NICOLA MARZARI: Wonderful.
OK.
So welcome to class 5.
We are actually counting the lab as class 4 in the numbering.
What we'll see today is really a wrap-up
on some of these ideas on the behavior of electrons
as quantum particles.
And we'll introduce really the first electronic structure
method in the modern sense--
that is Hartree-Fock theory--
that you'll see as a very simple, conceptual framework.
And it really derives directly from the Schrodinger equation
and from the reformulation of the Schrodinger equation
as a variational principle.
Next class, we'll start with density functional theory.
That is a completely different approach
to the problem than Hartree-Fock.
Although, in practice, it results
in equations for the wave function of the electron
that are formerly fairly simple.
Historically, Hartree-Fock has been the method
that has developed first, especially from the quantum
chemistry community.
And because it derives from the variational principle,
it can actually be augmented with perturbation theory.
So whenever you need very accurate calculations,
things like molar [INAUDIBLE] approaches
and other perturbation theory approaches
on top of Hartree-Fock can actually help you out.
But they tend to be very expensive.
OK.
Let me, in one slide, remind you of the most fundamental concept
from last class that we have seen.
That is the de Broglie relation.
This is something that tells us that if a particle has
a momentum mass times velocity p, well,
then there is a fundamental constant in nature that is
called the Planck constant-- that's written here--
such that this relation is satisfied
where lambda is the wavelength of that particle.
So basically, particles have wavelength properties.
But the real issue is that only electrons are light enough
to have wavelength comparable to the inter-atomic distance,
so to give rise to diffraction phenomena.
And so this is why we actually need
to treat the electrons with the Schrodinger equation
due to their wavelike properties.
And in general, we don't really need to nuclei
as a wave-like particle.
So this was one thing.
The other important thing that I'll repeat over and over again
is that when we move from a particle-like description,
where the only important quantity are
the position and the momentum, to a wave-like quantity, well,
we have a completely different computational content.
That is, a wave function is actually
defined by assigning the amplitude of the wave
at every point in space and at every time.
So in practice, it's a continuum field that in every, say,
computational calculation needs actually
to be discretized and approximated in different ways.
And then I introduced what is the equivalent
of the Newton's second law of dynamics for actually quantum
objects.
So classical particles evolve according to the law force
equals mass times acceleration, where
the acceleration is really the secondary derivative
of the trajectory.
Quantum particles obey a different fundamental law
that's basically been discovered by Erwin Schrodinger in 1925.
And the other major player where Werner Heisenberg, Paul Dirac,
and many others.
And I've written it here in the most general form
for the case of a single electron, so a particle
that is described by a wave function
psi, a function of space and time.
Very good.
The machine [INAUDIBLE] beep.
OK.
Professor Ceder will take care of this.
And let me continue.
So we have the descriptor that is the wave function.
And we have a potential.
A potential is the general concept.
It could be, say, the electromagnetic potential
that is coming from the nucleus of an atom.
So suppose that you are studying a hydrogen atom.
Well, the nucleus has an attractive potential
for the electron that is just 1 over the distance
between the electron and the nucleus.
So V here is really the same potential
that you have learned in classical dynamics
and electromagnetism.
But now, instead of acting on a particle,
it acts on the wave function.
And the way it does, it just multiplies the wave function.
So what the Schrodinger equation is telling
me is that whenever I have a potential V,
the solution for the dynamics of the electron
will be given by the solution of this second-order differential
equation, where here this is a Laplacian.
We have a second derivative with respect to space.
And so the wave function needs to satisfy this equation.
In most cases, to actually deal with problems
in which the potential does not depend on time--
suppose that you are studying a molecule.
What is your potential?
Think of, say, the benzene molecule.
Well, you have six carbon atoms in a ring and six hydrogen
atoms also in a ring.
And in each nucleus, there is a Z/r
that is an attractive potential for the electron.
And that we can either think of at a potential
that doesn't change in time--
or even if it changes, because the molecule is
a certain temperature and so atoms move around, what
it turns out will happen in practice
is that electrons are so much lighter than nuclei
that for all practical purposes they
move so much faster that they see the nuclei basically
as fixed in time.
So if you think of a particle like a nucleus that
has a huge mass and a particle like an electron that
has a very light mass, well, even if the nuclei move,
the electrons always are so much faster
to rearrange themselves to be in the minimum energy
configuration.
We'll actually see more of what is
called adiabatic approximation.
So the important thing here is that,
for all practical purposes, we consider in most problems
the potential as not really dependent on time
either because the atoms are not moving
or because even if they move, they move so much more
slowly than the electrons.
Then the electrons can always [INAUDIBLE]..
The form that the wave function can have.
And I'll briefly sketch out this derivation here.
So whenever the potential doesn't depend on time,
this is how we proceed.
We actually make an hypothesis-- often,
this is called an ansatz--
in which our wave function can actually
be separated into the product of two terms--
a term that depends only on space,
we'll call it phi; and a term that depends only on time.
So we are making this hypothesis.
We have decided that the potential does not
depend on time.
And we see what [AUDIO OUT].
What we have now is that whenever
we apply, say, the Laplacian secondary derivative
with respect to space, well, [INAUDIBLE]
the f component of the function, the time-dependent component.
So the Laplacian of psi can be written actually
as f times the Laplacian of phi.
And then I'm not writing the explicit dependence
on the variables.
But we have the potential times the wave function, phi f.
And then we have i h bar.
And now, the derivative with respect to time of phi--
well, a phase-dependent part is not affected.
So we write it as pih times df/dt.
OK.
So this is our first step.
And now, we take this expression.
And we just divide it by the product of phi times f.
And so when I do that, what I obtain
is minus h bar squared divided by 2m Laplacian of phi divided
by phi plus V equal to i h bar 1/f df/dt.
And this really is the end of the derivation
because something very important happens at this point.
Look for a moment at the right-hand side
of this equation.
