WEBVTT
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NICOLA MARZARI: --for
all our applications
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and for the lab sessions.
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I guess I keep using
Albert Einstein.
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This is the 100th anniversary of
his sort of famous year, 1905,
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so just a little celebration.
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One slide of a
reminder of what we
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have seen in the
previous lecture,
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we have really developed
the formalism leading
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to the Hartree-Fock equations.
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And the Hartree-Fock
equation follow
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from a set of very simple
and very beautiful path.
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We have the
Schrodinger equation,
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and we have reformulated the
Schrodinger equation in terms
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of the variational principle.
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So we have a functional.
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And we know that we can
throw into that functional
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any arbitrary wave function,
and it'll give us an expectation
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value of the energy.
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And sort of the closer we get
to the true ground state wave
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function, the lower that
energy is going to be.
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We are never going to go
below the ground state energy.
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And so it's sort of a
very powerful approach
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to try out sorts of
possibilities and solution.
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And in particular, sort
of Hartree and Fock
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took this approach.
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They wrote sort of the
most general many-body wave
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function that can be written
as a product of single particle
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orbitals.
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That was actually the
original Hartree solution.
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Wave functions
written as data do not
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satisfy a fundamental symmetry
of interacting fermions.
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That is they are
not anti-symmetric.
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And so what you do?
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You take this product of
single particle orbitals,
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and you sum it with all
the possible permutation,
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with all the possible
signs in front,
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so that the overall wave
function is anti-symmetric.
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And that can be sort
of written compactly
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as what is called as
later determinant here.
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And basically, now our unknowns
are the n orbitals phi.
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And so we need to determine the
shape of this n single particle
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orbitals.
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And we want to
determine them such
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that they minimize
the expectation value
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of the variational principle.
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And so that leads basically to
a set of differential equation
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is just functional analysis.
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And when you ask yourself
what are the conditions
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that those single
particle orbitals need
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to satisfy in order to minimize
the variational principle,
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well, this is it, the
Hartree-Fock equation.
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So each single particle
orbital phi of lambda
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need to satisfy basically a
Schrodinger-like equation.
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Again, as always, there is
a kinetic energy term here.
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There is the interaction
with the external potential
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that is just the
potential of the nuclei.
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And then come the
so-called mean field terms.
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So the electron lambda
here will interact
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with each and every
other electron
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mu via an electrostatic
interaction.
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You see phi star times phi
is the charge density coming
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from the orbital mu.
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And the field that the
electron lambda fills
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is the electrostatic
average density.
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And in these, we sum
over all the electrons
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including the electron lambda.
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So up to now, we have a system
that is self-interacting.
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An electron lambda fills the
electrostatic interaction
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with itself.
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That, in principle,
is not correct.
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But luckily, this next term
that is called the exchange term
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cancels that exactly.
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And the exchange term is
the direct consequence
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of having written the trial
wave function not just
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as a product of a
single particle orbital,
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because up to now
we would have sort
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of something closer to
the Hartree equation,
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but written as a proper
anti-symmetric wave
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function, summing on all
the permutation with them
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appropriate signs.
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And so, basically, we have
Schrodinger-like equation.
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A great advantage with respect
to the Hartree equation
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is now the operator doesn't
change depending on the index
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lambda because this sums
if you want to go over all
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the electrons including lambda.
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So our only constraint
here is that we
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need to find the n
lowest Eigen state
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of this single
differential equation.
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So if we have n
electrons, if you want,
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it's not that we have n
different differential
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equation, like it was the
case of the Hartree equation.
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But we have an identical
differential equation
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that is written here.
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And we need to find the
n lowest energy states.
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And those will be our
single-particle orbitals.
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In all of this, we have started
from a variational principle.
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So it's very easy to go
beyond the Hartree-Fock.
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We can say, in large
r variational cluster,
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we can add more
Slater determinants
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with sort of different
coefficients.
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We can try to construct a
more complex wave function.
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And that solution will
become better and better.
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Or we can sort of use
the perturbation theory.
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And so quantum
chemistry has developed
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a number of techniques that are
post-Hartree-Fock techniques
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that become systematically
more and more accurate.
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They are also more
and more expensive.
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And that's if you want, the main
limitation of that direction.
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What we'll see
today is something,
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as they say in Monty Python,
completely different.
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And that will be sort of
density functional theory.
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That's, if you
want a theory that
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starts from a very different set
of hypotheses, the net result
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will be, again, a set of
single-particle equations.
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The concept are very
similar actually, formally,
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to the Hartree-Fock
equation, but they
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have been derived in a
completely different spirit.
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Density functional
theory tends to be
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less expensive than
Hartree-Fock and, overall,
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tends to be more accurate.
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Especially for solid,
it's much more accurate.
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You'll see when we discuss
case studies the Hartree-Fock
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solution for, say,
interacting electron gas
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or in general for
metals tends to make
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them semiconducting
or insulating-like.
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So Hartree-Fock then works
very poorly for solids.
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And that's why, if you want
density functional theory,
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comes from the solid
state community,
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while Hartree-Fock that tends
to work very well for atoms
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comes from the quantum
chemistry community.
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And all the theory was developed
by Walter Kohn, and coworkers.
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So you'll see the
Hohenberg and Kohn
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theorem, the Kohn [INAUDIBLE]
mapping during the '60s.
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But I would say it's
only during the '70s
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that people started to
be able to actually solve
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interesting cases using
density functional theory.
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And it's really the
beginning of the '80s--
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you'll see some
cases here today--
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in which people started
calculating something that had
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sort of a direct application.
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So we'll see the phrase
diagram of silicon
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as a function of
pressure or volume
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and sort of the first principle
prediction of properties
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of solids.
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Walter Kohn, for the development
of this eventual theory,
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got the Nobel Prize
for chemistry in 1998
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together with John Pople.
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That has been the person
that's been fundamental
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in the development
of Hartree-Fock
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and post-Hartree-Fock
approaches in quantum chemistry.
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OK.
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So let's see sort of what is
the general idea behind density
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functional theory.
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And in many ways,
we'll sort of start
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from ideas that
had been developed
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at the end of the '20s and the
beginning of the '30s, what
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is nowadays called the
Thomas-Fermi approach.
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And again, the
basic idea here is
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that the wave function of a
many-body interacting problem
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is an object that is
too complex to treat.
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And it would be very, very
nice if we could instead try
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to deal with a simple object.
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And sort of one of the choices
could be the charge density.
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So if you want, Thomas
and Fermi independently
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were asking themselves,
well, could we
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try to solve not
really a Schrodinger
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equation in the
many-body wave function,
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but solve something else in
which our only unknown is
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the charge density?
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If you think for a
moment, the charge density
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is one of the sort of
fundamental variables
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in the description of our
interacting electron problem.
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And so this was the question.
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Can we do something just
with the charge density?
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And so what they
did is writing out
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what we would call a
heuristic function.
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That is trying to devise a set
of terms that would give us
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the energy of a set of
electrons in a potential
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just as a functional of
their charge density.
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And so, by now,
you could sort of
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think that some of
the relevant terms
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will be electron-electron
interactions, electron
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interaction.
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And we could write a sort
of electrostatic term,
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like the Hartree term in
the Hartree or Hartree-Fock
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equation, that is just a
functional of the charge
00:09:21.020 --> 00:09:21.640
density.
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So this is sort of fairly easy.
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It's also very easy
to sort of imagine
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what could be the
interaction of the electrons
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with an external potential
through the charge density.
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It will be just the integral
of that external potential
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times the charge density.
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What becomes really
critical is finding
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a functional that will give
us the quantum kinetic energy.
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If you think, in the
Schrodinger equation,
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the quantum kinetic energy is
really the second derivative
00:09:53.660 --> 00:09:55.310
of the wave function.
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And obtaining from
a charge density
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only some insight
into what could
00:10:03.400 --> 00:10:08.330
be the second derivative of the
wave function is very complex.
00:10:08.330 --> 00:10:12.630
If you think for a moment at the
extreme case of a plane wave,
00:10:12.630 --> 00:10:17.860
OK, so a sine and cosine sort
of in space, if you remember,
00:10:17.860 --> 00:10:22.000
the charge density given by
a plane wave is a constant.
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We just multiply the
imaginary exponential
00:10:26.590 --> 00:10:28.330
times its complex conjugate.
00:10:28.330 --> 00:10:30.230
That gives us a constant.
00:10:30.230 --> 00:10:33.250
So all plane waves
lead to a constant,
00:10:33.250 --> 00:10:36.550
but obviously the quantum
kinetic energy of a plane wave
00:10:36.550 --> 00:10:39.790
depends on the wavelength
of that plane wave
00:10:39.790 --> 00:10:43.130
because the second
derivative is what counts.
00:10:43.130 --> 00:10:45.010
So what I'm trying
to say is that, when
00:10:45.010 --> 00:10:51.090
we look at this as a possible
wave function, a function,
00:10:51.090 --> 00:10:56.340
say, of r and the charge
density that comes from this
00:10:56.340 --> 00:10:58.710
is going to be a constant,
this wave function
00:10:58.710 --> 00:11:00.630
times this complex conjugate.
00:11:00.630 --> 00:11:03.510
But the kinetic
energy of this object
00:11:03.510 --> 00:11:07.590
is going to be
minus 1/2 k square--
00:11:07.590 --> 00:11:09.870
sorry, plus 1/2 k square.
00:11:09.870 --> 00:11:13.170
And so there is
really not a good way
00:11:13.170 --> 00:11:16.770
for this extreme case
to correlate its charge
00:11:16.770 --> 00:11:18.330
density to its kinetic energy.
00:11:18.330 --> 00:11:19.870
It's an ill-defined problem.
00:11:19.870 --> 00:11:21.660
And this is really
the difficulty.
00:11:21.660 --> 00:11:23.790
So there isn't really
a good way if you
00:11:23.790 --> 00:11:26.010
wanted to extract
the information
00:11:26.010 --> 00:11:30.150
on the second derivative
from just the charge density.
00:11:30.150 --> 00:11:32.070
Now, [INAUDIBLE] sort
of this objection,
00:11:32.070 --> 00:11:37.620
they tried to find a
reasonable functional, so
00:11:37.620 --> 00:11:41.080
without sort of trying to
get the exact solution,
00:11:41.080 --> 00:11:45.210
but try to find a reasonable
function that would give us
00:11:45.210 --> 00:11:49.410
a good estimate to the quantum
kinetic energy starting
00:11:49.410 --> 00:11:51.750
from the charge density.
00:11:51.750 --> 00:11:54.000
And the solution to
this problem that
00:11:54.000 --> 00:11:56.280
is something very
important is what
00:11:56.280 --> 00:12:00.550
we could call a local
density approximation.
00:12:00.550 --> 00:12:05.850
So the problem here is that we
have a non-homogeneous charge
00:12:05.850 --> 00:12:08.430
density everywhere in space.
00:12:08.430 --> 00:12:11.010
And we try to figure
out what could
00:12:11.010 --> 00:12:16.260
be the quantum kinetic energy
of this non-homogeneous problem.
00:12:16.260 --> 00:12:20.130
And the approximation
that Thomas and Fermi did
00:12:20.130 --> 00:12:24.270
was, well, dividing this
non-homogeneous problem
00:12:24.270 --> 00:12:27.880
in a set of infinitesimal
volume in space.
00:12:27.880 --> 00:12:30.210
So it's a bit difficult
to draw, but suppose
00:12:30.210 --> 00:12:33.450
you have the charge
density coming
00:12:33.450 --> 00:12:36.390
from some atom or some molecule.
00:12:36.390 --> 00:12:41.070
This is a non-homogeneous
charge density distribution.
00:12:41.070 --> 00:12:45.210
Now, what you do is you
divide this in space
00:12:45.210 --> 00:12:47.970
and set a very small
infinitesimal, if you want,
00:12:47.970 --> 00:12:48.930
volume.
00:12:48.930 --> 00:12:53.400
And inside each volume,
the charge density
00:12:53.400 --> 00:12:56.040
can be approximated
as a constant.
00:12:56.040 --> 00:12:57.930
And what Thomas
and Fermi said is,
00:12:57.930 --> 00:13:04.170
well, the contribution coming
from this infinitesimal volume,
00:13:04.170 --> 00:13:08.790
say, the first one to the
overall quantum kinetic energy
00:13:08.790 --> 00:13:14.160
will be given by that volume
times the kinetic energy
00:13:14.160 --> 00:13:20.430
density of the homogeneous
electron gas at that density.
00:13:20.430 --> 00:13:23.790
So if, again, we
partition all space,
00:13:23.790 --> 00:13:27.720
we could have that the density
in this little cube is 0.5.
00:13:27.720 --> 00:13:28.780
Here is 0.6.
00:13:28.780 --> 00:13:30.450
Here is 0.7.
00:13:30.450 --> 00:13:34.170
Outside, it goes to 0.
00:13:34.170 --> 00:13:38.580
But we can actually calculate
in some other way what
00:13:38.580 --> 00:13:42.150
would be the quantum
kinetic energy
00:13:42.150 --> 00:13:44.770
of a homogeneous electron gas.
00:13:44.770 --> 00:13:46.430
That's a problem
that we can solve
00:13:46.430 --> 00:13:49.590
if the homogeneous electron
gas is not interacting.
00:13:49.590 --> 00:13:53.640
And we can solve it numerically
even if it is interacting.
00:13:53.640 --> 00:13:57.300
So we can know what is
the quantum kinetic energy
00:13:57.300 --> 00:14:00.450
of a homogeneous gas
with density 0.5,
00:14:00.450 --> 00:14:03.060
density 0.6, density 0.7.
00:14:03.060 --> 00:14:05.250
And so we can also
know what would
00:14:05.250 --> 00:14:10.380
be the quantum kinetic energy
per unit of volume of data.
