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NICOLA MARZARI:
Welcome, everyone.

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Lecture 8-- hopefully this
will be a somehow simpler class

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to follow than last class.

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That was sort of
heavily theoretical.

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What we are going to
do today is really,

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after two or three slides of
recap of what we have seen

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in the past lecture,
we'll go into really sort

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of practical application of
density functional theory.

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We'll first discuss how
you set up calculations.

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That is, what are
the parameters,

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and what are the
objects that you

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manipulate when
you do an electron

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[INAUDIBLE] calculation?

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And then we start seeing
a number of examples

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to give you the feeling
for the properties

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that we can calculate
using density function

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theory for the accuracy
and for the pitfalls

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you need to be careful with.

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And in the next
class on Thursday,

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we'll finish up with
more case studies.

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So let me actually
spend three slides

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to recap what we have
seen in the last lecture.

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Starting somehow from
this, that, I would say,

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is the most fundamental one.

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And it's really rewriting
quantum mechanics.

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So this is really the
Schrodinger equation

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from the '60s, density
functional theory.

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And this is what it says--
that we don't really

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need to solve a many-body
differential equation to find

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that the ground state charge
density, and the ground state

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energy, and the ground state
properties for a system.

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But we can actually solve it
using a variational principle

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on the charge density,
and the functional

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that has to be minimized
is written here.

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That is, for any given
charged density and prime,

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we have, at least in principle,
a well defined functional,

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and we need to change to vary
the charge density and prime.

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And the minimum value that
is functional will take

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is actually the
ground state energy.

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Somehow, sort of the
weakness of all of this

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is that everything is
well defined in principle,

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but it doesn't work in practice.

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To remind you what were the
sort of conceptual steps, what

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Hohenberg and Kohn proved
from their first theorem

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is that, given any charge
density and prime--

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more or less any charge
density and prime,

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an external potential v
prime is well defined,

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for which that charge density
is going to be the ground state

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charge density.

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This was sort of the inverse
part of the first Hohenberg

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and Kohn theorem.

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And then, at least in
principle, the solution

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to the Schrodinger equation
corresponding to v prime

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is well defined.

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And we call that c prime.

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And so one can construct a
functional f that is just

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the expectation value
of p-- psi prime

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over the many-body
kinetic energy

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operator plus the many-body
electron-electron interaction.

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And so this here is the well
defined, at least in theory,

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density functional
term that is here.

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And the other term
is just the integral

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of the external potential
times the charge density.

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So everything is well
defined in principle,

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and so in this, it's a
perfect reformulation

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of the Schrodinger equation.

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The real problem
is that we don't

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have an exact representation
of this universal density

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functional.

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And so what Walter Kohn
and Lu Sham did was they

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tried to figure out
an approximation

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to get a universal functional.

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And so in the total
energy functional,

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these were-- this was their
choice for the approximation.

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That is what they
said-- is, well,

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let's try to figure out what
are the most relevant physical

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terms in this functional, and
then we'll sort of leave--

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sweep under the rug all
the many-body complexity

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of this problem.

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And we'll call data an
exchange correlation function.

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It is really where the fine
detail of your solutions are.

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And so what they said, in
this universal functional,

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we can extract terms like the
Hartree electrostatic energy,

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written here, that is a
very simple and well defined

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functional of the
charge density.

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The sort of second term
was the tricky one,

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because they wanted
to figure out

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what was a significant
contribution

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to the overall quantum
kinetic energy.

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And as we have seen,
there isn't really

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a good way to extract kinetic
energy that's basically

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a second derivative of the
wave function-- of curvature

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of the wave function
from a charged density.

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The sort of most remarkable
case is that of a plane wave.

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A plane wave of any wavelength
gives you a constant charge

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density, but gives you a very
different second derivative,

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very different kinetic energy.

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And so what Kohn and Sham did--

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they introduced a related system
of non-interacting electrons

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that, for a given charge density
n, they would have the same--

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sorry, they would have
the same ground state

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energy of our original problem.

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But because these are
non-interacting electrons,

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one can define exactly what
is the quantum kinetic energy

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of this set of
non-interacting electrons.

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So if you want, this
is just a definition,

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but it turns out to be a
definition of the second piece

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that makes our third piece
here very, very small.

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So by extracting, selecting
the green and the red term,

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we were left with an
exchange correlational term

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that, if you want, still
contain all the complexity

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of this problem that it's
well defined in principle,

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but we don't know how to solve.

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But for this, then, they
applied that the same idea

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of Thomas and Fermi.

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That is, they said,
well, maybe we

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could try to approximate
this exchange correlation

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term with a local
density approximation.

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That is, we try to construct
this energy term by integrating

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an infinitesimal volume
by infinitesimal volume,

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and each infinitesimal
volume will contribute

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to this many-body problem
and exchange correlation

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energy density that is the
exchange correlation energy

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density of the interacting
homogeneous electron

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gas at that density.

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So when, in 1980,
Ceperley and Alder

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first did very complex
calculation-- quantum Monte

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Carlo calculation
of the interacting,

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but homogeneous electron gas
as a function of the density,

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this functional, at
least in principle,

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was, for the first
time, parameterized

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over a whole set of density.

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Often we call that the
Perdew-Zunger parameterization,

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and so we had finally
a working expression.

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Once we have a
working expression,

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then the problem of
finding the ground state

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becomes our electronic
structure computational problem.

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That is how to find the
minimum of this functional.

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Note the very significant
difference with respect

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to Hartree-Fock.

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In Hartree-Fock, we had
a variational principle,

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and that led us
to our expression

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for the energy for the
Hartree-Fock equation,

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in which there was a well
defined exchange term.

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It was written somehow
complicated-- complex, but well

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defined.

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But because it was
basically coming

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from a variational principle,
we had the possibility

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of making Hartree-Fock
better and better

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by extending our class
of search function

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from Slater determinant
to more flexible classes.

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We could just take,
say, combination

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of Slater determinant.

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So Hartree-Fock, in principle,
can be made better and better

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in a systematic way.

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And the computational cost that
you will pay is horrendous,

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but it gives you an avenue.

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Density functional theory
doesn't give you an avenue.

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It sort of monolithically states
that there is going to be,

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in principle, a well defined
universal functional,

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and in the Kohn and Sham
sort of the composition that

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is going to be in particular
one single given exchange

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correlation functional,
but it doesn't give us

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any sort of systematic
route to find

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better and better functional.

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And that's why for many
years there wasn't really

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much applied work using
density functional theory.

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Sort of in the
early '70s, people

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started studying
very simple system,

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like the surface of a
metal sort of represented

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as a step in a free
electron gas solution.

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And then it's only sort of the
beginning of the '80s, when

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we had the
parameterization of LDA

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from the calculation
of Ceperely and Alder,

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that people could sort of put
together an overall working

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algorithm.

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And I've shown you in the
last class sort of the phase

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diagram of silicon,
the first time

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that we see somehow that this
theory is going to give us

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a very accurate practical result
and will be able to give us

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the lattice parameters,
the bulk models of silicon,

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just by plotting the energy
as a function of the lattice

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parameter, or it will give
us even the phase transition

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to high pressure
phases of silicon.

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What I had written in the
previous slide was the total

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energy in this density
functional paradigm,

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and we had, as a
computational goal now--

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and you see how
we address this--

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the task of minimizing
that total energy.

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And there are two main
routes that we can take.

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Once the exchange
correlation potential

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is written in an
explicit form, this

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is a well defined, even
if nonlinear, functional

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of this independent
single particle

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orbitals-- the Kohn
and Sham orbitals

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of which the charge
density is just

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the sum of the square moduli.

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And so we can look at this as a
nonlinear minimization problem.

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We need to find the orbitals
that minimize this expression,

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or as it often happens, we
can write the associated

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Euler-Lagrange equation.

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We have a variational
principle so we

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can take the functional--
the functional

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differential of that,
and that gives us

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a set of differential equation
that is what we call the Kohn

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and Sham equation.

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And again, they are
written here-- sort

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of not very different.

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If you look at it, from
the Hartree-Fock equations,

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there are really the same terms.

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There is a quantum
kinetic energy.

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And now these are single
particle orbitals.

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There is a Hartree term.

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There is an external
potential term,

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and the only difference
is in the way

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the exchange or exchange
correlation term is calculated.

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In Hartree-Fock, it was
an explicit integral

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of the orbital.

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In density functional
theory, it's

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going to be some kind of complex
function of the charge density.

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And we are going
to try and find out

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what are the eigenstates of this
Hamiltonian, of which the most

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important part that
you need to remember

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is that this Hamiltonian is
what we call self-consistent.

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That is, it's an operator
that actually depends

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on its own solution, because
you see, what we have is

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that the charge density
is given by the sum

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of the square moduli.

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And the charge density goes into
the expression of the Hartree

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operator, and goes into the
expression of the exchange

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correlation operator.

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So the Hamiltonian, acting on
this single particle orbitals,

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depend on the charge density.

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And the charge
density is a function

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of the orbital themselves.

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And so the problem has
become slightly different

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from your usual problem.

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The Schrodinger
equation gives you

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an operator for which you
need to find the eigenvectors.

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Here, you have an operator for
which you find eigenvectors,

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but then these
eigenvectors give you

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a charge density that,
put back in here,

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gives you a different operator.

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And so exactly like
in Hartree-Fock,

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you have found your
ground state solution

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only once you have
become self-consistent.

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You have Hamiltonian
whose eigenstates give you

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a charge density that gives you
the same Hamiltonian you had

249
00:12:49,560 --> 00:12:51,930
calculated the eigenstates for.

250
00:12:51,930 --> 00:12:55,140
And so all the
computational approaches

251
00:12:55,140 --> 00:12:57,390
to solve the density
functional problem,

252
00:12:57,390 --> 00:13:00,810
as those that solve the
Hartree-Fock problem

253
00:13:00,810 --> 00:13:03,780
are iterative approaches.

254
00:13:03,780 --> 00:13:07,170
We can't just find a
solution to an Hamiltonian,

255
00:13:07,170 --> 00:13:12,360
but we really need to make
that problem self-consistent.

256
00:13:12,360 --> 00:13:16,500
So this is where we
concluded in last class.

257
00:13:16,500 --> 00:13:20,130
And so now we'll actually
go into practical feature

258
00:13:20,130 --> 00:13:21,970
of density function theory.

259
00:13:21,970 --> 00:13:27,150
So starting from reminding
you that all the quality

260
00:13:27,150 --> 00:13:30,150
of your calculation
depend ultimately

261
00:13:30,150 --> 00:13:32,400
on the quality of
your functional.

262
00:13:32,400 --> 00:13:35,130
And for many years, the only
functional that was used

263
00:13:35,130 --> 00:13:38,820
was really the local density
approximation functional.

264
00:13:38,820 --> 00:13:43,500
In the late '80s and
early '90s, people

265
00:13:43,500 --> 00:13:46,890
started to develop what are
called generalized gradient

266
00:13:46,890 --> 00:13:48,060
approximation.

267
00:13:48,060 --> 00:13:51,960
That is, they constructed
functionals of the charge

268
00:13:51,960 --> 00:13:55,320
density that didn't only depend
on the charge density itself,

269
00:13:55,320 --> 00:13:57,900
but also on its gradient.

270
00:13:57,900 --> 00:14:00,000
Not in the sense of
a Taylor expansion,

271
00:14:00,000 --> 00:14:02,370
because a Taylor expansion
wouldn't actually

272
00:14:02,370 --> 00:14:05,370
satisfy a number of
symmetry properties

273
00:14:05,370 --> 00:14:08,310
that we know that the
exact exchange correlation

274
00:14:08,310 --> 00:14:09,850
functional would do.

275
00:14:09,850 --> 00:14:12,690
And so in this sense, all
these new approximation

276
00:14:12,690 --> 00:14:15,750
are called generalized
gradient approximation.

277
00:14:15,750 --> 00:14:19,500
And there is a little menagerie
of acronyms and symbols

278
00:14:19,500 --> 00:14:22,260
that really are sort of
build up upon the names

279
00:14:22,260 --> 00:14:25,050
of the few people that have
really done a lot of work

280
00:14:25,050 --> 00:14:26,230
in this field.

281
00:14:26,230 --> 00:14:29,100
And so you'll see a lot
John Perdew in this,

282
00:14:29,100 --> 00:14:33,570
represented by a P, or
Axel Becke in Canada

283
00:14:33,570 --> 00:14:37,680
by a B, or Kieron Burke
in Rutgers by another B.

284
00:14:37,680 --> 00:14:40,450
So these are some of the
most popular functionals.

285
00:14:40,450 --> 00:14:44,420
I would say that, by now,
sort of almost every one

286
00:14:44,420 --> 00:14:49,090
is standardized on PBE as
being the most reasonable

287
00:14:49,090 --> 00:14:52,060
and the most accurate
advanced approximation

288
00:14:52,060 --> 00:14:53,470
beyond local density.

289
00:14:53,470 --> 00:14:56,020
And so this is what we'll use.

290
00:14:56,020 --> 00:15:00,430
There are classes of
more complex functionals.

291
00:15:00,430 --> 00:15:02,770
The quantum chemistry
has done-- community

292
00:15:02,770 --> 00:15:05,020
has done a lot of
work in looking

293
00:15:05,020 --> 00:15:09,510
at hybrid functionals,
functional in which

294
00:15:09,510 --> 00:15:14,050
a certain percentage of the
exchange correlation functional

295
00:15:14,050 --> 00:15:16,810
comes from the density
function approximation.

296
00:15:16,810 --> 00:15:18,880
It could be LDA,
or in most cases,

297
00:15:18,880 --> 00:15:20,920
could be something like PB.

298
00:15:20,920 --> 00:15:25,400
And some amount of Hartree-Fock
exchange is mixed in.

299
00:15:25,400 --> 00:15:27,670
So they really
take Kohn and Sham

300
00:15:27,670 --> 00:15:30,760
equation, in which they
exchange correlation term

301
00:15:30,760 --> 00:15:34,060
as both a component that is
density functional theory-like

302
00:15:34,060 --> 00:15:37,210
and a component that
is Hartree-Fock-like.