You see it's just a constant-- the Planck constant, i,
is the imaginary unit--
times 1/f df/dt.
So everything that's written on the right-hand side here
depends only on time.
The left-hand side, on the other hand,
we have a term that depends only on position.
Phi is a function of position.
And V-- remember, our initial hypothesis--
does not depend on time.
The potential does not depend on time.
So the left-hand side [AUDIO OUT]..
Now, the only way such a relation can be satisfied--
that is a term dependent solely on position can be equal
to a term that depends only on time--
is if each term separately.
That's the only way the overall function of r
can be [AUDIO OUT] overall function of t,
if they are constant.
Basically, because r and t are independent variables,
there is no relation.
If you want, in this ansatz [AUDIO OUT] down
the Schrodinger equation in two parts--
the left part being equal to a constant,
and the right part being equal to a constant.
So there is a simplification.
We can deal with separate equations
for the time-dependent part and for the space-dependent part.
What is the left-hand term equal to a constant becomes
what is called the stationary Schrodinger equation.
That is the equation that we'll try
to solve over and over again in our calculation.
And it's the term that I've written here.
So the stationery Schrodinger equation is written here.
And this is the differential equation
that, in principle, is complex to solve
because the potential can have an arbitrary complex shape.
But again, think of the potential
as being the attractive potential
for the electrons generated by all the nuclei in a molecule
or in a solid.
And the time [AUDIO OUT] has become trivial to integrate.
And we see this is the time part of the Schrodinger equation.
We see, actually, in the next slide
why this is simple because this is just
a first-order differential equation.
And we are looking, in general, for a function
whose first derivative with respect to time
is equal to the function itself times a constant.
[AUDIO OUT] equation to integrate
because the exponential function satisfies this condition.
When you take the exponential, you just
get the exponential itself times a constant.
And so really, there is no analysis [AUDIO OUT]
The other part, the space-dependent part
is the one that is complex to solve.
And we'll see here just a few examples [AUDIO OUT]
If we want the simplest example possible,
that is the case of a free particle.
What happens to an electron that doesn't feel any potential?
And again, its wave function is going
to be the product of a space-dependent part phi
times a time-dependent part f.
The f term [AUDIO OUT].
The space-dependent term here is the solution
of the stationary Schrodinger equation in the hypothesis
that the potential V is equal to 0.
So I've removed from the stationary [AUDIO OUT]..
Again, this is a fairly simple differential equation
to integrate because what we are looking for is basically
a function whose secondary derivative is
equal to the function itself times a constant.
And then again, the solution is just given by an exponential,
OK?
Now, I'm looking in particular at the problem in which I'm
trying to find a solution phi for a value of this constant E
that is positive.
And so when you actually work out the algebra,
you see that what you need as a term in front
of the x in the exponential is the square root
of a negative number, is the square root
of twice the mass times E divided by h bar squared.
And so that's why we get this imaginary unit here.
It's basically because we have rewritten this equation
as the square Laplacian of phi equal to minus 2mE divided
by h bar squared f.
And you see straightforwardly that if you
take the secondary whatever of the functions that
are written here, well, we'll get,
by taking the first derivative, a term
i square root of 20 divided by h bar.
And when we take the second derivative,
we get that term again.
And the square of that term is a square of i
that gives you my minus sign.
OK.
So this is the solution.
We have actually found the wave function for our free particle.
And the wave function is the product
of [AUDIO OUT] times the time-dependent part.
And in the next slide, I am actually
plotting this wave function in space.
Sorry, let me jump to this slide.
This is it.
So I am plotting it.
For simplicity, I've called it a different coefficient.
E over h bar, I've called it omega.
And square root of 2mE divided by h bar, I've called it t.
So this is the wave function for a free electron.
And this is what we call a plane wave.
And the reason why is that is simple
because if you think of what this term is,
the exponential of a complex number
is really a combination of sines and cosines.
And now, suppose that you look at the amplitude of this wave
function at a certain point in space--
that is at a certain x equal to x0.
What you see is that that amplitude
will have a kx factor that doesn't change because we
are sitting at a certain x0.
But what is changing with respect to time
is the second term, omega t.
And so you see an oscillating amplitude with a period
omega at any point in time.
Or in a different way, if for a moment
you think that what is this field of the wave function
at a frozen instant in time--
that is for a certain time t equal to t0--
well, it's nothing else than a complex exponential.
So it's a complex-valued function
that has a cosine or sine projection, depending
if we look at the real or complex axes.
So what you need to think of when
you think that at a free electron
is nothing else than this infinite oscillatory
field propagating in space.
But it has really no beginning and no end.
And one of the important things is that the wave function
really contains all the information that we might
want to know about an electron.
And we will see in the next slides
how we actually obtain the physical properties
about an electron from the function.
But the simplest property in [AUDIO OUT]
of finding an electron in a certain position
and a certain time.
And so [INAUDIBLE] that wave function amplitude
at a certain x and a certain t gives me
actually the physical probability
of finding an electron.
That means that if I have the wave function of an electron,
I can take at every point in space
the square models of its amplitude.
And so what I have now is a probability distribution.
And what quantum mechanics tells me
is that if I'm going to make a measurement
and try to see where the electron is,
well, I will find it in different points
in space with a probability [AUDIO OUT]
into these square models.
And you see one of the first conclusions from our analysis
of the Schrodinger equation in the case of a potential that
does not depend on time is a Schrodinger equation in which
the wave function can actually be written
as the product of a space part times a time part
is that the probability of finding the electron somewhere
does not depend on time basically
because [INAUDIBLE] square models
of psi that can also be written as psi
times its complex conjugate.
What you have is that the term exponential of minus i/h
gets multiplied by its complex conjugate.
That is the exponential of plus i/h.
And those terms cancel out.
So in the probability distribution for the electron,
we have only the square models of the space part.
