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PROFESSOR 1: --labs based
on electronic structure,

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and this will cover the
next two computational labs.

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And on Thursday,
we'll start looking

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at some of the
finite temperature

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ideas that are based
on the energy models

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that we have seen.

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This, by the way, is
[? Roberto ?] [? Bajo ?]

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missing a penalty shot.

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So a reminder of what we
have seen in the last class.

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We have introduced the
idea of pseudopotentials.

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In order to remove the cost of
carrying out calculation that

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include the core electrons
that are very many, especially

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in sort of larger atoms,
and that have exceedingly

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high oscillations
around the nucleus

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due to the orthogonality
constraint. what we have done

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is we have substituted
what would be the z over r

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Coulomb potential with--

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inside of the core
region of the nucleus--

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a pseudopotential.

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That is a potential
that reproduces

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the effect of the nucleus, and
of the frozen core electrons.

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And as you can see,
this pseudopotential

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tends to be repulsive close to
the origin, basically again,

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reproducing this sort
of angular momentum push

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outwards of the core electrons
to the valence electrons.

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And in order to make this
pseudopotential very accurate,

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there has been this idea
that has been developed

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of norm conserving
pseudopotential that

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will act differently on
the different components

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of a valence electron
wave function.

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So you have an
incoming electron,

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so you have a sort of ground
state valence wave function.

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You can decompose it.

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You can decide how much of
it is SP or the component,

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and you can act differently
on the different slices

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of this wave function.

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And so with this, basically,
we can solve a new problem

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in an effective external
potential in which the lowest

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energy ground
state will actually

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be identical in energy
to the valence eigenstate

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of the original so-called all
electron on the atom with all

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its electrons.

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And the wave function
for this pseudo-atom

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will be identical to the wave
function of the real atom

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outside of the core radius--
outside, in this case,

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three atomic units
in this slide.

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So this was sort of one of
the first important technical

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proofs that were introduced
in the '70s and '80s

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to make this calculation
really feasible.

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The other sort of
idea that I want

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to sort of remind
you is that whenever

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we deal with extended systems--

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solids, liquids,
and in particular,

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when we have periodic
boundary condition-- that

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is we have our unit
cell periodically

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repeated in all dimensions-- the
eigenstates of our Hamiltonian

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take the form of Bloch Theorem,
and get classified according

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to two quantum numbers.

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A discrete number-- the
bounding that's number n--

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and the continuous number k.

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And the overall
eigenstates can be written

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as a product of two functions.

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One is just a
plane wave that has

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the periodicity of the so-called
crystal momentum K, one

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of these quantum numbers.

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And the other is the periodic
part of this Bloch orbital

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written here as u.

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And that this is a function
with the same periodicity

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of your unit cell.

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So overall, the orbital itself--

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the [INAUDIBLE] orbital psi
can have any periodicity,

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but it can always be
decomposed into a part

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of a well-defined wavelength
times a periodic part.

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Periodic part is
going to be smooth.

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Again, we don't have
any more core electrons,

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so close to the nuclei,
it will look just

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as 1s, 2s, 2p orbitals.

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And that periodic
part can actually

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be expanded in a
set of plane waves--

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plane waves that need to
satisfy the periodic boundary

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condition of our system.

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And so we have seen that
we can write the functions

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e to the iGr, where the
G is a linear combination

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of primitive reciprocal
lattice vectors with integer

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coefficients, and
all those G vectors

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are such that e to the iGr
has the same periodicity

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of your direct lattice.

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And these are the coefficient
of the series expansion.

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And we sort of
can systematically

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improve the quality of
this basic set expansion

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by increasing the
number of G vectors

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that we use, and we can
do that systematically

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by taking G vectors with
longer and longer lengths,

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or as we say, with higher
and higher energy cutoff,

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because higher and
higher energy [INAUDIBLE]

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longer and longer moduli for
the G vectors corresponds

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to plane waves with finer
and finer periodicity,

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so with finer and
finer resolution.

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So this is a distinct
advantage of plane waves.

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And in addition, this
basis sets do not

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depend on atomic position.

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So when we calculate
forces, say,

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to do molecular dynamics
or a structural relaxation

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calculation, we don't
need to take into account

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the fact that the
basis set changes

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with the position of the atoms.

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Plane waves are not
the only choices.

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There are a number
of other choices

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that are very successful.

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In particular, a lot of
the quantum chemistry

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could use a Gaussian basis sets.

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So atom-centered orbitals
that decay as Gaussians.

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This is very convenient to do
calculations like the exchange

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term in Hartree-Fock.

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But of course, you could
have just a finite difference

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representation.

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So you could represent
your orbitals

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on a grid of points
in real space.

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That tends to be very efficient
to do parallel calculation,

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but it's very difficult
to do accurate calculation

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of the second derivative.