303
00:15:37,210 --> 00:15:39,490
And some of them are,
like these two mention

304
00:15:39,490 --> 00:15:43,510
here, can work very well,
especially for molecules.

305
00:15:43,510 --> 00:15:48,460
Somehow, again, Hartree-Fock
comes from atoms and molecules,

306
00:15:48,460 --> 00:15:51,880
and it tends to work
better in that limit.

307
00:15:51,880 --> 00:15:53,650
Density functional
theory is built

308
00:15:53,650 --> 00:15:56,230
on an approximation
like the LDA that

309
00:15:56,230 --> 00:15:59,650
ultimately comes from the
homogeneous electron gas.

310
00:15:59,650 --> 00:16:03,250
So it tends to work better
for solids, and as usual,

311
00:16:03,250 --> 00:16:05,320
the most difficult
cases in which you

312
00:16:05,320 --> 00:16:06,660
have a combination of the two.

313
00:16:06,660 --> 00:16:09,310
If you want to study a
molecule on a solid surface,

314
00:16:09,310 --> 00:16:11,680
then none of these
two approaches

315
00:16:11,680 --> 00:16:14,290
work really exceedingly well.

316
00:16:14,290 --> 00:16:18,310
There is a lot of
work in developing

317
00:16:18,310 --> 00:16:21,790
more complex
functionals that tend

318
00:16:21,790 --> 00:16:25,300
to work better than any of these
that I have mentioned here.

319
00:16:25,300 --> 00:16:28,210
They tend to be
exceedingly complex.

320
00:16:28,210 --> 00:16:30,550
And so again, there
are sort of meta-GGA

321
00:16:30,550 --> 00:16:34,150
functional-- there is a lot
of more complex functional

322
00:16:34,150 --> 00:16:36,850
that starts to depend
not only on the density

323
00:16:36,850 --> 00:16:40,180
and on the gradients, but
then maybe on the Laplacian.

324
00:16:40,180 --> 00:16:43,270
They might depend on
the orbital themselves.

325
00:16:43,270 --> 00:16:47,620
So you find terms like exact
exchange functional and so on.

326
00:16:47,620 --> 00:16:49,330
There is a lot of
current work, and there

327
00:16:49,330 --> 00:16:50,710
isn't a unique solution.

328
00:16:50,710 --> 00:16:55,030
And also they tend to be so much
more computationally expensive

329
00:16:55,030 --> 00:16:57,460
that, at this stage, I
would say they are still

330
00:16:57,460 --> 00:17:02,320
limited to a development
effort more than an application

331
00:17:02,320 --> 00:17:03,280
effort.

332
00:17:03,280 --> 00:17:06,130
But you see, once we
have a functional--

333
00:17:06,130 --> 00:17:10,210
and this is even done sort
of with standard LDA or TGA,

334
00:17:10,210 --> 00:17:13,810
we can actually describe
with remarkable accuracy

335
00:17:13,810 --> 00:17:16,609
a number of property for
very different materials.

336
00:17:16,609 --> 00:17:19,540
This is a table that
Chris Pickard in Cambridge

337
00:17:19,540 --> 00:17:24,460
had given me, and you see we
have a combination of metals,

338
00:17:24,460 --> 00:17:28,300
semiconductors, oxides, alloys.

339
00:17:28,300 --> 00:17:30,970
And what we have
here, say, is a list

340
00:17:30,970 --> 00:17:35,060
of what is the experimental
lattice parameter,

341
00:17:35,060 --> 00:17:37,490
and what is the
theoretical prediction.

342
00:17:37,490 --> 00:17:39,490
And you see we are sort
of in the range in which

343
00:17:39,490 --> 00:17:42,760
the errors are around 1%.

344
00:17:42,760 --> 00:17:45,310
And I would say,
for most materials,

345
00:17:45,310 --> 00:17:48,370
we are really in this ballpark.

346
00:17:48,370 --> 00:17:51,730
The error on a lattice
parameter of a material that

347
00:17:51,730 --> 00:17:55,630
doesn't have exotic
electronic properties is not

348
00:17:55,630 --> 00:17:59,110
a high-Tc superconductor,
is not as strongly

349
00:17:59,110 --> 00:18:01,600
correlated electronic
material can

350
00:18:01,600 --> 00:18:04,930
be expected to be in the
range of an error that

351
00:18:04,930 --> 00:18:07,960
is between 1% and 2%.

352
00:18:07,960 --> 00:18:09,370
So very, very good.

353
00:18:09,370 --> 00:18:11,080
You see predictive power.

354
00:18:11,080 --> 00:18:15,370
When we calculate with density
functional theory the lattice

355
00:18:15,370 --> 00:18:17,830
parameter of this, we
are really not trying

356
00:18:17,830 --> 00:18:20,500
to fit any potential at all.

357
00:18:20,500 --> 00:18:24,190
When we calculate the
lattice parameter of silver,

358
00:18:24,190 --> 00:18:25,960
we are just saying
there is going

359
00:18:25,960 --> 00:18:30,310
to be an array of Coulombic
potentials attracting

360
00:18:30,310 --> 00:18:33,280
the electrons with the
atomic number of silver.

361
00:18:33,280 --> 00:18:37,390
And we are looking at how the
total energy of this system

362
00:18:37,390 --> 00:18:39,460
varies with lattice parameters.

363
00:18:39,460 --> 00:18:43,090
And that's why, if you want,
electronic structure approaches

364
00:18:43,090 --> 00:18:46,360
tend to be extremely powerful,
because they are not fitted.

365
00:18:46,360 --> 00:18:52,180
We are just trying to find the
electronic structure solution.

366
00:18:52,180 --> 00:18:57,310
So how does it work in practice?

367
00:18:57,310 --> 00:19:00,010
And what I'm going
to describe here

368
00:19:00,010 --> 00:19:03,130
is what's become known by
now as the total-energy

369
00:19:03,130 --> 00:19:04,840
pseudopotential approach.

370
00:19:04,840 --> 00:19:07,930
This is really the approach
that has been developed

371
00:19:07,930 --> 00:19:11,668
to study solid systems.

372
00:19:11,668 --> 00:19:13,960
Again, you have to keep in
mind these two communities--

373
00:19:13,960 --> 00:19:18,460
one of sort of solid-state
extended studies of matter,

374
00:19:18,460 --> 00:19:20,890
and one of atoms and molecules.

375
00:19:20,890 --> 00:19:22,630
And really density
functional theory

376
00:19:22,630 --> 00:19:26,980
comes from these solid
state approaches.

377
00:19:26,980 --> 00:19:30,100
And for reasons that
we'll see in a moment that

378
00:19:30,100 --> 00:19:33,280
are really related
to what we use

379
00:19:33,280 --> 00:19:37,180
as a basis set to
describe electrons

380
00:19:37,180 --> 00:19:41,170
in a solid, what become
apparent very early on

381
00:19:41,170 --> 00:19:47,980
is that describing accurately
the core electron in an atom

382
00:19:47,980 --> 00:19:51,800
or in a solid would have
been exceedingly complex.

383
00:19:51,800 --> 00:19:54,430
If you think at it, whenever
you have an atom that

384
00:19:54,430 --> 00:19:57,550
forms a molecule or
a solid, you have

385
00:19:57,550 --> 00:19:59,770
a lot of electrons
that are going

386
00:19:59,770 --> 00:20:04,570
to be basically unaffected
by the chemical environment.

387
00:20:04,570 --> 00:20:10,060
If you take an iron atom, and
you put it in iron as a metal,

388
00:20:10,060 --> 00:20:13,420
you put as iron in a
transition metal oxide,

389
00:20:13,420 --> 00:20:16,990
or you put as iron as a center,
say, in a heme grouping--

390
00:20:16,990 --> 00:20:18,940
hemoglobin or myoglobin--

391
00:20:18,940 --> 00:20:21,880
what happens is that the
valence electrons or iron

392
00:20:21,880 --> 00:20:24,070
will redistribute themselves.

393
00:20:24,070 --> 00:20:27,550
But really, the core
electron of irons

394
00:20:27,550 --> 00:20:32,440
are so tightly bound to the
nucleus by order of magnitude

395
00:20:32,440 --> 00:20:34,120
in energy with
respect to the sort

396
00:20:34,120 --> 00:20:36,130
of typical energy
of valence electrons

397
00:20:36,130 --> 00:20:38,600
that they are
basically unaffected.

398
00:20:38,600 --> 00:20:41,530
And so in some ways,
we really don't

399
00:20:41,530 --> 00:20:45,970
want to carry all the
computational expense

400
00:20:45,970 --> 00:20:49,480
of describing core
electrons when

401
00:20:49,480 --> 00:20:52,030
we know that their
contribution is

402
00:20:52,030 --> 00:20:55,870
going to be a rigid contribution
that doesn't change depending

403
00:20:55,870 --> 00:20:57,700
on the chemical environment.

404
00:20:57,700 --> 00:20:59,950
The core electrons are
certainly important.

405
00:20:59,950 --> 00:21:00,850
They are there.

406
00:21:00,850 --> 00:21:02,980
They are bound to
the nucleus, and they

407
00:21:02,980 --> 00:21:04,225
screen the nucleus charge.

408
00:21:04,225 --> 00:21:08,500
In iron, the 1s electrons--
the two 1s electrons

409
00:21:08,500 --> 00:21:10,840
will be so tightly
bound to the nucleus

410
00:21:10,840 --> 00:21:13,150
that the nucleus
doesn't look to the 2s,

411
00:21:13,150 --> 00:21:17,500
or 2p, or 3s, 3p electrons
as saving 26 protons.

412
00:21:17,500 --> 00:21:20,260
But it really looks
like having 24 protons,

413
00:21:20,260 --> 00:21:23,960
because those two 1s
electrons screen completely.

414
00:21:23,960 --> 00:21:28,210
And so the 2s and 2p really
will also spin almost completely

415
00:21:28,210 --> 00:21:31,780
the nucleus from the point
of view of the 3s and the 3p.

416
00:21:31,780 --> 00:21:35,890
So we want to find out a way
of somehow taking into account

417
00:21:35,890 --> 00:21:37,900
the presence of
that core electrons,

418
00:21:37,900 --> 00:21:41,650
but we don't want to carry
that on in our calculation,

419
00:21:41,650 --> 00:21:43,750
because it's very expensive.

420
00:21:43,750 --> 00:21:49,060
Not to mention that the spatial
variation of core electrons

421
00:21:49,060 --> 00:21:51,010
is going to be extremely sharp.

422
00:21:51,010 --> 00:21:53,510
I'll show you an
example in a moment.

423
00:21:53,510 --> 00:21:56,710
And so we need a lot of
computational information

424
00:21:56,710 --> 00:22:00,820
to describe all the sharp
wiggles that core electrons

425
00:22:00,820 --> 00:22:02,570
will do around the nucleus.

426
00:22:02,570 --> 00:22:06,400
And this problem has been
solved that again from the late

427
00:22:06,400 --> 00:22:11,140
'70s to the early '80s by what
are called pseudopotential

428
00:22:11,140 --> 00:22:13,640
approaches-- in particular,
something that is called--

429
00:22:13,640 --> 00:22:15,400
and you'll see this term often--

430
00:22:15,400 --> 00:22:18,850
norm conserving pseudopotentials
that, in many ways,

431
00:22:18,850 --> 00:22:21,430
are some of the most
complex part of all

432
00:22:21,430 --> 00:22:23,390
these electron
structure approaches.

433
00:22:23,390 --> 00:22:27,520
And I'll just give you a
sort generic flavor of how

434
00:22:27,520 --> 00:22:29,410
they approach this problem.

435
00:22:34,800 --> 00:22:38,340
Once we have sort of
removed the core electrons

436
00:22:38,340 --> 00:22:41,310
from our problems, we
still need to find out

437
00:22:41,310 --> 00:22:44,640
what are the Kohn and Sham
orbitals that minimize

438
00:22:44,640 --> 00:22:46,470
our density functional problem.

439
00:22:46,470 --> 00:22:49,710
And as always-- and again, we'll
see this in the next slide--

440
00:22:49,710 --> 00:22:53,370
we need to find an appropriate
computational representation

441
00:22:53,370 --> 00:22:54,750
of these orbitals.

442
00:22:54,750 --> 00:22:58,920
And that is in particular
will expand those orbitals

443
00:22:58,920 --> 00:22:59,550
on a basis.

444
00:22:59,550 --> 00:23:04,830
That is, we'll represent any
possible orbital in a problem

445
00:23:04,830 --> 00:23:08,140
as a linear combination
of simple function.

446
00:23:08,140 --> 00:23:10,530
This is what is
called a basis set.

447
00:23:10,530 --> 00:23:13,650
So in our one dimensional
analysis problem,

448
00:23:13,650 --> 00:23:16,560
when you have got an arbitrary
function, you can expand--

449
00:23:16,560 --> 00:23:18,270
you can do, say,
Fourier analysis.

450
00:23:18,270 --> 00:23:21,810
You can expand it in a
series of sines and cosines.

451
00:23:21,810 --> 00:23:24,310
So the same general
problem applies here.

452
00:23:24,310 --> 00:23:27,090
We have arbitrary functions
in three dimensions that

453
00:23:27,090 --> 00:23:31,360
are our orbitals, and we want
to find an appropriate basis

454
00:23:31,360 --> 00:23:32,670
set that describe them.

455
00:23:32,670 --> 00:23:35,070
So this basis set
needs to be accurate--

456
00:23:35,070 --> 00:23:37,260
that this, needs to
be flexible enough

457
00:23:37,260 --> 00:23:40,800
to describe all the possible
wiggles of our orbitals,

458
00:23:40,800 --> 00:23:43,860
but also need to be
computationally convenient.

459
00:23:43,860 --> 00:23:47,200
And we'll go in
that in a moment.

