as possible. So the first thing we do is we imagine having two nuclear positions are one and our two and oops.
Okay. So I should tell you which problem we're analysing.
Right. We're going to do the so the problem we're going to look at look at look at H two plus.
So this is two protons and one electron, two protons and one electron.
I think you've looked at this problem before last term. Okay.
So we're going to see that allowing this electron to d localise between the two,
the two nuclei is going to lower its energy and therefore form a bond between the two nuclei.
Now, the first thing we're going to do is we're going to use Born Oppenheimer. Approximation born Oppenheimer.
J. Robert Oppenheimer was the father of the American Bond and Bomb Project upon Project Nation and Max Born.
We've discussed him already, Father, one of the fathers of quantum mechanics, and Olivia Newton-John's grandfather for one chocolate bar.
Does anyone know what a living in John's biggest hit song was? I'll give you a hint.
It has something to do with the subject we're studying. Yeah?
Yeah. Who said that? The same person. Okay, you get this one. I agree. Get physical.
Yeah. Ten weeks at number one. Okay. Anyway, so the Oppenheim approximation is basically fix.
Ah, one an hour to fix the position of the nuclei and let the electron move.
And that's okay because the nuclei are extremely heavy and they would move very slowly.
So the electron can come to its eigen state very quickly and you know,
and then you can sort of resolve it wherever you put the nuclei and look for wherever the nuclei have their lowest possible energy.
So then we write a Hamiltonian, Hamiltonian for our electron P squared over two M plus the interaction,
the Coulomb interaction with the first nucleus, plus the Coulomb interaction with the second nucleus.
Where V here v is the usual minus is squared over for pi epsilon, not r minus r one, so forth.
I'm going to abbreviate this thing as this term is K for kinetic energy.
This term is V one for interaction with the first nucleus.
This is V two for interaction with the second nucleus.
Now, this is a hard problem because there's both because of both nuclei, but it would be a lot simpler if we just threw away one of the nuclei.
Pretend it's not there. Okay.
So let's solve that, that problem first so we can then have K plus V one as our Hamiltonian and we'll call the Eigen State one epsilon, not one.
So what's what's the answer to that problem was just a hydrogen atom, you know, of on on or on nucleus one.
So one is the ground state of the electron on Nucleus one, the electron in the shell, if you like, you can do the same thing for two.
So this is if we only have Nucleus two, we would get an eigen state on Nucleus two.
So I can draw these if I want. So here is our one, here's our two.
Then there would be a wave function here of the electron sitting there as if this guy were there,
and then there would be an out wave function there too, sitting there as if this guy weren't there.
It's that clear.
So the two states, the two atomic orbitals I'm thinking about is the electron on orbit on Nucleus one, as if Nucleus two are not there at all.
And the electron on Nucleus two as if a nucleus one, we're not there at all.
Good, good. Yeah. Okay. So you can also think about one and two is being the eigen states if the two orbitals of the two nuclei taken very,
very far apart, in which case you have two possibilities.
The the electron sits on orbital one, on nucleus one, or the electron sits on nucleus two.
And what we expect is going to happen is that when we bring them together, there's going to be a lower energy state,
the bonding state, where the electron localises between the two, between the two nuclei.
And there may also be an anti bonding state, anti bonding, which is higher energy when we bring them together.
This kind of diagram is used a lot by chemists, is known as a molecular orbital diagram, molecular orbital diagram,
which tries to keep track of what happens to the eigen states when you start moving things together or apart.
All right. So how are we going to solve this problem?
Well, we're going to do it with a trial wave function, a variational trial, wave function, wave function.
And I believe you've studied variational methods before.
So the wave function we're going to use is of the form phi one times one plus PHI two times two.
And we're going to try to find the best coefficients, Phi one and PHI two, which will lower the energy as much as possible.
This is a variational principle. You try to get the best approximation to the ground state as you can by varying the coefficients arbitrarily.
Okay. Now, this actually gives the method its name.
The name of the method, linear combination of atomic orbitals. That's exactly what we're doing.
These are the atomic orbitals, and we're making a linear combination of them.
Okay. It's also known as tight binding because the electrons sitting on orbital one, we think of it as our nucleus one.
We think it is tightly bound to nucleus one and this one is tightly bound to Nucleus two.
And then we make some linear combination of those two.
One of the reasons people really like this method of analysing problems is because you can you can actually make the approximation
more and more accurate by adding more and more things to the right hand side with more and more variational parameters.