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If you remember, when you expand
a function in plane waves,

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you have an analytic expression
for the second derivative

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just because the first and
second derivative of plane

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waves are just ig or minus g
squared times the plane wave

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itself.

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And that's very important.

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So that's when sort of
real space representation

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becomes a little bit trickier.

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And then there are
a number of sort

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of more approximate
approaches based

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on a number of sort of
atomic-like localized orbitals.

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Or there are a number of
sort of accurate approaches

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that are based usually on a
combination of a basic set that

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has a plane wave-like
characteristic in the regions

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that are far away
from the nuclei,

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and then it has atomic like
characteristics in the region

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inside the nuclei.

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And this tends to be the most
accurate, but also slightly

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more expensive approaches.

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This concludes one of
the technical points

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that you need to be careful
in a practical calculation.

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The other point that sort
of comes over and over

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again has to do with
Brillouin Zone integration.

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Let me give you a first
example of a molecule.

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So if you are calculating
the electronic structure

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of a molecule, in the
course of your calculation,

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you'll need to calculate
integrated quantity

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like the charge density.

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The charge density
of a molecule is

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going to be the sum of
the square moduli of all

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the single particle orbitals.

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So if you are sort of studying
a molecule like just hydrogen 2,

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well, you just take the
sum of the square moduli

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of the first orbital--

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you have more orbital,
more complex molecule.

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You need to sum over
all the orbitals up

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to the [INAUDIBLE]---- the highest
occupied molecular orbital.

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You do a calculation
in a solid, you

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have to do exactly
the same thing.

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That is, you need to calculate
the charge density by summing

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over the occupied states.

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Now as I said, what
are the quantum numbers

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that describe a solid
are a [INAUDIBLE]

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and a continuous
index scale that we

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call the quasi-momentum.

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And we usually represent,
say, the energy of the states

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in a solid with a band diagram.

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And I've plotted here
on the left the band

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diagram for silicon.

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In particular, what
we are looking here

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is in the Brillouin
Zone of silicon

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along certain high
symmetry direction.

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I presume these are the
high symmetry points,

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like gamma would be 0, 0, 0.

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[? Alpha ?] would
be 1/2, 1/2, 1/2.

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Hope this is correct.

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This should be x,
and it is 1, 0, 0.

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So we plot the energies of all
the occupied bands of silicon

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in different directions
along the Brillouin Zone.

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Silicon has two
atoms per unit cell.

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Each atom has four
valence electrons,

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so we have eight valence
electrons per unit cell.

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It's a system in which there
is basically spin degeneracy.

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So there is really sort
of the same spatial part

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for spin up and spin down.

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So what we usually say is
that we can accommodate

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two electrons for
each space orbital,

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and they will just have
different spin quantum number.

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So with these eight
valence electrons,

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we end up with four valence.

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And sometimes, because of
symmetry, you have degeneracy,

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but you see that
somewhere like here

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at a sort of arbitrary low
symmetry point in the Brillouin

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Zone, you can clearly
see four bands.

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So if we want to calculate the
charge density of the system,

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we need to sum over all the
possible occupied states.

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That is, we need to sum
over all the four bands,

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and that's trivial.

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But we also need to integrate
over all the possible k

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vectors.

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That is, we need to make an
integral in the Brillouin Zone.

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That is, we need
to sort of really

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sum over all these
possible states.

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An integral is obviously
an analytical operation.

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In practice, on a computer,
what we do is just

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we discretize that
integral, and we take a sum.

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So it means that--

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and again, I'll use two
dimension as an example.

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It means that if this
were my Brillouin Zone--

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two-dimensional Brillouin
Zone, and my k vector

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can be anywhere in
this Brillouin Zone,

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what I would need to do
is an integral overall

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the possibilities inside there.

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But in practice,
what I'll do, I'll

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just take a discretization.

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And I'll calculate
my band structure

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at each one of these points,
and say a regular aqueous space

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dimension of k points.

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And that's an
expensive operation.

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So each calculation
at each k point

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will require a self-consistent
diagonalization

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of your problem.

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And so really, the cost
of your calculation

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is linearly scaling in
the number of k points.

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So in reality, you want to use
as few k points as possible.

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For a system like a
semiconductor or an insulator,

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in which if you want--

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the band gap structure
is very smooth--

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it becomes fairly easy to
integrate that smooth bands

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with very coarse measures.

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So for something like silicon,
you might be already very happy

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when you start to sample the
three-dimensional Brillouin

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Zone with a uniform mesh of k
points that could have four k

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points in each direction--

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4 by 4, by 4, 6 by 6,
by 6, 8 by 8, by 8.

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And this is sort of the order
of magnitude of the k points

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that you need to use.