00:14:10.380 --> 00:14:14.010
And so we'll say that this
non-homogeneous system in blue
00:14:14.010 --> 00:14:18.180
will have an overall quantum
kinetic energy that is given
00:14:18.180 --> 00:14:21.810
really by the integral
across space--
00:14:21.810 --> 00:14:23.280
and it's written here--
00:14:23.280 --> 00:14:29.250
of the quantum kinetic energy
of the homogeneous electron gas
00:14:29.250 --> 00:14:31.170
integrated over space.
00:14:31.170 --> 00:14:35.040
And, say, for the
non-interacting electron gas,
00:14:35.040 --> 00:14:36.960
it's actually very easy to do.
00:14:36.960 --> 00:14:42.120
So if you have a non-interacting
electron gas at a density rho,
00:14:42.120 --> 00:14:46.860
its quantum kinetic energy
is just the rho to the 2/3
00:14:46.860 --> 00:14:49.470
that then integrated
times the unit volume
00:14:49.470 --> 00:14:52.350
gives us rho to the 5/3.
00:14:52.350 --> 00:14:57.180
So by integrating this quantity,
we would get an approximation.
00:14:57.180 --> 00:14:59.670
This approximation
is basically exact
00:14:59.670 --> 00:15:03.780
in the limit of our
homogeneous system, obviously.
00:15:03.780 --> 00:15:07.140
And it will be
sort of quite good
00:15:07.140 --> 00:15:11.400
in the limit of our
non-homogeneous system that
00:15:11.400 --> 00:15:15.420
has a very slowly
changing charge density.
00:15:15.420 --> 00:15:18.300
The more, if you
want, inhomogeneous
00:15:18.300 --> 00:15:23.700
your system becomes, the less
accurate this approximation is.
00:15:23.700 --> 00:15:27.060
And of course, something
like an atom or a molecule
00:15:27.060 --> 00:15:28.950
is a very inhomogeneous system.
00:15:28.950 --> 00:15:31.050
You go with a
charge density that
00:15:31.050 --> 00:15:35.610
goes from 0 to very high
volumes close to the core
00:15:35.610 --> 00:15:36.270
of the nuclei.
00:15:40.600 --> 00:15:43.740
So this is basically
the overall answer
00:15:43.740 --> 00:15:47.430
for the overall expression
that Thomas and Fermi
00:15:47.430 --> 00:15:52.080
postulated for the energy
of an inhomogeneous system.
00:15:52.080 --> 00:15:54.330
They were saying,
well, suppose that we
00:15:54.330 --> 00:16:00.090
have a system that has a certain
distribution of charge rho.
00:16:00.090 --> 00:16:04.290
Without trying to solve the
Schrodinger equation finding
00:16:04.290 --> 00:16:07.860
out the wave function and sort
of go through that the very
00:16:07.860 --> 00:16:12.510
complex many-body route, we
can actually sort of postulate
00:16:12.510 --> 00:16:15.300
that the energy could
be written, again,
00:16:15.300 --> 00:16:17.340
as an electrostatic energy.
00:16:17.340 --> 00:16:20.790
You see sort of each
infinitesimal volume
00:16:20.790 --> 00:16:23.880
interacting with each
other infinitesimal volume
00:16:23.880 --> 00:16:27.850
times via 1 over R
electrostatic interaction.
00:16:27.850 --> 00:16:29.730
Then we have got an
external potential.
00:16:29.730 --> 00:16:32.950
Again, it's usually the
Coulombic field of the nuclei.
00:16:32.950 --> 00:16:34.890
And so the interaction
between the electron
00:16:34.890 --> 00:16:38.040
and that external potential
is just trivial given by rho
00:16:38.040 --> 00:16:38.970
times v.
00:16:38.970 --> 00:16:42.270
And the difficult term,
the quantum kinetic energy,
00:16:42.270 --> 00:16:46.650
has been calculated with a
local density approximation.
00:16:46.650 --> 00:16:49.920
And this is the term that's
not going to be very good,
00:16:49.920 --> 00:16:53.130
again, because it's very
difficult to figure out
00:16:53.130 --> 00:16:56.040
what could be the curvature
of our wave function
00:16:56.040 --> 00:16:59.670
just from the density that
that wave function produces.
00:16:59.670 --> 00:17:03.570
But anyhow, this is a very
simple expression to deal with.
00:17:03.570 --> 00:17:07.020
So for any external
potential v, we
00:17:07.020 --> 00:17:12.000
can try to find out the rho
that minimizes this expression.
00:17:12.000 --> 00:17:14.915
And this will be our
Thomas-Fermi solution.
00:17:23.010 --> 00:17:24.810
There are obviously
a number of problems.
00:17:24.810 --> 00:17:26.910
I'll show you in a
moment an example of what
00:17:26.910 --> 00:17:30.780
the Thomas-Fermi solution
would give to an atom.
00:17:30.780 --> 00:17:32.610
First of all, I
mean, there is really
00:17:32.610 --> 00:17:35.140
no theoretical basis to this.
00:17:35.140 --> 00:17:37.790
It's what we call a
heuristic derivation.
00:17:37.790 --> 00:17:39.690
Thomas and Fermi
just wrote out what
00:17:39.690 --> 00:17:42.900
could be a reasonable
energy functional,
00:17:42.900 --> 00:17:47.220
and then tried to sort of see
what results it would give.
00:17:47.220 --> 00:17:50.940
But there hasn't been any
kind of formal derivation
00:17:50.940 --> 00:17:52.050
of that functional.
00:17:52.050 --> 00:17:55.740
It's not like the Hartree-Fock
equation that sort of derive
00:17:55.740 --> 00:17:59.820
just with some analysis from
the variational principle.
00:17:59.820 --> 00:18:02.850
Another problem is that,
again, it doesn't really
00:18:02.850 --> 00:18:08.010
sort of introduce the concept
of anti-symmetry that fermions
00:18:08.010 --> 00:18:11.160
need to have, the fact that the
many-body wave function needs
00:18:11.160 --> 00:18:14.430
to be anti-symmetric
upon exchange.
00:18:14.430 --> 00:18:17.910
But you know, there is
no conceptual problem
00:18:17.910 --> 00:18:23.580
in adding an exchange energy
to the previous functional.
00:18:23.580 --> 00:18:26.520
Using the same
concept, the same idea
00:18:26.520 --> 00:18:29.610
of local density
approximation, suppose that we
00:18:29.610 --> 00:18:32.280
want to add an exchange term.
00:18:32.280 --> 00:18:37.380
Well, we could look at what
is the exchange energy coming
00:18:37.380 --> 00:18:39.840
from the Hartree-Fock
equation, say,
00:18:39.840 --> 00:18:42.690
for a homogeneous electron gas.
00:18:42.690 --> 00:18:47.040
And that gives us
rho to the 1/3 term.
00:18:47.040 --> 00:18:51.240
And that's basically the
exchange energy density.
00:18:51.240 --> 00:18:54.090
And so for an
inhomogeneous system,
00:18:54.090 --> 00:18:57.240
we are going to sort of
approximate its overall
00:18:57.240 --> 00:19:03.090
exchange energy just by taking
the integral of that energy
00:19:03.090 --> 00:19:08.250
density that is 1/3 times
the sort of local value
00:19:08.250 --> 00:19:09.570
of the charge density.
00:19:09.570 --> 00:19:11.640
And so we have a rho to the 4/3.
00:19:11.640 --> 00:19:17.610
And so, again, it's a local
density approximation.
00:19:17.610 --> 00:19:22.860
The great consequence of
having this energy functional
00:19:22.860 --> 00:19:25.080
that depends only
on r is that it
00:19:25.080 --> 00:19:28.260
is absolutely inexpensive
from the computational point
00:19:28.260 --> 00:19:28.890
of view.
00:19:28.890 --> 00:19:31.830
The only variable that we
need to be concerned with
00:19:31.830 --> 00:19:36.400
is just the a scalar as a
function of three coordinates.
00:19:36.400 --> 00:19:39.120
That is the density
as a function of rho.
00:19:39.120 --> 00:19:42.390
And it's what we call a
linear scaling system.
00:19:42.390 --> 00:19:45.360
If you double the
size of your system,
00:19:45.360 --> 00:19:49.380
the computational complexity
just becomes double.
00:19:49.380 --> 00:19:51.510
So it has a lot of
very good things,
00:19:51.510 --> 00:19:53.820
but it's got a
fundamental defect.
00:19:53.820 --> 00:19:57.090
Because of that approximation
in the kinetic energy,
00:19:57.090 --> 00:20:02.280
it actually does a very
poor job in describing
00:20:02.280 --> 00:20:04.800
a non-homogeneous system.
00:20:04.800 --> 00:20:09.510
So it would work reasonably
well for something like a metal.
00:20:09.510 --> 00:20:12.570
Suppose that you want
to describe sodium,
00:20:12.570 --> 00:20:15.360
or suppose you want
to describe aluminum.
00:20:15.360 --> 00:20:21.030
Those are system in which
the valence electron produce
00:20:21.030 --> 00:20:23.730
a charge density that
is very homogeneous.
00:20:23.730 --> 00:20:26.880
So a Thomas-Fermi approach
could actually work well.
00:20:26.880 --> 00:20:29.820
And it's actually been used
even very recently sort
00:20:29.820 --> 00:20:33.060
of quite successfully
to describe problems
00:20:33.060 --> 00:20:36.930
like the surfaces of lithium,
the surfaces of aluminum.
00:20:36.930 --> 00:20:41.280
What happens, say, when
these simple metals melt?
00:20:41.280 --> 00:20:43.110
What happens to the
sort of formation
00:20:43.110 --> 00:20:44.850
of defects in aluminum?
00:20:44.850 --> 00:20:46.890
So there are a
number of successes.
00:20:46.890 --> 00:20:50.070
But sort of clear example
of what goes wrong
00:20:50.070 --> 00:20:53.130
is, say, if we study an
inhomogeneous systems
00:20:53.130 --> 00:20:55.090
like the argon atom.
00:20:55.090 --> 00:20:59.340
And again, if we think at the
charge density of the argon
00:20:59.340 --> 00:21:02.640
atom as a function, say,
of the radial distance
00:21:02.640 --> 00:21:04.650
from the center
from the nucleus,
00:21:04.650 --> 00:21:06.490
well, it will look
something like this.
00:21:06.490 --> 00:21:11.280
We have first 1s, and then we
have the 2s, and the 2p shells.
00:21:11.280 --> 00:21:14.310
This is somewhat
a poor depiction
00:21:14.310 --> 00:21:15.750
of that charge density.
00:21:15.750 --> 00:21:18.870
If we try to solve
the argon atom
00:21:18.870 --> 00:21:22.110
with a Thomas-Fermi
approach, all these sort
00:21:22.110 --> 00:21:28.740
of fine structure of the
core shells in the atoms
00:21:28.740 --> 00:21:30.690
is completely washed out.
00:21:30.690 --> 00:21:31.410
OK.
00:21:31.410 --> 00:21:34.980
So it gives you a reasonable
approximation and sort
00:21:34.980 --> 00:21:37.740
of an appropriate decay
of the charge density
00:21:37.740 --> 00:21:41.130
as we move far away, but
a lot of those details
00:21:41.130 --> 00:21:43.330
have completely disappeared.
00:21:43.330 --> 00:21:47.340
And for this reason really
the Thomas-Fermi approach
00:21:47.340 --> 00:21:50.310
wasn't developed
beyond the '30s apart
00:21:50.310 --> 00:21:52.980
from some of these
recent applications
00:21:52.980 --> 00:21:55.890
for the very specific
case of solids
00:21:55.890 --> 00:21:59.340
that have a very
homogeneous charge density.
00:21:59.340 --> 00:22:01.290
The reason why we
describe it here
00:22:01.290 --> 00:22:03.240
is that because,
in many ways, it's
00:22:03.240 --> 00:22:06.720
the grandfather
of the ideas that
00:22:06.720 --> 00:22:10.170
were developed in the '60s
in density functional theory
00:22:10.170 --> 00:22:13.290
and, in particular, the
idea that for a moment
00:22:13.290 --> 00:22:16.680
that we should focus not
on the wave function,
00:22:16.680 --> 00:22:22.830
but on the charge density of the
system as the key ingredient.
00:22:22.830 --> 00:22:25.860
The great difference between
the Thomas-Fermi approach
00:22:25.860 --> 00:22:29.310
and density functional theory is
that density functional theory
00:22:29.310 --> 00:22:30.810
actually is a theory.
00:22:30.810 --> 00:22:33.810
It starts with some
theorems that are proven.
00:22:33.810 --> 00:22:38.700
And then it shows what are
the form of the equations
00:22:38.700 --> 00:22:42.300
that, say, a charge density
need to satisfy in order
00:22:42.300 --> 00:22:44.560
to solve exactly the problem.
00:22:44.560 --> 00:22:46.890
So in many way, density
functional theory
00:22:46.890 --> 00:22:50.490
is, in principle at
least, an exact theory.
00:22:50.490 --> 00:22:54.900
It's sort of writes out what
are the equation that the charge
00:22:54.900 --> 00:22:56.850
density needs to satisfy.
00:22:56.850 --> 00:22:59.670
And those are absolutely
equivalent to a Schrodinger
00:22:59.670 --> 00:23:01.860
equation for the wave function.
00:23:01.860 --> 00:23:03.160
There are some difficulties.
00:23:03.160 --> 00:23:06.900
And this is what we are going
to sort of go into right now.
00:23:06.900 --> 00:23:11.250
But let me first give you
the conceptual framework
00:23:11.250 --> 00:23:15.300
of density functional theory
and how it was derived.
00:23:15.300 --> 00:23:18.730
And as usual, we start from
the Schrodinger equation.
00:23:18.730 --> 00:23:19.440
OK.
