NICOLA MARZARI: --for all our applications
and for the lab sessions.
I guess I keep using Albert Einstein.
This is the 100th anniversary of his sort of famous year, 1905,
so just a little celebration.
One slide of a reminder of what we
have seen in the previous lecture,
we have really developed the formalism leading
to the Hartree-Fock equations.
And the Hartree-Fock equation follow
from a set of very simple and very beautiful path.
We have the Schrodinger equation,
and we have reformulated the Schrodinger equation in terms
of the variational principle.
So we have a functional.
And we know that we can throw into that functional
any arbitrary wave function, and it'll give us an expectation
value of the energy.
And sort of the closer we get to the true ground state wave
function, the lower that energy is going to be.
We are never going to go below the ground state energy.
And so it's sort of a very powerful approach
to try out sorts of possibilities and solution.
And in particular, sort of Hartree and Fock
took this approach.
They wrote sort of the most general many-body wave
function that can be written as a product of single particle
orbitals.
That was actually the original Hartree solution.
Wave functions written as data do not
satisfy a fundamental symmetry of interacting fermions.
That is they are not anti-symmetric.
And so what you do?
You take this product of single particle orbitals,
and you sum it with all the possible permutation,
with all the possible signs in front,
so that the overall wave function is anti-symmetric.
And that can be sort of written compactly
as what is called as later determinant here.
And basically, now our unknowns are the n orbitals phi.
And so we need to determine the shape of this n single particle
orbitals.
And we want to determine them such
that they minimize the expectation value
of the variational principle.
And so that leads basically to a set of differential equation
is just functional analysis.
And when you ask yourself what are the conditions
that those single particle orbitals need
to satisfy in order to minimize the variational principle,
well, this is it, the Hartree-Fock equation.
So each single particle orbital phi of lambda
need to satisfy basically a Schrodinger-like equation.
Again, as always, there is a kinetic energy term here.
There is the interaction with the external potential
that is just the potential of the nuclei.
And then come the so-called mean field terms.
So the electron lambda here will interact
with each and every other electron
mu via an electrostatic interaction.
You see phi star times phi is the charge density coming
from the orbital mu.
And the field that the electron lambda fills
is the electrostatic average density.
And in these, we sum over all the electrons
including the electron lambda.
So up to now, we have a system that is self-interacting.
An electron lambda fills the electrostatic interaction
with itself.
That, in principle, is not correct.
But luckily, this next term that is called the exchange term
cancels that exactly.
And the exchange term is the direct consequence
of having written the trial wave function not just
as a product of a single particle orbital,
because up to now we would have sort
of something closer to the Hartree equation,
but written as a proper anti-symmetric wave
function, summing on all the permutation with them
appropriate signs.
And so, basically, we have Schrodinger-like equation.
A great advantage with respect to the Hartree equation
is now the operator doesn't change depending on the index
lambda because this sums if you want to go over all
the electrons including lambda.
So our only constraint here is that we
need to find the n lowest Eigen state
of this single differential equation.
So if we have n electrons, if you want,
it's not that we have n different differential
equation, like it was the case of the Hartree equation.
But we have an identical differential equation
that is written here.
And we need to find the n lowest energy states.
And those will be our single-particle orbitals.
In all of this, we have started from a variational principle.
So it's very easy to go beyond the Hartree-Fock.
We can say, in large r variational cluster,
we can add more Slater determinants
with sort of different coefficients.
We can try to construct a more complex wave function.
And that solution will become better and better.
Or we can sort of use the perturbation theory.
And so quantum chemistry has developed
a number of techniques that are post-Hartree-Fock techniques
that become systematically more and more accurate.
They are also more and more expensive.
And that's if you want, the main limitation of that direction.
What we'll see today is something,
as they say in Monty Python, completely different.
And that will be sort of density functional theory.
That's, if you want a theory that
starts from a very different set of hypotheses, the net result
will be, again, a set of single-particle equations.
The concept are very similar actually, formally,
to the Hartree-Fock equation, but they
have been derived in a completely different spirit.
Density functional theory tends to be
less expensive than Hartree-Fock and, overall,
tends to be more accurate.
Especially for solid, it's much more accurate.
You'll see when we discuss case studies the Hartree-Fock
solution for, say, interacting electron gas
or in general for metals tends to make
them semiconducting or insulating-like.
So Hartree-Fock then works very poorly for solids.
And that's why, if you want density functional theory,
comes from the solid state community,
while Hartree-Fock that tends to work very well for atoms
comes from the quantum chemistry community.
And all the theory was developed by Walter Kohn, and coworkers.
So you'll see the Hohenberg and Kohn
theorem, the Kohn [INAUDIBLE] mapping during the '60s.
But I would say it's only during the '70s
that people started to be able to actually solve
interesting cases using density functional theory.
And it's really the beginning of the '80s--
you'll see some cases here today--
in which people started calculating something that had
sort of a direct application.
So we'll see the phrase diagram of silicon
as a function of pressure or volume
and sort of the first principle prediction of properties
of solids.
Walter Kohn, for the development of this eventual theory,
got the Nobel Prize for chemistry in 1998
together with John Pople.
That has been the person that's been fundamental
in the development of Hartree-Fock
and post-Hartree-Fock approaches in quantum chemistry.
OK.
So let's see sort of what is the general idea behind density
functional theory.
And in many ways, we'll sort of start
from ideas that had been developed
at the end of the '20s and the beginning of the '30s, what
is nowadays called the Thomas-Fermi approach.
And again, the basic idea here is
that the wave function of a many-body interacting problem
is an object that is too complex to treat.
And it would be very, very nice if we could instead try
to deal with a simple object.
And sort of one of the choices could be the charge density.
So if you want, Thomas and Fermi independently
were asking themselves, well, could we
try to solve not really a Schrodinger
equation in the many-body wave function,
but solve something else in which our only unknown is
the charge density?
If you think for a moment, the charge density
is one of the sort of fundamental variables
in the description of our interacting electron problem.
And so this was the question.
Can we do something just with the charge density?
And so what they did is writing out
what we would call a heuristic function.
That is trying to devise a set of terms that would give us
the energy of a set of electrons in a potential
just as a functional of their charge density.