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If you were to study for a
moment unit cell of silicon,

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you might want to study,
say, vacancy information

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energy like you were doing in
your first computational lab.

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Then you are not going
to use two atom units.

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You are going to use
a larger unit cell.

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And without sort of
dwelling that much on it,

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suppose that, again,
in two dimensions,

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you double the size
of your unit cell.

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What's really happening is that
the reciprocal lattice vectors

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will become one half in length.

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So remember this
general concept.

250
00:12:49,730 --> 00:12:53,180
When you sort of increase
the size of the unit cell,

251
00:12:53,180 --> 00:12:56,210
the reciprocal cell
becomes smaller.

252
00:12:56,210 --> 00:12:58,010
Suppose that we have doubled--

253
00:12:58,010 --> 00:13:01,640
we have gone from
two atoms in each--

254
00:13:01,640 --> 00:13:04,190
two atoms in our unit cell to--

255
00:13:04,190 --> 00:13:05,990
well, if we are
in two dimensions,

256
00:13:05,990 --> 00:13:07,430
eight atoms in the unit cell.

257
00:13:07,430 --> 00:13:10,880
We have doubled in real
space-- in reciprocal space.

258
00:13:10,880 --> 00:13:16,700
Our Brillouin Zone
becomes 4 times smaller.

259
00:13:16,700 --> 00:13:22,040
And so if we want to keep the
same quality of integration,

260
00:13:22,040 --> 00:13:26,030
actually, it means that our
k point sampling needs only

261
00:13:26,030 --> 00:13:29,840
to be one fourth the number
of points that we had before.

262
00:13:29,840 --> 00:13:33,170
That is we still
use the blue points

263
00:13:33,170 --> 00:13:38,450
that are included in this sort
of smaller blackish unit cell.

264
00:13:38,450 --> 00:13:43,460
So that also means that, say,
if 64 k points is a good number

265
00:13:43,460 --> 00:13:47,000
for a regular set of unit
cell of two atoms of silicon,

266
00:13:47,000 --> 00:13:49,310
if now you are
going to use a, say,

267
00:13:49,310 --> 00:13:52,580
unit cell that has
maybe 128 atoms--

268
00:13:52,580 --> 00:13:55,530
much larger to calculate the
vacancy formation energy--

269
00:13:55,530 --> 00:13:59,930
you really need only one k point
to calculate your total energy

270
00:13:59,930 --> 00:14:01,200
with the same accuracy.

271
00:14:01,200 --> 00:14:04,730
So there is this general
idea of scaling and folding.

272
00:14:04,730 --> 00:14:07,730
You make your real space
calculation larger.

273
00:14:07,730 --> 00:14:10,840
You actually need to
use fewer k points.

274
00:14:10,840 --> 00:14:13,340
And if you are familiar with
sort of some of the solid state

275
00:14:13,340 --> 00:14:15,860
ideas, that also
just means that when

276
00:14:15,860 --> 00:14:18,260
you double the size
of your unit cell,

277
00:14:18,260 --> 00:14:21,920
you are really refolding
some of this band

278
00:14:21,920 --> 00:14:24,480
structure in a smaller space.

279
00:14:24,480 --> 00:14:29,450
So you actually sum over a
number of bands that increases.

280
00:14:29,450 --> 00:14:32,270
The situation is
slightly more complex

281
00:14:32,270 --> 00:14:35,060
for something like a metal.

282
00:14:35,060 --> 00:14:37,478
The sort of
fundamental difference

283
00:14:37,478 --> 00:14:39,020
when you deal with
a metal as opposed

284
00:14:39,020 --> 00:14:41,300
to a semiconductor
or an insulator

285
00:14:41,300 --> 00:14:44,900
is that there is something
called a Fermi energy.

286
00:14:44,900 --> 00:14:48,680
That is now there
isn't any more a gap,

287
00:14:48,680 --> 00:14:51,050
so the total charge
density is again

288
00:14:51,050 --> 00:14:56,990
given by sum of the states,
sum over all the bands,

289
00:14:56,990 --> 00:15:01,090
and integral over
all the k space.

290
00:15:01,090 --> 00:15:04,810
But really, the total
charge density--

291
00:15:04,810 --> 00:15:07,300
so the ultimate
integral that would

292
00:15:07,300 --> 00:15:10,180
be the number of
electrons depends really

293
00:15:10,180 --> 00:15:12,130
on where we stop.

294
00:15:12,130 --> 00:15:14,830
And there isn't any more
sort of natural separation

295
00:15:14,830 --> 00:15:18,080
between empty and
occupied states.

296
00:15:18,080 --> 00:15:22,120
So there is going to be
in a metal an energy level

297
00:15:22,120 --> 00:15:26,290
that determines what is
occupied and what is empty,

298
00:15:26,290 --> 00:15:28,060
and that's the Fermi energy.