So instead of just keeping track of two possibilities,
the electron is on over the one the ground state of nucleus one and ground state of Nucleus two.
You can start also adding coefficients and other possible states.
The electron can be in excited states on an orbital one, a nucleus one excited states on nucleus two.
And you make a bigger and bigger basis set.
And as you make the the more possibilities on the right hand side, more and more you get closer and closer to the exact wave function on on the left.
Okay. So now we have to actually write down our effective Schrödinger equation for this trial wave function,
and that's something you're actually going to do for homework.
I'll write out. The answer is, it looks very much like a shorthand equation.
It's something you probably believe without too much justification.
It looks entirely like a shorthand equation. The only difference is that this h h i j is a two by two matrix.
I. H j.
I forgot to tell you something that we're assuming we put it over here. So this is a bad assumption.
Bad assumption? Things out of order here.
The bad assumption is that i j is delta.
I j in other words, one two equals zero.
That one and two are orthogonal wave functions.
Now that's a good approximation if the way if the nuclei are far apart.
But if you put the nuclei really close together, it's no longer a good assumption.
The reason we're making this bad assumption is because it makes our life a lot easier.
And as in many cases, making this bad assumption isn't going to throw out the crucial physics.
So we're going to keep this bad assumption for now. There's an exercise in the book if you want to go through it and do it more properly,
but you'll get the crucial physics right by making this bad assumption.
So we're going to go ahead and do it. I believe when you did it last term, you did not use this bad assumption.
The algebra gets much worse. So it's going to try to keep the algebra as simple as possible by making that bad assumption.
Given that bad assumption, this is the Schrödinger equation that you have to solve and it looks pretty much like a Schrödinger equation.
You know, there's a Hamiltonian wave function, energy wave function.
The only difference is it's not the full Hamiltonian, it's just the way the Hamiltonian projected to our restricted space,
which includes only the wave function one and the wave function two. So it's a projected Schrödinger equation, if you like.
So to solve this, all we need to do is we need to come up with this a two by two matrix two by two.
In our case, we need to write out the matrix elements here. So let's do that one.
H one is one.
K plus v one. Do I still have the Hamiltonian?
Hamiltonian still up there? I hope it doesn't go off the board. One plus 1v21.
This term is easy because one the cette one is an eigen state of k plus v one.
That's how we defined it. And its eigenvalue is epsilon not.
So in fact one on one is one we can find it to be.
So this piece here just gives us epsilon, not this part here.
We don't know what that is. I'm going to call it V Cross and physically, what is it?
V Cross? We have an electron sitting on orbital one over here.
And so here's the electron sitting and here's Orbital two and V cross is the interaction between the electron sitting on state one,
sitting on Nucleus one interacting with Nucleus two. That's when V crosses.
I believe last term you may have called it direct interaction.
I may have called it J. Does that sound familiar from last year?
Yeah. Okay. The reason I'm not calling it direct and J is because in condensed matter, those words are used for something somewhat different.
And I'm trying to avoid confusion about that. So I'm going to call it V Cross instead.
I mean, there's some relation between the two nomenclature, but not enough is worth keeping it.
So throw that out. We're going to call it V Cross. Now on to H2.
We need that also very similar to K plus V two, two plus two, v one, two.
And again this thing is epsilon nought plus v cross.
The more interesting piece is 1h2 which is one K plus v22 plus 1v12.
Okay. So this thing is easy actually because k plus v two is epsilon not two and then two and one have been assumed orthogonal.
So that gives zero. Okay. So that's simplifies our life a lot.
This one here, I'm going to call hopping. Hopping, and generally it's called minus T.
I believe last year you call the exchange. And I'd probably called it tapas or something.
So again, I'm not going to use that nomenclature because we use an exchange for something else.
Now, why is it called hopping? If you think in terms of the time dependent Schrödinger equation.
It's this kind of term that's sort of off diagonal, takes you from 2 to 1.
So a hopping term can give dynamics to an electron can.
Let's take an electron that starts on on nucleus two and lands it on Nucleus one, so allows it to hop from one nucleus to the other.
Hence, name hopping, reasonable name. Okay, so given this, we're almost there.
We have our two by two Hamiltonian matrix epsilon nought plus V cross minus T minus T.
If it's if it's complex, it would get a t star here, then epsilon last.
But the cross is a nice little formation matrix.
We can diagonals it we get the eigenvalues I guess equal epsilon not plus v cross plus or minus absolute value of T.
Now. I left off one little piece, which I'm now going to put back in.