That is to say that the probability of finding
an electron does not depend on time.
It's just a function of space if the potential does not
depend on time.
You see, [INAUDIBLE] our plane wave.
The potential does not depend on time.
If we take the square models of this function,
well, what we have is really a constant
because we are just taking an imaginary exponential
times its complex conjugate.
And so that product is a constant.
So what we discover really is that the electron
is free as a probability distribution that is constant.
That is, you can find a free electron anywhere in space.
Yes?
AUDIENCE: [INAUDIBLE].
NICOLA MARZARI: So this is actually the solution
in three dimensions.
So what the difference is, if you work out the algebra--
and we can look at it after class--
is that what you have is--
in the space term, you have really a scholar product
in three dimensions.
That is, what you have is that now
a point in your three-dimensional space
is described by a three-dimensional vector r.
And now, you can have solutions that are classified--
if you want, this k is actually called the quantum number.
It's what classifies all this possible continuum
of solutions.
And if you think, k is telling you
what the wavelength of this plane wave is
and what is its orientation.
So really, what I am plotting here
is specific plane waves that, if you want,
is a k vector that is ordered in this direction.
So if we were in one dimension, you have, if you want,
the equivalent of a sine or a cosine.
And your only degree of freedom is the wavelength.
But if you are in three dimensions,
your degrees of freedom are not only the wavelength--
that is the distance between these crests--
but also this direction.
And any arbitrary k vector is actually valid.
So you can have plane waves with all possible wavelengths
and all possible orientations.
And actually, this gives me a good chance
to mention another sort of physical observable
that comes from the wave function.
The first one that we have seen is the probability
distribution.
And that's obtained by taking the square models.
Another one is what is called the quantum kinetic energy
of that particle.
And the quantum kinetic energy is really just
given by minus h bar squared divided by 2m.
So the quantum kinetic energy is given-- well,
I'll describe it in more detail in a moment.
But it is obtained by really taking,
in a slightly more complex way than what I am saying now,
the second derivative of the wave function.
So the second derivative here with respect
to the space of this wave function
would give me a k square term.
That basically means that a plane wave with all wavelengths
corresponds to electrons with all possible energies.
But there is something other that
is very subtle that is reminiscent of what
in quantum mechanics people call the indetermination principle.
So this free electron, as you see,
is an electron that is distributed equally in space.
So we can find it with equal probability
here, here, or everywhere else.
And it is actually a perfectly well-defined second derivative
with respect to space.
It has a perfectly well-defined kinetic energy,
has a perfectly well-defined momentum.
And that's really one example of the indetermination principle.
That is, there is a fundamental rule
that quantum particles satisfy.
That is, you can't actually measure
their momentum and their position
simultaneously with arbitrary accuracy.
And so a free electron, if you want,
is an electron that has perfectly well-defined
momentum, perfectly well-defined kinetic energy.
And for this reason, it has perfectly undefined position.
That is, the electron can be found anywhere.
And so this is, if you want, an exotic result
coming from the Heisenberg indetermination principle.
I wanted to give you another physical example,
getting closer to reality.
Suppose that, for a moment, we are studying a metal surface.
This wouldn't be very different from what you have studied
in your first laboratory.
What you are studying is basically a slab.
And you have been finding quantities
like what is the energy, what is the lattice parameter, what
is the surface energy.
Suppose that now you are using quantum mechanics.
Well, what do the electrons in the metal look like?
And we'll see in next classes that an electron inside a metal
really feels and behaves very much like a free particle,
especially in very simple metals, in metals
like sodium, potassium, even aluminum to a certain point.
There is a very delicate cancellation
of attractive and repulsive terms
for the electron in the metal.
So the net result, to a very good approximation,
is that the high energy electrons,
the valence electrons, really feel almost free.
I'm saying almost free because they are really
confined inside the box that is your metal
crystal inside the box that is given by your metal slab.
And so to have a simple but very good approximation,
we can think of the potential felt by an electron in a metal
as a type of potential.
I'm thinking of a case in which, on this side, on the left,
I have really my semi-infinite slab.
And so the potential that is drawn by this thick line
is actually 0 for x smaller than 0.
And then there is a confinement.
So there is a step.
This is a little bit like a well.
There is a potential difference v0
that contains the electron inside the slab.
So you really need to pay an energy v0
to get out of the slab.
But this is in physical terms what people would
think of as the work function.
The work function is really the energy
that you need to pay to pick up the highest energy
electron in the metal and bring it out, bring it very far
away from your slab.
And so to a good approximation, the potential
that these electrons close, as we say,
to the Fermi energy feel is this.
And what we are trying to find now
is a solution for the Schrodinger equation
for a case of an electron having an energy.
Actually, the E, that constant in the Schrodinger equation,
is actually the total energy of the particle.
So we are looking for a solution for an electron having
an energy that is basically between 0 and v0.
That means that that electron classically
is really confined to the left side of the potential.
And so what do we have here?
Well, what we have is that, on the left, we have again,
a very simple Schrodinger equation
that is minus h bar squared over 2m d
squared over phi equal to E times phi.
So this is, on the left side, the Schrodinger equation
for the free particle.
And we know the solution.
And it's just being given by one of these plane waves.
But now, the difference is that, on the right,
we have all of a sudden a potential.
So the equation on the right is going
to be minus h bar squared over 2m times delta squared phi.
Remember now, in the stationary Schrodinger equation,
we have a term that is given by the potential that,
in this case, is just a constant times the wave function.
And this is going to be equal to d phi.
What's the difference between these two equations?
Well, let me actually rewrite the one on the right side
as minus h bar squared over 2m delta
squared phi equal to E minus v0 phi.
Well, in both cases, we are trying
to find a wave function whose second derivative
is basically equal to the wave function itself
times a constant.
But the big difference is that we are actually
looking for physical solutions for an electron that
has an energy lying somewhere in between 0--
that is our conventional minimum--
and v0.