299
00:15:28,060 --> 00:15:33,850
So for copper I think has
11 electrons per unit cell

300
00:15:33,850 --> 00:15:37,360
sort of FCC metal
with one atom per--

301
00:15:37,360 --> 00:15:39,460
one atom in each unit cell.

302
00:15:39,460 --> 00:15:43,420
What we really need to do
in our electrons calculation

303
00:15:43,420 --> 00:15:46,960
is find what is
this energy level.

304
00:15:46,960 --> 00:15:50,380
That is find what is
this Fermi energy, such

305
00:15:50,380 --> 00:15:55,150
that the integral over all
the state below that level

306
00:15:55,150 --> 00:15:57,910
gives us the right charge
density-- in particular,

307
00:15:57,910 --> 00:16:00,520
the right number of electrons.

308
00:16:00,520 --> 00:16:03,010
And this is sort of one of
the difficulties that come out

309
00:16:03,010 --> 00:16:05,890
in metals, because now
what we are trying to do

310
00:16:05,890 --> 00:16:11,740
is we are trying to integrate
the bands below a level--

311
00:16:11,740 --> 00:16:13,150
below the black line.

312
00:16:13,150 --> 00:16:17,270
And so you really introduce a
discontinuity in your integral.

313
00:16:17,270 --> 00:16:19,720
And so to calculate
integrals of discontinuous

314
00:16:19,720 --> 00:16:24,010
function usually requires
a much finer accuracy

315
00:16:24,010 --> 00:16:25,790
in your k point sampling.

316
00:16:25,790 --> 00:16:30,400
And so the calculation tends
to become much more expensive.

317
00:16:30,400 --> 00:16:32,080
Nothing else.

318
00:16:32,080 --> 00:16:38,110
So the general solution to this
problem besides using a larger

319
00:16:38,110 --> 00:16:41,800
number of k points,
and particular accuracy

320
00:16:41,800 --> 00:16:45,670
to this sampling issue is that
of introducing what is called

321
00:16:45,670 --> 00:16:48,230
a finite electron temperature.

322
00:16:48,230 --> 00:16:52,480
So what is actually done in sort
of every practical calculation

323
00:16:52,480 --> 00:16:56,270
is introduce a small
amount of temperature

324
00:16:56,270 --> 00:16:59,840
so that in reality, there's
a sharp discontinuity become

325
00:16:59,840 --> 00:17:00,430
smoother.

326
00:17:00,430 --> 00:17:02,410
Because when you have
an electron temperature,

327
00:17:02,410 --> 00:17:06,369
states above the Fermi energy
can be slightly occupied

328
00:17:06,369 --> 00:17:09,040
with the Fermi [INAUDIBLE]
occupation, and states

329
00:17:09,040 --> 00:17:13,240
just below the Fermi energy
can be slightly empty.

330
00:17:13,240 --> 00:17:15,339
So to summarize, I
mean, we need to be

331
00:17:15,339 --> 00:17:18,910
careful in sort of sampling.

332
00:17:18,910 --> 00:17:21,730
And that's the other sort
of fundamental parameter

333
00:17:21,730 --> 00:17:23,109
of your calculation.

334
00:17:23,109 --> 00:17:24,730
If you are dealing
with a metal, you

335
00:17:24,730 --> 00:17:27,280
need to be particularly
careful, and you

336
00:17:27,280 --> 00:17:30,880
need to use electron
temperature techniques.

337
00:17:30,880 --> 00:17:34,900
If you are using sort of-- if
you are studying an insulator

338
00:17:34,900 --> 00:17:37,390
a semiconductor,
what you usually do

339
00:17:37,390 --> 00:17:41,890
is just use a regular
equi-spaced mesh of k points.

340
00:17:41,890 --> 00:17:44,290
And that sort of
known in the community

341
00:17:44,290 --> 00:17:45,760
with a technical name.

342
00:17:45,760 --> 00:17:49,370
They are called [INAUDIBLE]
for some very good reason.

343
00:17:49,370 --> 00:17:51,160
Actually, there
are some actually

344
00:17:51,160 --> 00:17:53,230
fairly beautiful
symmetry thoughts

345
00:17:53,230 --> 00:17:58,690
on why equi-spaced coarse
mesh can work very well.

346
00:17:58,690 --> 00:18:00,700
But in practice,
it's nothing else

347
00:18:00,700 --> 00:18:04,690
than choosing the blue points
inside of the green Brillouin

348
00:18:04,690 --> 00:18:05,800
zone.

349
00:18:05,800 --> 00:18:09,520
If you are studying a
really large system,

350
00:18:09,520 --> 00:18:13,240
you can actually reduce
yourself to sampling only one

351
00:18:13,240 --> 00:18:14,410
point in the Brillouin Zone.