00:23:19.440 --> 00:23:23.580
So we start from the idea
that, in quantum mechanics,
00:23:23.580 --> 00:23:26.940
for any given
external potential,
00:23:26.940 --> 00:23:30.320
you have a well-defined
differential equation.
00:23:30.320 --> 00:23:30.900
OK.
00:23:30.900 --> 00:23:33.120
It's sort of very complex.
00:23:33.120 --> 00:23:35.780
It describes a
many-body wave function.
00:23:35.780 --> 00:23:40.110
So in most practical cases, we
might not be able to solve it,
00:23:40.110 --> 00:23:42.480
but everything is well-defined.
00:23:42.480 --> 00:23:44.370
You have an external potential.
00:23:44.370 --> 00:23:46.230
You have the
differential equation
00:23:46.230 --> 00:23:48.810
that the many-body wave
function needs to satisfy.
00:23:48.810 --> 00:23:51.910
And so, in principle,
you have the solution.
00:23:51.910 --> 00:23:54.270
And so in that sense, sort
of the first statement
00:23:54.270 --> 00:23:55.530
here is summarized.
00:23:55.530 --> 00:23:59.970
For a given external potential
and knowing how many electrons
00:23:59.970 --> 00:24:03.540
are going to fill
this potential,
00:24:03.540 --> 00:24:07.770
our quantum problem is
formally completely defined.
00:24:07.770 --> 00:24:11.110
In principle, the
solution exists unique.
00:24:11.110 --> 00:24:14.610
We might not be able to
calculate it, but it exists.
00:24:14.610 --> 00:24:18.270
And once we know the many-body
wave function, that solution,
00:24:18.270 --> 00:24:22.980
we know everything about
our quantum system.
00:24:22.980 --> 00:24:23.700
OK.
00:24:23.700 --> 00:24:27.960
So this is, if you want, the
trivial part of the conclusion.
00:24:27.960 --> 00:24:31.110
That is, given an
external potential,
00:24:31.110 --> 00:24:35.400
we find, by the Schrodinger
equation, the wave function.
00:24:35.400 --> 00:24:39.000
The wave function determine all
the properties of our system
00:24:39.000 --> 00:24:42.600
and, in particular, determine
the ground state charge
00:24:42.600 --> 00:24:43.780
density.
00:24:43.780 --> 00:24:46.710
So there is a
unique pathway that
00:24:46.710 --> 00:24:49.560
starts from the
external potential
00:24:49.560 --> 00:24:52.530
and leads us to the charge
density, the ground state
00:24:52.530 --> 00:24:53.550
charge density.
00:24:53.550 --> 00:24:56.640
Once you have defined
the potential,
00:24:56.640 --> 00:25:00.090
you, in principle,
have uniquely defined
00:25:00.090 --> 00:25:04.180
what is the ground state's
charge density of your system.
00:25:04.180 --> 00:25:07.860
And so in that sense, we say
that the ground state charge
00:25:07.860 --> 00:25:11.250
density, the ground state
energy, and all the properties
00:25:11.250 --> 00:25:14.670
of our system are,
in some complex way,
00:25:14.670 --> 00:25:18.330
a functional of our
external potential
00:25:18.330 --> 00:25:20.070
and the number of electrons.
00:25:20.070 --> 00:25:22.500
Functional, again,
can be anything.
00:25:22.500 --> 00:25:25.200
And in this case, it goes
through the Schrodinger
00:25:25.200 --> 00:25:30.180
equation, nothing sort
of complex at this point.
00:25:30.180 --> 00:25:34.860
The sort of remarkable result
that no one had sort of figured
00:25:34.860 --> 00:25:41.490
out between 1964 and 1965
is that the opposite is also
00:25:41.490 --> 00:25:44.020
true and is not trivial at all.
00:25:44.020 --> 00:25:47.740
So what Hohenberg and Kohn
stated first, actually,
00:25:47.740 --> 00:25:53.310
in 1964, was this, that
the ground state charge
00:25:53.310 --> 00:25:58.710
density is a
fundamental quantity,
00:25:58.710 --> 00:26:01.860
as fundamental as the
external potential.
00:26:01.860 --> 00:26:07.020
And in particular, not only
the external potential,
00:26:07.020 --> 00:26:11.460
the terms uniquely the
ground state's charge density
00:26:11.460 --> 00:26:15.510
of your system, but also
the vice versa is true.
00:26:15.510 --> 00:26:20.250
That is, given a ground
state charge density,
00:26:20.250 --> 00:26:23.760
in principle, one
can prove that there
00:26:23.760 --> 00:26:29.730
is a unique external potential
for which that ground state's
00:26:29.730 --> 00:26:33.270
charge density is the
ground state solution
00:26:33.270 --> 00:26:35.470
for that external potential.
00:26:35.470 --> 00:26:38.160
So if you have the
external potential,
00:26:38.160 --> 00:26:41.940
conceptually it's trivial to
go through the Schrodinger
00:26:41.940 --> 00:26:45.120
equation and its solution
to the charge density.
00:26:45.120 --> 00:26:47.760
What Hohenberg and
Kohn are telling us--
00:26:47.760 --> 00:26:51.060
and I'll just show you a sketch
of the proof in a moment--
00:26:51.060 --> 00:26:56.250
is that in principle, if someone
is giving you a chance density
00:26:56.250 --> 00:26:58.770
and is telling you
this charge density
00:26:58.770 --> 00:27:01.350
is the ground state's
charge density
00:27:01.350 --> 00:27:04.290
of a number of
electrons and electrons
00:27:04.290 --> 00:27:07.000
in an external
potential, in principle,
00:27:07.000 --> 00:27:11.220
what is that external potential
is an information that
00:27:11.220 --> 00:27:15.510
is completely contained
into the charged density.
00:27:15.510 --> 00:27:18.300
And it's not contained
in a trivial way.
00:27:18.300 --> 00:27:21.220
It's not that you can look at
the ground state charge density
00:27:21.220 --> 00:27:24.220
and guess what the
external potentially is.
00:27:24.220 --> 00:27:27.910
And that's where all the
complexity of practical densely
00:27:27.910 --> 00:27:29.680
functional theory comes.
00:27:29.680 --> 00:27:33.050
But from the conceptual and
mathematical point of view,
00:27:33.050 --> 00:27:35.950
these two quantities are
absolutely equivalent.
00:27:35.950 --> 00:27:39.190
From one, you get the
other and vice versa.
00:27:39.190 --> 00:27:45.370
And this sort of vice
versa was not trivial.
00:27:45.370 --> 00:27:47.920
And that is sort of
what is contained
00:27:47.920 --> 00:27:52.120
in the so-called first
Hohenberg and Kohn problem.
00:27:52.120 --> 00:27:54.220
I won't go through
the derivation.
00:27:54.220 --> 00:27:55.540
It's actually very simple.
00:27:55.540 --> 00:27:59.420
I've printed it here in case
you sort of want to read it.
00:27:59.420 --> 00:28:01.870
But it's basically is a
derivation ad absurdum.
00:28:01.870 --> 00:28:03.730
What they are saying
there is that,
00:28:03.730 --> 00:28:08.200
if that external
potential were not unique,
00:28:08.200 --> 00:28:11.440
if there were two
external potential that
00:28:11.440 --> 00:28:15.640
were different and would give
the same ground state energy,
00:28:15.640 --> 00:28:18.070
we would get to absurdum.
00:28:18.070 --> 00:28:20.750
So typical mathematical
demonstration,
00:28:20.750 --> 00:28:23.860
we suppose that there are two
different external potential
00:28:23.860 --> 00:28:26.650
that give the same
ground states as density
00:28:26.650 --> 00:28:29.410
and we show that we arrive
to a conclusion that
00:28:29.410 --> 00:28:30.710
doesn't make sense.
00:28:30.710 --> 00:28:34.630
So there can be only a
single external potential.
00:28:34.630 --> 00:28:35.800
And that's the proof.
00:28:35.800 --> 00:28:37.330
And again, it wasn't trivial.
00:28:37.330 --> 00:28:40.030
I mean, if you want, this
is a very basic statement.
00:28:40.030 --> 00:28:43.450
But it took 40 years
to be formulated.
00:28:43.450 --> 00:28:47.500
And it's actually not
true in other cases
00:28:47.500 --> 00:28:51.370
that to first glance
look very similar.
00:28:51.370 --> 00:28:56.210
Suppose that for a moment we
want to discuss excited states.
00:28:56.210 --> 00:28:58.480
You could say, well, if
I have a charge density
00:28:58.480 --> 00:29:01.840
and I say this is the
charge density of an excited
00:29:01.840 --> 00:29:04.600
electronic state,
maybe I could also
00:29:04.600 --> 00:29:07.720
recover the potential
that has generated there.
00:29:07.720 --> 00:29:09.160
And that's not true, actually.
00:29:09.160 --> 00:29:11.080
So there are sort
of a number of cases
00:29:11.080 --> 00:29:12.640
in which this is not true.
00:29:12.640 --> 00:29:16.360
But for this fundamental sort
of relation between the charge
00:29:16.360 --> 00:29:19.090
density of the ground state
and the external potential,
00:29:19.090 --> 00:29:20.420
this is true.
00:29:20.420 --> 00:29:24.250
So we have sort of moved
away now our attention.
00:29:24.250 --> 00:29:26.440
It's not any more
the many-body wave
00:29:26.440 --> 00:29:29.910
function that we want to focus,
but is the charge density.
00:29:29.910 --> 00:29:34.060
The charge density is as
much a fundamental variable
00:29:34.060 --> 00:29:34.960
of our problem.
00:29:34.960 --> 00:29:36.470
It's not a derived variable.
00:29:36.470 --> 00:29:38.740
It's not something that
comes from the wave function,
00:29:38.740 --> 00:29:40.282
but is something
that we can actually
00:29:40.282 --> 00:29:44.050
focus all our attention into.
00:29:44.050 --> 00:29:49.750
And now, we need to find the
equivalent of the Schrodinger
00:29:49.750 --> 00:29:51.900
equation for the
charged density.
00:29:51.900 --> 00:29:55.600
This is what Schrodinger had
done in the '20s, in 1925.
00:29:55.600 --> 00:29:59.500
He said, this is the equation
that quantum objects satisfy.
00:29:59.500 --> 00:30:02.110
And I'll call it the
Schrodinger equation.
00:30:02.110 --> 00:30:05.110
Now, Hohenberg
and Kohn has shown
00:30:05.110 --> 00:30:08.920
that we don't need to think
in terms of the wave function.
00:30:08.920 --> 00:30:11.170
We can think in terms
of the charge density
00:30:11.170 --> 00:30:13.930
as being the fundamental
descriptor of our quantum
00:30:13.930 --> 00:30:14.710
system.
00:30:14.710 --> 00:30:15.640
What is left?
00:30:15.640 --> 00:30:18.220
They need to show me that
there is an equivalent
00:30:18.220 --> 00:30:19.570
of the Schrodinger equation.
00:30:19.570 --> 00:30:24.700
That is we can write a
density equation that
00:30:24.700 --> 00:30:28.540
is sort of what will give me
the ground state and sort of all
00:30:28.540 --> 00:30:30.230
the properties of the system.
00:30:30.230 --> 00:30:35.590
And that's really the second
Hohenberg and Kohn theorem.
00:30:35.590 --> 00:30:40.960
That is really writing
out the equivalent concept
00:30:40.960 --> 00:30:44.110
of the Schrodinger equation
for the charge density.
00:30:44.110 --> 00:30:47.320
And now, sort of, it
becomes fairly conceptual.
00:30:47.320 --> 00:30:48.100
OK.
00:30:48.100 --> 00:30:52.010
So this is the procedure.
00:30:52.010 --> 00:30:55.300
And all of this in
the next few slides
00:30:55.300 --> 00:30:58.130
is still a conceptual procedure.
00:30:58.130 --> 00:31:03.280
It will describe objects that
are well-defined in principle,
00:31:03.280 --> 00:31:05.560
that are conceptually
well-defined,
00:31:05.560 --> 00:31:09.220
but we still don't
have a clue on what
00:31:09.220 --> 00:31:11.200
they look like in practice.
00:31:11.200 --> 00:31:15.370
And all the sort of density
functional application
00:31:15.370 --> 00:31:17.200
goes through a
procedure that we'll
00:31:17.200 --> 00:31:19.870
see later on that is the
sort of [INAUDIBLE] mapping
00:31:19.870 --> 00:31:23.950
that gives a hint of what
these objects look like.
00:31:23.950 --> 00:31:26.770
But up to now, we are
going to introduce objects
00:31:26.770 --> 00:31:29.380
that are well-defined
in principle,
00:31:29.380 --> 00:31:31.710
but we don't know
how they look like.
00:31:31.710 --> 00:31:34.240
And so that's why somehow
density functional theory
00:31:34.240 --> 00:31:37.180
is a much less intuitive
theory than something
00:31:37.180 --> 00:31:38.750
like Hartree-Fock.
00:31:38.750 --> 00:31:39.250
OK.
00:31:39.250 --> 00:31:42.830
So this is going towards the
second Hohenberg and Kohn
00:31:42.830 --> 00:31:46.840
theorem, defining the
fundamental equation
00:31:46.840 --> 00:31:48.740
for the charge density.
00:31:48.740 --> 00:31:52.000
And this is the step.
00:31:52.000 --> 00:31:57.190
For any charge density
rho, so someone gives you,
00:31:57.190 --> 00:32:01.000
someone draws you, an
arbitrary charge density.
00:32:01.000 --> 00:32:06.160
Well, we know that there
is an external potential
00:32:06.160 --> 00:32:09.640
of which that charge
density is the ground state.
00:32:09.640 --> 00:32:11.290
We don't know what
it is, honestly,
00:32:11.290 --> 00:32:16.450
but we have proven that there
is a unique external potential.
00:32:16.450 --> 00:32:17.320
OK.