The thing I left off was the e nuclear nuclear that the energy, the two nuclei are interacting with each other as well.
So if I change the distance between the two nuclei, they would have some Coulomb interaction.
But it turns out that in nuclear, a nuclear is very approximately equal to minus V cross.
And that actually should be obvious from what you know about electrostatic.
If you have a spherical charge, a cloud of charge around another spherical cloud in charge of cloud of charge,
you can considering the whole charge as being at the origin. And the whole charge is that the origin is zero.
The E cancels the plus one, the minus one cancels plus one.
So this ends up these two pieces cancel each other.
So I'm just going to write E plus or minus. The two possible energies are in OT plus or minus absolute T.
So this is something that seems fairly natural that the amount of gaining or lowering of the energy we get when we go,
when we allow the electron to hop between the two orbitals is given by how much we allow them to hop.
So by spreading out the electron wave function, you lower its energy.
The more you spin out the wave function, the more you lower the energy. So we can even I mean, we do it here.
Yeah. We can even write down the two eigen states of this.
If you diagonals that that matrix and find its eigenvectors, you would find that the sy plus one over root two one plus one minus two.
This is the anti bonding, the higher energy anti bond, the higher energy wave function,
and then the lower energy wave function, one over two, one plus two.
And in fact, those forms of the wave function should look a lot like what we had and the particle in the box.
So the particle in the box, we had a bonding orbital bond here and we have an anti bonding orbital here and here with a node in the middle.
And now we have two nuclei. The bonding orbital looks like this.
It's symmetric bond. Bond, and the anti bond looks like this.
Plus plus anti bond.
So it's very similar to what we have in the particle box. In both cases, the existence of the node raises the energy.
Okay, so since the amount of energy you gain or lose depends on the amount of hopping.
As a function of r one minus R2 the distance between the nuclei, you might expect that the hopping goes up as you put the two nuclei closer together.
So you would expect that the anti bonding looks like this, the bonding looks like this, and that's more or less right.
But actually more accurate is something that looks like this.
er1 minus r two anti bonding goes like this.
And bonding goes like this. Again, when the two nuclei get too close together, that all of a sudden becomes extremely repulsive.
And the reason it becomes repulsive is because a lot of our approximations break down.
This approximation breaks down into the two nuclear.
The two atomic orbitals we're using are no longer orthogonal and further that statement I made up there
about the two charge clouds cancelling breaks down once the positive charge gets inside the other electrons.
You know, once the other nucleus gets inside the charged cloud of the electron, then you can no longer count.
And you remember this rule that if you're inside some charge, you don't count it.
And if you're outside the charge, if you count it when you're talking about a spherical charge, distribution,
anything that's inside you closer to the centre, you count and you put all the at the centre and anything is outside you that you throw away.
So if the nucleus gets inside the charged cloud of the electron,
then it starts to see the other nucleus and it starts to be repelled by the other nucleus.
So the energy of the bond goes goes way up. One more thing here before we before we end, you can also consider a very similar physics.
If you have two different atoms, so you have hydrogen and fluorine.
And here I've drawn one up here and one down here because they're energies.
If you just have an electron hydrogen, it's much higher energy than the electron on fluorine.
Or maybe I should add sodium and fluorine, sodium and fluorine.
It doesn't matter. Just, you know, on something that has a low ionisation energy or wants the electron a lot less and something
with a high electron affinity or a high ionisation energy wants the electron a lot more.
And you can allow an electron to do localised between the sodium and the fluorine.
If you do that, the only thing that changes here is that these two energies on the diagonal become e sodium and e fluorine like that.
And those two can be different from each other. If you diagnose this matrix now, you'll find two different energies again.
So we'll draw that as a bonding and an anti bonding orbital.
So as antibodies are bonding down here, again, allowing the electron to go back and forth changes the total energy.
And here's anti bonding up here. But these coefficients will not be the same that for the antibody orbital.
This this coefficient being a minus alpha will be bigger.
Better. It will be bigger than alpha. There'll be a higher probability that the electron will be over here for the antibody orbital,
whereas for the bonding orbital beta alpha like this, the the most of the weight of the wave function will be over here on fluorine.
It's sort of a simple two by two matrix diagonal ization problem. There's an exercise in the book on this as well.
So the idea is that but just by looking at this two by two matrix,
you can see how you go from a covalent bond where the electron is distributed equally between the two,
the two nuclei to an ionic bond where really the lower energy orbital puts the electron almost entirely over here on on the lower energy species.
Okay, I guess I stop there. I'll see you tomorrow.