352
00:18:14,410 --> 00:18:16,570
Brillouin Zone has
become so small

353
00:18:16,570 --> 00:18:20,986
that just taking sort of--

354
00:18:20,986 --> 00:18:22,090
how do you call it?

355
00:18:22,090 --> 00:18:25,840
A mean-- the
theorem of the mean.

356
00:18:25,840 --> 00:18:28,030
That is, you can
substitute the integral

357
00:18:28,030 --> 00:18:31,150
with the value of the
function at one point.

358
00:18:31,150 --> 00:18:34,160
And usually, you
have two choices.

359
00:18:34,160 --> 00:18:36,010
You can just use
the gamma point.

360
00:18:36,010 --> 00:18:38,110
This means that 0, 0, 0--

361
00:18:38,110 --> 00:18:39,880
that has a
computational advantage.

362
00:18:39,880 --> 00:18:42,070
When you sample things
at the gamma point,

363
00:18:42,070 --> 00:18:45,820
you can choose your functions
to be real instead of complex.

364
00:18:45,820 --> 00:18:49,360
And so you have right away
your computational cost.

365
00:18:49,360 --> 00:18:53,610
Or you can choose sort of what
could be the best single point

366
00:18:53,610 --> 00:18:55,840
for your given symmetry
that's sometimes

367
00:18:55,840 --> 00:18:57,890
called the Baldereschi point.

368
00:18:57,890 --> 00:19:00,070
And that can be
again sort of useful

369
00:19:00,070 --> 00:19:03,880
if you need to do an accurate
calculation in a large scale

370
00:19:03,880 --> 00:19:06,040
system.

371
00:19:06,040 --> 00:19:10,960
Once you have set these
two fundamental parameters,

372
00:19:10,960 --> 00:19:13,120
you are really
ready to do actually

373
00:19:13,120 --> 00:19:14,980
a practical calculation.

374
00:19:14,980 --> 00:19:18,730
And so as I said, we have a
self-consistent Hamiltonian,

375
00:19:18,730 --> 00:19:20,560
and so what you
need to do is you

376
00:19:20,560 --> 00:19:24,940
need to iterate your problem
until the eigenstates

377
00:19:24,940 --> 00:19:27,670
that you find give you
a charge density that

378
00:19:27,670 --> 00:19:29,770
is identical to
the charge density

379
00:19:29,770 --> 00:19:31,610
that you have done before.

380
00:19:31,610 --> 00:19:35,240
So in practice, how would your
electronic structure code work?

381
00:19:35,240 --> 00:19:40,690
Well, first you would tell
your code where the atoms are.

382
00:19:40,690 --> 00:19:43,710
So you need to specify
the position of the atom.

383
00:19:43,710 --> 00:19:47,200
Suppose that you are studying
silicon, you could sort of--

384
00:19:47,200 --> 00:19:49,840
since you know actually what
is the structure of silicon,

385
00:19:49,840 --> 00:19:52,450
you could already put
them in the origin,

386
00:19:52,450 --> 00:19:56,140
and in the position one
fourth, one fourth, one fourth.

387
00:19:56,140 --> 00:19:58,760
And then you need to
specify in particular

388
00:19:58,760 --> 00:20:01,330
which flavor of
non-local pseudopotential

389
00:20:01,330 --> 00:20:02,800
you are going to use.

390
00:20:02,800 --> 00:20:05,440
That is, there will be a
library of pseudopotential

391
00:20:05,440 --> 00:20:09,760
that basically represents a
silicon atom with all the core

392
00:20:09,760 --> 00:20:11,070
electrons frozen.

393
00:20:11,070 --> 00:20:13,570
And there are sort of a number
of technicalities that you'll

394
00:20:13,570 --> 00:20:16,180
see in the lab on which
one you should choose,

395
00:20:16,180 --> 00:20:19,420
but they are sort of more or
less all the same, at least

396
00:20:19,420 --> 00:20:21,080
from this point of view.

397
00:20:21,080 --> 00:20:23,720
Once the code knows
where the atoms are--

398
00:20:23,720 --> 00:20:27,700
that is, knows the position of
the atoms inside the unit cell,

399
00:20:27,700 --> 00:20:29,950
knows what is the
shape of the unit cell,

400
00:20:29,950 --> 00:20:31,810
and what is the length
of the dotted lattice

401
00:20:31,810 --> 00:20:35,590
vectors, this infinite
array of atoms,

402
00:20:35,590 --> 00:20:38,110
the infinite crystalline,
or amorphous,

403
00:20:38,110 --> 00:20:41,320
or disordered extended
system is set.