And so, by now, you could sort of
think that some of the relevant terms
will be electron-electron interactions, electron
interaction.
And we could write a sort of electrostatic term,
like the Hartree term in the Hartree or Hartree-Fock
equation, that is just a functional of the charge
density.
So this is sort of fairly easy.
It's also very easy to sort of imagine
what could be the interaction of the electrons
with an external potential through the charge density.
It will be just the integral of that external potential
times the charge density.
What becomes really critical is finding
a functional that will give us the quantum kinetic energy.
If you think, in the Schrodinger equation,
the quantum kinetic energy is really the second derivative
of the wave function.
And obtaining from a charge density
only some insight into what could
be the second derivative of the wave function is very complex.
If you think for a moment at the extreme case of a plane wave,
OK, so a sine and cosine sort of in space, if you remember,
the charge density given by a plane wave is a constant.
We just multiply the imaginary exponential
times its complex conjugate.
That gives us a constant.
So all plane waves lead to a constant,
but obviously the quantum kinetic energy of a plane wave
depends on the wavelength of that plane wave
because the second derivative is what counts.
So what I'm trying to say is that, when
we look at this as a possible wave function, a function,
say, of r and the charge density that comes from this
is going to be a constant, this wave function
times this complex conjugate.
But the kinetic energy of this object
is going to be minus 1/2 k square--
sorry, plus 1/2 k square.
And so there is really not a good way
for this extreme case to correlate its charge
density to its kinetic energy.
It's an ill-defined problem.
And this is really the difficulty.
So there isn't really a good way if you
wanted to extract the information
on the second derivative from just the charge density.
Now, [INAUDIBLE] sort of this objection,
they tried to find a reasonable functional, so
without sort of trying to get the exact solution,
but try to find a reasonable function that would give us
a good estimate to the quantum kinetic energy starting
from the charge density.
And the solution to this problem that
is something very important is what
we could call a local density approximation.
So the problem here is that we have a non-homogeneous charge
density everywhere in space.
And we try to figure out what could
be the quantum kinetic energy of this non-homogeneous problem.
And the approximation that Thomas and Fermi did
was, well, dividing this non-homogeneous problem
in a set of infinitesimal volume in space.
So it's a bit difficult to draw, but suppose
you have the charge density coming
from some atom or some molecule.
This is a non-homogeneous charge density distribution.
Now, what you do is you divide this in space
and set a very small infinitesimal, if you want,
volume.
And inside each volume, the charge density
can be approximated as a constant.
And what Thomas and Fermi said is,
well, the contribution coming from this infinitesimal volume,
say, the first one to the overall quantum kinetic energy
will be given by that volume times the kinetic energy
density of the homogeneous electron gas at that density.
So if, again, we partition all space,
we could have that the density in this little cube is 0.5.
Here is 0.6.
Here is 0.7.
Outside, it goes to 0.
But we can actually calculate in some other way what
would be the quantum kinetic energy
of a homogeneous electron gas.
That's a problem that we can solve
if the homogeneous electron gas is not interacting.
And we can solve it numerically even if it is interacting.
So we can know what is the quantum kinetic energy
of a homogeneous gas with density 0.5,
density 0.6, density 0.7.
And so we can also know what would
be the quantum kinetic energy per unit of volume of data.
And so we'll say that this non-homogeneous system in blue
will have an overall quantum kinetic energy that is given
really by the integral across space--
and it's written here--
of the quantum kinetic energy of the homogeneous electron gas
integrated over space.
And, say, for the non-interacting electron gas,
it's actually very easy to do.
So if you have a non-interacting electron gas at a density rho,
its quantum kinetic energy is just the rho to the 2/3
that then integrated times the unit volume
gives us rho to the 5/3.
So by integrating this quantity, we would get an approximation.
This approximation is basically exact
in the limit of our homogeneous system, obviously.
And it will be sort of quite good
in the limit of our non-homogeneous system that
has a very slowly changing charge density.
The more, if you want, inhomogeneous
your system becomes, the less accurate this approximation is.
And of course, something like an atom or a molecule
is a very inhomogeneous system.
You go with a charge density that
goes from 0 to very high volumes close to the core
of the nuclei.
So this is basically the overall answer
for the overall expression that Thomas and Fermi
postulated for the energy of an inhomogeneous system.
They were saying, well, suppose that we
have a system that has a certain distribution of charge rho.
Without trying to solve the Schrodinger equation finding
out the wave function and sort of go through that the very
complex many-body route, we can actually sort of postulate
that the energy could be written, again,
as an electrostatic energy.
You see sort of each infinitesimal volume
interacting with each other infinitesimal volume
times via 1 over R electrostatic interaction.
Then we have got an external potential.
Again, it's usually the Coulombic field of the nuclei.
And so the interaction between the electron
and that external potential is just trivial given by rho
times v.
And the difficult term, the quantum kinetic energy,
has been calculated with a local density approximation.
And this is the term that's not going to be very good,
again, because it's very difficult to figure out
what could be the curvature of our wave function
just from the density that that wave function produces.
But anyhow, this is a very simple expression to deal with.
So for any external potential v, we
can try to find out the rho that minimizes this expression.
And this will be our Thomas-Fermi solution.
There are obviously a number of problems.
I'll show you in a moment an example of what
the Thomas-Fermi solution would give to an atom.
First of all, I mean, there is really
no theoretical basis to this.
It's what we call a heuristic derivation.
Thomas and Fermi just wrote out what
could be a reasonable energy functional,
and then tried to sort of see what results it would give.
But there hasn't been any kind of formal derivation
of that functional.
It's not like the Hartree-Fock equation that sort of derive
just with some analysis from the variational principle.
Another problem is that, again, it doesn't really
sort of introduce the concept of anti-symmetry that fermions
need to have, the fact that the many-body wave function needs
to be anti-symmetric upon exchange.
But you know, there is no conceptual problem
in adding an exchange energy to the previous functional.
Using the same concept, the same idea
of local density approximation, suppose that we
want to add an exchange term.
Well, we could look at what is the exchange energy coming
from the Hartree-Fock equation, say,
for a homogeneous electron gas.
And that gives us rho to the 1/3 term.