460
00:23:47,200 --> 00:23:51,630
So once we have sort of put
together these elements,

461
00:23:51,630 --> 00:23:56,490
we have sort of our external
potential being represented

462
00:23:56,490 --> 00:23:59,700
by this set of
nuclei, and somehow,

463
00:23:59,700 --> 00:24:01,680
via the pseudopotential
approach,

464
00:24:01,680 --> 00:24:02,970
also the core electrons.

465
00:24:02,970 --> 00:24:06,240
And once we have decided
what is our basic set,

466
00:24:06,240 --> 00:24:10,830
we really need to go and
solve self-consistently

467
00:24:10,830 --> 00:24:12,780
either the Kohn
and Sham equation

468
00:24:12,780 --> 00:24:14,760
that you have seen
before, or we want

469
00:24:14,760 --> 00:24:18,430
to minimize the nonlinear
energy functional.

470
00:24:18,430 --> 00:24:21,450
And so there are sort
of several steps.

471
00:24:21,450 --> 00:24:27,040
Usually, what we'll do is we'll
start from a trial solution.

472
00:24:27,040 --> 00:24:29,730
Remember that, since
the Hamiltonian depends

473
00:24:29,730 --> 00:24:32,070
on the charge density,
we need to have

474
00:24:32,070 --> 00:24:34,860
a guess to our
initial charge density

475
00:24:34,860 --> 00:24:38,940
to even construct our operator,
because our operator depends

476
00:24:38,940 --> 00:24:41,650
on the ground state
charge density.

477
00:24:41,650 --> 00:24:43,800
So since that we don't have
the ground state charge

478
00:24:43,800 --> 00:24:45,510
density when we
start our problem,

479
00:24:45,510 --> 00:24:48,120
we need to start with
a trial solution.

480
00:24:48,120 --> 00:24:50,040
It could be a trial
charge density.

481
00:24:50,040 --> 00:24:51,720
Could be trial orbitals.

482
00:24:51,720 --> 00:24:57,220
But once we have that, all
our operator is well defined.

483
00:24:57,220 --> 00:25:00,090
So we can calculate all
the different terms--

484
00:25:00,090 --> 00:25:03,220
the quantum kinetic
energy, the Hartree energy,

485
00:25:03,220 --> 00:25:05,730
the exchange correlation terms.

486
00:25:05,730 --> 00:25:08,880
And we can try to
solve that Hamiltonian

487
00:25:08,880 --> 00:25:10,200
from that Hamiltonian.

488
00:25:10,200 --> 00:25:14,880
We'll find new orbitals,
and with those new orbitals,

489
00:25:14,880 --> 00:25:18,540
we'll calculate the ground
state charge density,

490
00:25:18,540 --> 00:25:21,120
and we'll obtain a
new Hamiltonian that

491
00:25:21,120 --> 00:25:23,910
then will sort of
keep iterating,

492
00:25:23,910 --> 00:25:27,990
finding new orbitals, new
charge density, new Hamiltonian,

493
00:25:27,990 --> 00:25:30,810
until it reaches a
fixed point, until we

494
00:25:30,810 --> 00:25:32,490
reach self-consistency.

495
00:25:32,490 --> 00:25:36,150
Or we can just take the approach
of minimizing the total energy

496
00:25:36,150 --> 00:25:38,140
functional to self-consistency.

497
00:25:38,140 --> 00:25:43,690
I've written in a more
compact way the problem here.

498
00:25:43,690 --> 00:25:48,610
Well, it'll come in a moment.

499
00:25:48,610 --> 00:25:55,510
So let's describe first
our first computational

500
00:25:55,510 --> 00:25:57,460
approximation in this problem--

501
00:25:57,460 --> 00:26:01,030
that is, the introduction
of pseudopotentials.

502
00:26:01,030 --> 00:26:03,850
And as I said,
what we want to do

503
00:26:03,850 --> 00:26:08,920
is we want to get rid of the
electrons in the inner shells,

504
00:26:08,920 --> 00:26:11,770
in the core of the atoms,
because they really

505
00:26:11,770 --> 00:26:16,270
don't have any contribution
to the valence chemical bonds.

506
00:26:16,270 --> 00:26:17,215
They are just there.

507
00:26:17,215 --> 00:26:19,030
They're important
because they screen

508
00:26:19,030 --> 00:26:22,690
the nucleus in a very specific
quantum mechanical term,

509
00:26:22,690 --> 00:26:26,200
but they really don't
change much when

510
00:26:26,200 --> 00:26:29,620
the valence
coordination changes.

511
00:26:29,620 --> 00:26:33,490
We call that, actually, a
frozen core approximation.

512
00:26:33,490 --> 00:26:38,020
That is, we take an atom--
again, we take an iron atom.

513
00:26:38,020 --> 00:26:42,100
We solve the density functional
problem for the iron atom.

514
00:26:42,100 --> 00:26:46,020
That is, we find a density
functional orbital for the 1s,

515
00:26:46,020 --> 00:26:50,740
2s, 2p, 3s, 3p,
and so on orbitals.

516
00:26:50,740 --> 00:26:55,510
For an atom, it's reasonable to
do that [INAUDIBLE] solution,

517
00:26:55,510 --> 00:26:57,280
even for the core
electrons, because you

518
00:26:57,280 --> 00:26:58,730
have spherical symmetry.

519
00:26:58,730 --> 00:27:01,540
And the problem is still
sort of fairly simple.

520
00:27:01,540 --> 00:27:04,810
And at this point, we say,
well, from now on, we are not

521
00:27:04,810 --> 00:27:08,710
going to describe
the full iron atom,

522
00:27:08,710 --> 00:27:13,060
but we are going always
to deal with a pseudo iron

523
00:27:13,060 --> 00:27:21,100
atom in which the nucleus and
the 1s, 2s, and 2p electrons

524
00:27:21,100 --> 00:27:23,290
have been frozen.

525
00:27:23,290 --> 00:27:27,400
And so what really the
valence electron need to see

526
00:27:27,400 --> 00:27:30,400
is a pseudo nucleus
that is not given

527
00:27:30,400 --> 00:27:32,770
just by the bare
Coulombic potential,

528
00:27:32,770 --> 00:27:36,640
but by the bare Coulombic
potential screened

529
00:27:36,640 --> 00:27:41,410
by this 1s, and 2s,
and 2p electron frozen

530
00:27:41,410 --> 00:27:43,570
in their atomic configuration.

531
00:27:43,570 --> 00:27:46,420
And there is a theorem that
we won't dwell into by-- from

532
00:27:46,420 --> 00:27:48,280
Barth and Gelatt from the '80s--

533
00:27:48,280 --> 00:27:52,210
that says that this freezing
of the core electrons

534
00:27:52,210 --> 00:27:54,310
is actually a very
good approximation.

535
00:27:54,310 --> 00:27:55,390
It doesn't really matter.

536
00:27:55,390 --> 00:27:57,790
It's been verified
over and over again.

537
00:27:57,790 --> 00:28:00,820
It Is the last thing
we have to worry about.

538
00:28:00,820 --> 00:28:03,220
To be precise, the
only thing that we

539
00:28:03,220 --> 00:28:05,680
have to worry about in
density functional theory,

540
00:28:05,680 --> 00:28:09,940
apart from making sure that
our technical approximations--

541
00:28:09,940 --> 00:28:13,150
computation approximation
are all accurate

542
00:28:13,150 --> 00:28:15,100
is the exchange
correlation functional.

543
00:28:15,100 --> 00:28:18,200
That is truly the
only source of error.

544
00:28:18,200 --> 00:28:20,710
And so here, if you
want, I've represented

545
00:28:20,710 --> 00:28:24,460
the idea of a pseudo potential.

546
00:28:24,460 --> 00:28:27,790
This would have been the
standard solution for an atom.

547
00:28:27,790 --> 00:28:30,940
Actually, here I've chosen
something simpler than iron.

548
00:28:30,940 --> 00:28:32,860
I've chosen aluminum.

549
00:28:32,860 --> 00:28:38,590
And so we would have had a
Kohn and Sham set of equations

550
00:28:38,590 --> 00:28:42,670
for aluminum in which,
again, this term

551
00:28:42,670 --> 00:28:46,990
here contains the Hartree
potential, the exchange

552
00:28:46,990 --> 00:28:51,220
correlation potential,
and external potential

553
00:28:51,220 --> 00:28:53,620
that would be just the
bare nuclear potential.

554
00:28:53,620 --> 00:28:54,940
We solve this problem.

555
00:28:54,940 --> 00:28:55,800
What we find?

556
00:28:55,800 --> 00:28:58,570
Well, we find this
hierarchy of states.

557
00:28:58,570 --> 00:29:02,890
You see, we find two
electrons in the 1s state.

558
00:29:02,890 --> 00:29:06,850
We find two electrons in the
2s, two electrons in the 2p.

559
00:29:06,850 --> 00:29:09,880
But really, the electrons
that do all the chemistry

560
00:29:09,880 --> 00:29:12,910
are the 3s and 3p electrons.

561
00:29:12,910 --> 00:29:17,320
And you see the enormous
difference in energy scales.

562
00:29:17,320 --> 00:29:21,610
So the binding energy of the
1s electrons to the nucleus

563
00:29:21,610 --> 00:29:24,880
is 1,500 electron volts.

564
00:29:24,880 --> 00:29:27,100
There is no way,
unless you are sort

565
00:29:27,100 --> 00:29:31,360
of throwing X-rays of very
high energy to those electrons,

566
00:29:31,360 --> 00:29:36,040
that you are going to affect or
perturb in any significant way

567
00:29:36,040 --> 00:29:37,540
these 1s electrons.

568
00:29:37,540 --> 00:29:41,560
Remember, the energy
of a hydrogen bond

569
00:29:41,560 --> 00:29:45,920
is 0.29 electrons-- is a
fraction of an electronvolt.

570
00:29:45,920 --> 00:29:49,490
So this is four orders
of magnitude larger.

571
00:29:49,490 --> 00:29:53,320
So what we want is we want
to say this set of electrons

572
00:29:53,320 --> 00:29:56,200
are so tightly
bound that we don't

573
00:29:56,200 --> 00:29:59,770
need to consider how they
change during the formation

574
00:29:59,770 --> 00:30:01,250
of a chemical bond.

575
00:30:01,250 --> 00:30:06,550
And so what we want to
find is a new potential

576
00:30:06,550 --> 00:30:10,240
that we call our
pseudopotential, such

577
00:30:10,240 --> 00:30:14,950
that the eigenstates
in the presence

578
00:30:14,950 --> 00:30:22,270
of this pseudopotential give us
solutions that really reproduce

579
00:30:22,270 --> 00:30:27,290
exactly the valence electron
solution of the original atom.

580
00:30:27,290 --> 00:30:32,680
So you see, what we want is
we want to go from 13 over r,

581
00:30:32,680 --> 00:30:37,240
the Coulombic potential
that is contained in here

582
00:30:37,240 --> 00:30:43,900
for the bare aluminum nucleus,
to a new potential that somehow

583
00:30:43,900 --> 00:30:48,250
contains both the bare
potential of the nucleus,

584
00:30:48,250 --> 00:30:51,290
the screening from
the core electrons,

585
00:30:51,290 --> 00:30:56,560
and it is constructed according
to a well-defined prescription

586
00:30:56,560 --> 00:31:00,130
so that the eigenstates of
this new set of Kohn and Sham

587
00:31:00,130 --> 00:31:03,070
equation are in order.

588
00:31:03,070 --> 00:31:07,120
The lowest eigenstates
are actually

589
00:31:07,120 --> 00:31:11,620
the valence eigenstates
of our regional problem.

590
00:31:11,620 --> 00:31:15,460
We get the same eigenvalues,
and we get, in a way

591
00:31:15,460 --> 00:31:21,340
that I'll specify in a moment,
the same eigenfunctions.

592
00:31:21,340 --> 00:31:25,270
And so here, it's how we would
actually look at this problem.

593
00:31:25,270 --> 00:31:30,070
And this is a figure courtesy
of Chris [INAUDIBLE]..

594
00:31:30,070 --> 00:31:33,760
So you see, what we
usually ever in an atom

595
00:31:33,760 --> 00:31:37,730
is a Coulombic potential,
is this thin red line here.

596
00:31:37,730 --> 00:31:40,420
This is the potential
as a function

597
00:31:40,420 --> 00:31:42,190
of the radial distance.

598
00:31:42,190 --> 00:31:43,840
So it diverges.

599
00:31:43,840 --> 00:31:46,660
It goes to minus
infinity Coulombically.

600
00:31:46,660 --> 00:31:53,610
For aluminum, it
would go as 13 over r.

601
00:31:53,610 --> 00:32:00,600
What are the solutions, say,
for the 1s state in this--

602
00:32:00,600 --> 00:32:05,730
sorry, what would be the
solution for a valence electron

603
00:32:05,730 --> 00:32:07,110
in this potential?

604
00:32:07,110 --> 00:32:10,770
Well, there are going to be
all the sort of core electrons,

605
00:32:10,770 --> 00:32:15,280
but then a valence electron
would look something like this.

606
00:32:15,280 --> 00:32:19,740
It has-- you see this enormous
number of oscillations.

607
00:32:19,740 --> 00:32:22,620
The reason why those
oscillations are there

608
00:32:22,620 --> 00:32:26,940
is that eigenfunction of
the Kohn and Sham equation

609
00:32:26,940 --> 00:32:29,880
need to be orthogonal
to each other.

610
00:32:29,880 --> 00:32:31,320
This is another
one of these sort

611
00:32:31,320 --> 00:32:34,770
of fundamental quantum
mechanical rules.

612
00:32:34,770 --> 00:32:36,620
It's basically the
Pauli principles.

613
00:32:36,620 --> 00:32:40,530
You can't have two electrons
in the same quantum states.

614
00:32:40,530 --> 00:32:43,380
And in particular,
electrons corresponding

615
00:32:43,380 --> 00:32:47,110
to different quantum states need
to be orthogonal to each other.