404
00:20:41,320 --> 00:20:44,380
And what we really need to do is
throw the electron [? seeds ?]

405
00:20:44,380 --> 00:20:48,110
and let the electron find
their own ground state.

406
00:20:48,110 --> 00:20:51,830
And so we need to make sure
that we have the right basis set

407
00:20:51,830 --> 00:20:52,330
cutoff.

408
00:20:52,330 --> 00:20:53,705
That is, we are
going to describe

409
00:20:53,705 --> 00:20:56,710
the orbitals [? anchored. ?]
We have the right sampling.

410
00:20:56,710 --> 00:20:59,170
And at this point,
we can sort of

411
00:20:59,170 --> 00:21:02,650
start the self-consistent
procedure.

412
00:21:02,650 --> 00:21:05,110
And in the sort
of simplest form,

413
00:21:05,110 --> 00:21:07,780
well, we first need
to figure out what

414
00:21:07,780 --> 00:21:09,880
is our Hamiltonian operator.

415
00:21:09,880 --> 00:21:12,970
The Hamiltonian, remember,
depends on the charge density

416
00:21:12,970 --> 00:21:15,730
itself, because some of the
terms in the Hamiltonian,

417
00:21:15,730 --> 00:21:18,310
like the Hartree energy,
the Hartree potential

418
00:21:18,310 --> 00:21:20,140
of the exchange
correlation potential

419
00:21:20,140 --> 00:21:21,620
depends on the density.

420
00:21:21,620 --> 00:21:25,000
So we need to pick an initial
guess for a trial charge

421
00:21:25,000 --> 00:21:26,080
density.

422
00:21:26,080 --> 00:21:31,360
It could just be a superposition
of atomic charge density.

423
00:21:31,360 --> 00:21:33,400
Once we have the
charge density, we

424
00:21:33,400 --> 00:21:37,870
can construct the Hamiltonian,
the [INAUDIBLE] Hamiltonian.

425
00:21:37,870 --> 00:21:39,640
The kinetic energy operator is--

426
00:21:39,640 --> 00:21:40,540
we always know it.

427
00:21:40,540 --> 00:21:44,140
But we can construct
this Hartree and exchange

428
00:21:44,140 --> 00:21:47,050
correlation terms that
depend on the charge density,

429
00:21:47,050 --> 00:21:49,870
and then we'll have
the external potential

430
00:21:49,870 --> 00:21:54,640
that is given by this array
of non-local pseudopotential.

431
00:21:54,640 --> 00:21:57,580
At this point, we
have the Hamiltonian,

432
00:21:57,580 --> 00:22:00,800
and we try to find
the lowest energy

433
00:22:00,800 --> 00:22:02,950
eigenstates for Hamiltonian.

434
00:22:02,950 --> 00:22:07,900
In particular, we just need to
calculate a number of states

435
00:22:07,900 --> 00:22:11,750
that is equal to the
number of occupied orbitals

436
00:22:11,750 --> 00:22:14,080
if we are dealing
with a semiconductor,

437
00:22:14,080 --> 00:22:18,820
or it's equal to the
number of sort of electrons

438
00:22:18,820 --> 00:22:23,080
plus 20%, 30% in a
metal to make sure

439
00:22:23,080 --> 00:22:25,810
that at different points
in the Brillouin Zone,

440
00:22:25,810 --> 00:22:28,960
we calculate all
the bends that could

441
00:22:28,960 --> 00:22:31,930
be below our Fermi energy.

442
00:22:31,930 --> 00:22:33,350
So we solve this.

443
00:22:33,350 --> 00:22:36,250
And this is really
the expensive step--

444
00:22:36,250 --> 00:22:39,940
and I mean very expensive step
in any electronic structure

445
00:22:39,940 --> 00:22:40,900
calculation.

446
00:22:40,900 --> 00:22:44,470
And there are sort
of a number of ways

447
00:22:44,470 --> 00:22:48,340
of diagonalizing a
matrix, of solving

448
00:22:48,340 --> 00:22:50,200
this eigenstate equation.

449
00:22:50,200 --> 00:22:53,380
In a basic set, it
tends to be very large.

450
00:22:53,380 --> 00:22:58,390
When you do sort of a realistic
calculation, even for silicon,

451
00:22:58,390 --> 00:23:02,350
you could have hundreds
of plane waves, So.

452
00:23:02,350 --> 00:23:04,750
Hundreds of basic set elements.

453
00:23:04,750 --> 00:23:08,080
And large scale calculation
like you would doing in research

454
00:23:08,080 --> 00:23:12,400
would contain tens of
thousands of plane waves.

455
00:23:12,400 --> 00:23:15,970
And actually, you don't--

456
00:23:15,970 --> 00:23:19,880
you can't really diagonalize
on a regular computer

457
00:23:19,880 --> 00:23:22,180
even a matrix that
has [INAUDIBLE]..