00:32:17.320 --> 00:32:21.400
So because there is a
unique external potential,
00:32:21.400 --> 00:32:25.540
there is a many-body
Schrodinger equation
00:32:25.540 --> 00:32:27.790
with that potential in there.
00:32:27.790 --> 00:32:31.480
And there is a
wave function that
00:32:31.480 --> 00:32:33.700
is going to be the
ground state wave
00:32:33.700 --> 00:32:37.220
function of that many-body
Schrodinger equation.
00:32:37.220 --> 00:32:43.930
So given a certain rho, we
know that an external potential
00:32:43.930 --> 00:32:45.040
exists.
00:32:45.040 --> 00:32:47.230
And it's unique.
00:32:47.230 --> 00:32:49.630
It determines a
Schrodinger equation.
00:32:49.630 --> 00:32:53.575
And that Schrodinger equation
determines our ground state
00:32:53.575 --> 00:32:57.130
wave function that we call psi.
00:32:57.130 --> 00:33:03.520
So what we are saying is that,
given a rho, in principle
00:33:03.520 --> 00:33:08.590
that psi, the ground state wave
function of the Schrodinger
00:33:08.590 --> 00:33:10.660
equation in the
external potential
00:33:10.660 --> 00:33:12.610
that is uniquely
defined by the rho,
00:33:12.610 --> 00:33:15.010
is also a well-defined object.
00:33:15.010 --> 00:33:18.520
Again, we don't know what it
is, but it is well-defined.
00:33:18.520 --> 00:33:21.640
And because it's a
well-defined object,
00:33:21.640 --> 00:33:25.510
we can calculate the
expectation value
00:33:25.510 --> 00:33:31.750
of that well-defined object
of the quantum kinetic energy
00:33:31.750 --> 00:33:36.670
minus 1/2 sum over all i
of the second derivatives
00:33:36.670 --> 00:33:40.870
and the electron-electron
interaction, just the 1
00:33:40.870 --> 00:33:44.080
over ri minus rj term.
00:33:44.080 --> 00:33:50.080
So again, this term is, in
principle, well-defined.
00:33:50.080 --> 00:33:56.050
And we call this term the
universal density functional.
00:33:56.050 --> 00:34:02.380
That is for any given
arbitrary rho, i, in principle,
00:34:02.380 --> 00:34:06.970
can define a number that
is this number here.
00:34:06.970 --> 00:34:08.560
I have the rho.
00:34:08.560 --> 00:34:12.429
In principle, from the rho, I
have the external potential.
00:34:12.429 --> 00:34:15.730
From the external potential, I
have the Schrodinger equation.
00:34:15.730 --> 00:34:20.139
In principle, I'm able to solve
that Schrodinger equation found
00:34:20.139 --> 00:34:23.500
in principle the many-body
ground state wave function.
00:34:23.500 --> 00:34:24.880
That will be psi.
00:34:24.880 --> 00:34:28.540
And I can calculate
the expectation value
00:34:28.540 --> 00:34:32.139
of psi of the quantum
kinetic energy
00:34:32.139 --> 00:34:35.739
and of the electron-electron
interaction term,
00:34:35.739 --> 00:34:37.400
all well-defined.
00:34:37.400 --> 00:34:40.810
We have really no clue on how
to calculate because we can't
00:34:40.810 --> 00:34:43.429
really do in practice
any of the steps,
00:34:43.429 --> 00:34:49.880
but this universal function of
the density is well-defined.
00:34:49.880 --> 00:34:52.989
So with this
universal functional
00:34:52.989 --> 00:34:59.350
that is now well-defined,
we can write out something.
00:34:59.350 --> 00:35:08.580
We can write to an energy for
any given external potential
00:35:08.580 --> 00:35:11.820
and for any given
charge density.
00:35:11.820 --> 00:35:14.700
And we write it as this.
00:35:14.700 --> 00:35:19.560
So for any given
charge density, there
00:35:19.560 --> 00:35:25.530
will be a well-defined number
that is this universal density
00:35:25.530 --> 00:35:28.960
functional of beta rho prime.
00:35:28.960 --> 00:35:32.520
And then we add
another term that
00:35:32.520 --> 00:35:37.020
is just trivially the
integral of this v,
00:35:37.020 --> 00:35:42.600
this external potential, times
the charge density rho prime.
00:35:42.600 --> 00:35:46.920
So again, this new
expression that we written
00:35:46.920 --> 00:35:48.960
is well-defined.
00:35:48.960 --> 00:35:54.120
For any rho prime and for
any external potential,
00:35:54.120 --> 00:35:57.120
we can calculate
trivially this term.
00:35:57.120 --> 00:36:01.770
And in principle, we
know what this number is.
00:36:01.770 --> 00:36:08.400
And this is, if you want,
1964, 1965, the reformulation
00:36:08.400 --> 00:36:10.110
of quantum mechanics.
00:36:10.110 --> 00:36:12.870
Because, now,
Hohenberg and Kohn are
00:36:12.870 --> 00:36:17.520
able to prove that there
is a variational principle.
00:36:17.520 --> 00:36:20.700
That is, for this
expression written here,
00:36:20.700 --> 00:36:24.090
for this functional
of rho prime,
00:36:24.090 --> 00:36:29.430
we can prove that
for any rho prime
00:36:29.430 --> 00:36:34.980
that we can throw in the
overall numerical value
00:36:34.980 --> 00:36:37.650
of this expression
is always going
00:36:37.650 --> 00:36:46.110
to be either greater or equal
to the ground state energy
00:36:46.110 --> 00:36:49.300
that we would obtain from
the Schrodinger equation
00:36:49.300 --> 00:36:51.910
in the presence of this
external potential.
00:36:51.910 --> 00:36:56.380
So now, we have a well-defined
density functional.
00:36:56.380 --> 00:36:58.350
So if you have an
external potential,
00:36:58.350 --> 00:37:04.110
the z over r of your atom,
you can try out now not
00:37:04.110 --> 00:37:06.480
wave functions that
are very difficult,
00:37:06.480 --> 00:37:10.140
but you can try
out charge density.
00:37:10.140 --> 00:37:12.900
And the charged
density that gives you
00:37:12.900 --> 00:37:16.410
the lowest expectation
value, the lowest
00:37:16.410 --> 00:37:19.405
value for this functional, will
be the ground state, the charge
00:37:19.405 --> 00:37:19.905
density.
00:37:23.080 --> 00:37:26.530
Small problem, we
have no clue what
00:37:26.530 --> 00:37:29.700
this looks like as a
function of rho prime.
00:37:29.700 --> 00:37:35.170
But if we knew, we would have
a wonderfully simple approach
00:37:35.170 --> 00:37:37.040
to quantum mechanics.
00:37:37.040 --> 00:37:40.960
Now, we don't need to deal
with the many-body complexity.
00:37:40.960 --> 00:37:47.630
We just minimize this expression
as a function of rho prime.
00:37:47.630 --> 00:37:49.720
And again, it's
sort of fairly easy
00:37:49.720 --> 00:37:52.808
to prove this
variational principle.
00:37:52.808 --> 00:37:54.100
But one needs probably to sit--
00:37:54.100 --> 00:37:55.390
I've given you some reading.
00:37:55.390 --> 00:37:57.790
So you're welcome, if you are
really interested in this,
00:37:57.790 --> 00:38:00.610
to go back and read the first
Hohenberg and Kohn theorem
00:38:00.610 --> 00:38:03.460
and read the second
Hohenberg and Kohn theorem.
00:38:03.460 --> 00:38:08.065
But in many ways, the proof
of this second Hohenberg
00:38:08.065 --> 00:38:11.140
and Kohn theorem
can be done again
00:38:11.140 --> 00:38:13.210
through the
variational principle.
00:38:13.210 --> 00:38:16.960
That is, if we have
a certain rho prime,
00:38:16.960 --> 00:38:21.670
well, that, again, uniquely
determines the ground state
00:38:21.670 --> 00:38:22.690
wave function.
00:38:22.690 --> 00:38:26.920
Rho prime will determine
an external potential
00:38:26.920 --> 00:38:29.650
that, in principle, is
different from this.
00:38:29.650 --> 00:38:33.160
But rho prime will determine
an external potential
00:38:33.160 --> 00:38:35.170
and will determine
our wave function
00:38:35.170 --> 00:38:38.750
that is the solution of the
many-body Schrodinger equation.
00:38:38.750 --> 00:38:45.010
And if we take the expectation
value of our Hamiltonian
00:38:45.010 --> 00:38:47.410
with this external
potential in this,
00:38:47.410 --> 00:38:51.100
but evaluated on the
wave function of c prime
00:38:51.100 --> 00:38:53.890
that comes from this
charge density rho prime,
00:38:53.890 --> 00:38:58.150
well, we can show that
this expectation value here
00:38:58.150 --> 00:39:02.470
is just identical to functional
that I have just written.
00:39:02.470 --> 00:39:04.630
And for the
variational principle,
00:39:04.630 --> 00:39:08.380
then it needs to be
greater or equal than E0.
00:39:08.380 --> 00:39:10.620
I won't sort of dwell into that.
00:39:10.620 --> 00:39:13.600
And again, you can look at the
sort of detailed description
00:39:13.600 --> 00:39:16.480
and in sort of some of the
many references that I've given
00:39:16.480 --> 00:39:19.850
or that I've also
posted on the website.
00:39:19.850 --> 00:39:21.970
But what is
conceptually important
00:39:21.970 --> 00:39:24.530
is that we have a new equation.
00:39:24.530 --> 00:39:25.030
OK.
00:39:25.030 --> 00:39:29.860
So 1964, '65, quantum
mechanics turned around.
00:39:29.860 --> 00:39:33.250
We don't have to think at
many-body wave functions.
00:39:33.250 --> 00:39:36.490
We can think just
at charge density.
00:39:36.490 --> 00:39:42.820
And all would be well
apart from this detail,
00:39:42.820 --> 00:39:48.010
that we don't know what
that functional f of rho is.
00:39:48.010 --> 00:39:50.560
And so we have a
conceptual approach,
00:39:50.560 --> 00:39:54.360
but we don't have a
practical approach
00:39:54.360 --> 00:39:57.640
to solve the density functional
reformulation of quantum
00:39:57.640 --> 00:39:58.700
mechanics.
00:39:58.700 --> 00:40:01.540
And this is, if you
wanted, true to this day.
00:40:01.540 --> 00:40:07.090
We don't know what is the
exact form of f of rho.
00:40:07.090 --> 00:40:11.620
If we knew it, sort
of most of our sort
00:40:11.620 --> 00:40:14.170
of quantum mechanical
computational problems
00:40:14.170 --> 00:40:15.460
would be trivially solved.
00:40:15.460 --> 00:40:19.840
Because solving that variational
principle in the charge density
00:40:19.840 --> 00:40:24.070
would be most likely
a trivial thing to do.
00:40:24.070 --> 00:40:27.850
The issue is that not
only we don't know,
00:40:27.850 --> 00:40:31.240
but we have understood a
lot of what that exchange
00:40:31.240 --> 00:40:34.960
correlation-- of what that
universal density functional
00:40:34.960 --> 00:40:35.890
is.
00:40:35.890 --> 00:40:38.500
And it's very complex.
00:40:38.500 --> 00:40:46.150
So it's unlikely that there is
a simple analytical expression
00:40:46.150 --> 00:40:49.750
of it as a function of
the charge density only.
00:40:49.750 --> 00:40:53.950
But, you know, the other
great piece of, if you want,
00:40:53.950 --> 00:40:56.170
quantum engineering
by Walter Kohn
00:40:56.170 --> 00:40:59.950
was finding out a very
good approximation
00:40:59.950 --> 00:41:01.930
to that density functional.
00:41:01.930 --> 00:41:04.210
We don't know what
the exact one is.
00:41:04.210 --> 00:41:06.790
But now, what they
are doing is, well,
00:41:06.790 --> 00:41:12.190
finding out one that is going
to be very, very closely
00:41:12.190 --> 00:41:14.780
similar to the exact one.
00:41:14.780 --> 00:41:18.040
And so they are going to throw
in some physical intuition
00:41:18.040 --> 00:41:20.530
to this problem that
up to now, if you want,
00:41:20.530 --> 00:41:23.860
has been a mathematical problem.
00:41:23.860 --> 00:41:26.480
It's another layer of
complexity in this discussion,
00:41:26.480 --> 00:41:28.120
so I hope I'm not losing you.
00:41:28.120 --> 00:41:31.240
But sort of what
Walter Kohn did--
00:41:31.240 --> 00:41:34.750
I think he had a young postdoc
arriving from Cambridge.
00:41:34.750 --> 00:41:39.562
Lu Sham had just done his PhD
in England and came there.
00:41:39.562 --> 00:41:41.020
And sort of, you
know, he told him,
00:41:41.020 --> 00:41:43.570
I have this new
variational principle.
00:41:43.570 --> 00:41:48.580
Let's see what we can do to make
it into a practical solution.
00:41:48.580 --> 00:41:52.270
I think they were in Santa
Barbara, in San Diego probably,
00:41:52.270 --> 00:41:53.590
at that time.
00:41:53.590 --> 00:41:54.400
OK.
00:41:54.400 --> 00:41:58.480
So this is what they
are going to do.
00:41:58.480 --> 00:42:00.520
Remember, sort of,
what is the problem.
00:42:00.520 --> 00:42:07.510
We need to figure out what
is a reasonable approximation
00:42:07.510 --> 00:42:10.310
to this functional here.
00:42:10.310 --> 00:42:13.270
So what they say
is, well, suppose
00:42:13.270 --> 00:42:17.060
that someone has given
us this charge density.
00:42:17.060 --> 00:42:20.320
So we need, in
principle, to find out
00:42:20.320 --> 00:42:23.260
what would be the many-body
wave function that
00:42:23.260 --> 00:42:27.100
is solution of this external
potential that corresponds
00:42:27.100 --> 00:42:29.810
to this charge density.