616
00:32:47,110 --> 00:32:52,060
So the integral of the psi
star of theta 3s electron

617
00:32:52,060 --> 00:32:56,070
in aluminum times the psi of
the 1s electron in aluminum

618
00:32:56,070 --> 00:32:57,360
needs to be 0.

619
00:32:57,360 --> 00:33:02,440
And so what it means is that the
higher you go, the more wiggles

620
00:33:02,440 --> 00:33:02,940
you have.

621
00:33:02,940 --> 00:33:05,910
And these wiggles-- that
is, these changes of signs

622
00:33:05,910 --> 00:33:09,000
is what allows
you orthogonality.

623
00:33:09,000 --> 00:33:16,575
So if you have a 1s electron
that looks like this,

624
00:33:16,575 --> 00:33:19,260
a 1s electron will
have actually sort

625
00:33:19,260 --> 00:33:23,940
of exponential to the minus
r decaying wave function.

626
00:33:23,940 --> 00:33:28,290
The wave function say-- already
the 2s wave function needs

627
00:33:28,290 --> 00:33:31,200
to be orthogonal
to this electron,

628
00:33:31,200 --> 00:33:36,720
and so it needs to change
sign to allow orthogonality

629
00:33:36,720 --> 00:33:43,630
when you take the product
of these two orbitals.

630
00:33:43,630 --> 00:33:48,930
So orthogonality creates
a lot of wiggles,

631
00:33:48,930 --> 00:33:55,680
and these wiggles also
make the charge density

632
00:33:55,680 --> 00:33:58,950
coming from a valence
electron more spread

633
00:33:58,950 --> 00:34:01,830
towards the outside
of the nucleus.

634
00:34:01,830 --> 00:34:06,660
There is, if you want,
another quantum mechanical

635
00:34:06,660 --> 00:34:10,860
derived effect in which
valence electrons are

636
00:34:10,860 --> 00:34:14,820
moved outwards because of
this orthogonality constraint.

637
00:34:14,820 --> 00:34:20,100
And so you see this would be,
in light blue, the exact wave

638
00:34:20,100 --> 00:34:23,920
function for a valence electron.

639
00:34:23,920 --> 00:34:28,860
And what we want to find
is a pseudopotential that

640
00:34:28,860 --> 00:34:33,690
substitutes for the original
bare Coulombic potential--

641
00:34:33,690 --> 00:34:37,440
and it's written here in sort
of-- with a thick red line--

642
00:34:37,440 --> 00:34:42,690
that is constructed so that its
ground state wave function--

643
00:34:42,690 --> 00:34:45,449
that is here in a thick blue--

644
00:34:45,449 --> 00:34:50,340
has the same energy eigenvalue
of the valence electron.

645
00:34:50,340 --> 00:34:54,870
And in many ways, it's identical
to the original wave function.

646
00:34:54,870 --> 00:34:58,290
And the way we make it
identical is actually

647
00:34:58,290 --> 00:35:01,620
we only require
it to be identical

648
00:35:01,620 --> 00:35:05,760
to our original solution
outside the core,

649
00:35:05,760 --> 00:35:09,090
because outside of
the core of the atom

650
00:35:09,090 --> 00:35:11,820
is where chemistry takes place.

651
00:35:11,820 --> 00:35:18,750
Chemistry will be here when
atoms bind with other atoms.

652
00:35:18,750 --> 00:35:24,330
Inside here, this is the region
where core electron sits,

653
00:35:24,330 --> 00:35:29,790
and that's a region that will
never overlap with other atoms,

654
00:35:29,790 --> 00:35:32,520
again, because there are
sort of already these very

655
00:35:32,520 --> 00:35:34,710
strongly bound electrons.

656
00:35:34,710 --> 00:35:38,940
And to bring two atoms to
have their core regions to

657
00:35:38,940 --> 00:35:43,260
overlap requires
enormous pressure,

658
00:35:43,260 --> 00:35:44,640
hundreds of gigapascals.

659
00:35:44,640 --> 00:35:46,810
So we never really
have to worry.

660
00:35:46,810 --> 00:35:48,990
So what we are
trying to do here is

661
00:35:48,990 --> 00:35:53,730
we want to construct a
pseudopotential in thick red

662
00:35:53,730 --> 00:35:58,840
such that its ground state
wave functions in thick blue

663
00:35:58,840 --> 00:36:04,710
have the same eigenvalues of
my original eigenfunctions

664
00:36:04,710 --> 00:36:06,900
in thin blue lines.

665
00:36:06,900 --> 00:36:10,290
And in many ways, they
represent the same physics

666
00:36:10,290 --> 00:36:12,450
and the same
chemistry, because they

667
00:36:12,450 --> 00:36:16,950
are sort of identical
outside the core.

668
00:36:16,950 --> 00:36:21,840
So the thick red line is
how we describe an atom.

669
00:36:21,840 --> 00:36:23,970
The bare Coulombic potential--

670
00:36:23,970 --> 00:36:28,110
but we also built all
the screening action

671
00:36:28,110 --> 00:36:30,540
from the core
electrons, and you see

672
00:36:30,540 --> 00:36:32,670
the central feature
of the pseudopotential

673
00:36:32,670 --> 00:36:37,320
is that, if you want, it becomes
repulsive around the center

674
00:36:37,320 --> 00:36:39,180
of the atom because
we really need

675
00:36:39,180 --> 00:36:45,180
to keep the valence electron
to be mostly delocalized

676
00:36:45,180 --> 00:36:48,060
in the valence region.

677
00:36:48,060 --> 00:36:50,130
How to build this
pseudopotential

678
00:36:50,130 --> 00:36:53,400
is actually a complex
and somehow dark art.

679
00:36:53,400 --> 00:36:58,090
By now, it's been perfected,
and that-- there are basically

680
00:36:58,090 --> 00:37:00,700
tables of pseudopotential.

681
00:37:00,700 --> 00:37:05,340
So it's a problem you almost
never have to worry anymore.

682
00:37:05,340 --> 00:37:07,230
An electronic
structure code will

683
00:37:07,230 --> 00:37:09,510
have a set of pseudopotential.

684
00:37:09,510 --> 00:37:11,970
And in the course
of the years, there

685
00:37:11,970 --> 00:37:15,180
have been two flavors
that have been developed.

686
00:37:15,180 --> 00:37:19,050
The first one that was
actually, I would say,

687
00:37:19,050 --> 00:37:23,390
invented at Bell Labs at the
end of the '70s by Hamann,

688
00:37:23,390 --> 00:37:25,710
Schluter, and Chiang
is what are called

689
00:37:25,710 --> 00:37:28,620
norm conserving
pseudopotential, of which I'll

690
00:37:28,620 --> 00:37:31,650
describe this plot in a moment.

691
00:37:31,650 --> 00:37:35,780
There was a sort of improvement
of this norm conserving

692
00:37:35,780 --> 00:37:39,300
pseudopotential that are called
ultrasoft pseudopotential,

693
00:37:39,300 --> 00:37:42,870
developed by David Vanderbilt
at Rutgers University.

694
00:37:42,870 --> 00:37:48,630
They tend to be much
less expensive to be--

695
00:37:48,630 --> 00:37:51,930
to use in practical
calculation, basically

696
00:37:51,930 --> 00:37:54,540
because they are
smoother and smoother,

697
00:37:54,540 --> 00:37:59,970
and so they can be described
with sort of a smaller

698
00:37:59,970 --> 00:38:02,340
basis set, and they
tend to be more

699
00:38:02,340 --> 00:38:08,040
accurate over a broader range
of coordination and energies.

700
00:38:08,040 --> 00:38:10,770
But so the general idea
of the pseudopotential

701
00:38:10,770 --> 00:38:13,050
from a different way--

702
00:38:13,050 --> 00:38:17,190
in the previous slide, I
showed you how it looks like.

703
00:38:17,190 --> 00:38:20,000
But the genetic idea
of a pseudopotential

704
00:38:20,000 --> 00:38:22,730
is that we want to
create something

705
00:38:22,730 --> 00:38:26,480
that contains both the nucleus
and these core electrons.

706
00:38:26,480 --> 00:38:30,140
And that basically will
act on the valence electron

707
00:38:30,140 --> 00:38:34,910
in the same way as the true
problem of core electron

708
00:38:34,910 --> 00:38:36,830
explicitly and nucleus.

709
00:38:36,830 --> 00:38:40,220
And so what we say is
that we want a pseudo

710
00:38:40,220 --> 00:38:44,990
potential to scatter
an incoming wave,

711
00:38:44,990 --> 00:38:48,440
an incoming electron
in the same way

712
00:38:48,440 --> 00:38:54,610
as our real set of core
electrons in an atom

713
00:38:54,610 --> 00:38:56,180
would do that.

714
00:38:56,180 --> 00:38:59,920
And in order to do
that very accurately,

715
00:38:59,920 --> 00:39:03,040
for this pseudopotential,
what we need to do

716
00:39:03,040 --> 00:39:06,460
is actually to have
the pseudopotential

717
00:39:06,460 --> 00:39:10,990
not just be a single local
form, as in the previous slide.

718
00:39:10,990 --> 00:39:15,160
It's shown you the sort of
red thick line that was just

719
00:39:15,160 --> 00:39:18,700
a unique form of the
potential as a function of r,

720
00:39:18,700 --> 00:39:22,510
but we actually want
our pseudopotential

721
00:39:22,510 --> 00:39:27,760
to be different, depending on
the angular momentum component

722
00:39:27,760 --> 00:39:29,750
of the electron coming.

723
00:39:29,750 --> 00:39:31,390
So if you have a
valence electron,

724
00:39:31,390 --> 00:39:33,530
and you have a
valence wave function,

725
00:39:33,530 --> 00:39:36,700
you can actually say that that
valence wave function will

726
00:39:36,700 --> 00:39:43,570
have a 20% angular symmetry of
the S type, 30% of the P type,

727
00:39:43,570 --> 00:39:47,000
30% of the D type, and
so on and so forth.

728
00:39:47,000 --> 00:39:51,460
And what you do is
you act differently

729
00:39:51,460 --> 00:39:54,070
on the S component
of the wave function,

730
00:39:54,070 --> 00:39:57,610
on the P component of the wave
function, on the D component,

731
00:39:57,610 --> 00:39:59,240
and so on and so forth.

732
00:39:59,240 --> 00:40:05,080
And so this would be the
different radial parts

733
00:40:05,080 --> 00:40:07,090
of the pseudopotential,
depending on which

734
00:40:07,090 --> 00:40:08,590
components they act on.

735
00:40:08,590 --> 00:40:12,070
They all have this
overall 1 over r--

736
00:40:12,070 --> 00:40:14,710
or z over r asymptotic trend.

737
00:40:14,710 --> 00:40:18,680
But as we get more close to the
nucleus, they start deferring.

738
00:40:18,680 --> 00:40:21,760
You see, from the
core region inwards,

739
00:40:21,760 --> 00:40:23,540
they are all different.

740
00:40:23,540 --> 00:40:27,760
And in particular, they are
all, or at least in part,

741
00:40:27,760 --> 00:40:33,360
repulsive to again represent
this frozen set of electrons.

742
00:40:33,360 --> 00:40:36,730
And this is how the
corresponding wave functions

743
00:40:36,730 --> 00:40:37,910
would look like.

744
00:40:37,910 --> 00:40:41,140
And I've made-- again,
this is for an indium atom,

745
00:40:41,140 --> 00:40:47,350
a comparison between, say, the
p all electron wave function

746
00:40:47,350 --> 00:40:51,340
and the pseudo wave function.

747
00:40:51,340 --> 00:40:55,600
And you see they are identical
in the valence region,

748
00:40:55,600 --> 00:40:59,740
but in the core region, the
true all electron valence wave

749
00:40:59,740 --> 00:41:01,780
function is a lot
of oscillation,

750
00:41:01,780 --> 00:41:04,990
while the pseudo wave
function is much smoother.

751
00:41:04,990 --> 00:41:08,290
And this smoothness
means that we

752
00:41:08,290 --> 00:41:14,200
can describe this pseudo wave
function in sines and cosines

753
00:41:14,200 --> 00:41:21,200
with a much smaller set of waves
at different wavelengths then

754
00:41:21,200 --> 00:41:23,750
we would have to
do if we were to--

755
00:41:23,750 --> 00:41:27,590
if we had the need to describe
all this high frequency

756
00:41:27,590 --> 00:41:28,640
oscillation.

757
00:41:28,640 --> 00:41:32,690
So at the end,
pseudopotential are just there

758
00:41:32,690 --> 00:41:37,760
to sort of give us a
chance to avoid describing

759
00:41:37,760 --> 00:41:41,540
all these high frequency
oscillation of the true valence

760
00:41:41,540 --> 00:41:43,760
wave function, and
also to give us

761
00:41:43,760 --> 00:41:48,320
the possibility of avoiding
the explicit description

762
00:41:48,320 --> 00:41:50,390
of the core
electrons that really

763
00:41:50,390 --> 00:41:54,200
don't change their shape
in going from the atom

764
00:41:54,200 --> 00:41:54,830
to the solid.

765
00:41:58,070 --> 00:42:02,360
So now with this, we have
an exchange correlation

766
00:42:02,360 --> 00:42:03,320
functional.

767
00:42:03,320 --> 00:42:07,190
We have this new approximation
to the external potential

768
00:42:07,190 --> 00:42:11,420
given by a sort of
array of pseudopotential

769
00:42:11,420 --> 00:42:13,250
centered on each nucleus.

770
00:42:13,250 --> 00:42:17,540
And so we are back to this
sort of self-consistent problem

771
00:42:17,540 --> 00:42:22,610
of having to solve these
Kohn and Sham differential

772
00:42:22,610 --> 00:42:23,870
equations.

773
00:42:23,870 --> 00:42:27,590
And again, the way we want
to attack this problem

774
00:42:27,590 --> 00:42:34,550
on a computer is by making this
problem into a numerical matrix

775
00:42:34,550 --> 00:42:36,020
algebra problem.

776
00:42:36,020 --> 00:42:41,180
And the way we go about this
is by choosing a busy set

777
00:42:41,180 --> 00:42:44,390
and expanding our
orbitals in a basis.