That is the energy cost to get that electron out.
So what we have is that, for the left side equation, what
we are trying to find is a second derivative
in which, you see, the constant is
going to be minus 2mE over h bar squared.
So it's a negative number.
So we need to take a second derivative
and get a function times a negative number.
In the other case, remember, what we are looking
is for a solution in which the energy is between 0 and V0.
So this the sign of the coefficient here has changed.
Here, before, it was positive.
Now, it's negative.
And with the negative sign on the other side,
it means that what we are looking
is for a function whose secondary what
is equal to a positive constant times the function itself.
And all this long story is just to say
that the solution on the left will
be given by an exponential that has an i term in there.
So it's a complex exponential.
Well, the solution on the right-hand side
will be given by an exponential that
has a real term coefficient.
And so what we have is on the left side
a periodic oscillatory solution typical of the free electron.
And what we have on the right side
is actually a decaying exponential.
And so what we have is that our free electron is
going to have a wave function that really looks like this--
free particle like inside the oscillatory and exponentially
decaying outside the surface.
And this is how truly electrons in a metal--
and in most metals, it looks like the wave function
decays exponentially.
That's actually why people, say in 1981,
could invent something as fancy as the scanning tunneling
microscope.
You see, the STM is one of the fanciest instruments
that we have that can actually be used to see atoms.
And what does an STM do?
Well, an STM comes with really a nanometer-sized tip.
And what the STM measures is the current between, say,
the metal and the tip.
And it's very sensitive because, you
see, since the wave function decays exponentially,
moving the tip by a small amount will change exponentially
the current going through the scanning tunneling microscope.
So we can actually move this paper
and have an enormous response in the current
to a very small displacement.
So we can actually measure the topography
on the top of a metal surface with great accuracy
because basically what the scanning tunneling
microscopy is doing is really filling up
these exponentially decaying tails coming out
from the metal, coming out from the metal surface.
OK.
So I'll show you just a few examples so
that we get more and more the feeling of how
wave functions look, getting closer and closer
to physical problems.
Keep in mind this general idea--
if we have got zero potential, we
have this wave-like solution.
So a typical problem that is very simple to actually solve
analytically would be the one in which we
have an infinite square well.
That is, we have a particle that feels
zero potential between, say, minus a and a
and infinite potential outside.
And this qualitatively starts to be very similar to what
an electron in an atom feels.
And electron in an atom doesn't really
feel a potential that is 0 in a region and infinite outside.
It really feels an attractive potential that is 1/r.
So the potential is actually shaped very differently.
It's going to be a Coulombic cusp ending up in 0.
But you see, the solutions are very similar.
And we can understand them better
if we actually look at the solution for this case.
And again, what we have is a problem in which,
between minus a and a, we have 0 potential.
So the solutions are going to be free electron-like, wave-like.
But now, we have a new condition.
That is, what we have is that being the potential
is infinite, say from this point onwards on the right
and from this point downwards on the left,
what we need to have is that the wave
function needs to have 0 amplitude outside this region.
If the potential is infinite, you
can think physically that there is 0 probability of finding
the electron.
It's truly infinite.
So it's fully constraining the electron.
So the wave function needs to be 0
from here onwards and from here onwards.
And so another great discovery of quantum mechanics
is that the only sine or cosine-like solutions
of the Schrodinger equation that are
such that they have 0 amplitude at this point and at this point
there are sines or cosines with a wavelength that is
compatible with this distance.
So all of a sudden, our solutions that,
in the case of the free electron,
could have any wavelength in the world
have now become quantized.
There is only a discrete set of solutions with wavelengths
that are compatible with the distance between the two
infinite, confining sides.
And this is the same concept of an organ pipe and the music
that it plays.
When you play music in an organ pipe, you are really exciting
either a sound wave with a wavelength corresponding
to the fundamental harmonica.
Or the only other notes that your organ pipe can produce
are actually higher harmonics.
And so they are, again, sound waves
that will have 0 amplitudes at the borders.
Or same thing, say, with the chord of a guitar.
And so the wave functions for a confined potential
are actually quantized.
And they have corresponding energies--
that is, in this specific case, go actually as the square.
Although, this is less relevant for our case.
So you will see that, in most cases,
like when we look at the solution
for an atom in a confining Coulombic potential,
we'll actually find only a discrete set of wave functions
with discrete energy levels.
And again, we can get some hints on how
this solution would look like.
If you think for a moment at what
would happen if our square well, instead of having
infinite confining walls, would have only finite confining
walls with a finite height, it's really
the same case as the metal surface
because now you have a finite wall.
You have a finite wave function decaying exponentially
outside in that region.
And so what was, you remember, our fundamental harmonic--
the second harmonic, third harmonic, fourth harmonic,
what were those oscillatory solutions inside the box
now start getting these exponential tails decaying
in the vacuum.
And this is really how solutions for the Schrodinger
equation of an atom look like.
And I don't want really to bore you with the algebra.
Some of you might actually have seen this over and over again,
if you have taken a quantum class.
I'll just give you one slide with the conceptual framework
of what we should do if we were to solve the Schrodinger
equation for a single electron in what we
call a central potential.
This is a general concept.
Central potential is a potential that
depends only on the distance from a center,
but not on the angle of distribution.
And so central potential would be
written as V of r, where r is the modulus of our vector,
is just the distance.
And the algebra is fairly complex.
But the general idea is that this is now
our Schrodinger equation.
We need to find a wave function such
that the application of the second derivative plus
this potential is equal to the wave
function itself times a constant.
And because somehow we are dealing
with the geometry of a central potential,
it's convenient to write the Laplacian,
the second derivative, not in Cartesian coordinate,
but in spherical coordinates-- that is, in coordinates which,
instead of giving the x, y, and z components of a vector,
we give the distance from the origin and the two
angles, theta and phi, that that vector makes
with respect to different axes.