00:42:29.810 --> 00:42:31.670
This is going to
be very complex.
00:42:31.670 --> 00:42:38.770
Let's invent a problem
in which electrons do not
00:42:38.770 --> 00:42:41.020
interact between each other.
00:42:41.020 --> 00:42:41.860
OK.
00:42:41.860 --> 00:42:43.840
So electrons-- so
that's the sort
00:42:43.840 --> 00:42:46.840
of main problem in the
Schrodinger equation,
00:42:46.840 --> 00:42:49.720
that electrons interacting
with each other
00:42:49.720 --> 00:42:53.710
introduce the two-body
electrostatic repulsion
00:42:53.710 --> 00:42:55.610
in the Schrodinger equation.
00:42:55.610 --> 00:42:57.310
And that's what
makes it difficult.
00:42:57.310 --> 00:42:59.710
Well, what Kohn and
Sham say is let's
00:42:59.710 --> 00:43:04.030
for a moment suppose that
there is a system of electrons
00:43:04.030 --> 00:43:05.020
that don't interact.
00:43:05.020 --> 00:43:08.410
So the only thing that those
so-called Kohn and Sham
00:43:08.410 --> 00:43:13.000
electrons fill is the
external potential.
00:43:13.000 --> 00:43:13.990
OK.
00:43:13.990 --> 00:43:17.920
So those Kohn and Sham
electrons will solve,
00:43:17.920 --> 00:43:23.110
will satisfy, a Schrodinger
equation that is much simpler.
00:43:23.110 --> 00:43:26.260
Because there is no
electron-electron interaction.
00:43:26.260 --> 00:43:29.770
Those Kohn and Sham electron,
the only thing that they fill
00:43:29.770 --> 00:43:33.130
is a new potential.
00:43:33.130 --> 00:43:36.620
And they will have their
own quantum kinetic energy.
00:43:36.620 --> 00:43:45.040
So what they are saying is, for
any given charge density rho,
00:43:45.040 --> 00:43:49.930
there is going to be
a non-interacting set
00:43:49.930 --> 00:43:56.440
of electrons who's ground
state charge density is
00:43:56.440 --> 00:43:58.750
identical to rho.
00:43:58.750 --> 00:44:02.260
So we have said, if we
have a charge density rho,
00:44:02.260 --> 00:44:06.190
you can all go through, find
out the external potential that
00:44:06.190 --> 00:44:08.260
comes from rho, the
Schrodinger equation,
00:44:08.260 --> 00:44:12.340
the many-body interacting
electrons solution.
00:44:12.340 --> 00:44:13.960
But now, what we
are going to say
00:44:13.960 --> 00:44:20.080
is we can also think at a system
of non-interacting electrons.
00:44:20.080 --> 00:44:24.490
And we want those
non-interacting electrons
00:44:24.490 --> 00:44:31.450
to fill a potential that is
such that their ground state is
00:44:31.450 --> 00:44:34.090
going to give us a
charge density that
00:44:34.090 --> 00:44:38.050
is identical to the charge
density I'm dealing with.
00:44:38.050 --> 00:44:38.740
OK.
00:44:38.740 --> 00:44:41.710
And we call that
external potential
00:44:41.710 --> 00:44:43.810
the Kohn-Sham potential.
00:44:43.810 --> 00:44:44.590
OK.
00:44:44.590 --> 00:44:48.490
So now, for a
charge density, you
00:44:48.490 --> 00:44:51.010
don't only have to think
at all the complexity
00:44:51.010 --> 00:44:53.230
that I've discussed up to now.
00:44:53.230 --> 00:44:56.650
But you have also to think
that, for a charge density,
00:44:56.650 --> 00:44:58.750
there is going to be
this set of Kohn and Sham
00:44:58.750 --> 00:45:00.310
known interacting electrons.
00:45:00.310 --> 00:45:03.790
And there is going to
be a potential that
00:45:03.790 --> 00:45:07.030
is called the Kohn and
Sham potential that is such
00:45:07.030 --> 00:45:10.480
that the ground state
of the Schrodinger
00:45:10.480 --> 00:45:12.820
equation for non-interacting
electron, that
00:45:12.820 --> 00:45:15.497
is without electron-electron
interaction, in that Kohn
00:45:15.497 --> 00:45:19.240
and Sham potential will give
us a wave function and a ground
00:45:19.240 --> 00:45:24.010
state that leads to a charge
density identical to the charge
00:45:24.010 --> 00:45:27.250
density I'm sort
of dealing with.
00:45:27.250 --> 00:45:27.910
OK.
00:45:27.910 --> 00:45:29.750
What do we do with this?
00:45:29.750 --> 00:45:35.980
Well, at this stage, there is
a sort of great simplification
00:45:35.980 --> 00:45:39.040
that, for the
Schrodinger equation
00:45:39.040 --> 00:45:43.330
of non-interacting
electron, we actually know
00:45:43.330 --> 00:45:45.500
what is the exact solution.
00:45:45.500 --> 00:45:47.950
So it's actually
very simple to solve
00:45:47.950 --> 00:45:52.630
a Schrodinger equation in which
the electrons do not interact.
00:45:52.630 --> 00:45:56.320
Because, now, this
later determinant
00:45:56.320 --> 00:45:59.720
is actually the exact solution.
00:45:59.720 --> 00:46:02.530
So if you have a set of
non-interacting electrons,
00:46:02.530 --> 00:46:05.020
you don't have the
electron-electron term
00:46:05.020 --> 00:46:08.380
in that Schrodinger equation,
this later determinant is not
00:46:08.380 --> 00:46:11.410
only a good approximation,
but it's actually
00:46:11.410 --> 00:46:14.650
the exact solution.
00:46:14.650 --> 00:46:18.670
So for this non-interactive
set of electrons,
00:46:18.670 --> 00:46:21.970
we can solve everything exactly.
00:46:21.970 --> 00:46:25.030
And in particular, we
can calculate, say,
00:46:25.030 --> 00:46:28.090
what is the kinetic
energy of this set
00:46:28.090 --> 00:46:30.880
of non-interacting electrons.
00:46:30.880 --> 00:46:31.810
OK.
00:46:31.810 --> 00:46:39.070
So now, we can sort of have
a somehow pseudo-physical way
00:46:39.070 --> 00:46:43.510
of decompose this
mysterious density
00:46:43.510 --> 00:46:48.010
functional into different terms.
00:46:48.010 --> 00:46:50.800
So what we are actually
doing via the Kohn and Sham
00:46:50.800 --> 00:46:56.440
mapping is extracting from
here terms that are very large
00:46:56.440 --> 00:47:00.430
and that we know how to write,
we know how to calculate.
00:47:00.430 --> 00:47:07.060
And then, hopefully, once we
have extracted all these terms
00:47:07.060 --> 00:47:10.960
that we know how to define,
we remain with something
00:47:10.960 --> 00:47:14.200
that is very small
and that we'll
00:47:14.200 --> 00:47:18.430
find another numerical
approximation for it.
00:47:18.430 --> 00:47:22.410
So Kohn and Sham say, well, we
have this well-defined density
00:47:22.410 --> 00:47:23.400
functional.
00:47:23.400 --> 00:47:27.240
We extract two terms
that are well-defined.
00:47:27.240 --> 00:47:31.320
And these two terms that's
sort of the great achievement
00:47:31.320 --> 00:47:35.160
actually contain most of
the physics of our problem.
00:47:35.160 --> 00:47:38.710
And the sort of small
term that is left over,
00:47:38.710 --> 00:47:41.260
we are going to approximate
in some simple way.
00:47:41.260 --> 00:47:43.500
And actually, the
approximation that they found
00:47:43.500 --> 00:47:44.775
worked very well.
00:47:44.775 --> 00:47:46.650
And that's why some of
this functional theory
00:47:46.650 --> 00:47:48.540
became a practical theory.
00:47:48.540 --> 00:47:52.410
And so in this sort
of density functional,
00:47:52.410 --> 00:47:57.300
the first physical large
term that they extract
00:47:57.300 --> 00:48:00.510
is the quantum
kinetic energy that we
00:48:00.510 --> 00:48:03.720
call Ts not of the real system.
00:48:03.720 --> 00:48:06.790
Because, again, even
if it's well-defined,
00:48:06.790 --> 00:48:09.220
we don't know how to do that.
00:48:09.220 --> 00:48:14.130
But what they were able to write
is the quantum kinetic energy
00:48:14.130 --> 00:48:18.250
of this non-interacting problem.
00:48:18.250 --> 00:48:20.490
So for a given
charge density, there
00:48:20.490 --> 00:48:24.540
is this set of Kohn and Sham
non-interacting electrons
00:48:24.540 --> 00:48:30.060
that lives in a potential, such
that they have the same ground
00:48:30.060 --> 00:48:31.650
state charge density.
00:48:31.650 --> 00:48:33.900
And their kinetic
energy is trivial
00:48:33.900 --> 00:48:36.480
because it's going to be
just the kinetic energy
00:48:36.480 --> 00:48:39.870
of this later determinant, just
the sum of a single particle
00:48:39.870 --> 00:48:40.650
term.
00:48:40.650 --> 00:48:42.870
So for a charged
density now, there
00:48:42.870 --> 00:48:46.680
is a well-defined
quantum kinetic energy
00:48:46.680 --> 00:48:50.140
that is not the true quantum
kinetic energy of the system,
00:48:50.140 --> 00:48:52.080
but is the quantum
kinetic energy
00:48:52.080 --> 00:48:55.890
of this sort of
associated system
00:48:55.890 --> 00:48:57.900
of non-interacting electrons.
00:48:57.900 --> 00:49:00.720
But this term is
now well-defined.
00:49:00.720 --> 00:49:04.590
They say, well, let's
extract another term that
00:49:04.590 --> 00:49:08.880
is well-defined that is just
a Hartree electrostatic energy
00:49:08.880 --> 00:49:10.920
of a charge density
distribution.
00:49:10.920 --> 00:49:11.670
OK.
00:49:11.670 --> 00:49:14.880
So if we look at the charge
density distribution in which
00:49:14.880 --> 00:49:16.950
each infinitesimal
volume interacts
00:49:16.950 --> 00:49:19.320
with each other
infinitesimal volume
00:49:19.320 --> 00:49:21.180
with an electrostatic
interaction,
00:49:21.180 --> 00:49:23.430
that's going to be the term.
00:49:23.430 --> 00:49:27.960
And you know, what we
are left is now something
00:49:27.960 --> 00:49:31.560
that they call the
exchange correlation term.
00:49:31.560 --> 00:49:33.430
That is everything else.
00:49:33.430 --> 00:49:34.230
OK.
00:49:34.230 --> 00:49:38.010
So F, in principle,
is an exact quantity.
00:49:38.010 --> 00:49:43.290
We are now able to define our
quantum kinetic energy term.
00:49:43.290 --> 00:49:47.280
That is an exact quantity,
but is not really
00:49:47.280 --> 00:49:49.710
the quantum kinetic
energy of the true system.
00:49:49.710 --> 00:49:52.020
But we sort of
say, you know, this
00:49:52.020 --> 00:49:55.050
is going to be equal
to a well-defined term
00:49:55.050 --> 00:49:58.650
plus another well-defined
term plus a third term
00:49:58.650 --> 00:49:59.730
that we don't know.
00:49:59.730 --> 00:50:01.980
So we have sort of
decomposed a quantity
00:50:01.980 --> 00:50:06.780
that we have no clue what it is
into three terms of which two
00:50:06.780 --> 00:50:08.430
terms are well-defined.
00:50:08.430 --> 00:50:12.840
And all our cluelessness is
contained in the third term.
00:50:12.840 --> 00:50:16.650
And we call this third term
the exchange correlation,
00:50:16.650 --> 00:50:19.320
but the sort of
physical advantage
00:50:19.320 --> 00:50:22.200
of having done this
is that it turns out
00:50:22.200 --> 00:50:26.850
that these two terms capture
a lot of the complexity
00:50:26.850 --> 00:50:28.230
of your problem.
00:50:28.230 --> 00:50:31.810
And this term tends
to be fairly small.
00:50:31.810 --> 00:50:32.580
OK.
00:50:32.580 --> 00:50:34.680
So that's all, actually.
00:50:34.680 --> 00:50:37.770
That's why it works very
well because somehow they
00:50:37.770 --> 00:50:42.430
manage to capture the
complexity of our system.
00:50:42.430 --> 00:50:53.470
And so once that exchange
correlation term is defined
00:50:53.470 --> 00:50:59.370
and it's approximated in some
way that we'll see in a moment,
00:50:59.370 --> 00:51:04.950
our problem is now
well-defined because we really
00:51:04.950 --> 00:51:06.900
have a variational principle.
00:51:06.900 --> 00:51:09.780
Remember, the universal
density functional
00:51:09.780 --> 00:51:12.870
plus the external
potential plus the charge
00:51:12.870 --> 00:51:15.330
density in the field of
the external potential
00:51:15.330 --> 00:51:18.750
minimizes the sort of
new variational principle
00:51:18.750 --> 00:51:21.450
that comes from the
Hohenberg and Kohn theorem.
00:51:21.450 --> 00:51:24.120
And so we can write it,
our variational principle.
00:51:24.120 --> 00:51:28.320
That is this quantity
with the constraint
00:51:28.320 --> 00:51:31.020
that the number of
electrons should be
00:51:31.020 --> 00:51:33.420
equal to n should be minimum.
00:51:33.420 --> 00:51:37.350
And as usual, when you sort of
write a variational principle,
00:51:37.350 --> 00:51:40.680
you are saying that sort of the
differential of that quantity
00:51:40.680 --> 00:51:42.330
needs to be equal to 0.