778
00:42:44,390 --> 00:42:48,980
That is, we say by saying that
any sort of generating function

779
00:42:48,980 --> 00:42:51,860
can be thought of as
a linear combination

780
00:42:51,860 --> 00:42:54,200
of a set of simple function.

781
00:42:54,200 --> 00:42:58,340
Simple function, for which we
can do analytical operation--

782
00:42:58,340 --> 00:43:04,100
that is, we are able, say, to
calculate the matrix elements,

783
00:43:04,100 --> 00:43:07,700
say, of the kinetic energy
between this simple function,

784
00:43:07,700 --> 00:43:10,340
or we are able to
calculate analytically

785
00:43:10,340 --> 00:43:14,660
the second derivative
of this basis function.

786
00:43:14,660 --> 00:43:18,890
Well, by making this ansatz,
by saying that a generic wave

787
00:43:18,890 --> 00:43:21,050
function is a linear
combination of this,

788
00:43:21,050 --> 00:43:25,790
our problem transforms from a
differential analysis problem

789
00:43:25,790 --> 00:43:28,960
to a problem of finding
these coefficients.

790
00:43:28,960 --> 00:43:33,620
And again, sort of the
basis set that we'll

791
00:43:33,620 --> 00:43:37,460
choose in all sort of our
practical computational lab

792
00:43:37,460 --> 00:43:40,730
will really be a basis
set of sines and cosines,

793
00:43:40,730 --> 00:43:42,620
or what we call plane waves.

794
00:43:42,620 --> 00:43:45,590
That is especially
adapted and especially

795
00:43:45,590 --> 00:43:48,110
useful for the case of solids.

796
00:43:48,110 --> 00:43:50,520
And I've given you
an example here.

797
00:43:50,520 --> 00:43:53,000
Suppose that what
we want to describe

798
00:43:53,000 --> 00:43:58,070
is a Gaussian charge
density, something

799
00:43:58,070 --> 00:44:03,350
that is a little bit similar to
the charge density in an atom.

800
00:44:03,350 --> 00:44:07,440
Well, this localized
charge density

801
00:44:07,440 --> 00:44:12,850
can be actually expressed
fairly accurately

802
00:44:12,850 --> 00:44:19,780
as a linear combination of
very few sines or cosines

803
00:44:19,780 --> 00:44:21,220
with different wavelengths.

804
00:44:21,220 --> 00:44:23,920
You see, I go from
this wavelength

805
00:44:23,920 --> 00:44:25,930
to smaller and
smaller wavelength.

806
00:44:25,930 --> 00:44:29,080
And if I choose my
coefficients such

807
00:44:29,080 --> 00:44:32,080
that I have constructive
interference here

808
00:44:32,080 --> 00:44:36,070
in the center and destructive
interference outside,

809
00:44:36,070 --> 00:44:39,940
I actually get just by
something nine terms here,

810
00:44:39,940 --> 00:44:43,040
something that basically
describes perfectly

811
00:44:43,040 --> 00:44:44,570
my charge density.

812
00:44:44,570 --> 00:44:47,680
So this will be our general
sort of approach to the problem.

813
00:44:47,680 --> 00:44:50,260
We choose a basis set.

814
00:44:50,260 --> 00:44:54,130
That is, we choose a set
of elementary functions

815
00:44:54,130 --> 00:44:58,330
that are particularly suited
to describe our problem,

816
00:44:58,330 --> 00:45:02,230
and for which we can do all
the analysis that we want.

817
00:45:02,230 --> 00:45:04,630
That is, we are able to
calculate second derivatives.

818
00:45:04,630 --> 00:45:08,710
We are able to calculate
matrix elements analytically.

819
00:45:08,710 --> 00:45:11,410
And then our
computational problem

820
00:45:11,410 --> 00:45:16,550
becomes really just a problem
of finding the coefficients.

821
00:45:16,550 --> 00:45:20,530
And this is one of those
approximations for which you

822
00:45:20,530 --> 00:45:22,370
need to test the accuracy.

823
00:45:22,370 --> 00:45:26,260
That is, if you use a
very small business set,

824
00:45:26,260 --> 00:45:30,220
if you use only 10 plain waves
with different wavelengths,

825
00:45:30,220 --> 00:45:32,920
your calculation will
be very inexpensive,

826
00:45:32,920 --> 00:45:36,070
because you have only 10
coefficients to work with.

827
00:45:36,070 --> 00:45:39,130
But it probably won't
be very accurate.

828
00:45:39,130 --> 00:45:41,620
If you use one
million plane waves,

829
00:45:41,620 --> 00:45:44,260
your calculation will
be extremely accurate,

830
00:45:44,260 --> 00:45:46,760
but it will be also
very expensive.

831
00:45:46,760 --> 00:45:50,320
So this is an approximation,
but an approximation-- and this

832
00:45:50,320 --> 00:45:53,680
is fundamental-- for which
you can control the accuracy.

833
00:45:53,680 --> 00:45:56,200
That is, you just need to
make sure that your business

834
00:45:56,200 --> 00:45:58,180
set is good enough.

835
00:45:58,180 --> 00:46:02,860
And that is something that
you can always check and test.

836
00:46:02,860 --> 00:46:06,640
You can't test, apart from
comparison with experiment,

837
00:46:06,640 --> 00:46:09,690
if your exchange correlation
functional is good enough.

838
00:46:09,690 --> 00:46:11,890
That's why we put
so much importance,

839
00:46:11,890 --> 00:46:15,760
and you should never forget that
really all the accuracy or all

840
00:46:15,760 --> 00:46:19,270
the errors that ultimately
come in your calculation

841
00:46:19,270 --> 00:46:21,550
come from the exchange
correlational function.

842
00:46:21,550 --> 00:46:25,480
All the other computational
numerical approximation

843
00:46:25,480 --> 00:46:29,680
using pseudopotentials,
using a finite basis set

844
00:46:29,680 --> 00:46:32,200
are all approximations
that can be tested.

845
00:46:32,200 --> 00:46:35,920
And we always assume that a
well done electronic structure

846
00:46:35,920 --> 00:46:38,650
calculation will have
those under control.

847
00:46:38,650 --> 00:46:41,050
You are using enough basis sets.

848
00:46:41,050 --> 00:46:44,020
You are using accurate
pseudopotential, and so on.

849
00:46:46,660 --> 00:46:49,120
As I said, in solids,
for a reason that you'll

850
00:46:49,120 --> 00:46:52,870
see in a moment, really the
universal basis of choice

851
00:46:52,870 --> 00:46:55,060
is that of plane
waves-- basically

852
00:46:55,060 --> 00:46:58,450
the e to the I
complex exponential,

853
00:46:58,450 --> 00:46:59,830
and you'll see why.

854
00:46:59,830 --> 00:47:02,290
But in a sort of--

855
00:47:02,290 --> 00:47:07,600
atoms and molecules-- their
almost universal basis

856
00:47:07,600 --> 00:47:11,980
choice is that a
combination of Gaussians.

857
00:47:11,980 --> 00:47:14,800
And the reason why
Gaussians have been chosen,

858
00:47:14,800 --> 00:47:17,590
and actually why the most
famous quantum chemistry

859
00:47:17,590 --> 00:47:20,950
code is called Gaussian
is that it's very easy

860
00:47:20,950 --> 00:47:24,460
to do the analytical
integrals that

861
00:47:24,460 --> 00:47:28,450
are present in the exchange
term for Hartree-Fock.

862
00:47:28,450 --> 00:47:31,320
So if you want an exchange
term for Hartree-Fock,

863
00:47:31,320 --> 00:47:34,270
the term means the
choice of Gaussian

864
00:47:34,270 --> 00:47:39,880
as a particularly
convenient basis set.

865
00:47:39,880 --> 00:47:43,120
And again, the product of
two Gaussians is a Gaussian,

866
00:47:43,120 --> 00:47:48,430
and that's why this is
ultimately a good choice.

867
00:47:48,430 --> 00:47:51,340
This is what has developed.

868
00:47:51,340 --> 00:47:56,230
At the end, you can still use
Gaussians to study solids,

869
00:47:56,230 --> 00:47:58,960
or you can use plane
waves to study molecules.

870
00:47:58,960 --> 00:48:03,310
And there is even a sort
of richer phenomenology.

871
00:48:03,310 --> 00:48:07,300
But generally speaking,
these two approaches,

872
00:48:07,300 --> 00:48:10,780
that of localized
functions in chemistry

873
00:48:10,780 --> 00:48:16,630
and of delocalized function
in extended systems,

874
00:48:16,630 --> 00:48:23,110
are the basic categories.

875
00:48:27,550 --> 00:48:32,050
We'll start from the
point of view of solids,

876
00:48:32,050 --> 00:48:33,700
and then you'll
see why we actually

877
00:48:33,700 --> 00:48:37,210
describe-- we use plain waves
to describe these orbitals.

878
00:48:37,210 --> 00:48:39,640
And when you deal
with solid, you

879
00:48:39,640 --> 00:48:44,560
need to be aware of one of
the fundamental symmetries

880
00:48:44,560 --> 00:48:48,520
that the eigenstates
of our solution

881
00:48:48,520 --> 00:48:52,180
in our periodic potential,
as what happens in solids,

882
00:48:52,180 --> 00:48:53,470
satisfy.

883
00:48:53,470 --> 00:48:57,160
That is, in particular,
the Hamiltonian of a solid

884
00:48:57,160 --> 00:49:00,190
is periodically invariant.

885
00:49:00,190 --> 00:49:02,290
That's the characteristic
of a solid.

886
00:49:02,290 --> 00:49:07,570
It's a regular periodic array of
externals pseudopotential that

887
00:49:07,570 --> 00:49:08,770
[INAUDIBLE].

888
00:49:08,770 --> 00:49:11,440
So in the language
of quantum mechanics,

889
00:49:11,440 --> 00:49:15,340
we say that the
Hamiltonian commutes

890
00:49:15,340 --> 00:49:18,040
with the translational operator.

891
00:49:18,040 --> 00:49:21,280
Translation operator
by a vector--

892
00:49:21,280 --> 00:49:23,750
that is, a direct
lattice vector.

893
00:49:23,750 --> 00:49:30,100
So if you take out cubic system
and you displace, you translate

894
00:49:30,100 --> 00:49:34,750
your set of potentials by
1 cubic lattice vector,

895
00:49:34,750 --> 00:49:38,060
you have the same operator.

896
00:49:38,060 --> 00:49:39,800
That is, trivially
to say that, if you

897
00:49:39,800 --> 00:49:45,770
have a regular array of blue
dots that is infinite in space,

898
00:49:45,770 --> 00:49:50,660
and you displace them, and
the displaced one r crosses,

899
00:49:50,660 --> 00:49:55,190
and they have been displaced
by this lattice vector r here,

900
00:49:55,190 --> 00:49:57,470
and both the red dots
and the blue dots

901
00:49:57,470 --> 00:49:59,960
are infinite,
well, by displacing

902
00:49:59,960 --> 00:50:01,910
the blue dot into
the red dots, you

903
00:50:01,910 --> 00:50:05,120
have the same identical problem.

904
00:50:05,120 --> 00:50:09,170
When two operator commuting
quantum mechanics, what we know

905
00:50:09,170 --> 00:50:14,420
is that we can find eigenstates
that are common eigenstates.

906
00:50:14,420 --> 00:50:18,080
That is, they are eigenstates
both of the first operator,

907
00:50:18,080 --> 00:50:19,760
and then on the second operator.

908
00:50:19,760 --> 00:50:22,610
I won't dwell on
this in case you

909
00:50:22,610 --> 00:50:26,450
are not familiar with parts
of this quantum mechanical

910
00:50:26,450 --> 00:50:27,210
problem.

911
00:50:27,210 --> 00:50:31,310
But what I want to highlight
is that the net result,

912
00:50:31,310 --> 00:50:34,850
the fact that we can
choose eigenvectors that

913
00:50:34,850 --> 00:50:37,025
are common eigenvectors
of the Hamiltonians

914
00:50:37,025 --> 00:50:40,280
and of the translation
operator, tells us

915
00:50:40,280 --> 00:50:44,300
that a genetic eigenvector
for aperiodic potential

916
00:50:44,300 --> 00:50:49,610
needs to have a very well
defined symmetry and a very

917
00:50:49,610 --> 00:50:51,350
well defined symmetric form.

918
00:50:51,350 --> 00:50:55,040
And that's sort of summarized
in the Bloch theorem.

919
00:50:55,040 --> 00:50:59,330
And that tells us
that an eigenvector

920
00:50:59,330 --> 00:51:04,540
will be given by a term--

921
00:51:04,540 --> 00:51:10,420
a function that has the same
periodicity of the crystal

922
00:51:10,420 --> 00:51:16,510
times a plane wave--

923
00:51:16,510 --> 00:51:19,510
again, a complex
exponential that

924
00:51:19,510 --> 00:51:22,960
modulates these
periodic parts and that

925
00:51:22,960 --> 00:51:25,370
can have any wavelength.

926
00:51:25,370 --> 00:51:29,590
So in other words, what the
Bloch theorem here is telling

927
00:51:29,590 --> 00:51:33,640
us is that the solutions,
the eigenstates

928
00:51:33,640 --> 00:51:37,960
of aperiodic Hamiltonian can
have any possible periodicity.

929
00:51:37,960 --> 00:51:41,920
You don't have to think at
the electronic eigenstates

930
00:51:41,920 --> 00:51:46,480
of aperiodic Hamiltonian as
having the same periodicity

931
00:51:46,480 --> 00:51:48,010
of your lattice.

932
00:51:48,010 --> 00:51:51,790
But the shape that they
have is actually this.

933
00:51:51,790 --> 00:51:54,400
That is the product
of two function--

934
00:51:54,400 --> 00:52:00,250
one that has any arbitrary
wavelength, and another term--

935
00:52:00,250 --> 00:52:04,310
the so-called periodic part
of our orbitals-- that does

936
00:52:04,310 --> 00:52:09,260
have the same periodicity
of the lattice.