Again, the solution is fairly complicated.
So I won't really go into this.
But this is how our Hamiltonian--
this is how the left part of the stationary equation
would look for this central potential,
having just rewritten basically this second derivative
in Cartesian coordinates.
And when one goes through all the algebra,
one finds out that the function--
the space part of the wave function
can be written as a term that depends only
on the distance from the center, from the place where
the nucleus is, times an angular part.
And if you are familiar with this,
we call this angular part spherical harmonics.
And this would be the radial part of the wave function.
That is a solution of a slightly different, one-dimensional
Schrodinger equation.
Again, this is just the start, if you
are interested to take this.
But somehow, this is all the algebra that gives rise
to the periodic table.
That is, when we study actually what
are the solutions of the Schrodinger
equation for a central potential,
well, we find that we can classify the solution as we
were classifying the solution in the particle in a box
depending on, if you want, the fundamental harmonic
and the higher harmonics.
Well, we can still classify the solution depending
on the angular distribution--
so what are actually the L and M indices
of the spherical harmonics.
And we can also classify the radial distribution.
And so all these possible solutions
give us all the possible states of an electron in a hydrogen
atom.
And you probably have seen this over and over again.
But it's really the fundamental alphabet that gives rise
to the periodic table.
So for the hydrogen atom, the fundamental solution
is one in which the electron is distributed
with spherical symmetry around the nucleus.
And if we look at the probability density
as a function of the distance from the center,
it's something that really looks a little bit like this.
So the probability of finding the electron in the hydrogen
atom at a certain distance really
has a maximum at a distance that is, I believe, at 1.5 bohrs.
I need to check actually this number.
The classical radius of the hydrogen atom is 1 bohr.
It's 0.529 Angstrom.
And the maximum, I think, is at 1.5.
But basically, there is a maximum of probability.
And then it decays again.
But then there is a whole zoology
of excited states that can have, say, a more
complex radial part.
And we classify these-- you probably remember
this from your chemistry.
We call this state a 1s state where
1 really refers to the radial part
and s refers to the angular part.
And the next state would be a 2s-like state,
in which, being s, the angular part is still the same.
It's spherically symmetric.
But now, the radial part has changed.
And it actually changed sign.
So it goes inside.
Then it goes back to 0.
Then it goes again.
And it goes back to 0.
And then there are states like the 2p states
in which the angular part starts to play a role.
And so this would be a specific p state that, if you want,
has two lobes with opposite signs
across a plane of symmetry.
And then there are more complex states like the d states
and so on and so forth.
And a very important thing is that really this angular
symmetry comes from just having considered a central potential.
That is a potential that doesn't depend
from the angular variable, but just from the distance.
And so these solutions have the same qualitative difference
not only for a single electron in a hydrogen atom,
but, say, in general for an electron in an atom where
really that electron will feel not
only the attractive potential of the nucleus,
but also the repulsive potential from all the other electrons.
But to a great extent, that repulsive potential
of all the other electrons is going
to be a fairly spherical symmetry kind of potential
because all the other electrons, when you take their charge
density, are going to be roughly spherically
symmetrically distributed.
And so the solution really, for one electron
in a central potential, does not only apply
to the hydrogen atom, but applies
to really any atom in the periodic table
to some approximation.
That is to the approximation that we consider.
The repulsive potential that all the other electrons
have for this, say, top valance electron in a generic atom, say
iron, is that the only approximation
is that these repulsive potential is also
spherically symmetric.
That is not true but is true to a very large extent.
OK.
Now, this concludes, if you want, our general panorama
on what is quantum mechanics.
And now, really, we want to go into electronic structure
methods.
That is, we want to arrive at what is called the Hartree-Fock
solution.
That is, again, one of the fundamental techniques,
especially in quantum chemistry.
And to do this, I need to give you
a little bit of nomenclature.
Again, if you have never seen a set of quantum mechanics
or if you have never taken a quantum mechanics class,
it would be very useful to have a look at some of the books
that I've given you in the bibliography, either
some fundamental books like Quantum Mechanics
or Physics of Atoms and Molecules,
both by Bransden and Joachain, that are very good quantum
mechanics books, or some of the electronic structure books,
like the Phoenix books on inter-atomic forces
or the Richard Martin book, or the Kaxiras book.
But the nomenclature that we'll use over and over again
is what is called the bracket nomenclature of the Dirac.
And so in general, a wave function
is actually represented by this sign here.
This is called [INAUDIBLE].
Again, that means nothing else than our function.
That is an amplitude defined everywhere
in space or everywhere in space and time.
And in general, it's a complex amplitude.
And now, one of the fundamentals of the rules of quantum
mechanics is that, say, a wave function
corresponding to different states of an electron--
say you are considering the function for the ground state
of the hydrogen atom, the 1s.
And then you are considering the function for an excited state,
like the 2s.
So the wave function for different states
are going to be orthonormal.
What does it mean?
It means that when we take the scalar product,
that scalar product is going to be either 0 or 1,
depending if we are taking the scalar product with the wave
function itself or with the wave function of a different state.
So this delta ij is what is called usually a chronicle
delta.
It's a symbol that means this is going to be equal to 1
if i is equal to j or is going to be equal to 0
if i is different from j.
And the scalar product between two wave functions
is written in shorthand notation as this.
It's written as a bra times a ket, a bracket.
But again, that's nothing else than a shorthand
form for the proper definition of the scalar product.
Again, you are probably not heard of Hilbert spaces.
But this is a vector space of wave function
that has a metric defined.
And so we can define something like a scalar product
by taking the integral of the wave function
on the left that has been complex
conjugated times the wave function on the right.
So all wave functions that are solutions of a Schrodinger
equation satisfy this orthonormality normality
property.
If you take this integral, that is going to be equal to 1
if you are just taking the square of the wave function.