And that's basically the exchange energy density.
And so for an inhomogeneous system,
we are going to sort of approximate its overall
exchange energy just by taking the integral of that energy
density that is 1/3 times the sort of local value
of the charge density.
And so we have a rho to the 4/3.
And so, again, it's a local density approximation.
The great consequence of having this energy functional
that depends only on r is that it
is absolutely inexpensive from the computational point
of view.
The only variable that we need to be concerned with
is just the a scalar as a function of three coordinates.
That is the density as a function of rho.
And it's what we call a linear scaling system.
If you double the size of your system,
the computational complexity just becomes double.
So it has a lot of very good things,
but it's got a fundamental defect.
Because of that approximation in the kinetic energy,
it actually does a very poor job in describing
a non-homogeneous system.
So it would work reasonably well for something like a metal.
Suppose that you want to describe sodium,
or suppose you want to describe aluminum.
Those are system in which the valence electron produce
a charge density that is very homogeneous.
So a Thomas-Fermi approach could actually work well.
And it's actually been used even very recently sort
of quite successfully to describe problems
like the surfaces of lithium, the surfaces of aluminum.
What happens, say, when these simple metals melt?
What happens to the sort of formation
of defects in aluminum?
So there are a number of successes.
But sort of clear example of what goes wrong
is, say, if we study an inhomogeneous systems
like the argon atom.
And again, if we think at the charge density of the argon
atom as a function, say, of the radial distance
from the center from the nucleus,
well, it will look something like this.
We have first 1s, and then we have the 2s, and the 2p shells.
This is somewhat a poor depiction
of that charge density.
If we try to solve the argon atom
with a Thomas-Fermi approach, all these sort
of fine structure of the core shells in the atoms
is completely washed out.
OK.
So it gives you a reasonable approximation and sort
of an appropriate decay of the charge density
as we move far away, but a lot of those details
have completely disappeared.
And for this reason really the Thomas-Fermi approach
wasn't developed beyond the '30s apart
from some of these recent applications
for the very specific case of solids
that have a very homogeneous charge density.
The reason why we describe it here
is that because, in many ways, it's
the grandfather of the ideas that
were developed in the '60s in density functional theory
and, in particular, the idea that for a moment
that we should focus not on the wave function,
but on the charge density of the system as the key ingredient.
The great difference between the Thomas-Fermi approach
and density functional theory is that density functional theory
actually is a theory.
It starts with some theorems that are proven.
And then it shows what are the form of the equations
that, say, a charge density need to satisfy in order
to solve exactly the problem.
So in many way, density functional theory
is, in principle at least, an exact theory.
It's sort of writes out what are the equation that the charge
density needs to satisfy.
And those are absolutely equivalent to a Schrodinger
equation for the wave function.
There are some difficulties.
And this is what we are going to sort of go into right now.
But let me first give you the conceptual framework
of density functional theory and how it was derived.
And as usual, we start from the Schrodinger equation.
OK.
So we start from the idea that, in quantum mechanics,
for any given external potential,
you have a well-defined differential equation.
OK.
It's sort of very complex.
It describes a many-body wave function.
So in most practical cases, we might not be able to solve it,
but everything is well-defined.
You have an external potential.
You have the differential equation
that the many-body wave function needs to satisfy.
And so, in principle, you have the solution.
And so in that sense, sort of the first statement
here is summarized.
For a given external potential and knowing how many electrons
are going to fill this potential,
our quantum problem is formally completely defined.
In principle, the solution exists unique.
We might not be able to calculate it, but it exists.
And once we know the many-body wave function, that solution,
we know everything about our quantum system.
OK.
So this is, if you want, the trivial part of the conclusion.
That is, given an external potential,
we find, by the Schrodinger equation, the wave function.
The wave function determine all the properties of our system
and, in particular, determine the ground state charge
density.
So there is a unique pathway that
starts from the external potential
and leads us to the charge density, the ground state
charge density.
Once you have defined the potential,
you, in principle, have uniquely defined
what is the ground state's charge density of your system.
And so in that sense, we say that the ground state charge
density, the ground state energy, and all the properties
of our system are, in some complex way,
a functional of our external potential
and the number of electrons.
Functional, again, can be anything.
And in this case, it goes through the Schrodinger
equation, nothing sort of complex at this point.
The sort of remarkable result that no one had sort of figured
out between 1964 and 1965 is that the opposite is also
true and is not trivial at all.
So what Hohenberg and Kohn stated first, actually,
in 1964, was this, that the ground state charge
density is a fundamental quantity,
as fundamental as the external potential.
And in particular, not only the external potential,
the terms uniquely the ground state's charge density
of your system, but also the vice versa is true.
That is, given a ground state charge density,
in principle, one can prove that there
is a unique external potential for which that ground state's
charge density is the ground state solution
for that external potential.
So if you have the external potential,
conceptually it's trivial to go through the Schrodinger
equation and its solution to the charge density.
What Hohenberg and Kohn are telling us--
and I'll just show you a sketch of the proof in a moment--
is that in principle, if someone is giving you a chance density
and is telling you this charge density
is the ground state's charge density
of a number of electrons and electrons
in an external potential, in principle,
what is that external potential is an information that
is completely contained into the charged density.
And it's not contained in a trivial way.
It's not that you can look at the ground state charge density
and guess what the external potentially is.
And that's where all the complexity of practical densely
functional theory comes.
But from the conceptual and mathematical point of view,
these two quantities are absolutely equivalent.
From one, you get the other and vice versa.
And this sort of vice versa was not trivial.
And that is sort of what is contained
in the so-called first Hohenberg and Kohn problem.
I won't go through the derivation.
It's actually very simple.
I've printed it here in case you sort of want to read it.
But it's basically is a derivation ad absurdum.
What they are saying there is that,
if that external potential were not unique,
if there were two external potential that
were different and would give the same ground state energy,
we would get to absurdum.
So typical mathematical demonstration,
we suppose that there are two different external potential
that give the same ground states as density
and we show that we arrive to a conclusion that
doesn't make sense.
So there can be only a single external potential.
And that's the proof.
And again, it wasn't trivial.
I mean, if you want, this is a very basic statement.