937
00:52:09,260 --> 00:52:12,100
This can be proven
in a variety of ways,

938
00:52:12,100 --> 00:52:16,390
and somehow I summarized
here why ultimately this

939
00:52:16,390 --> 00:52:18,940
is the shape that
the eigenfunction

940
00:52:18,940 --> 00:52:21,100
of the Hamiltonian
needs to have,

941
00:52:21,100 --> 00:52:24,820
basically because if
we have a translation

942
00:52:24,820 --> 00:52:26,650
and the Hamiltonian
is invariant,

943
00:52:26,650 --> 00:52:31,330
the charge density of your
solid must also be invariant.

944
00:52:31,330 --> 00:52:34,150
And this solution
satisfies that,

945
00:52:34,150 --> 00:52:38,440
because if you look at the
wave function-- not at r,

946
00:52:38,440 --> 00:52:41,410
but r plus a direct
lattice vector--

947
00:52:41,410 --> 00:52:44,050
this term will be
unchanged by definition

948
00:52:44,050 --> 00:52:46,330
because this term
has the periodicity.

949
00:52:46,330 --> 00:52:50,800
This term will become
e to the ik times

950
00:52:50,800 --> 00:52:54,597
scalar product r plus a
direct lattice vector.

951
00:52:54,597 --> 00:52:56,680
But when you look at the
charge density-- that is,

952
00:52:56,680 --> 00:52:58,100
the square modulus--

953
00:52:58,100 --> 00:52:59,580
this term disappears.

954
00:52:59,580 --> 00:53:03,700
So only this remains
in the charge density.

955
00:53:03,700 --> 00:53:05,950
And you also want
two translations

956
00:53:05,950 --> 00:53:08,590
being equivalent to the
sum of the other two.

957
00:53:08,590 --> 00:53:12,820
And again, if you translate
that by two consecutive lattice

958
00:53:12,820 --> 00:53:15,070
vectors, this term
doesn't change.

959
00:53:15,070 --> 00:53:19,210
And the exponential of a
sum of two lattice vectors

960
00:53:19,210 --> 00:53:22,150
is going to be just the
product of two exponentials

961
00:53:22,150 --> 00:53:23,920
for each lattice vector.

962
00:53:23,920 --> 00:53:26,920
But the fundamental
concept here is

963
00:53:26,920 --> 00:53:29,320
this-- that the
wave function are

964
00:53:29,320 --> 00:53:33,830
modulated by this wavelength
and we actually classify them.

965
00:53:33,830 --> 00:53:36,730
We do these two quantum numbers.

966
00:53:36,730 --> 00:53:40,150
As we classify orbitals
in the periodic table,

967
00:53:40,150 --> 00:53:41,830
we give them quantum numbers.

968
00:53:41,830 --> 00:53:46,120
We say this is going to be
a 3s electron with spin-up.

969
00:53:46,120 --> 00:53:49,960
We are saying what
the quantum number nlm

970
00:53:49,960 --> 00:53:52,030
and spin are for that electron.

971
00:53:52,030 --> 00:53:55,840
In a solid, these are
our quantum numbers.

972
00:53:55,840 --> 00:53:58,240
We have a bond index.

973
00:53:58,240 --> 00:54:02,290
That is, if you wanted solid
equivalent of your energy

974
00:54:02,290 --> 00:54:07,570
levels, and it's actually
a discrete integer index

975
00:54:07,570 --> 00:54:11,860
that somehow classifies
the different eigenvectors,

976
00:54:11,860 --> 00:54:18,620
but those eigenvectors can
have also any wavelength.

977
00:54:18,620 --> 00:54:21,730
And so the wavelength
of the block orbital

978
00:54:21,730 --> 00:54:26,270
is another quantum number
that characterize our system.

979
00:54:26,270 --> 00:54:28,990
So when you actually
look at a molecule,

980
00:54:28,990 --> 00:54:31,960
you think that in
terms of energy levels,

981
00:54:31,960 --> 00:54:34,900
and you draw an energy
diagram in which you will

982
00:54:34,900 --> 00:54:38,110
have a 1s, 2s, 3s energy level.

983
00:54:38,110 --> 00:54:41,110
When you think of the
possible energy levels

984
00:54:41,110 --> 00:54:44,050
from four electrons
in a solid, you'll

985
00:54:44,050 --> 00:54:48,850
draw a more complex band
diagram in which you

986
00:54:48,850 --> 00:54:52,600
will sort of classify
levels, not only in terms

987
00:54:52,600 --> 00:54:57,320
of discrete integers, but also
in terms of the wavelength.

988
00:54:57,320 --> 00:55:02,620
And so this is what is usually
band energy diagram, of which I

989
00:55:02,620 --> 00:55:03,880
have an example here.

990
00:55:15,030 --> 00:55:19,760
So this is-- would be the
solid equivalent of the energy

991
00:55:19,760 --> 00:55:20,490
levels.

992
00:55:20,490 --> 00:55:22,880
So for a molecule,
you would have

993
00:55:22,880 --> 00:55:26,360
a discrete set of energy level.

994
00:55:26,360 --> 00:55:31,100
For a solid, you don't have
only a set of discrete quantum

995
00:55:31,100 --> 00:55:36,140
numbers n, but you have also a
continuous set of wavelengths.

996
00:55:36,140 --> 00:55:40,580
So you have bands
that show you how

997
00:55:40,580 --> 00:55:43,280
your energy of that
eigenvector changes

998
00:55:43,280 --> 00:55:47,000
depending on the wavelength
of that eigenvector.

999
00:55:47,000 --> 00:55:50,810
And what I've plotted
here is actually

1000
00:55:50,810 --> 00:55:56,060
the band energy diagram for
a free electron in an FCC

1001
00:55:56,060 --> 00:55:57,290
lattice.

1002
00:55:57,290 --> 00:56:00,050
What I'm saying is
that, suppose that I'm

1003
00:56:00,050 --> 00:56:05,000
considering an electron that
doesn't feel any potential.

1004
00:56:05,000 --> 00:56:08,130
What would be its band energy?

1005
00:56:08,130 --> 00:56:10,070
Well, it's band
energy is actually

1006
00:56:10,070 --> 00:56:12,000
going to be a parabola.

1007
00:56:12,000 --> 00:56:17,800
So it's something that
we would represent

1008
00:56:17,800 --> 00:56:21,860
as this sort of
energy, as being really

1009
00:56:21,860 --> 00:56:24,740
proportional to its wavelength.

1010
00:56:24,740 --> 00:56:30,470
That is, again, because a
free electron is represented

1011
00:56:30,470 --> 00:56:35,600
by a plain wave e to the ikr.

1012
00:56:35,600 --> 00:56:39,320
And so the second
derivative of a plain wave--

1013
00:56:39,320 --> 00:56:41,510
that is, the quantum
kinetic energy--

1014
00:56:41,510 --> 00:56:44,150
is just the k squared term.

1015
00:56:44,150 --> 00:56:48,030
And if an electron is free,
there is no potential.

1016
00:56:48,030 --> 00:56:53,690
So we can think of the energy
as a function of wavelength

1017
00:56:53,690 --> 00:56:58,350
for a free electron to
be perfectly parabolic.

1018
00:56:58,350 --> 00:57:01,080
But now what we can
think for a moment

1019
00:57:01,080 --> 00:57:06,660
is also how to
represent this parabola.

1020
00:57:06,660 --> 00:57:10,740
In principle, this parabola
is a single parabola infinite

1021
00:57:10,740 --> 00:57:12,930
as a function of wavelength.

1022
00:57:12,930 --> 00:57:20,550
But we can also, for a moment,
represent the data in an FCC

1023
00:57:20,550 --> 00:57:24,090
or in any arbitrary
periodic lattice.

1024
00:57:24,090 --> 00:57:28,110
So what we could be saying is
that suppose that now there

1025
00:57:28,110 --> 00:57:32,430
is an imaginary geometric
structure that is,

1026
00:57:32,430 --> 00:57:38,220
say, in real space Bravais
lattice with FCC periodicity,

1027
00:57:38,220 --> 00:57:41,220
and that, in reciprocal
space, will correspond

1028
00:57:41,220 --> 00:57:43,470
to a BCC reciprocal lattice.

1029
00:57:43,470 --> 00:57:48,870
And what I want to do is I
want to take this parabola

1030
00:57:48,870 --> 00:57:51,510
and fold it.

1031
00:57:51,510 --> 00:57:57,600
So instead of having a
single parabolic branch that

1032
00:57:57,600 --> 00:58:04,140
extends to infinity, every
time I hit in Brillouin zone

1033
00:58:04,140 --> 00:58:07,050
in the reciprocal
space a boundary,

1034
00:58:07,050 --> 00:58:10,650
I fold this parabola back.

1035
00:58:10,650 --> 00:58:14,670
So what I've shown here is
really the band energy diagram

1036
00:58:14,670 --> 00:58:18,720
that is a parabola,
supposing for a moment

1037
00:58:18,720 --> 00:58:24,390
that my electron leaves in the
geometry of an FCC lattice.

1038
00:58:24,390 --> 00:58:26,460
And this is just to
give you the feeling

1039
00:58:26,460 --> 00:58:30,390
that sometimes something
that looks very complex

1040
00:58:30,390 --> 00:58:35,550
is just a representation
of something very simple

1041
00:58:35,550 --> 00:58:37,560
in a specific geometry.

1042
00:58:37,560 --> 00:58:40,350
And now you see in a
moment what happens

1043
00:58:40,350 --> 00:58:47,760
if we study the electron in a
system that truly has the FCC

1044
00:58:47,760 --> 00:58:49,630
Bravais lattice in real space.

1045
00:58:49,630 --> 00:58:54,250
And so it is a BCC Bravais
lattice in reciprocal space.

1046
00:58:54,250 --> 00:58:55,830
And that is silicon.

1047
00:58:55,830 --> 00:59:00,210
And you see that the
band energy dispersions

1048
00:59:00,210 --> 00:59:05,160
for electrons in silicon
looked, actually very, very

1049
00:59:05,160 --> 00:59:07,200
similar to free electrons.

1050
00:59:07,200 --> 00:59:09,210
You see now for a
moment something

1051
00:59:09,210 --> 00:59:11,160
that you thought it
was always extremely

1052
00:59:11,160 --> 00:59:14,340
complex, like the band
diagram for silicon,

1053
00:59:14,340 --> 00:59:18,780
turns out to be just a
parabola, just a free electron

1054
00:59:18,780 --> 00:59:20,760
slightly modified.

1055
00:59:20,760 --> 00:59:23,880
And you can clearly see
all the same pieces,

1056
00:59:23,880 --> 00:59:25,920
but obviously now
these electrons,

1057
00:59:25,920 --> 00:59:28,080
these valence
electrons in silicon

1058
00:59:28,080 --> 00:59:30,820
feel the periodic potential.

1059
00:59:30,820 --> 00:59:33,120
And so the overall
solution is modified.

1060
00:59:33,120 --> 00:59:36,480
And for those of you
familiar with solids,

1061
00:59:36,480 --> 00:59:39,840
there is a band gap
between valence electrons

1062
00:59:39,840 --> 00:59:41,320
and conduction electrons.

1063
00:59:41,320 --> 00:59:43,650
And here, we would
be at a gamma point.

1064
00:59:43,650 --> 00:59:45,630
This would be the
lowest valence band,

1065
00:59:45,630 --> 00:59:50,490
and here we would have the
heavy and the light whole bands

1066
00:59:50,490 --> 00:59:52,380
for the top valence--

1067
00:59:52,380 --> 00:59:54,180
for the top valence bands.

1068
00:59:54,180 --> 01:00:00,150
But what this is telling us
is that electrons in a solid

1069
01:00:00,150 --> 01:00:04,050
really look like free
electrons with a little bit

1070
01:00:04,050 --> 01:00:09,840
of a perturbation that is enough
to transform this parabola

1071
01:00:09,840 --> 01:00:12,450
into something different.

1072
01:00:12,450 --> 01:00:15,670
And again, because they
are almost free electrons,

1073
01:00:15,670 --> 01:00:19,860
and because they have
the periodicity--

1074
01:00:19,860 --> 01:00:21,930
the periodic part of
the Bloch orbitals

1075
01:00:21,930 --> 01:00:24,810
need to have the periodicity
of the crystal lattice

1076
01:00:24,810 --> 01:00:26,790
that we are studying--
it will actually

1077
01:00:26,790 --> 01:00:32,640
be extremely appropriate
to use as a basis set

1078
01:00:32,640 --> 01:00:37,860
to describe at each k
point in the Brillouin.

1079
01:00:37,860 --> 01:00:39,990
The periodic part of
the Bloch orbitals

1080
01:00:39,990 --> 01:00:43,800
will be extremely convenient
to use plane waves.

1081
01:00:43,800 --> 01:00:45,600
And this is the
reason, ultimately,

1082
01:00:45,600 --> 01:00:49,830
why I written here,
sort of again reminding

1083
01:00:49,830 --> 01:00:52,890
some basics of crystallography.

1084
01:00:52,890 --> 01:00:57,370
I've taken, say, the
zinc blende structure.

1085
01:00:57,370 --> 01:01:04,230
So again, an FCC lattice here,
where these three here in green

1086
01:01:04,230 --> 01:01:09,040
would be the primitive lattice
vectors of the FCC lattice--

1087
01:01:09,040 --> 01:01:14,910
so what I am calling
here a1, a2 and a3.

1088
01:01:14,910 --> 01:01:16,950
And these three lattices--

1089
01:01:16,950 --> 01:01:23,400
three vector lattices in 1/2,
1/2, 0, 0, 1/2, 1/2, and 1/2,

1090
01:01:23,400 --> 01:01:28,620
0, 1/2 really represents
my Bravais lattice

1091
01:01:28,620 --> 01:01:34,260
by repeating blue atoms at any
linear combination of three--

1092
01:01:34,260 --> 01:01:35,700
of these three vectors.