00:51:42.330 --> 00:51:45.120
Or if you want, I mean,
this is a generic term.
00:51:45.120 --> 00:51:48.360
You have a set of what are
called Euler-Lagrange equation,
00:51:48.360 --> 00:51:49.050
basically.
00:51:49.050 --> 00:51:51.120
That is nothing else than
differential analysis.
00:51:51.120 --> 00:51:54.030
That is you are
asking yourself, what
00:51:54.030 --> 00:51:56.370
are going to be
the conditions that
00:51:56.370 --> 00:51:59.730
need to be satisfied
by the charge density
00:51:59.730 --> 00:52:03.270
in order to satisfy the
variational principle?
00:52:03.270 --> 00:52:06.030
There is always this sort
of 1 to 1 correspondence.
00:52:06.030 --> 00:52:07.710
You have a
variational principle.
00:52:07.710 --> 00:52:09.840
It gives you
differential equation.
00:52:09.840 --> 00:52:11.550
Or you have
differential equation.
00:52:11.550 --> 00:52:14.010
You can rewrite them in
a variational principle.
00:52:14.010 --> 00:52:16.230
We have seen that for
the Schrodinger equation.
00:52:16.230 --> 00:52:21.660
And we see this, in particular,
now explicitly for the Kohn
00:52:21.660 --> 00:52:23.100
and Sham orbital.
00:52:23.100 --> 00:52:28.230
So I'll actually go directly
to the explicit expression
00:52:28.230 --> 00:52:31.290
of the Kohn and Sham orbitals.
00:52:31.290 --> 00:52:33.750
Again, remember that
what we have done
00:52:33.750 --> 00:52:38.820
is we have defined a
variational principle that
00:52:38.820 --> 00:52:41.970
acts on a universal
density functional
00:52:41.970 --> 00:52:46.320
F plus the charge density
and the external potential.
00:52:46.320 --> 00:52:48.780
And we have decomposed,
we have extracted,
00:52:48.780 --> 00:52:51.660
from this universal
functional sort of terms
00:52:51.660 --> 00:52:53.770
that are large and physical.
00:52:53.770 --> 00:52:57.720
And we have sort of pushed
all the many-body complexity
00:52:57.720 --> 00:53:01.050
of the problem in something
that we call the exchange
00:53:01.050 --> 00:53:02.760
correlation functional.
00:53:02.760 --> 00:53:05.680
That is, again, a functional
of the charge density.
00:53:05.680 --> 00:53:08.580
We don't know yet what that
function of the charge density
00:53:08.580 --> 00:53:09.120
is.
00:53:09.120 --> 00:53:10.940
But luckily, it's
going to be small.
00:53:10.940 --> 00:53:13.360
So in a moment,
we'll approximate it.
00:53:13.360 --> 00:53:19.050
And then we ask ourselves,
what are the differential
00:53:19.050 --> 00:53:22.620
equations that derive from
this variational principle?
00:53:22.620 --> 00:53:25.540
Well, in principle, I
had written them here.
00:53:25.540 --> 00:53:26.470
OK.
00:53:26.470 --> 00:53:28.500
We just need to take the
variation with respect
00:53:28.500 --> 00:53:31.380
to the charge density
and imposing the Lagrange
00:53:31.380 --> 00:53:33.180
multiplication constraint.
00:53:33.180 --> 00:53:38.670
And so this would be basically
that the charge density needs
00:53:38.670 --> 00:53:42.780
to satisfy this set of
equation, the sort of functional
00:53:42.780 --> 00:53:45.510
derivative of this
non-interacting quantum
00:53:45.510 --> 00:53:50.100
kinetic energy plus a
number of terms that
00:53:50.100 --> 00:53:52.590
really contain the
external potential,
00:53:52.590 --> 00:53:53.940
the Hartree interaction.
00:53:53.940 --> 00:53:56.100
And the exchange
correlation need
00:53:56.100 --> 00:53:58.770
to be equal to the
Lagrange multiplier that
00:53:58.770 --> 00:54:03.870
fixes the number of electrons.
00:54:03.870 --> 00:54:08.940
We are not able to calculate
this functional derivative
00:54:08.940 --> 00:54:11.940
because, remember, the
quantum kinetic energy
00:54:11.940 --> 00:54:14.340
of the non-interacting
system is again
00:54:14.340 --> 00:54:16.823
written as a later determinant.
00:54:16.823 --> 00:54:18.240
And so there is
sort of, you know,
00:54:18.240 --> 00:54:23.730
this type of back in which, even
if we had written everything
00:54:23.730 --> 00:54:26.970
in terms of a charge
density, we are not
00:54:26.970 --> 00:54:32.040
able to explicitly
calculate not only
00:54:32.040 --> 00:54:36.450
the derivative of the true
interacting electron's
00:54:36.450 --> 00:54:38.190
kinetic energy with
respect to rho,
00:54:38.190 --> 00:54:40.680
but we are not even able
to calculate the functional
00:54:40.680 --> 00:54:43.590
derivative of the
non-interacting kinetic energy
00:54:43.590 --> 00:54:44.890
with respect to rho.
00:54:44.890 --> 00:54:49.530
But what we are able is actually
to calculate the derivative
00:54:49.530 --> 00:54:53.640
of that non-interacting
kinetic energy with respect
00:54:53.640 --> 00:54:57.940
to the orbitals that describe
the Kohn and Sham electrons.
00:54:57.940 --> 00:55:00.660
Remember that these
non-independent Kohn and Sham
00:55:00.660 --> 00:55:04.530
electrons have an exact solution
that is a later determinant.
00:55:04.530 --> 00:55:07.560
And so we know there are trivial
many-body wave function is
00:55:07.560 --> 00:55:12.210
a later determinant composed
by a single particle orbitals.
00:55:12.210 --> 00:55:17.490
And the functional derivative
of that independent
00:55:17.490 --> 00:55:20.070
non-interacting
electron's kinetic energy
00:55:20.070 --> 00:55:23.160
with respect to the
single-particle orbital
00:55:23.160 --> 00:55:28.710
is now trivial and is
just minus 1/2 del square.
00:55:28.710 --> 00:55:34.050
So at the end of all these sort
of complex formulation, what
00:55:34.050 --> 00:55:36.810
we are left with is
something very simple
00:55:36.810 --> 00:55:39.720
and probably something you
should focus your attention
00:55:39.720 --> 00:55:40.860
from now on.
00:55:40.860 --> 00:55:45.210
We have now a set of
Kohn and Sham equation
00:55:45.210 --> 00:55:47.480
that are the
differential equation
00:55:47.480 --> 00:55:51.260
that the electrons need
to satisfy in order
00:55:51.260 --> 00:55:55.690
to satisfy the variational
principle with the caveat
00:55:55.690 --> 00:55:58.430
that, in this Kohn
and Sham equation,
00:55:58.430 --> 00:56:02.210
there is at term, an
exchange correlation term,
00:56:02.210 --> 00:56:04.460
that we still don't
know what it is.
00:56:04.460 --> 00:56:07.430
It's sort the formally defined
as the functional derivative
00:56:07.430 --> 00:56:09.170
of the exchange
correlation energy
00:56:09.170 --> 00:56:11.000
with respect to
the charge density.
00:56:11.000 --> 00:56:14.330
But we'll need to
approximate somewhere.
00:56:14.330 --> 00:56:19.610
And what this equation
describes is not anymore
00:56:19.610 --> 00:56:21.890
the true electrons
in your system,
00:56:21.890 --> 00:56:25.970
but they describe this
cousins of the true electrons.
00:56:25.970 --> 00:56:31.190
They describe this Kohn and
Sham non-interacting electrons
00:56:31.190 --> 00:56:34.670
that have their
own orbital psi i.
00:56:34.670 --> 00:56:38.870
And that will give us
a ground state charge
00:56:38.870 --> 00:56:42.470
density that, if the exchange
correlation functional was
00:56:42.470 --> 00:56:46.940
exact, would be not only this,
as obviously the same ground
00:56:46.940 --> 00:56:50.930
state energy of our
interacting electron system,
00:56:50.930 --> 00:56:55.920
but it would be the exact
solution of the problem.
00:56:55.920 --> 00:56:56.550
OK.
00:56:56.550 --> 00:57:02.040
So this equation look a lot
like a Schrodinger equation.
00:57:02.040 --> 00:57:06.057
They look a lot, if you want,
like the Hartree-Fock equation
00:57:06.057 --> 00:57:07.140
that we've written before.
00:57:07.140 --> 00:57:12.370
Because what we are seeing
is that the a Kohn and Sham
00:57:12.370 --> 00:57:18.210
electron i fills a quantum
kinetic energy operator,
00:57:18.210 --> 00:57:23.370
fills a Hartree operator,
fills the external potential,
00:57:23.370 --> 00:57:28.590
and then fills this
sort of remaining term
00:57:28.590 --> 00:57:31.830
that is the exchange
correlation potential.
00:57:31.830 --> 00:57:36.900
Again, if we knew what were
this exact exchange correlation
00:57:36.900 --> 00:57:40.750
potential, we would have an
exact solution to the problem.
00:57:40.750 --> 00:57:43.120
But we know a very
good approximation.
00:57:43.120 --> 00:57:46.290
And then if you want
finding the ground state,
00:57:46.290 --> 00:57:48.690
it's not very different
from finding the ground
00:57:48.690 --> 00:57:52.320
state of the Hartree-Fock
equation with the caveat
00:57:52.320 --> 00:57:55.080
that actually this
term here is going
00:57:55.080 --> 00:57:59.580
to be much simpler
than the exchange term
00:57:59.580 --> 00:58:01.350
of the Hartree-Fock equation.
00:58:01.350 --> 00:58:03.330
If you go back to
the first slide
00:58:03.330 --> 00:58:06.420
to the Hartree-Fock
equation, the last term
00:58:06.420 --> 00:58:10.290
is that numerically
very complex expression
00:58:10.290 --> 00:58:13.260
in which we sort of
take the orbital,
00:58:13.260 --> 00:58:18.720
and we put it inside an
integral differential operator.
00:58:18.720 --> 00:58:21.015
Now, it's become simpler.
00:58:21.015 --> 00:58:22.140
And that's all if you want.
00:58:22.140 --> 00:58:25.560
So the Kohn and Sham
equation look very similar.
00:58:25.560 --> 00:58:28.380
In practice, they
are simpler to solve.
00:58:28.380 --> 00:58:32.010
They tend to be more
accurate in most cases.
00:58:32.010 --> 00:58:34.690
And that's, at the end,
what leads to the success.
00:58:34.690 --> 00:58:37.140
But what is critical
for all of this
00:58:37.140 --> 00:58:40.470
is having a reasonable
approximation to the exchange
00:58:40.470 --> 00:58:41.910
correlation potential.
00:58:41.910 --> 00:58:44.610
If we had the exact exchange
correlation potential,
00:58:44.610 --> 00:58:48.420
everything would be exact
in this formulation.
00:58:48.420 --> 00:58:53.130
We would find the Kohn and
Sham independent electrons
00:58:53.130 --> 00:58:58.920
that were sort of the ground
state electrons for that charge
00:58:58.920 --> 00:59:01.740
density that is ultimately
equal to the charge
00:59:01.740 --> 00:59:04.500
density of the
interacting electrons
00:59:04.500 --> 00:59:06.470
in this external potential.
00:59:10.800 --> 00:59:11.670
OK.
00:59:11.670 --> 00:59:19.080
And we have the Euler-Lagrange
or Kohn and Sham differential
00:59:19.080 --> 00:59:20.940
equation in the previous page.
00:59:20.940 --> 00:59:24.480
I written here sort of, you
know, just for reference
00:59:24.480 --> 00:59:29.440
also what would be the
total energy of the system.
00:59:29.440 --> 00:59:34.410
And usually, if you had
an independent electron,
00:59:34.410 --> 00:59:37.710
the total energy of
the system is trivially
00:59:37.710 --> 00:59:42.890
the sum of each of the
single particle energies.
00:59:42.890 --> 00:59:43.440
OK.
00:59:43.440 --> 00:59:45.690
If you have 10
electrons and they
00:59:45.690 --> 00:59:47.430
don't interact with
each other, you
00:59:47.430 --> 00:59:50.970
can calculate what is the energy
of each of these 10 electrons.
00:59:50.970 --> 00:59:54.630
Sum all of them, and that's
the total energy of the system.
00:59:54.630 --> 00:59:57.960
In this case, it's more complex.
00:59:57.960 --> 01:00:00.270
And the total
energy of the system
01:00:00.270 --> 01:00:02.490
can't be really written
as that, but it's
01:00:02.490 --> 01:00:05.790
got other terms that depend
on the charge density.
01:00:05.790 --> 01:00:08.670
That's sort of this
is, in summary, what
01:00:08.670 --> 01:00:10.300
your total energy is.
01:00:10.300 --> 01:00:13.800
And again, there's nothing
else than kinetic energy term
01:00:13.800 --> 01:00:17.490
sort of a Hartree term
functional charge density,
01:00:17.490 --> 01:00:19.470
this exchange
correlation functional,
01:00:19.470 --> 01:00:23.010
and the interaction between
the external potential
01:00:23.010 --> 01:00:24.280
and the charge density.
01:00:24.280 --> 01:00:30.560
But this is actually
different from the sum
01:00:30.560 --> 01:00:32.600
of the eigenvalues.
01:00:32.600 --> 01:00:40.610
That would be the sum of the
expectation values of psi i
01:00:40.610 --> 01:00:48.330
calculated on the
single-particle orbital where
01:00:48.330 --> 01:00:53.020
T is, again, just the simple
quantum kinetic energy.
01:00:53.020 --> 01:00:56.590
And the VKS is this
Kohn and Sham potential.