But it took 40 years to be formulated.
And it's actually not true in other cases
that to first glance look very similar.
Suppose that for a moment we want to discuss excited states.
You could say, well, if I have a charge density
and I say this is the charge density of an excited
electronic state, maybe I could also
recover the potential that has generated there.
And that's not true, actually.
So there are sort of a number of cases
in which this is not true.
But for this fundamental sort of relation between the charge
density of the ground state and the external potential,
this is true.
So we have sort of moved away now our attention.
It's not any more the many-body wave
function that we want to focus, but is the charge density.
The charge density is as much a fundamental variable
of our problem.
It's not a derived variable.
It's not something that comes from the wave function,
but is something that we can actually
focus all our attention into.
And now, we need to find the equivalent of the Schrodinger
equation for the charged density.
This is what Schrodinger had done in the '20s, in 1925.
He said, this is the equation that quantum objects satisfy.
And I'll call it the Schrodinger equation.
Now, Hohenberg and Kohn has shown
that we don't need to think in terms of the wave function.
We can think in terms of the charge density
as being the fundamental descriptor of our quantum
system.
What is left?
They need to show me that there is an equivalent
of the Schrodinger equation.
That is we can write a density equation that
is sort of what will give me the ground state and sort of all
the properties of the system.
And that's really the second Hohenberg and Kohn theorem.
That is really writing out the equivalent concept
of the Schrodinger equation for the charge density.
And now, sort of, it becomes fairly conceptual.
OK.
So this is the procedure.
And all of this in the next few slides
is still a conceptual procedure.
It will describe objects that are well-defined in principle,
that are conceptually well-defined,
but we still don't have a clue on what
they look like in practice.
And all the sort of density functional application
goes through a procedure that we'll
see later on that is the sort of [INAUDIBLE] mapping
that gives a hint of what these objects look like.
But up to now, we are going to introduce objects
that are well-defined in principle,
but we don't know how they look like.
And so that's why somehow density functional theory
is a much less intuitive theory than something
like Hartree-Fock.
OK.
So this is going towards the second Hohenberg and Kohn
theorem, defining the fundamental equation
for the charge density.
And this is the step.
For any charge density rho, so someone gives you,
someone draws you, an arbitrary charge density.
Well, we know that there is an external potential
of which that charge density is the ground state.
We don't know what it is, honestly,
but we have proven that there is a unique external potential.
OK.
So because there is a unique external potential,
there is a many-body Schrodinger equation
with that potential in there.
And there is a wave function that
is going to be the ground state wave
function of that many-body Schrodinger equation.
So given a certain rho, we know that an external potential
exists.
And it's unique.
It determines a Schrodinger equation.
And that Schrodinger equation determines our ground state
wave function that we call psi.
So what we are saying is that, given a rho, in principle
that psi, the ground state wave function of the Schrodinger
equation in the external potential
that is uniquely defined by the rho,
is also a well-defined object.
Again, we don't know what it is, but it is well-defined.
And because it's a well-defined object,
we can calculate the expectation value
of that well-defined object of the quantum kinetic energy
minus 1/2 sum over all i of the second derivatives
and the electron-electron interaction, just the 1
over ri minus rj term.
So again, this term is, in principle, well-defined.
And we call this term the universal density functional.
That is for any given arbitrary rho, i, in principle,
can define a number that is this number here.
I have the rho.
In principle, from the rho, I have the external potential.
From the external potential, I have the Schrodinger equation.
In principle, I'm able to solve that Schrodinger equation found
in principle the many-body ground state wave function.
That will be psi.
And I can calculate the expectation value
of psi of the quantum kinetic energy
and of the electron-electron interaction term,
all well-defined.
We have really no clue on how to calculate because we can't
really do in practice any of the steps,
but this universal function of the density is well-defined.
So with this universal functional
that is now well-defined, we can write out something.
We can write to an energy for any given external potential
and for any given charge density.
And we write it as this.
So for any given charge density, there
will be a well-defined number that is this universal density
functional of beta rho prime.
And then we add another term that
is just trivially the integral of this v,
this external potential, times the charge density rho prime.
So again, this new expression that we written
is well-defined.
For any rho prime and for any external potential,
we can calculate trivially this term.
And in principle, we know what this number is.
And this is, if you want, 1964, 1965, the reformulation
of quantum mechanics.
Because, now, Hohenberg and Kohn are
able to prove that there is a variational principle.
That is, for this expression written here,
for this functional of rho prime,
we can prove that for any rho prime
that we can throw in the overall numerical value
of this expression is always going
to be either greater or equal to the ground state energy
that we would obtain from the Schrodinger equation
in the presence of this external potential.
So now, we have a well-defined density functional.
So if you have an external potential,
the z over r of your atom, you can try out now not
wave functions that are very difficult,
but you can try out charge density.
And the charged density that gives you
the lowest expectation value, the lowest
value for this functional, will be the ground state, the charge
density.
Small problem, we have no clue what
this looks like as a function of rho prime.
But if we knew, we would have a wonderfully simple approach
to quantum mechanics.
Now, we don't need to deal with the many-body complexity.
We just minimize this expression as a function of rho prime.
And again, it's sort of fairly easy
to prove this variational principle.
But one needs probably to sit--
I've given you some reading.
So you're welcome, if you are really interested in this,
to go back and read the first Hohenberg and Kohn theorem
and read the second Hohenberg and Kohn theorem.
But in many ways, the proof of this second Hohenberg
and Kohn theorem can be done again
through the variational principle.
That is, if we have a certain rho prime,
well, that, again, uniquely determines the ground state
wave function.
Rho prime will determine an external potential
that, in principle, is different from this.
But rho prime will determine an external potential
and will determine our wave function
that is the solution of the many-body Schrodinger equation.
And if we take the expectation value of our Hamiltonian
with this external potential in this,
but evaluated on the wave function of c prime
that comes from this charge density rho prime,
well, we can show that this expectation value here
is just identical to functional that I have just written.
And for the variational principle,
then it needs to be greater or equal than E0.
I won't sort of dwell into that.
And again, you can look at the sort of detailed description
and in sort of some of the many references that I've given
or that I've also posted on the website.