And it's going to be equal to 0 if i is different to j.
And you see why it's equal to 1 actually
if i is equal to j because remember
what is psi i star times psi i?
Well, it's just a square modulus of psi i.
So psi i star times psi i in a certain point r in space
gives me the probability of finding the electron
in point in space.
And when I integrate that probability all over space,
it becomes obvious that that number must be 1
because what is the probability of finding an electron anywhere
in space?
Well, you'll find it somewhere.
So the probability of finding it anywhere--
that is the integral of psi squared as a function of r--
must be equal to 1.
So that's the physical connection.
And the last definition that I need
is that of expectation values.
That is, once we have found a wave
function for this electron--
as I said, we know everything that we need
to know about that electron.
And so we can extract all the physical properties
that we want.
It's like, in classical mechanics,
if you have the trajectory of your electrons,
you can calculate at any instant in time what is, say, 1/2 times
the mass times the square velocity.
That would be the kinetic energy of that particle.
Similarly, if you know the wave function of that electron,
you can calculate any kind of physical quantity.
And one of the most important physical quantities,
apart from the probability of finding it somewhere,
is what is the energy of that electron.
And this is the shorthand term in the bracket nomenclature
of what would be the energy of an electron.
We call this an expectation value.
So in quantum mechanics, for every physical observable,
like the energy, there is going to be an operator that
acts on the wave function and is such that its expectation
value gives me the physical quantity I'm interested in.
So if I want to know the energy of an electron that has a wave
function psi i, I need to calculate this expectation
value, psi h psi i--
that will give me the energy.
And the quantum mechanical operator
that gives me the energy is what is called the Hamiltonian
operator, h, that I've written here.
So the Hamiltonian operator is nothing else.
And I forgot it here.
And sorry, there is a Laplacina.
The Hamiltonian operator is nothing
else what we have seen in the Schrodinger equation.
It's minus h bar squared divided by 2m times
the Laplacian plus the potential.
So if you want to calculate what is
the energy of this electron, what you need to do
is to apply the energy operator, that is the Hamiltonian,
to the wave function.
So you need to take the second derivative with respect
to the wave function.
With the appropriate coefficient,
you need to sum the potential times the wave function.
And now, to this new quantity that
is again a field that is a function of r,
well, you need to take that multiplied
by the complex conjugate of the wave function
and integrate it all over space.
So if you want, it's a much more complex operation.
But that's exactly what are the rules for quantum objects.
You have a Schrodinger equation.
You find its solutions.
And then the solutions are the ground state or the excited
state wave function.
And if you want to calculate what
are the physical values of what we
say are observable quantities, like the energy,
you need to take an expectation value
of the appropriate operator.
And if you want the most fundamental operator,
it's the Hamiltonian h that is just written here.
And the expectation value that is written in shorthand form
here is nothing else but the integral overall space
of the complex conjugate of the wave
function times what we obtain by applying to the wave
function the Hamiltonian operator.
And we integrate that in space.
Now, it's actually simpler than it
looks because remember for a moment
that this psi is actually a solution of the Schrodinger
equation.
This psi is a solution of the stationary Schrodinger
equation.
So we know already, because the Hamiltonian is really--
remember, it's the left and term that
is in this stationary equation.
So we know already that, by the fact of this
being the solution, applying this operator to this wave
function is going to give me a constant times
this wave function.
And so the concept, we call it nothing else than E
with a subscript i.
So this times the wave function gives me a constant times
the wave function itself.
And I can take the constant out of the integral sine.
And I'm left with constant E of phi times the integral of psi i
star times Ci.
And that's nothing else than 1.
And so this is, if you want, the explicit solution
of why this is epsilon i.
Of course, if you have never seen this,
it might look very arcane.
And this is why I urge you to read some introductory quantum
mechanics textbooks.
If you have seen it already, of course, it might seem trivial.
It might seem trivial to you.
OK.
And now, in the next three or four slides,
we go really into the two fundamental concepts
of computational electronic structure.
That involves two things.
One, it involves finding the solution of the Schrodinger
equation on a computer-- that is, on a system
that is really not able to do analysis, if you want,
is not able to do analytic derivatives
and analytic integrals.
And so we'll how we actually deal in practice
with this issue.
And then in the second set of slides,
we'll give you the fundamental concept
that is a reformulation of the differential equation,
of the Schrodinger equation, in a different form that
is actually computer-friendly and algorithmic-friendly.
So we have a Schrodinger equation
that we know we need to solve.
But we can actually rewrite it in a form
that is much more convenient.
And respectively, you can think of them
as the general concept of expanding wave function
in a basis that transforms all our analysis and integrals
and differentials in matrix algebra.
And I'll show you in a moment why that is the case.
And in the second part, we'll see
how the Schrodinger equation is transformed
into a variational principle.
So really, what you are seeing now is, I would say,
the central content of today's class.
OK.
Let me first go back to the Schrodinger equation
as you have seen it before.
And now, I'm using actually the shorthand term
H, this Hamiltonian operator, that is again nothing
less than minus H bar squared over 2m times
the Laplacian plus the potential.
And so the Schrodinger equation tells us
that, for a given potential, we need to find a wave function
psi such that when we apply on it
those derivatives and those multiplication
contained in the Hamiltonian operator, that
is H, what we obtain is nothing else
but the wave function itself times a constant.
So this is the Schrodinger equation in shorthand form.
And we can even be more compact and write it
in the bracket formulation on the right side
as H psi equal to epsilon psi.
So this is our Schrodinger equation.
And now, this is the fundamental step.
And this is what really makes all
of this solvable on a computer.
What we say now is that any generic wave function--
OK.
So we are still considering one electron
in three dimensional space.
So what we have is really an amplitude field all over space.
Well, that amplitude field can actually
be written as a linear combination
of an infinite number of simple functions times a coefficient.