1093
01:01:35,700 --> 01:01:39,720
I span all the infinite
blue sublattice

1094
01:01:39,720 --> 01:01:41,850
of the cations in a zinc blende.

1095
01:01:41,850 --> 01:01:46,840
And the zinc blende is just a
parallel compound of silicon.

1096
01:01:46,840 --> 01:01:50,080
It's just that the second atom
in the unit cell is different.

1097
01:01:50,080 --> 01:01:52,860
And again, by
applying translations

1098
01:01:52,860 --> 01:01:55,170
that are linear combination
of these green vectors,

1099
01:01:55,170 --> 01:01:57,560
I span all the blue lattice.

1100
01:01:57,560 --> 01:01:59,570
And silicon would be
identical to this.

1101
01:01:59,570 --> 01:02:02,580
Just the red and the
blue would be identical.

1102
01:02:02,580 --> 01:02:07,700
So having defined
direct vector lattices,

1103
01:02:07,700 --> 01:02:15,170
the a1, a2, and a3, what I can
define is a set of dual vectors

1104
01:02:15,170 --> 01:02:21,380
that we call reciprocal lattice
vectors g1, g2, and g3 that

1105
01:02:21,380 --> 01:02:26,780
are such that the scalar
product between g1 and g2

1106
01:02:26,780 --> 01:02:29,000
is either 1 or 0.

1107
01:02:29,000 --> 01:02:32,810
And this is the [INAUDIBLE]
delta times 2 pi.

1108
01:02:32,810 --> 01:02:34,100
This is a definition.

1109
01:02:34,100 --> 01:02:37,070
This is what we know
is the definition

1110
01:02:37,070 --> 01:02:42,050
of the reciprocal
lattice vector,

1111
01:02:42,050 --> 01:02:45,290
but what is very
important for us

1112
01:02:45,290 --> 01:02:53,120
is that these primitive
reciprocal lattice vectors are

1113
01:02:53,120 --> 01:02:57,980
the fundamental descriptor of
all the plane waves that I want

1114
01:02:57,980 --> 01:03:03,410
to use in describing a solid
and the reason for that

1115
01:03:03,410 --> 01:03:12,560
is that a plane wave written
like this exponential of iGr,

1116
01:03:12,560 --> 01:03:17,180
where the G here, where
this vector is just

1117
01:03:17,180 --> 01:03:19,880
a linear combination
with integer

1118
01:03:19,880 --> 01:03:23,810
number of my primitive
lattice vectors--

1119
01:03:23,810 --> 01:03:29,360
so it's going to be l
with l integer G1 times G2

1120
01:03:29,360 --> 01:03:33,600
to times n G3.

1121
01:03:33,600 --> 01:03:39,180
So well defined by the
triplet of numbers l, m, n.

1122
01:03:39,180 --> 01:03:42,720
Well, these plane
waves, e to the iGr,

1123
01:03:42,720 --> 01:03:46,440
for any linear combination
of g1, g2, and g3

1124
01:03:46,440 --> 01:03:50,820
is a function,
defined in real space,

1125
01:03:50,820 --> 01:03:56,130
that is a periodicity
compatible with the periodicity

1126
01:03:56,130 --> 01:03:58,170
of my direct lattice.

1127
01:03:58,170 --> 01:04:02,760
So suppose that we were in
one dimension for a moment,

1128
01:04:02,760 --> 01:04:08,340
and I would have a system
that has this periodicity.

1129
01:04:08,340 --> 01:04:12,030
And remember, the periodic
part of the Bloch orbitals

1130
01:04:12,030 --> 01:04:15,460
will have the same
periodicity in real space.

1131
01:04:15,460 --> 01:04:19,530
So maybe the
periodic part of that

1132
01:04:19,530 --> 01:04:23,460
will look something like this,
with the same periodicity.

1133
01:04:26,680 --> 01:04:32,890
Well, then what I want
to do is really expand

1134
01:04:32,890 --> 01:04:37,060
this periodic function
as a linear combination

1135
01:04:37,060 --> 01:04:40,270
with appropriate
coefficient of plane waves

1136
01:04:40,270 --> 01:04:42,490
with different
wavelengths, but I

1137
01:04:42,490 --> 01:04:45,730
need to choose these
plane waves to all

1138
01:04:45,730 --> 01:04:50,860
have the same periodicity
of the green lattice.

1139
01:04:50,860 --> 01:04:57,010
So my plane waves will
only look like this,

1140
01:04:57,010 --> 01:05:00,430
and this will be the sort
of fundamental harmonic

1141
01:05:00,430 --> 01:05:01,420
if you want.

1142
01:05:01,420 --> 01:05:08,250
And then another plane wave
could have this periodicity,

1143
01:05:08,250 --> 01:05:12,840
and so on and so forth, going
to higher and higher wavelength.

1144
01:05:12,840 --> 01:05:17,400
But the compact mathematical
representation of this

1145
01:05:17,400 --> 01:05:18,540
is in here.

1146
01:05:18,540 --> 01:05:21,630
That is, given a
direct lattice defined

1147
01:05:21,630 --> 01:05:24,780
by the three vectors
a1, a2, and a3,

1148
01:05:24,780 --> 01:05:27,540
I can define a
reciprocal lattice

1149
01:05:27,540 --> 01:05:31,650
that is represented by the
three vector G1, G1, and G3.

1150
01:05:31,650 --> 01:05:36,300
And any linear combination with
integer numbers of this G1, G2,

1151
01:05:36,300 --> 01:05:40,680
and G3 will give me a vector
G such that the complex

1152
01:05:40,680 --> 01:05:44,430
exponential e to
the iGr is actually

1153
01:05:44,430 --> 01:05:48,810
a periodicity that is compatible
with my direct lattice.

1154
01:05:48,810 --> 01:05:53,580
That is, the moment I
create a direct lattice,

1155
01:05:53,580 --> 01:05:56,340
there are an infinite
number of wavelengths

1156
01:05:56,340 --> 01:05:58,980
that are not compatible
with the periodicity

1157
01:05:58,980 --> 01:06:00,960
that I've thrown into my system.

1158
01:06:00,960 --> 01:06:02,160
Those disappear.

1159
01:06:02,160 --> 01:06:04,440
I don't want to have
anything to deal with.

1160
01:06:04,440 --> 01:06:09,240
What I'm left is with
a numerable infinity

1161
01:06:09,240 --> 01:06:13,590
of plane waves that are all
compatible with the direct

1162
01:06:13,590 --> 01:06:15,700
lattice that I'm choosing from.

1163
01:06:15,700 --> 01:06:18,480
And you see now there
is a natural way

1164
01:06:18,480 --> 01:06:21,450
of both choosing plane waves--

1165
01:06:21,450 --> 01:06:22,650
they will be here--

1166
01:06:22,650 --> 01:06:26,580
and also a natural way to
choose the most important one,

1167
01:06:26,580 --> 01:06:30,660
because when we choose our
basis set for our calculation,

1168
01:06:30,660 --> 01:06:35,130
we'll start from the plane
waves of longer wavelength.

1169
01:06:35,130 --> 01:06:38,430
That is, that the
blue wavelength

1170
01:06:38,430 --> 01:06:39,810
that I've plotted here--

1171
01:06:39,810 --> 01:06:42,480
and will decrease
the wavelength.

1172
01:06:42,480 --> 01:06:45,720
That will include more
and more plane wave

1173
01:06:45,720 --> 01:06:49,260
that sort of have finer
and finer wavelength, finer

1174
01:06:49,260 --> 01:06:50,760
and finer resolution.

1175
01:06:50,760 --> 01:06:53,790
And that can be
just naturally done

1176
01:06:53,790 --> 01:06:58,350
by including plane waves
with G vectors that

1177
01:06:58,350 --> 01:07:02,280
have a larger square modulus.

1178
01:07:02,280 --> 01:07:07,470
That is the larger this G vector
here, the finer the resolution,

1179
01:07:07,470 --> 01:07:11,020
the higher the wavelength
of this plane wave.

1180
01:07:11,020 --> 01:07:13,560
So there is truly a
natural way to choose

1181
01:07:13,560 --> 01:07:16,740
basis set now,
because, again, what

1182
01:07:16,740 --> 01:07:19,660
we have is a reciprocal space.

1183
01:07:19,660 --> 01:07:22,890
And so we'll have a
Brillouin zone in red.

1184
01:07:22,890 --> 01:07:25,650
We'll have, if I'm
in two dimension,

1185
01:07:25,650 --> 01:07:28,690
the two G1 and G2 vectors.

1186
01:07:28,690 --> 01:07:32,640
And so the linear combination
G's that I've described

1187
01:07:32,640 --> 01:07:40,710
will be given by the infinite,
but discrete set of G vectors

1188
01:07:40,710 --> 01:07:43,060
represented here.

1189
01:07:43,060 --> 01:07:48,060
And so only the G vectors
denoted here with a cross

1190
01:07:48,060 --> 01:07:50,580
give rise to a
plane wave that has

1191
01:07:50,580 --> 01:07:54,870
the compatible periodicity
with my periodic boundary

1192
01:07:54,870 --> 01:07:56,200
conditions.

1193
01:07:56,200 --> 01:08:00,900
And so in order to choose
a basis set, what I do,

1194
01:08:00,900 --> 01:08:06,270
I just draw a circle, or in
three dimensions, a sphere,

1195
01:08:06,270 --> 01:08:11,130
and I decide to include
all the vector that

1196
01:08:11,130 --> 01:08:13,140
sit inside that sphere.

1197
01:08:13,140 --> 01:08:16,410
And by making that
sphere larger and larger,

1198
01:08:16,410 --> 01:08:18,930
I'm going to include
G vectors that

1199
01:08:18,930 --> 01:08:22,200
have larger and larger modules
that is finer and finer

1200
01:08:22,200 --> 01:08:23,130
wavelength.

1201
01:08:23,130 --> 01:08:27,810
And that is what is called
your cutoff in your basis set.

1202
01:08:27,810 --> 01:08:32,040
So one of the fundamental
parameters of your calculation

1203
01:08:32,040 --> 01:08:36,210
will be choosing your cutoff
for your plane wave basis set--

1204
01:08:36,210 --> 01:08:40,859
that is, choosing the radius of
that sphere that includes plane

1205
01:08:40,859 --> 01:08:46,050
waves with wavelength compatible
to the periodic boundary

1206
01:08:46,050 --> 01:08:49,920
conditions up to a
certain resolution.

1207
01:08:49,920 --> 01:08:53,310
You make this cutoff
sphere larger and larger.

1208
01:08:53,310 --> 01:08:57,510
You systematically include
finer and finer resolutions,

1209
01:08:57,510 --> 01:09:00,750
so you become better and better
at describing your problem.

1210
01:09:00,750 --> 01:09:04,260
And you'll always need to
check in your first calculation

1211
01:09:04,260 --> 01:09:06,120
that your basis
set is good enough.

1212
01:09:06,120 --> 01:09:10,240
That is, by increasing
your cutoff radius,

1213
01:09:10,240 --> 01:09:15,210
you will not see any
significant physical change

1214
01:09:15,210 --> 01:09:18,210
in the quantities that
you are calculating.

1215
01:09:18,210 --> 01:09:21,510
You don't see a physical
change in the energy

1216
01:09:21,510 --> 01:09:22,529
as a function of volume.

1217
01:09:22,529 --> 01:09:26,430
You don't see a change in the
forces acting on atoms, and so

1218
01:09:26,430 --> 01:09:27,300
on and so forth.

1219
01:09:30,590 --> 01:09:32,300
This will take forever.

1220
01:09:32,300 --> 01:09:36,500
Again, this is not the
only choice of basis set,

1221
01:09:36,500 --> 01:09:40,160
but as you can see, it's
really the appropriate one

1222
01:09:40,160 --> 01:09:42,350
for periodic systems.

1223
01:09:42,350 --> 01:09:48,770
And it has sort of a set of nice
advantages, of which I would

1224
01:09:48,770 --> 01:09:53,660
say the most important
is the fact that it's

1225
01:09:53,660 --> 01:09:58,970
systematic in the sense that
you can continuously improve

1226
01:09:58,970 --> 01:10:05,810
the resolution in your problem
by including more plane waves.

1227
01:10:05,810 --> 01:10:15,310
There is a negative
side to this that

1228
01:10:15,310 --> 01:10:20,110
says that it doesn't have
any information on how

1229
01:10:20,110 --> 01:10:23,860
valence electrons should
look like close to a nucleus

1230
01:10:23,860 --> 01:10:25,390
or close to a core.

1231
01:10:25,390 --> 01:10:29,230
It requires a very large
number of basis elements.

1232
01:10:29,230 --> 01:10:31,210
So plane wave
calculation tend to have

1233
01:10:31,210 --> 01:10:33,580
a lot of elements in that.

1234
01:10:33,580 --> 01:10:39,610
There are a number of
other important practical

1235
01:10:39,610 --> 01:10:40,630
consequences.

1236
01:10:40,630 --> 01:10:43,720
In particular, as
much as Gaussians,

1237
01:10:43,720 --> 01:10:46,960
they allow for very
easy evaluation

1238
01:10:46,960 --> 01:10:49,400
of some of the analytical
terms that we need.

1239
01:10:49,400 --> 01:10:54,220
So it's very easy to take
the gradient of a plane wave

1240
01:10:54,220 --> 01:10:56,470
or the Laplacian
of a plane wave,

1241
01:10:56,470 --> 01:11:00,100
because basically the derivative
of a complex exponential,

1242
01:11:00,100 --> 01:11:04,360
if you have e the iGr, the
derivative with respect

1243
01:11:04,360 --> 01:11:08,020
to r of this is just
the function itself

1244
01:11:08,020 --> 01:11:10,450
times the vector G. This
would be the gradient,

1245
01:11:10,450 --> 01:11:13,360
and the Laplacian-- that is,
the second derivative-- is just

1246
01:11:13,360 --> 01:11:16,220
G squared times
the wave function.