But what is conceptually important
is that we have a new equation.
OK.
So 1964, '65, quantum mechanics turned around.
We don't have to think at many-body wave functions.
We can think just at charge density.
And all would be well apart from this detail,
that we don't know what that functional f of rho is.
And so we have a conceptual approach,
but we don't have a practical approach
to solve the density functional reformulation of quantum
mechanics.
And this is, if you wanted, true to this day.
We don't know what is the exact form of f of rho.
If we knew it, sort of most of our sort
of quantum mechanical computational problems
would be trivially solved.
Because solving that variational principle in the charge density
would be most likely a trivial thing to do.
The issue is that not only we don't know,
but we have understood a lot of what that exchange
correlation-- of what that universal density functional
is.
And it's very complex.
So it's unlikely that there is a simple analytical expression
of it as a function of the charge density only.
But, you know, the other great piece of, if you want,
quantum engineering by Walter Kohn
was finding out a very good approximation
to that density functional.
We don't know what the exact one is.
But now, what they are doing is, well,
finding out one that is going to be very, very closely
similar to the exact one.
And so they are going to throw in some physical intuition
to this problem that up to now, if you want,
has been a mathematical problem.
It's another layer of complexity in this discussion,
so I hope I'm not losing you.
But sort of what Walter Kohn did--
I think he had a young postdoc arriving from Cambridge.
Lu Sham had just done his PhD in England and came there.
And sort of, you know, he told him,
I have this new variational principle.
Let's see what we can do to make it into a practical solution.
I think they were in Santa Barbara, in San Diego probably,
at that time.
OK.
So this is what they are going to do.
Remember, sort of, what is the problem.
We need to figure out what is a reasonable approximation
to this functional here.
So what they say is, well, suppose
that someone has given us this charge density.
So we need, in principle, to find out
what would be the many-body wave function that
is solution of this external potential that corresponds
to this charge density.
This is going to be very complex.
Let's invent a problem in which electrons do not
interact between each other.
OK.
So electrons-- so that's the sort
of main problem in the Schrodinger equation,
that electrons interacting with each other
introduce the two-body electrostatic repulsion
in the Schrodinger equation.
And that's what makes it difficult.
Well, what Kohn and Sham say is let's
for a moment suppose that there is a system of electrons
that don't interact.
So the only thing that those so-called Kohn and Sham
electrons fill is the external potential.
OK.
So those Kohn and Sham electrons will solve,
will satisfy, a Schrodinger equation that is much simpler.
Because there is no electron-electron interaction.
Those Kohn and Sham electron, the only thing that they fill
is a new potential.
And they will have their own quantum kinetic energy.
So what they are saying is, for any given charge density rho,
there is going to be a non-interacting set
of electrons who's ground state charge density is
identical to rho.
So we have said, if we have a charge density rho,
you can all go through, find out the external potential that
comes from rho, the Schrodinger equation,
the many-body interacting electrons solution.
But now, what we are going to say
is we can also think at a system of non-interacting electrons.
And we want those non-interacting electrons
to fill a potential that is such that their ground state is
going to give us a charge density that
is identical to the charge density I'm dealing with.
OK.
And we call that external potential
the Kohn-Sham potential.
OK.
So now, for a charge density, you
don't only have to think at all the complexity
that I've discussed up to now.
But you have also to think that, for a charge density,
there is going to be this set of Kohn and Sham
known interacting electrons.
And there is going to be a potential that
is called the Kohn and Sham potential that is such
that the ground state of the Schrodinger
equation for non-interacting electron, that
is without electron-electron interaction, in that Kohn
and Sham potential will give us a wave function and a ground
state that leads to a charge density identical to the charge
density I'm sort of dealing with.
OK.
What do we do with this?
Well, at this stage, there is a sort of great simplification
that, for the Schrodinger equation
of non-interacting electron, we actually know
what is the exact solution.
So it's actually very simple to solve
a Schrodinger equation in which the electrons do not interact.
Because, now, this later determinant
is actually the exact solution.
So if you have a set of non-interacting electrons,
you don't have the electron-electron term
in that Schrodinger equation, this later determinant is not
only a good approximation, but it's actually
the exact solution.
So for this non-interactive set of electrons,
we can solve everything exactly.
And in particular, we can calculate, say,
what is the kinetic energy of this set
of non-interacting electrons.
OK.
So now, we can sort of have a somehow pseudo-physical way
of decompose this mysterious density
functional into different terms.
So what we are actually doing via the Kohn and Sham
mapping is extracting from here terms that are very large
and that we know how to write, we know how to calculate.
And then, hopefully, once we have extracted all these terms
that we know how to define, we remain with something
that is very small and that we'll
find another numerical approximation for it.
So Kohn and Sham say, well, we have this well-defined density
functional.
We extract two terms that are well-defined.
And these two terms that's sort of the great achievement
actually contain most of the physics of our problem.
And the sort of small term that is left over,
we are going to approximate in some simple way.
And actually, the approximation that they found
worked very well.
And that's why some of this functional theory
became a practical theory.
And so in this sort of density functional,
the first physical large term that they extract
is the quantum kinetic energy that we
call Ts not of the real system.
Because, again, even if it's well-defined,
we don't know how to do that.
But what they were able to write is the quantum kinetic energy
of this non-interacting problem.
So for a given charge density, there
is this set of Kohn and Sham non-interacting electrons
that lives in a potential, such that they have the same ground
state charge density.
And their kinetic energy is trivial
because it's going to be just the kinetic energy
of this later determinant, just the sum of a single particle
term.
So for a charged density now, there
is a well-defined quantum kinetic energy
that is not the true quantum kinetic energy of the system,
but is the quantum kinetic energy
of this sort of associated system
of non-interacting electrons.
But this term is now well-defined.
They say, well, let's extract another term that
is well-defined that is just a Hartree electrostatic energy
of a charge density distribution.
OK.
So if we look at the charge density distribution in which
each infinitesimal volume interacts
with each other infinitesimal volume
with an electrostatic interaction,
that's going to be the term.
And you know, what we are left is now something
that they call the exchange correlation term.
That is everything else.
OK.
So F, in principle, is an exact quantity.
We are now able to define our quantum kinetic energy term.