So we are saying, well, we can expand every possible field
into a linear combination of orthogonal functions
that would be our basis function.
Say, if you have seen Fourier analysis,
you have seen that, in most cases,
a well-behaved function can actually
be expanded as a linear combination of sines
and cosines.
Or in general, if you have seen more of analysis,
you have seen that, in a lot of these Hilbert spaces,
you can define what are called complete sets
of orthogonal functions.
Sines and cosines are a complete set of orthogonal functions.
If you take sines or cosines with different wavelengths
and you integrate them over a space,
you get 0 unless the wavelength is the same.
And by using an infinite number of sines and cosines
with different wavelengths, you can describe any function
that is, in some ways, smooth enough, that doesn't, say,
infinite discontinuities.
And so what we do really, in most
of our computational application,
is that we define a basis set.
Let's say, for the case of solids,
it's actually going exactly to be the basis set of plane
waves, those E to the i kr with all possible wavelength
and with all possible directions.
That is an orthogonal set.
And it's complete.
And so once we decide what is our basis set,
say sines and cosines, and plane waves--
and there are a lot more that I'll describe in the following.
Well, at that point, our function
needs to be specified by a set of coefficients.
And in all practical implementation,
we won't use infinite sets.
If you have got a well-behaved function,
you don't need to describe--
it probably won't have oscillation
with extremely thin wavelength.
So you can expand a wave function
into a combination of sines and cosines
that have progressively shorter and shorter wavelengths.
But at a certain point, you can say, well, I'll stop here.
I don't need a resolution that is thinner than what
is physically reasonable.
I mean, if you are looking at electrons,
well, they are going to have oscillations
that are of the order of the inter-atomic distance.
But they are not going to have oscillations
that are of the order, say, of the diameter of a nucleus.
It just doesn't make any physical sense.
So you can understand, for every physical problem at hand,
when you can actually stop this infinite expansion
in your orthogonal basis.
And at that point, your wave function
becomes defined just by a finite number of coefficients.
And so suppose that for a moment you
have only 10 basis functions.
It could be sine of x, sine of 2x, sine of 3x, sine of 4x,
up to sine of 10x in one dimension.
Well, once you have decided that those 10 are your good basis
function, your wave function is described by 10 numbers.
It's just a vector.
And so what you need to do is find those 10 numbers.
So if you want, what was an analysis problem
now becomes a bit of a linear algebra problem in the ways
you'll see in a moment, in which we really need
to find those 10 coefficients.
We don't need to find any more integrals
and derivatives of functions.
And so let's actually see how this develops.
Remember, what we need to solve is the Schrodinger equation
that I've written over here.
And now, I have decided that I've got an orthogonal basis.
And I am going to do some algebra.
In particular, I take this Schrodinger equation above.
Remember, it's operator applied on the wave
function equal to a constant times the wave function.
And what I do is I say I multiply this relation
on the left and on the right by the complex conjugate of phi m.
So in going from here to here, what I do
is I multiply this times the complex conjugate of phi m.
So it's phi m star.
And then I integrate everything over space.
And so you see, when I take H psi,
I multiply it by phi m star.
And I integrate.
What I obtain is something that, in shorthand bracket
formulation, can be written as phi m H psi
This is nothing less than a shorthand
for the integral of overall space of phi star m times
what results from applying H to psi.
Same to the right-hand side--
I multiply E times by phi m star.
And I integrate over space.
Now, E is just a constant.
So it can come out of the integral sine.
And what I am left with is the integral
of phi m star times psi.
And again, it's very important that you go back just
after this class and you rewrite explicitly
all these algebraic terms to make sure that you
become comfortable with them.
So I've done this operation.
This is what I have obtained.
And now, what I do I exploit my hypothesis.
I've said that my wave function can
be written as a linear combination of my basis
function with certain coefficients.
So in this integral, now I substitute for psi
at sum of wave function with appropriate coefficients.
And as usual, I can take the sum and the coefficient
out of the integral sine.
So what I'm left with is a sum of the Cn that
multiplies the integral of phi star m H phi star n.
This is what I have on the left-hand side.
And now, on the right-hand side, this is likely trickier to see.
Again, I'm substituting for psi that sum.
So what I obtain here is really--
let me write it as an integral for a moment--
is the integral over all space of phi m star times
the sum over n of Cn phi m.
I can take a--
sorry, it's a sum over--
it's a bit difficult to see.
Let me write it more clearly.
It's a sum over n Cn phi star n.
OK.
These are just coefficients.
I can take them out of the integral sine.
So what I'm left with is sum over n Cn times
the integral over the r of phi n star times phi m.
For some reason, I wrote this down here.
So it's just a phi n.
And now, the integral of dr phi m star
times phi n is going to be equal either to 1 or to 0,
depending if m is equal to Min or m is different from n
because, in our hypothesis, we have said that these basis
functions are orthogonal.
And that orthogonal means we take the integral of them.
If they are identical, it gives 1.
If they are different, it gives 0.
So in all the sum over the n, there
is only one index that will be identical to this.
So for that case, the integral gives 1.
Otherwise, it gives 0.
So what is saved in all of this is
only the coefficient that corresponds
to this m coefficient here.
So what we obtain from this is the Cm times a constant.
OK.
So again, I've rewritten the Schrodinger equation
by multiplying it by phi m star, integrating it.
And what I obtain is a new formulation
of the Schrodinger equation that I have also in the next slide.
And that we are actually going to analyze.
So this is nothing else than your last slide.
And again, what we need to find are these coefficiencies.
Those have become now our solution.
And we can actually write this explicitly using
a bit of nomenclature.
That is, we can call that scalar, that expectation
value--
phi m H phi n, we can call it that shorthand H m n.
We call it actually the matrix element
of the Hamiltonian over the two basis phi m and phi n.
And so again, our Schrodinger equation
has been rewritten in compact form as this.
That is, the physical problem tells
us what is the potential that the electron feels.