458
00:23:22,180 --> 00:23:26,290
If you think a matrix that
has a dimension of 1,000

459
00:23:26,290 --> 00:23:30,490
requires one million elements.

460
00:23:30,490 --> 00:23:36,460
And one number-- a complex
number requires 16 bytes.

461
00:23:36,460 --> 00:23:40,780
So just a matrix that
has 1,000 sides that

462
00:23:40,780 --> 00:23:45,790
will require 16 megabytes to
be described, and this number

463
00:23:45,790 --> 00:23:49,070
explodes quadratically
very quickly.

464
00:23:49,070 --> 00:23:51,550
So you can construct
the full Hamiltonian,

465
00:23:51,550 --> 00:23:53,440
and you don't want
to calculate--

466
00:23:53,440 --> 00:23:56,620
if you have a matrix
of dimension 1,000,

467
00:23:56,620 --> 00:23:59,020
it will have 1,000 eigenstates.

468
00:23:59,020 --> 00:24:02,290
But you only care, say, if
you are studying silicon,

469
00:24:02,290 --> 00:24:04,930
on the lowest four eigenstates.

470
00:24:04,930 --> 00:24:07,450
So you want to have
numerical techniques that

471
00:24:07,450 --> 00:24:10,350
calculate for you only the
lowest energy eigenstates.

472
00:24:10,350 --> 00:24:13,840
And there are a number of them
that are well-established.

473
00:24:13,840 --> 00:24:17,230
So now you have obtained
with one of these techniques

474
00:24:17,230 --> 00:24:21,940
your lowest energy eigenstates,
and you sum their square moduli

475
00:24:21,940 --> 00:24:24,160
to obtain the new
charge density.

476
00:24:24,160 --> 00:24:26,650
And with this, you
go back to the step.

477
00:24:26,650 --> 00:24:29,170
You construct the
Hamiltonian operator again,

478
00:24:29,170 --> 00:24:34,420
you diagonalize new
density, and iterate.

479
00:24:34,420 --> 00:24:37,570
Of course, naturally,
a recipe like this

480
00:24:37,570 --> 00:24:41,140
would most likely
never converge.

481
00:24:41,140 --> 00:24:47,380
So one needs to develop
a mixing approach--

482
00:24:47,380 --> 00:24:51,040
mixing approaches that sort
make the change in the charge

483
00:24:51,040 --> 00:24:55,000
density at every
iterative step smoother

484
00:24:55,000 --> 00:24:56,720
than what I've described.

485
00:24:56,720 --> 00:24:58,510
So if you calculate
a new charge density

486
00:24:58,510 --> 00:25:02,560
and you diagonalize it again,
your second new charge density

487
00:25:02,560 --> 00:25:04,570
will be even more
different from anything

488
00:25:04,570 --> 00:25:06,470
that you have obtained before.

489
00:25:06,470 --> 00:25:08,770
So what you really
do is you need

490
00:25:08,770 --> 00:25:13,660
to find some schemes to
evolve in a very smooth way

491
00:25:13,660 --> 00:25:14,930
your charge density.

492
00:25:14,930 --> 00:25:17,530
So maybe once you
have calculated

493
00:25:17,530 --> 00:25:20,260
the new sum of
eigenstates, you don't

494
00:25:20,260 --> 00:25:22,810
take that as huge as
density, but you just

495
00:25:22,810 --> 00:25:27,310
update your old charge
density with 10%

496
00:25:27,310 --> 00:25:31,210
of what you have calculated now
to try to make the iteration

497
00:25:31,210 --> 00:25:33,730
to self-consistent very smooth.

498
00:25:33,730 --> 00:25:36,610
And I have to say a
lot of the know-how

499
00:25:36,610 --> 00:25:38,350
in electronic
structure calculation

500
00:25:38,350 --> 00:25:41,590
in the '90s has really
gone into trying

501
00:25:41,590 --> 00:25:46,090
to find sort of mixing
approaches that evolve

502
00:25:46,090 --> 00:25:48,400
our charge density
to solve consistency,

503
00:25:48,400 --> 00:25:52,870
and that converge under a
large variety of circumstances

504
00:25:52,870 --> 00:25:58,030
for large or complex systems,
especially for metals.

505
00:25:58,030 --> 00:26:02,080
So stripped to
the bare elements,

506
00:26:02,080 --> 00:26:05,830
an electronic structure code
really needs to do two things.

507
00:26:05,830 --> 00:26:10,270
It needs to diagonalize
inexpensively

508
00:26:10,270 --> 00:26:14,260
a Hamiltonian that expressed
on a plane wave basis

509
00:26:14,260 --> 00:26:16,270
is a very large order.