1247
01:11:16,220 --> 01:11:19,000
And so all these terms
are easy to calculate,

1248
01:11:19,000 --> 01:11:24,490
and there is a sort of more
subtle conclusion of this,

1249
01:11:24,490 --> 01:11:29,290
that if you start to have a
calculation in which atoms

1250
01:11:29,290 --> 01:11:34,360
move, like a molecular dynamic
calculation in which you need

1251
01:11:34,360 --> 01:11:36,610
to calculate things
like forces, you

1252
01:11:36,610 --> 01:11:39,520
need to calculate the derivative
of the energy with respect

1253
01:11:39,520 --> 01:11:41,290
to the position of an atom.

1254
01:11:41,290 --> 01:11:46,240
But now the energy in
itself is an expression

1255
01:11:46,240 --> 01:11:49,270
that involves linear
combination of your basis set.

1256
01:11:49,270 --> 01:11:52,660
Well, this basis
set does not depend

1257
01:11:52,660 --> 01:11:54,830
on the position of the atom.

1258
01:11:54,830 --> 01:11:58,780
So there is no term in the
force in the derivative

1259
01:11:58,780 --> 01:12:01,090
of the energy with
respect to the position

1260
01:12:01,090 --> 01:12:02,500
that comes from this.

1261
01:12:02,500 --> 01:12:06,100
If, on the other hand, you are
using, say, a Gaussian that

1262
01:12:06,100 --> 01:12:09,700
was centered on an atom, a
Gaussian centered on an atom

1263
01:12:09,700 --> 01:12:13,010
would have in there the
position of the atom.

1264
01:12:13,010 --> 01:12:15,340
And so in order to take
the derivative, the force,

1265
01:12:15,340 --> 01:12:18,550
you would also need to take a
derivative of your basis set.

1266
01:12:18,550 --> 01:12:21,310
And these are what are
called Pulay terms.

1267
01:12:21,310 --> 01:12:24,430
And that just add another
layer of complexity

1268
01:12:24,430 --> 01:12:25,760
to your calculation.

1269
01:12:25,760 --> 01:12:27,530
And again, this is,
I think, ultimately,

1270
01:12:27,530 --> 01:12:31,330
is one of the reasons why
in the solid-state community

1271
01:12:31,330 --> 01:12:36,700
problems like molecular
dynamics and forced relaxation

1272
01:12:36,700 --> 01:12:40,060
were developed earlier,
basically because it's much

1273
01:12:40,060 --> 01:12:42,400
simpler to calculate forces.

1274
01:12:42,400 --> 01:12:47,400
And I want to give
you an example on how

1275
01:12:47,400 --> 01:12:50,340
certain parts of
your problem become

1276
01:12:50,340 --> 01:12:53,370
very easy in your solution.

1277
01:12:53,370 --> 01:12:56,640
In particular, remember
sort of in the construction

1278
01:12:56,640 --> 01:12:59,730
of the self-consistent operator
from the charge density

1279
01:12:59,730 --> 01:13:03,120
we needed to construct
the Hartree operator

1280
01:13:03,120 --> 01:13:07,200
from the charge density integral
of n over r minus r prime.

1281
01:13:07,200 --> 01:13:09,150
Well, that's
basically a solution

1282
01:13:09,150 --> 01:13:11,910
of what is called
the Poisson equation.

1283
01:13:11,910 --> 01:13:17,400
That is, given a charge
density, the Hartree potential

1284
01:13:17,400 --> 01:13:23,620
is really coming from the
solution of this differential

1285
01:13:23,620 --> 01:13:24,280
equation.

1286
01:13:24,280 --> 01:13:26,710
That is, the Laplacian,
the second derivative

1287
01:13:26,710 --> 01:13:29,890
of the Hartree potential
is equal to minus,

1288
01:13:29,890 --> 01:13:35,320
in atomic units, 4 pi the
n electronic charge density

1289
01:13:35,320 --> 01:13:38,110
taken as positive here
in particular, just

1290
01:13:38,110 --> 01:13:40,210
to get the signs right.

1291
01:13:40,210 --> 01:13:41,950
This is a differential equation.

1292
01:13:41,950 --> 01:13:43,810
We need to solve
it in the course

1293
01:13:43,810 --> 01:13:46,420
of our self-consistent
problem, because remember,

1294
01:13:46,420 --> 01:13:48,670
we have sort of
diagonalized Hamiltonian.

1295
01:13:48,670 --> 01:13:50,020
We have gotten some orbitals.

1296
01:13:50,020 --> 01:13:52,515
From those orbitals, we have
calculated a charge density.

1297
01:13:52,515 --> 01:13:53,890
But now, from the
charge density,

1298
01:13:53,890 --> 01:13:56,500
we need to calculate the
electrostatic potential.

1299
01:13:56,500 --> 01:13:58,330
And that's how it works.

1300
01:13:58,330 --> 01:14:01,000
Well, this differential
equation is actually

1301
01:14:01,000 --> 01:14:04,360
trivial to solve
if, for a moment,

1302
01:14:04,360 --> 01:14:07,090
you think at a
plane wave solution.

1303
01:14:07,090 --> 01:14:11,490
That is, you think
that your potential--

1304
01:14:11,490 --> 01:14:14,220
that is, a function
of r is now being

1305
01:14:14,220 --> 01:14:20,120
written as a linear combination
where the coefficients are

1306
01:14:20,120 --> 01:14:25,520
called v of g of plane waves.

1307
01:14:25,520 --> 01:14:28,070
So I'm taking my potential.

1308
01:14:28,070 --> 01:14:29,990
It's going to have
the same periodicity

1309
01:14:29,990 --> 01:14:32,120
of the reciprocal
lattice, and I'm

1310
01:14:32,120 --> 01:14:35,570
writing it out as a linear
combination of waves.

1311
01:14:35,570 --> 01:14:39,990
And I'm doing the same for
the charge density here.

1312
01:14:39,990 --> 01:14:43,730
So the charge
density in itself is

1313
01:14:43,730 --> 01:14:51,270
going to be given by
a linear combination

1314
01:14:51,270 --> 01:14:55,410
with coefficient
G of plane waves.

1315
01:14:55,410 --> 01:15:00,000
So this is my expansion in
plane waves of this real space

1316
01:15:00,000 --> 01:15:01,110
functions.

1317
01:15:01,110 --> 01:15:03,540
And then the algebra is
trivial, because to take

1318
01:15:03,540 --> 01:15:07,050
the derivative of this red
term here, what I obtain

1319
01:15:07,050 --> 01:15:11,970
is nothing else
than the sum over g.

1320
01:15:11,970 --> 01:15:19,960
Second derivative will give me
r minus g squared v Hartree of G

1321
01:15:19,960 --> 01:15:22,930
e to the iGr.

1322
01:15:22,930 --> 01:15:30,070
And that is-- needs to be
equal to, well, minus 4 pi sum

1323
01:15:30,070 --> 01:15:34,420
over G n of G e to the iGr.

1324
01:15:34,420 --> 01:15:37,300
So I have done nothing
else than inserting

1325
01:15:37,300 --> 01:15:39,580
the explicit expansion
in plane waves

1326
01:15:39,580 --> 01:15:42,070
of my potential of
my charge density

1327
01:15:42,070 --> 01:15:45,020
in the differential equation.

1328
01:15:45,020 --> 01:15:49,330
And now, well, sort
of mathematically I

1329
01:15:49,330 --> 01:15:53,290
can actually sort of multiply
the left and the right hand

1330
01:15:53,290 --> 01:15:58,570
term for something like e to
the minus iG prime times r.

1331
01:15:58,570 --> 01:16:01,810
And then I can
integrate in the r.

1332
01:16:01,810 --> 01:16:04,150
And this is just the
mathematical operation

1333
01:16:04,150 --> 01:16:06,910
that on an
orthonormal basis set,

1334
01:16:06,910 --> 01:16:11,920
as this is, tells me that,
in order for this equality

1335
01:16:11,920 --> 01:16:15,310
to be satisfied, what I
really need to have is

1336
01:16:15,310 --> 01:16:19,690
that each coefficient
corresponding to the same G

1337
01:16:19,690 --> 01:16:22,510
vector on the left hand side
and on the right hand side

1338
01:16:22,510 --> 01:16:24,340
is separately equal.

1339
01:16:24,340 --> 01:16:31,020
So the solution to the
Poisson-Boltzmann equation

1340
01:16:31,020 --> 01:16:34,290
is trivial, because
what I need to have is

1341
01:16:34,290 --> 01:16:41,520
that G square vG is
equal to 4 pi nG.

1342
01:16:41,520 --> 01:16:46,350
That is, each coefficient
needs to be identical G by G.

1343
01:16:46,350 --> 01:16:49,830
So if I have a charge
density in real space,

1344
01:16:49,830 --> 01:16:53,010
I can expand it in plane waves.

1345
01:16:53,010 --> 01:16:55,290
And this coefficient and
nothing else than a Fourier

1346
01:16:55,290 --> 01:16:58,470
transform, so a computer
is very good, given

1347
01:16:58,470 --> 01:17:00,240
a periodic function
in real space,

1348
01:17:00,240 --> 01:17:02,310
to give me these coefficients.

1349
01:17:02,310 --> 01:17:05,520
And once I have this
coefficient n of G,

1350
01:17:05,520 --> 01:17:09,690
I will know instantly
what are the coefficients

1351
01:17:09,690 --> 01:17:13,590
of the potential that is
the solution of this Poisson

1352
01:17:13,590 --> 01:17:16,980
equation, because those
vG coefficients are really

1353
01:17:16,980 --> 01:17:21,750
just my charge density
coefficient multiplied by 4 pi

1354
01:17:21,750 --> 01:17:23,550
and divided by G square.

1355
01:17:23,550 --> 01:17:25,680
So you see with
plane waves, a lot

1356
01:17:25,680 --> 01:17:28,500
of the analytical work
in a transaction problem

1357
01:17:28,500 --> 01:17:31,230
becomes trivial.

1358
01:17:31,230 --> 01:17:37,440
And here, it's contained
one of the trickiest part

1359
01:17:37,440 --> 01:17:39,420
of electronic
structure calculation.

1360
01:17:39,420 --> 01:17:42,930
And the reason why electronic
structure calculation

1361
01:17:42,930 --> 01:17:47,370
of large system becomes more
and more computationally

1362
01:17:47,370 --> 01:17:50,700
ill-defined, and more
and more difficult to do

1363
01:17:50,700 --> 01:17:54,390
as the size of the
system becomes larger,

1364
01:17:54,390 --> 01:17:57,840
because as the size of
the system becomes larger,

1365
01:17:57,840 --> 01:18:02,520
your cell in real space
will become larger.

1366
01:18:02,520 --> 01:18:05,430
And your Brillouin zone
in reciprocal space

1367
01:18:05,430 --> 01:18:07,410
will become smaller and smaller.

1368
01:18:07,410 --> 01:18:10,350
As the three a1, a2,
a3 vectors becomes

1369
01:18:10,350 --> 01:18:13,590
larger, the three dual
reciprocal space vector--

1370
01:18:13,590 --> 01:18:16,360
G1, G1, and G3--
becomes smaller.

1371
01:18:16,360 --> 01:18:21,330
So the smallest of the G vector
will become smaller and smaller

1372
01:18:21,330 --> 01:18:23,490
as the supercell grows.

1373
01:18:23,490 --> 01:18:26,700
And so then you see that
there is an instability,

1374
01:18:26,700 --> 01:18:30,930
because this component
of the Hartree potential

1375
01:18:30,930 --> 01:18:35,610
is given by this n coefficient
divided by G square.

1376
01:18:35,610 --> 01:18:39,660
As your system becomes larger,
G square becomes smaller.

1377
01:18:39,660 --> 01:18:43,260
And so as smaller
numerical instability

1378
01:18:43,260 --> 01:18:47,580
in your charge density gives
you a large instability

1379
01:18:47,580 --> 01:18:48,940
in your potential.

1380
01:18:48,940 --> 01:18:51,300
And so you need to be
more and more careful.

1381
01:18:51,300 --> 01:18:53,640
You need to become
more and more careful

1382
01:18:53,640 --> 01:18:57,930
in your electrostatic solution
as the problem becomes

1383
01:18:57,930 --> 01:18:58,875
larger and larger.

1384
01:19:02,170 --> 01:19:04,380
This will take forever to say.

1385
01:19:04,380 --> 01:19:07,620
I think that I need to
stop here for today,

1386
01:19:07,620 --> 01:19:11,640
and what we'll see in the
next class on Thursday

1387
01:19:11,640 --> 01:19:14,790
will sort of wrap up the
last few technical details

1388
01:19:14,790 --> 01:19:18,780
that you need to go into the
computational lab on Tuesday,

1389
01:19:18,780 --> 01:19:23,040
and we'll start discussing
case studies for our density

1390
01:19:23,040 --> 01:19:24,780
functional problem.

1391
01:19:24,780 --> 01:19:28,440
As a reminder-- and I've
emailed and hope all of you have

1392
01:19:28,440 --> 01:19:30,030
received my email--

1393
01:19:30,030 --> 01:19:32,940
in order to do the
computational lab on Tuesday,

1394
01:19:32,940 --> 01:19:36,150
it is essential that
each and every one of you

1395
01:19:36,150 --> 01:19:39,090
has an independent
personal computer

1396
01:19:39,090 --> 01:19:42,780
account on the computer class,
because this calculation will

1397
01:19:42,780 --> 01:19:44,100
become expensive.

1398
01:19:44,100 --> 01:19:48,330
And we need to run them not
on the main a central node.

1399
01:19:48,330 --> 01:19:52,440
That will collapse if more than
two or three of you run on it.

1400
01:19:52,440 --> 01:19:55,260
But we need to spool
it via queuing system

1401
01:19:55,260 --> 01:20:01,020
on the nodes of the cluster that
lie beneath the master nose.

1402
01:20:01,020 --> 01:20:02,320
This is all for today.

1403
01:20:02,320 --> 01:20:06,890
Enjoy the snow, and see
you on Thursday morning.