01:00:56.590 --> 01:01:00.450
So if you want to calculate the
total energy of your system,
01:01:00.450 --> 01:01:03.210
even if it's made of
independent electron,
01:01:03.210 --> 01:01:07.470
you can't sum just a
single particle orbitals.
01:01:07.470 --> 01:01:10.260
But you have to sort of
deal with this expression.
01:01:10.260 --> 01:01:13.920
Nothing complex in this, it's
just sort of a caveat that
01:01:13.920 --> 01:01:15.795
is relevant when you
want to sort of-- you
01:01:15.795 --> 01:01:18.180
know, this is the reason
why we can't really
01:01:18.180 --> 01:01:20.280
find out the equivalent
of the [INAUDIBLE]
01:01:20.280 --> 01:01:22.060
theorems for Hartree-Fock.
01:01:22.060 --> 01:01:27.690
This is why at the end these
single-particle energies
01:01:27.690 --> 01:01:32.010
are ultimately not
physically meaningful.
01:01:32.010 --> 01:01:35.850
They sort of don't give us
the total energy of the system
01:01:35.850 --> 01:01:40.250
just by taking the
sum over all of them.
01:01:40.250 --> 01:01:41.030
OK.
01:01:41.030 --> 01:01:45.470
So in order to make this
into a practical algorithm,
01:01:45.470 --> 01:01:51.440
the only part that remains
is finding an approximation
01:01:51.440 --> 01:01:55.250
to that exchange correlation
term, to that last term.
01:01:55.250 --> 01:01:58.250
Remember, we had sort of
defined this density functional.
01:01:58.250 --> 01:02:01.520
We have been able to extract
two meaningful terms,
01:02:01.520 --> 01:02:06.290
the Hartree electrostatic energy
and the non-interacting Kohn
01:02:06.290 --> 01:02:07.850
and Sham kinetic energy.
01:02:07.850 --> 01:02:11.390
And we have said what is left
is a function of the charge
01:02:11.390 --> 01:02:12.860
density that we
call the exchange
01:02:12.860 --> 01:02:15.110
correlation functional.
01:02:15.110 --> 01:02:17.540
How we are going to
approximate that?
01:02:17.540 --> 01:02:21.270
Well, we go back to
the Thomas-Fermi idea.
01:02:21.270 --> 01:02:24.260
We are going to
do a local density
01:02:24.260 --> 01:02:29.370
approximation to that exchange
correlation functional.
01:02:29.370 --> 01:02:32.150
So again, what we
want to calculate
01:02:32.150 --> 01:02:38.360
is the exchange correlation
energy for any arbitrary charge
01:02:38.360 --> 01:02:39.560
density.
01:02:39.560 --> 01:02:41.420
Sometimes I call the
charge density n.
01:02:41.420 --> 01:02:43.430
Sometimes I call the
charge density rho,
01:02:43.430 --> 01:02:45.240
but they are always the same.
01:02:45.240 --> 01:02:47.100
So how do we do this?
01:02:47.100 --> 01:02:51.090
Well, we don't have
the full solution.
01:02:51.090 --> 01:02:53.120
But what we can
say, again, is that,
01:02:53.120 --> 01:02:58.760
for an inhomogeneous charge
density that changes values
01:02:58.760 --> 01:03:02.660
and then drops to 0, I
can calculate the exchange
01:03:02.660 --> 01:03:06.860
correlation energy for this
charge density distribution
01:03:06.860 --> 01:03:12.260
by sort of decomposing it
in infinitesimal volumes.
01:03:12.260 --> 01:03:15.140
Inside each
infinitesimal volume,
01:03:15.140 --> 01:03:18.800
I can say the charge
density is constant.
01:03:18.800 --> 01:03:22.790
And you see, I make a local
density approximation.
01:03:22.790 --> 01:03:28.640
That is I say the contribution
to the overall exchange
01:03:28.640 --> 01:03:32.600
correlation energy of
this inhomogeneous system
01:03:32.600 --> 01:03:34.130
can be broken down.
01:03:34.130 --> 01:03:39.110
And each infinitesimal volume
will give its own contribution
01:03:39.110 --> 01:03:42.352
to the total exchange
correlation density.
01:03:42.352 --> 01:03:44.060
You know, in principle,
it's not correct.
01:03:44.060 --> 01:03:48.500
I mean, our problem doesn't
have to be local in any way.
01:03:48.500 --> 01:03:51.440
Actually, as people say,
this exchange correlation
01:03:51.440 --> 01:03:53.300
functional, the true
one, although we
01:03:53.300 --> 01:03:57.680
don't know what it is, we
know that is ultra non-local.
01:03:57.680 --> 01:04:01.310
So it can't be
decomposed into terms
01:04:01.310 --> 01:04:04.040
that independently sum up.
01:04:04.040 --> 01:04:06.380
So in principle,
we can't do this.
01:04:06.380 --> 01:04:09.260
But in practice, it tends
to be a good approximation
01:04:09.260 --> 01:04:11.110
for a lot of cases.
01:04:11.110 --> 01:04:15.740
And so what is going to be the
contribution to the exchange
01:04:15.740 --> 01:04:19.730
correlation energy from
this infinitesimal volume?
01:04:19.730 --> 01:04:24.500
Let's say the charge
density there is 0.5.
01:04:24.500 --> 01:04:27.590
Well, what we need
to do is we need
01:04:27.590 --> 01:04:32.270
to find out what is the
exchange correlation
01:04:32.270 --> 01:04:35.750
energy for the
homogeneous electron
01:04:35.750 --> 01:04:37.780
gas that is at this density.
01:04:37.780 --> 01:04:41.630
That's something that, with
some advanced computational
01:04:41.630 --> 01:04:45.860
techniques, we can actually
find out almost exactly.
01:04:45.860 --> 01:04:51.260
So we would know, if we had
a homogeneous charge density
01:04:51.260 --> 01:04:54.320
0.5 everywhere, what
would be the charge
01:04:54.320 --> 01:04:56.480
density per unit volume.
01:04:56.480 --> 01:05:01.400
And we can find out what is
the exchange correlation charge
01:05:01.400 --> 01:05:06.270
density per unit volume not only
for 0.5, 0.6, 0.7, anything.
01:05:06.270 --> 01:05:08.990
And what we are saying is
that, in this non-homogeneous
01:05:08.990 --> 01:05:12.110
problem, we construct the
overall exchange correlation
01:05:12.110 --> 01:05:15.990
energy by summing up
these different pieces.
01:05:15.990 --> 01:05:20.360
And so this is what Ceperley
and Alder did in 1980.
01:05:20.360 --> 01:05:27.200
They basically found out what
was the almost exact sort
01:05:27.200 --> 01:05:30.950
of closely to numerical
exact solution
01:05:30.950 --> 01:05:33.560
for the homogeneous
electron gas.
01:05:33.560 --> 01:05:38.690
That is for a system in
which you have only electrons
01:05:38.690 --> 01:05:42.320
homogeneously, so the
charge density is identical
01:05:42.320 --> 01:05:43.230
everywhere.
01:05:43.230 --> 01:05:45.270
And those electrons interact.
01:05:45.270 --> 01:05:49.040
So you can calculate the energy
of this interacting electron
01:05:49.040 --> 01:05:52.530
problem exactly as a
function of the density.
01:05:52.530 --> 01:05:53.030
OK.
01:05:53.030 --> 01:05:56.810
So you change the density
in your sort of volume,
01:05:56.810 --> 01:05:58.850
and you find out
what is this energy.
01:05:58.850 --> 01:06:02.060
And then you can
calculate, for any
01:06:02.060 --> 01:06:06.140
of this density, what is
the Kohn and Sham quantum
01:06:06.140 --> 01:06:06.890
kinetic energy.
01:06:06.890 --> 01:06:11.880
You can find out what is the
Hartree electrostatic energy.
01:06:11.880 --> 01:06:15.980
And so you can also find
out, for this specific case
01:06:15.980 --> 01:06:19.850
of the homogeneous
gas, numerically
01:06:19.850 --> 01:06:23.210
what would be the exchange
correlation density.
01:06:23.210 --> 01:06:26.050
And so that's
basically a function.
01:06:26.050 --> 01:06:29.240
So for the homogeneous gas,
that is for the case in which n
01:06:29.240 --> 01:06:33.710
doesn't depend on
r, people found out
01:06:33.710 --> 01:06:39.650
what was basically this
exchange correlation energy.
01:06:39.650 --> 01:06:43.200
It was calculated as a function.
01:06:43.200 --> 01:06:49.370
This is a function of
what people call rs.
01:06:49.370 --> 01:06:55.080
rs is the radius of a sphere
that contains one electron.
01:06:55.080 --> 01:06:57.560
So it's a sort of
inverse quantity
01:06:57.560 --> 01:06:59.240
with respect to the density.
01:06:59.240 --> 01:07:02.630
So numerical calculation, what
are called quantum Monte Carlo
01:07:02.630 --> 01:07:07.490
calculation, really solved
the interacting Schrodinger
01:07:07.490 --> 01:07:08.790
equation problem.
01:07:08.790 --> 01:07:12.830
But for the specific case
of an electron gas that
01:07:12.830 --> 01:07:15.200
has a homogeneous
density, they were
01:07:15.200 --> 01:07:17.880
able to do that for
various density.
01:07:17.880 --> 01:07:22.760
And so, now, we have a function
for the homogeneous problem.
01:07:22.760 --> 01:07:27.080
For the non-homogeneous
problem, we take a local density
01:07:27.080 --> 01:07:30.530
approximation, and we say
that the overall exchange
01:07:30.530 --> 01:07:33.830
correlation energy is given
by the integral over all
01:07:33.830 --> 01:07:35.270
the infinitesimal volume.
01:07:35.270 --> 01:07:38.660
And each infinitesimal volume
will have a certain density
01:07:38.660 --> 01:07:43.085
and will contribute
with its own density.
01:07:43.085 --> 01:07:45.980
If the density is going
to be equal to here,
01:07:45.980 --> 01:07:47.990
this will be the value
of the contribution
01:07:47.990 --> 01:07:49.730
of that infinitesimal volume.
01:07:49.730 --> 01:07:52.460
If the density somewhere
else corresponds to this,
01:07:52.460 --> 01:07:54.030
this will be the corresponding.
01:07:54.030 --> 01:07:59.390
So we really match up this
overall exchange correlation
01:07:59.390 --> 01:08:03.050
term from all the little
infinitesimal volume
01:08:03.050 --> 01:08:05.600
exactly as
Thomas-Fermi had done,
01:08:05.600 --> 01:08:13.490
but now we do it for a term
that is a much smaller term
01:08:13.490 --> 01:08:14.900
in our problem.
01:08:14.900 --> 01:08:19.100
Thomas and Fermi had done it
for the quantum kinetic energy.
01:08:19.100 --> 01:08:21.560
Instead, what Kohn
and Sham do, they
01:08:21.560 --> 01:08:25.550
do it for what is left from
the universal functional
01:08:25.550 --> 01:08:28.340
once you have taken
out the electrostatic
01:08:28.340 --> 01:08:31.189
and once you have taken out
the quantum kinetic energy
01:08:31.189 --> 01:08:34.100
of the non-interacting
electrons.
01:08:34.100 --> 01:08:39.590
At this point, if you
want 1980 and even
01:08:39.590 --> 01:08:42.380
before without the
computation, with some sort
01:08:42.380 --> 01:08:45.140
of analytical approximations
to this curve,
01:08:45.140 --> 01:08:48.260
density functional theory
becomes not only a theory,
01:08:48.260 --> 01:08:50.870
but also a practical algorithm.
01:08:50.870 --> 01:08:54.380
We have a set of expression for
the exchange correlation term.
01:08:54.380 --> 01:08:56.569
And so, now, it's just
a matter of trying
01:08:56.569 --> 01:09:00.560
to find out what the solution
to these problems are.
01:09:00.560 --> 01:09:03.529
And because somehow
conceptually we
01:09:03.529 --> 01:09:07.189
start from the
homogeneous electron gas,
01:09:07.189 --> 01:09:12.080
it turns out that this
approach worked especially well
01:09:12.080 --> 01:09:13.250
for solids.
01:09:13.250 --> 01:09:17.029
I mean, the valence
electrons in a solid
01:09:17.029 --> 01:09:22.790
are much less structured than
the electrons in a molecule
01:09:22.790 --> 01:09:24.740
that they need to drop to 0.
01:09:24.740 --> 01:09:27.500
So the charge density
in a solid overall
01:09:27.500 --> 01:09:31.130
varies less dramatically
as a function of space
01:09:31.130 --> 01:09:34.130
than the electron density
in atoms and molecules.
01:09:34.130 --> 01:09:36.200
And these are
actually sort of what
01:09:36.200 --> 01:09:41.189
were summarized the numerical
result of Ceperley and Alder.
01:09:41.189 --> 01:09:44.720
So they had calculated this
exchange correlation energy
01:09:44.720 --> 01:09:46.560
as a function of the density.
01:09:46.560 --> 01:09:48.979
And that was actually
a computational curve,
01:09:48.979 --> 01:09:50.359
a set of dots.
01:09:50.359 --> 01:09:54.620
And this is often cited,
again, per Perdew and Zunger
01:09:54.620 --> 01:09:58.340
in a sort of paper of
theirs, among other things
01:09:58.340 --> 01:10:03.830
sort of suggested the
analytical interpolation of all
01:10:03.830 --> 01:10:05.180
the numerical data.
01:10:05.180 --> 01:10:07.970
And so you see it's
something somehow exotic.
01:10:07.970 --> 01:10:13.010
But while it's defined, this
is just not even a functional.
01:10:13.010 --> 01:10:15.950
It's just a function
of the charge density.
01:10:15.950 --> 01:10:17.900
So it's something
that is very simple
01:10:17.900 --> 01:10:20.160
to calculate in practice.
01:10:20.160 --> 01:10:23.180
And so at this point,
density functional theory,
01:10:23.180 --> 01:10:25.770
is a well-defined theory.