That is an exact quantity, but is not really
the quantum kinetic energy of the true system.
But we sort of say, you know, this
is going to be equal to a well-defined term
plus another well-defined term plus a third term
that we don't know.
So we have sort of decomposed a quantity
that we have no clue what it is into three terms of which two
terms are well-defined.
And all our cluelessness is contained in the third term.
And we call this third term the exchange correlation,
but the sort of physical advantage
of having done this is that it turns out
that these two terms capture a lot of the complexity
of your problem.
And this term tends to be fairly small.
OK.
So that's all, actually.
That's why it works very well because somehow they
manage to capture the complexity of our system.
And so once that exchange correlation term is defined
and it's approximated in some way that we'll see in a moment,
our problem is now well-defined because we really
have a variational principle.
Remember, the universal density functional
plus the external potential plus the charge
density in the field of the external potential
minimizes the sort of new variational principle
that comes from the Hohenberg and Kohn theorem.
And so we can write it, our variational principle.
That is this quantity with the constraint
that the number of electrons should be
equal to n should be minimum.
And as usual, when you sort of write a variational principle,
you are saying that sort of the differential of that quantity
needs to be equal to 0.
Or if you want, I mean, this is a generic term.
You have a set of what are called Euler-Lagrange equation,
basically.
That is nothing else than differential analysis.
That is you are asking yourself, what
are going to be the conditions that
need to be satisfied by the charge density
in order to satisfy the variational principle?
There is always this sort of 1 to 1 correspondence.
You have a variational principle.
It gives you differential equation.
Or you have differential equation.
You can rewrite them in a variational principle.
We have seen that for the Schrodinger equation.
And we see this, in particular, now explicitly for the Kohn
and Sham orbital.
So I'll actually go directly to the explicit expression
of the Kohn and Sham orbitals.
Again, remember that what we have done
is we have defined a variational principle that
acts on a universal density functional
F plus the charge density and the external potential.
And we have decomposed, we have extracted,
from this universal functional sort of terms
that are large and physical.
And we have sort of pushed all the many-body complexity
of the problem in something that we call the exchange
correlation functional.
That is, again, a functional of the charge density.
We don't know yet what that function of the charge density
is.
But luckily, it's going to be small.
So in a moment, we'll approximate it.
And then we ask ourselves, what are the differential
equations that derive from this variational principle?
Well, in principle, I had written them here.
OK.
We just need to take the variation with respect
to the charge density and imposing the Lagrange
multiplication constraint.
And so this would be basically that the charge density needs
to satisfy this set of equation, the sort of functional
derivative of this non-interacting quantum
kinetic energy plus a number of terms that
really contain the external potential,
the Hartree interaction.
And the exchange correlation need
to be equal to the Lagrange multiplier that
fixes the number of electrons.
We are not able to calculate this functional derivative
because, remember, the quantum kinetic energy
of the non-interacting system is again
written as a later determinant.
And so there is sort of, you know,
this type of back in which, even if we had written everything
in terms of a charge density, we are not
able to explicitly calculate not only
the derivative of the true interacting electron's
kinetic energy with respect to rho,
but we are not even able to calculate the functional
derivative of the non-interacting kinetic energy
with respect to rho.
But what we are able is actually to calculate the derivative
of that non-interacting kinetic energy with respect
to the orbitals that describe the Kohn and Sham electrons.
Remember that these non-independent Kohn and Sham
electrons have an exact solution that is a later determinant.
And so we know there are trivial many-body wave function is
a later determinant composed by a single particle orbitals.
And the functional derivative of that independent
non-interacting electron's kinetic energy
with respect to the single-particle orbital
is now trivial and is just minus 1/2 del square.
So at the end of all these sort of complex formulation, what
we are left with is something very simple
and probably something you should focus your attention
from now on.
We have now a set of Kohn and Sham equation
that are the differential equation
that the electrons need to satisfy in order
to satisfy the variational principle with the caveat
that, in this Kohn and Sham equation,
there is at term, an exchange correlation term,
that we still don't know what it is.
It's sort the formally defined as the functional derivative
of the exchange correlation energy
with respect to the charge density.
But we'll need to approximate somewhere.
And what this equation describes is not anymore
the true electrons in your system,
but they describe this cousins of the true electrons.
They describe this Kohn and Sham non-interacting electrons
that have their own orbital psi i.
And that will give us a ground state charge
density that, if the exchange correlation functional was
exact, would be not only this, as obviously the same ground
state energy of our interacting electron system,
but it would be the exact solution of the problem.
OK.
So this equation look a lot like a Schrodinger equation.
They look a lot, if you want, like the Hartree-Fock equation
that we've written before.
Because what we are seeing is that the a Kohn and Sham
electron i fills a quantum kinetic energy operator,
fills a Hartree operator, fills the external potential,
and then fills this sort of remaining term
that is the exchange correlation potential.
Again, if we knew what were this exact exchange correlation
potential, we would have an exact solution to the problem.
But we know a very good approximation.
And then if you want finding the ground state,
it's not very different from finding the ground
state of the Hartree-Fock equation with the caveat
that actually this term here is going
to be much simpler than the exchange term
of the Hartree-Fock equation.
If you go back to the first slide
to the Hartree-Fock equation, the last term
is that numerically very complex expression
in which we sort of take the orbital,
and we put it inside an integral differential operator.
Now, it's become simpler.
And that's all if you want.
So the Kohn and Sham equation look very similar.
In practice, they are simpler to solve.
They tend to be more accurate in most cases.
And that's, at the end, what leads to the success.
But what is critical for all of this
is having a reasonable approximation to the exchange
correlation potential.
If we had the exact exchange correlation potential,
everything would be exact in this formulation.
We would find the Kohn and Sham independent electrons
that were sort of the ground state electrons for that charge
density that is ultimately equal to the charge
density of the interacting electrons
in this external potential.
OK.
And we have the Euler-Lagrange or Kohn and Sham differential
equation in the previous page.
I written here sort of, you know, just for reference
also what would be the total energy of the system.
And usually, if you had an independent electron,
the total energy of the system is trivially
the sum of each of the single particle energies.
OK.