The potential specifies what the Hamiltonian operator
H is because it's, again, just the Laplacian
plus the potential.
And so once we have decided our basis set
onto which we develop our wave function, what
we need to calculate are all these matrix elements.
Now, these H m n are explicit numbers.
Once my potential is well-defined,
once my bases sector is defined, those H are just numbers.
And they actually can be written out in a matrix.
And my Schrodinger equation has been rewritten in this form.
That is, I have that this matrix times the vector of coefficient
of my expansion needs to be equal to a number
times the same vector.
And if you want, I have two things to find out.
I need to find out what are the values of E
for which this linear algebra system is a solution at all.
And we know from linear algebra that there
aren't an infinite number of values of E
for which this system has a solution.
And I'll comment in that on a moment.
And once we have found those values of E
for which there is a solution, we need to find the solution.
That is, we need to find this vector.
And so again, our problem is matrix times
vector needs to be equal to a constant times vector.
And we actually-- again, I don't have the time
to go into linear algebra.
But we know this can actually be rewritten as a Hamiltonian
matrix, an H matrix minus an E on all the diagonal terms
times this vector equal to 0.
And we know that this linear algebra solution--
this linear algebra problem has a solution
only for values of E such that the determinant of the matrix H
minus E times the identity is 0.
So our goal is first finding those values of E for which
this determinant is 0.
And we call those eigenvalues, are the only values
of the energy that are allowed in our quantized Schrodinger
equation.
And then for each of these eigenvalues, E,
there is going to be a vector.
There is going to be a set of coefficient from C1 and Ck
that satisfies this linear algebra problem.
And we call that specific vector of coefficients an eigenvector.
So this is an eigenvalue equation
that only certain values of phi which
this equation is a solution.
And those values of phi for which it has a solution
will give rise to the eigenvector
that satisfies this equation.
And in a computer, you see the only thing that we need to do
is calculate this matrix element of the Hamiltonian,
find its eigenvalue, and find its eigenvectors that
are all well-defined linear algebra problems.
And so this is the first fundamental conclusion.
We can always rewrite our quantum mechanical problem
into a linear algebra problem once we specify
a basis set for our function.
And depending on the problem at hand,
we can choose basis sets that are more appropriate or less
appropriate.
If we are studying, say, electrons in a metal
that would look like free electron-like,
a basis set of plane waves with different wavelengths
is a very good choice.
If we are studying solutions, say, in a molecule,
maybe a basis set in which each state looks more
like atomic orbitals would be a more appropriate choice.
And again, we'll see this in the next classes.
But now, the problem has become a linear algebra problem.
And there is another conceptual and last step for today's class
that I want to highlight.
That is very powerful because, especially when
we move from trying to find the solution for a single electron
to finding the solution for many interacting electrons,
it will give us a systematic handle
to move in a problem that becomes conceptually
more and more complex.
We have discussed how there is this exponentially exploding
complexity.
And there is what is called a variational principle.
That is nothing else but reformulation
of the Schrodinger equation.
And what is this?
Well, what I am saying is that there
is what we call a functional of any arbitrary function.
What is a functional?
Well, it's something that, given a function,
operates on that function and produces a number.
So we call our functional E of phi.
It's an algorithm that eats as input a function
and gives as output a number.
And the explicitly definition of this algorithm
is given here on the right-hand side.
And what is it?
Well, for any arbitrary function, what we need
is to calculate the expectation value of the Hamiltonian.
So we need to calculate again the integral
over space of phi star times the Hamiltonian operator applied
onto phi.
And we divide that this by the integral of phi star times phi.
OK.
This is an operative definition.
I gave you a function.
What you obtain back by calculating this integral
and taking the ratio is a number.
But now, the important thing is, what can we prove?
Well, we can prove this fundamental theorem that tells
us, for a given Hamiltonian-- that is,
for a given potential--
you can throw into this functional any arbitrary
function.
You can try to calculate this ratio for any phi.
And what you'll find is that the number
that you obtain for any phi that you throw in
is going to be always greater or equal than the ground state
energy of your electron.
Not only that, but in particular this number
is equal to the ground state energy of your electron,
the function that you have used is really the ground state wave
function, is the wave function corresponding
to the lowest energy of the Schrodinger equation.
And so you see this gives us a sort of trial and error
practical recipe to try and find out what is the ground state
energy of an electron in very complex potential
because the only thing that we have to do
is try a lot of wave functions.
And what we will obtain as our best
guess is the wave function that has the minimum expectation
value.
It will never be an exact solution,
unless we examine all the possible wave
functions in the world.
But it's a solution that becomes better and better
the more systematic we become.
The more wave functions we try, the better
our solution will be, the closer it
will be to the ground state energy.
And I think I'll leave it as an exercise--
and this is my last slide--
to be proven by you for next time.
And that is, I'm asking you to go back and try to prove
this variational principle.
And the only operation that you really need to do
is, again, expand your wave function phi
into a linear combination now of states
that I can call chi of n that are actually
the eigenstates of the Hamiltonian.
That is, we have a well-defined Hamiltonian operator.
So in principle, at least formally,
we know that it will have a set of solutions
that we write as H applied to this eigenfunction
is going to be a number times the eigenfunction itself.
And the set of solutions of a Hamiltonian
are actually in themselves a complete orthonormal set.
So we use that complete orthonormal
set to expand our wave function as a linear combination
of these eigenvectors.
And with this ansatz, wave function expanded
in a combination of eigenfunctions
of the Hamiltonian, you can go back and prove by yourself
the variational principle in the previous slide.
And that, again, is probably the most powerful principle
that we have to develop all the many-body solutions
to the electronic structure problem.
And we'll see that in next class on Thursday.
And again, all of this, if you have never
seen quantum mechanics, might seem very Arcane.
So either read some of the readings
that I have given you or just come and see me.
Fix up an appointment.