01:10:25.770 --> 01:10:29.420
So you see 1980, Ceperley and
Alder do this quantum Monte
01:10:29.420 --> 01:10:33.230
Carlo calculation, find out
sort of what is this exchange
01:10:33.230 --> 01:10:34.430
correlation energy.
01:10:34.430 --> 01:10:37.130
Perdew and Zunger write
out a simple interpolation.
01:10:37.130 --> 01:10:40.370
1982, sort of the
first time that I
01:10:40.370 --> 01:10:44.000
think we see sort of where
all of this is going,
01:10:44.000 --> 01:10:48.080
Marvin Cohen in Berkeley
sort of has been
01:10:48.080 --> 01:10:49.970
working for two or three years.
01:10:49.970 --> 01:10:51.980
Alex Zunger was there.
01:10:51.980 --> 01:10:54.750
[INAUDIBLE] him, a
number of his students,
01:10:54.750 --> 01:10:56.810
they have been able
to actually write out
01:10:56.810 --> 01:10:58.940
all the electronic
structure codes
01:10:58.940 --> 01:11:03.140
that are able to solve the
density functional equation
01:11:03.140 --> 01:11:05.330
for the case of
a periodic solid.
01:11:05.330 --> 01:11:07.460
And so they address
the case of silicon,
01:11:07.460 --> 01:11:11.510
sort of the most important
material in electronics.
01:11:11.510 --> 01:11:13.790
And so what they do
is they are able now
01:11:13.790 --> 01:11:17.480
to calculate the
energy of that system
01:11:17.480 --> 01:11:21.080
as a function of the atomic
position and, in particular,
01:11:21.080 --> 01:11:23.810
as a function of the
lattice parameter.
01:11:23.810 --> 01:11:25.760
So you know, first
thing that they do
01:11:25.760 --> 01:11:31.190
is they take silicon in
its diamond structure,
01:11:31.190 --> 01:11:35.180
so the FCC lattice with
two atoms as a basis.
01:11:35.180 --> 01:11:38.150
And they calculate that
energy as a function
01:11:38.150 --> 01:11:39.720
of the lattice parameter.
01:11:39.720 --> 01:11:41.610
And it looks
something like this.
01:11:41.610 --> 01:11:44.460
And then obviously, as
you have learned by now,
01:11:44.460 --> 01:11:46.650
you look at what is the
minimum of that energy.
01:11:46.650 --> 01:11:48.510
And that is the
theoretical prediction
01:11:48.510 --> 01:11:49.710
of the lattice parameter.
01:11:49.710 --> 01:11:52.440
And there is [INAUDIBLE],,
you know, 1% error.
01:11:52.440 --> 01:11:55.890
They look at the
second derivative.
01:11:55.890 --> 01:11:58.350
This curvature here is
really the bulk models
01:11:58.350 --> 01:12:02.770
of your problem,
again, 5% 10% error.
01:12:02.770 --> 01:12:05.340
And then they say, well, let's
suppose that we have silicon
01:12:05.340 --> 01:12:07.650
not in the diamond
phase, but let's suppose
01:12:07.650 --> 01:12:11.280
that we have silicon
in the beta tin phase.
01:12:11.280 --> 01:12:13.590
And so this is also
experimentally known.
01:12:13.590 --> 01:12:15.930
And we know in the beta
tin what is the lattice
01:12:15.930 --> 01:12:17.100
parameter of silicon.
01:12:17.100 --> 01:12:19.620
And we know from the
Maxwell construction
01:12:19.620 --> 01:12:24.390
what is the pressure
at which we would
01:12:24.390 --> 01:12:29.310
have a transition from,
say, diamond to beta tin.
01:12:29.310 --> 01:12:32.160
And again, you know, I can't
remember what was the error,
01:12:32.160 --> 01:12:34.140
but it's substantially correct.
01:12:34.140 --> 01:12:38.310
And they were able to
actually sort of calculate
01:12:38.310 --> 01:12:42.450
the sort of complex zoology
of all the high pressure
01:12:42.450 --> 01:12:43.650
phases of silicon.
01:12:43.650 --> 01:12:46.530
And it was in remarkable
agreement with the experiment.
01:12:46.530 --> 01:12:50.040
So 1982, this is
[INAUDIBLE] and Cohen.
01:12:50.040 --> 01:12:57.375
But in particular, Marvin Cohen
in Berkeley showed that for a--
01:12:57.375 --> 01:13:01.620
Marvin Cohen.
01:13:01.620 --> 01:13:05.520
For a realistic case,
density functional theory
01:13:05.520 --> 01:13:09.330
is able really to give us
quantity of prediction.
01:13:09.330 --> 01:13:11.010
Marvin Cohen has
actually become,
01:13:11.010 --> 01:13:15.060
this year, the president of
the American Physical Society.
01:13:15.060 --> 01:13:15.750
OK.
01:13:15.750 --> 01:13:19.380
So this is really the beginning
of density functional theory
01:13:19.380 --> 01:13:21.750
as a practical approach.
01:13:21.750 --> 01:13:27.420
And in many ways, what has
happened between 1982 and today
01:13:27.420 --> 01:13:30.870
is that we have become
better and better
01:13:30.870 --> 01:13:35.370
at solving the algorithm
for this, overall, still
01:13:35.370 --> 01:13:37.440
complex computational problem.
01:13:37.440 --> 01:13:41.940
And you see a lot of this in
the next lecture that follows.
01:13:41.940 --> 01:13:46.740
And we have become somewhat
better, not really dramatically
01:13:46.740 --> 01:13:50.970
better, in calculating that
exchange correlation energy.
01:13:50.970 --> 01:13:55.110
In a way, sort of the
ideas of Kohn and Sham
01:13:55.110 --> 01:13:59.790
from 1965 of having a
local density approximation
01:13:59.790 --> 01:14:01.440
is still very good.
01:14:01.440 --> 01:14:06.660
I mean, it's not used
nowadays any more that much,
01:14:06.660 --> 01:14:09.690
but it's as close as--
01:14:09.690 --> 01:14:13.710
what we can do now is not
really that much better.
01:14:13.710 --> 01:14:15.630
And as you can
imagine, sort of what
01:14:15.630 --> 01:14:18.510
people have done
that was a bit better
01:14:18.510 --> 01:14:21.770
was introducing gradients
in your problem.
01:14:21.770 --> 01:14:24.840
So you're trying to
guess what the energy
01:14:24.840 --> 01:14:28.710
of an inhomogeneous
system comes starting
01:14:28.710 --> 01:14:31.620
from what you know about the
homogeneous electron gas.
01:14:31.620 --> 01:14:35.640
Well, maybe you should somehow
throw in into your problem also
01:14:35.640 --> 01:14:38.530
the first derivative of the
gradient of the density.
01:14:38.530 --> 01:14:42.300
And so people did that fairly
soon in the early '80s.
01:14:42.300 --> 01:14:47.130
And sort of using the gradients
was actually much worse.
01:14:47.130 --> 01:14:51.480
There was a miracle in the
local density approximation
01:14:51.480 --> 01:14:55.080
in which the actual expression
of the local density
01:14:55.080 --> 01:15:00.390
approximation satisfies a
lot of symmetry properties
01:15:00.390 --> 01:15:03.750
and scaling properties of what
would be the exact exchange
01:15:03.750 --> 01:15:05.220
correlation functional.
01:15:05.220 --> 01:15:07.860
At the time, people
put in gradients.
01:15:07.860 --> 01:15:11.730
All these sort of symmetries
and scaling properties
01:15:11.730 --> 01:15:13.830
were sort of thrown to the dogs.
01:15:13.830 --> 01:15:15.990
And actually the GGAs--
01:15:15.990 --> 01:15:18.840
sorry, the gradient
approximation,
01:15:18.840 --> 01:15:21.370
were working much, much worse.
01:15:21.370 --> 01:15:23.610
And so people need
to realize, sort of
01:15:23.610 --> 01:15:28.650
in the late '80s, the work
of Axel Becke, of John Perdew
01:15:28.650 --> 01:15:31.770
especially, a lot
of it, you sort of
01:15:31.770 --> 01:15:35.640
need to introduce
gradients in ways
01:15:35.640 --> 01:15:41.100
that still satisfy a lot
of these analytical forms.
01:15:41.100 --> 01:15:46.320
And in many ways, by now,
there is a sort of generalized
01:15:46.320 --> 01:15:50.100
exchange correlation functional
that's been sort of developed
01:15:50.100 --> 01:15:55.200
in the mid-'90s by Perdew,
Kieron Burke now at Rutgers,
01:15:55.200 --> 01:15:57.540
and Matthias Ernzerhof.
01:15:57.540 --> 01:15:59.250
That is called the
PBE functional.
01:15:59.250 --> 01:16:01.620
That has become sort
of the workhorse.
01:16:01.620 --> 01:16:03.750
So a lot of the
time, you'll see sort
01:16:03.750 --> 01:16:05.460
of density functional
calculation done
01:16:05.460 --> 01:16:09.420
in the PBE, GGA approximation.
01:16:09.420 --> 01:16:13.680
But again, these are
important improvements,
01:16:13.680 --> 01:16:16.470
but if you want
just sort of very
01:16:16.470 --> 01:16:18.900
little on top of
the local density
01:16:18.900 --> 01:16:22.170
approximation of
the [INAUDIBLE]..
01:16:22.170 --> 01:16:25.620
The chemistry community
has also sort of done
01:16:25.620 --> 01:16:29.940
a number of very
intriguing developments.
01:16:29.940 --> 01:16:34.020
In particular, there are
things that the Hartree-Fock
01:16:34.020 --> 01:16:35.340
does very well.
01:16:35.340 --> 01:16:37.350
In particular, because
you have the sort
01:16:37.350 --> 01:16:41.370
of exchange term in
Hartree-Fock, you cancel,
01:16:41.370 --> 01:16:44.610
remember, the
self-interaction say,
01:16:44.610 --> 01:16:47.860
in the single-electron problem
coming from the Hartree,
01:16:47.860 --> 01:16:50.130
the electrostatic problem.
01:16:50.130 --> 01:16:53.080
Density functional
theory, in theory,
01:16:53.080 --> 01:16:56.640
in its exact formulation, would
be self-interaction corrected.
01:16:56.640 --> 01:16:58.470
But in practice, it is not.
01:16:58.470 --> 01:17:02.820
If you solve the hydrogen atom
with density functional theory,
01:17:02.820 --> 01:17:04.740
you have that the
electron interacts
01:17:04.740 --> 01:17:07.950
with the charge density
created by the same,
01:17:07.950 --> 01:17:09.970
by the electron itself.
01:17:09.970 --> 01:17:14.880
And so what sort of the quantum
chemistry community has done
01:17:14.880 --> 01:17:20.400
is, well, they said let's take
LDAs, let's actually take GGAs
01:17:20.400 --> 01:17:22.140
that seem to work very well.
01:17:22.140 --> 01:17:26.520
But let's actually construct an
exchange correlation functional
01:17:26.520 --> 01:17:29.535
that has a little
bit of this, but it's
01:17:29.535 --> 01:17:32.280
got also a little bit of
what we know worthwhile
01:17:32.280 --> 01:17:34.150
in the Hartree-Fock equation.
01:17:34.150 --> 01:17:36.780
So they construct
hybrid functional
01:17:36.780 --> 01:17:40.410
which there are sort of
pure density financial terms
01:17:40.410 --> 01:17:45.080
and the sort of Hartree-Fock
exchange term mixed in.
01:17:45.080 --> 01:17:47.770
It makes the equation
much more complex.
01:17:47.770 --> 01:17:52.260
And if you want, it's a sort
of less pure formulation
01:17:52.260 --> 01:17:54.780
of density functional
theory, but it
01:17:54.780 --> 01:17:58.590
can work reasonably well or
very well especially, again,
01:17:58.590 --> 01:18:00.990
for atoms and molecules.
01:18:00.990 --> 01:18:04.950
And this is where we are,
basically, with exchange
01:18:04.950 --> 01:18:06.660
correlation functional.
01:18:06.660 --> 01:18:09.150
I think I'll stop here for
today because that's actually
01:18:09.150 --> 01:18:10.300
a lot of work.
01:18:10.300 --> 01:18:12.690
What we'll start seeing
in the next class
01:18:12.690 --> 01:18:17.010
is how we actually solve
this equation in practice.
01:18:17.010 --> 01:18:20.310
On March 8th, you're going to
your second lab in which you
01:18:20.310 --> 01:18:23.250
will actually study the
energy of a solid using
01:18:23.250 --> 01:18:24.900
density functional theory.
01:18:24.900 --> 01:18:27.180
What I said today
is probably the last
01:18:27.180 --> 01:18:28.710
of the conceptual lectures.
01:18:28.710 --> 01:18:30.945
And I understand that some
of it is very complex.
01:18:33.540 --> 01:18:36.880
There is reading material
posted on the Stella website.
01:18:36.880 --> 01:18:39.030
There is the
Kohanoff-Gidopoulos paper
01:18:39.030 --> 01:18:40.590
on density functional theory.
01:18:40.590 --> 01:18:44.280
And some of the readings that
I've given are very useful.
01:18:44.280 --> 01:18:46.800
The two best books
that are also cited
01:18:46.800 --> 01:18:49.410
at the end of this lecture
are probably the one
01:18:49.410 --> 01:18:51.990
by [INAUDIBLE] or the
one by [INAUDIBLE]..
01:18:51.990 --> 01:18:54.990
And they are both called Density
Functional Theory or Density
01:18:54.990 --> 01:18:56.610
Functional Theory In Practice.
01:18:56.610 --> 01:18:59.520
And they are cited
on the last page.
01:18:59.520 --> 01:19:04.880
Otherwise, this is it for
today and see you next week.