If you have 10 electrons and they
don't interact with each other, you
can calculate what is the energy of each of these 10 electrons.
Sum all of them, and that's the total energy of the system.
In this case, it's more complex.
And the total energy of the system
can't be really written as that, but it's
got other terms that depend on the charge density.
That's sort of this is, in summary, what
your total energy is.
And again, there's nothing else than kinetic energy term
sort of a Hartree term functional charge density,
this exchange correlation functional,
and the interaction between the external potential
and the charge density.
But this is actually different from the sum
of the eigenvalues.
That would be the sum of the expectation values of psi i
calculated on the single-particle orbital where
T is, again, just the simple quantum kinetic energy.
And the VKS is this Kohn and Sham potential.
So if you want to calculate the total energy of your system,
even if it's made of independent electron,
you can't sum just a single particle orbitals.
But you have to sort of deal with this expression.
Nothing complex in this, it's just sort of a caveat that
is relevant when you want to sort of-- you
know, this is the reason why we can't really
find out the equivalent of the [INAUDIBLE]
theorems for Hartree-Fock.
This is why at the end these single-particle energies
are ultimately not physically meaningful.
They sort of don't give us the total energy of the system
just by taking the sum over all of them.
OK.
So in order to make this into a practical algorithm,
the only part that remains is finding an approximation
to that exchange correlation term, to that last term.
Remember, we had sort of defined this density functional.
We have been able to extract two meaningful terms,
the Hartree electrostatic energy and the non-interacting Kohn
and Sham kinetic energy.
And we have said what is left is a function of the charge
density that we call the exchange
correlation functional.
How we are going to approximate that?
Well, we go back to the Thomas-Fermi idea.
We are going to do a local density
approximation to that exchange correlation functional.
So again, what we want to calculate
is the exchange correlation energy for any arbitrary charge
density.
Sometimes I call the charge density n.
Sometimes I call the charge density rho,
but they are always the same.
So how do we do this?
Well, we don't have the full solution.
But what we can say, again, is that,
for an inhomogeneous charge density that changes values
and then drops to 0, I can calculate the exchange
correlation energy for this charge density distribution
by sort of decomposing it in infinitesimal volumes.
Inside each infinitesimal volume,
I can say the charge density is constant.
And you see, I make a local density approximation.
That is I say the contribution to the overall exchange
correlation energy of this inhomogeneous system
can be broken down.
And each infinitesimal volume will give its own contribution
to the total exchange correlation density.
You know, in principle, it's not correct.
I mean, our problem doesn't have to be local in any way.
Actually, as people say, this exchange correlation
functional, the true one, although we
don't know what it is, we know that is ultra non-local.
So it can't be decomposed into terms
that independently sum up.
So in principle, we can't do this.
But in practice, it tends to be a good approximation
for a lot of cases.
And so what is going to be the contribution to the exchange
correlation energy from this infinitesimal volume?
Let's say the charge density there is 0.5.
Well, what we need to do is we need
to find out what is the exchange correlation
energy for the homogeneous electron
gas that is at this density.
That's something that, with some advanced computational
techniques, we can actually find out almost exactly.
So we would know, if we had a homogeneous charge density
0.5 everywhere, what would be the charge
density per unit volume.
And we can find out what is the exchange correlation charge
density per unit volume not only for 0.5, 0.6, 0.7, anything.
And what we are saying is that, in this non-homogeneous
problem, we construct the overall exchange correlation
energy by summing up these different pieces.
And so this is what Ceperley and Alder did in 1980.
They basically found out what was the almost exact sort
of closely to numerical exact solution
for the homogeneous electron gas.
That is for a system in which you have only electrons
homogeneously, so the charge density is identical
everywhere.
And those electrons interact.
So you can calculate the energy of this interacting electron
problem exactly as a function of the density.
OK.
So you change the density in your sort of volume,
and you find out what is this energy.
And then you can calculate, for any
of this density, what is the Kohn and Sham quantum
kinetic energy.
You can find out what is the Hartree electrostatic energy.
And so you can also find out, for this specific case
of the homogeneous gas, numerically
what would be the exchange correlation density.
And so that's basically a function.
So for the homogeneous gas, that is for the case in which n
doesn't depend on r, people found out
what was basically this exchange correlation energy.
It was calculated as a function.
This is a function of what people call rs.
rs is the radius of a sphere that contains one electron.
So it's a sort of inverse quantity
with respect to the density.
So numerical calculation, what are called quantum Monte Carlo
calculation, really solved the interacting Schrodinger
equation problem.
But for the specific case of an electron gas that
has a homogeneous density, they were
able to do that for various density.
And so, now, we have a function for the homogeneous problem.
For the non-homogeneous problem, we take a local density
approximation, and we say that the overall exchange
correlation energy is given by the integral over all
the infinitesimal volume.
And each infinitesimal volume will have a certain density
and will contribute with its own density.
If the density is going to be equal to here,
this will be the value of the contribution
of that infinitesimal volume.
If the density somewhere else corresponds to this,
this will be the corresponding.
So we really match up this overall exchange correlation
term from all the little infinitesimal volume
exactly as Thomas-Fermi had done,
but now we do it for a term that is a much smaller term
in our problem.
Thomas and Fermi had done it for the quantum kinetic energy.
Instead, what Kohn and Sham do, they
do it for what is left from the universal functional
once you have taken out the electrostatic
and once you have taken out the quantum kinetic energy
of the non-interacting electrons.
At this point, if you want 1980 and even
before without the computation, with some sort
of analytical approximations to this curve,
density functional theory becomes not only a theory,
but also a practical algorithm.
We have a set of expression for the exchange correlation term.
And so, now, it's just a matter of trying
to find out what the solution to these problems are.
And because somehow conceptually we
start from the homogeneous electron gas,
it turns out that this approach worked especially well
for solids.
I mean, the valence electrons in a solid
are much less structured than the electrons in a molecule
that they need to drop to 0.
So the charge density in a solid overall
varies less dramatically as a function of space
than the electron density in atoms and molecules.
And these are actually sort of what
were summarized the numerical result of Ceperley and Alder.
So they had calculated this exchange correlation energy
as a function of the density.
And that was actually a computational curve,
a set of dots.
And this is often cited, again, per Perdew and Zunger
in a sort of paper of theirs, among other things
sort of suggested the analytical interpolation of all
the numerical data.
And so you see it's something somehow exotic.
But while it's defined, this is just not even a functional.
It's just a function of the charge density.
So it's something that is very simple
to calculate in practice.
And so at this point, density functional theory,
is a well-defined theory.
So you see 1980, Ceperley and Alder do this quantum Monte
Carlo calculation, find out sort of what is this exchange
correlation energy.
Perdew and Zunger write out a simple interpolation.
1982, sort of the first time that I
think we see sort of where all of this is going,
Marvin Cohen in Berkeley sort of has been
working for two or three years.
Alex Zunger was there.
[INAUDIBLE] him, a number of his students,
they have been able to actually write out
all the electronic structure codes
that are able to solve the density functional equation
for the case of a periodic solid.
And so they address the case of silicon,
sort of the most important material in electronics.
And so what they do is they are able now
to calculate the energy of that system
as a function of the atomic position and, in particular,
as a function of the lattice parameter.
So you know, first thing that they do
is they take silicon in its diamond structure,
so the FCC lattice with two atoms as a basis.
And they calculate that energy as a function
of the lattice parameter.
And it looks something like this.
And then obviously, as you have learned by now,
you look at what is the minimum of that energy.
And that is the theoretical prediction
of the lattice parameter.
And there is [INAUDIBLE],, you know, 1% error.
They look at the second derivative.
This curvature here is really the bulk models
of your problem, again, 5% 10% error.
And then they say, well, let's suppose that we have silicon
not in the diamond phase, but let's suppose
that we have silicon in the beta tin phase.
And so this is also experimentally known.
And we know in the beta tin what is the lattice
parameter of silicon.
And we know from the Maxwell construction
what is the pressure at which we would
have a transition from, say, diamond to beta tin.
And again, you know, I can't remember what was the error,
but it's substantially correct.
And they were able to actually sort of calculate
the sort of complex zoology of all the high pressure
phases of silicon.
And it was in remarkable agreement with the experiment.
So 1982, this is [INAUDIBLE] and Cohen.
But in particular, Marvin Cohen in Berkeley showed that for a--
Marvin Cohen.
For a realistic case, density functional theory
is able really to give us quantity of prediction.
Marvin Cohen has actually become,
this year, the president of the American Physical Society.
OK.
So this is really the beginning of density functional theory
as a practical approach.
And in many ways, what has happened between 1982 and today
is that we have become better and better
at solving the algorithm for this, overall, still
complex computational problem.
And you see a lot of this in the next lecture that follows.
And we have become somewhat better, not really dramatically
better, in calculating that exchange correlation energy.
In a way, sort of the ideas of Kohn and Sham
from 1965 of having a local density approximation
is still very good.
I mean, it's not used nowadays any more that much,
but it's as close as--
what we can do now is not really that much better.
And as you can imagine, sort of what
people have done that was a bit better
was introducing gradients in your problem.
So you're trying to guess what the energy
of an inhomogeneous system comes starting
from what you know about the homogeneous electron gas.
Well, maybe you should somehow throw in into your problem also
the first derivative of the gradient of the density.
And so people did that fairly soon in the early '80s.
And sort of using the gradients was actually much worse.
There was a miracle in the local density approximation
in which the actual expression of the local density
approximation satisfies a lot of symmetry properties
and scaling properties of what would be the exact exchange
correlation functional.
At the time, people put in gradients.
All these sort of symmetries and scaling properties
were sort of thrown to the dogs.
And actually the GGAs--
sorry, the gradient approximation,
were working much, much worse.
And so people need to realize, sort of
in the late '80s, the work of Axel Becke, of John Perdew
especially, a lot of it, you sort of
need to introduce gradients in ways
that still satisfy a lot of these analytical forms.
And in many ways, by now, there is a sort of generalized
exchange correlation functional that's been sort of developed
in the mid-'90s by Perdew, Kieron Burke now at Rutgers,
and Matthias Ernzerhof.
That is called the PBE functional.
That has become sort of the workhorse.
So a lot of the time, you'll see sort
of density functional calculation done
in the PBE, GGA approximation.
But again, these are important improvements,
but if you want just sort of very
little on top of the local density
approximation of the [INAUDIBLE]..
The chemistry community has also sort of done
a number of very intriguing developments.
In particular, there are things that the Hartree-Fock
does very well.
In particular, because you have the sort
of exchange term in Hartree-Fock, you cancel,
remember, the self-interaction say,
in the single-electron problem coming from the Hartree,
the electrostatic problem.
Density functional theory, in theory,
in its exact formulation, would be self-interaction corrected.
But in practice, it is not.
If you solve the hydrogen atom with density functional theory,
you have that the electron interacts
with the charge density created by the same,
by the electron itself.
And so what sort of the quantum chemistry community has done
is, well, they said let's take LDAs, let's actually take GGAs
that seem to work very well.
But let's actually construct an exchange correlation functional
that has a little bit of this, but it's
got also a little bit of what we know worthwhile
in the Hartree-Fock equation.
So they construct hybrid functional
which there are sort of pure density financial terms
and the sort of Hartree-Fock exchange term mixed in.
It makes the equation much more complex.
And if you want, it's a sort of less pure formulation
of density functional theory, but it
can work reasonably well or very well especially, again,
for atoms and molecules.
And this is where we are, basically, with exchange
correlation functional.
I think I'll stop here for today because that's actually
a lot of work.
What we'll start seeing in the next class
is how we actually solve this equation in practice.
On March 8th, you're going to your second lab in which you
will actually study the energy of a solid using
density functional theory.
What I said today is probably the last
of the conceptual lectures.
And I understand that some of it is very complex.
There is reading material posted on the Stella website.
There is the Kohanoff-Gidopoulos paper
on density functional theory.
And some of the readings that I've given are very useful.
The two best books that are also cited
at the end of this lecture are probably the one
by [INAUDIBLE] or the one by [INAUDIBLE]..
And they are both called Density Functional Theory or Density
Functional Theory In Practice.
And they are cited on the last page.
Otherwise, this is it for today and see you next week.