510
00:26:16,270 --> 00:26:18,472
And I just mentioned
the names of some

511
00:26:18,472 --> 00:26:20,680
of the [? algorithm. ?]
Things like the [INAUDIBLE],,

512
00:26:20,680 --> 00:26:23,440
and the [INAUDIBLE],, or some
of the conjugate gradient

513
00:26:23,440 --> 00:26:25,810
algorithms are all
algorithms that

514
00:26:25,810 --> 00:26:30,850
give us reasonable cost the
lowest energy eigenstates

515
00:26:30,850 --> 00:26:32,380
of that Hamiltonian.

516
00:26:32,380 --> 00:26:34,930
And then once you
have that eigenstates,

517
00:26:34,930 --> 00:26:37,000
you need to calculate
charge density,

518
00:26:37,000 --> 00:26:39,190
and you need to have
a mixing strategy.

519
00:26:39,190 --> 00:26:43,570
You need to have a strategy
to evolve your charge density

520
00:26:43,570 --> 00:26:45,950
towards self-consistency.

521
00:26:45,950 --> 00:26:50,350
And that is also a
very tricky approach.

522
00:26:50,350 --> 00:26:53,640
There is a sort of
completely different approach

523
00:26:53,640 --> 00:27:01,270
to the problem that sees the
ground state solution not

524
00:27:01,270 --> 00:27:04,950
as a self-consistent
iteration, but

525
00:27:04,950 --> 00:27:09,580
as a nonlinear direct
minimization of the functional.

526
00:27:09,580 --> 00:27:12,590
If you remember, we have
the energy function--

527
00:27:12,590 --> 00:27:15,220
or I think it's written
here in the next slide.

528
00:27:15,220 --> 00:27:16,150
No, it's not.

529
00:27:16,150 --> 00:27:19,600
We have sort of written the
density functional of theory

530
00:27:19,600 --> 00:27:24,280
energy functional, and it
is a well-defined expression

531
00:27:24,280 --> 00:27:25,810
of the orbitals only.

532
00:27:25,810 --> 00:27:27,070
It will be in one of the--

533
00:27:27,070 --> 00:27:30,250
so the following slides
will pick it up again.

534
00:27:30,250 --> 00:27:33,250
And so we can also
see the problem

535
00:27:33,250 --> 00:27:40,000
as the problem of minimization
of that functional in a space

536
00:27:40,000 --> 00:27:42,970
that is very large, because
the sort of variables that we

537
00:27:42,970 --> 00:27:47,020
really deal with are the
coefficients of our plane wave

538
00:27:47,020 --> 00:27:48,100
expansion.

539
00:27:48,100 --> 00:27:50,800
But in principle-- and I'll
show you that in a moment--

540
00:27:50,800 --> 00:27:54,610
we can actually write out
a minimization algorithm.

541
00:27:54,610 --> 00:27:58,420
The advantage of this approach,
if it's done properly,

542
00:27:58,420 --> 00:28:01,930
it has always a solution.

543
00:28:01,930 --> 00:28:06,730
If you keep minimizing
your energy, at the end,

544
00:28:06,730 --> 00:28:10,580
you will get to a global
or to a local minimum.

545
00:28:10,580 --> 00:28:14,260
So this approach tends
to sort of converge

546
00:28:14,260 --> 00:28:17,740
under every circumstances
if done properly, and done

547
00:28:17,740 --> 00:28:20,410
properly is not trivial.

548
00:28:20,410 --> 00:28:23,920
But then sort of the efficiency
of the different things

549
00:28:23,920 --> 00:28:27,590
is really system dependent.

550
00:28:27,590 --> 00:28:31,175
And I guess without wanting to
bore you with sort of math--

551
00:28:31,175 --> 00:28:34,130
so just I wanted
to remind you again

552
00:28:34,130 --> 00:28:36,020
what happens in our computer.

553
00:28:36,020 --> 00:28:39,830
That is what happens when
we say we want to solve

554
00:28:39,830 --> 00:28:42,500
this eigenstate equation.

555
00:28:42,500 --> 00:28:47,570
Supposing let's say we are in a
self-consistent diagonalization

556
00:28:47,570 --> 00:28:48,410
approach.

557
00:28:48,410 --> 00:28:52,700
And as always, you have to
remember we expand our wave

558
00:28:52,700 --> 00:28:58,310
function in a well-defined
set of orbitals--

559
00:28:58,310 --> 00:28:59,420
that is our basis set.

560
00:28:59,420 --> 00:29:02,180
I represented here them as phi.

561
00:29:02,180 --> 00:29:05,070
And it could be plane waves,
it could be atomic orbitals.

562
00:29:05,070 --> 00:29:06,950
We use plane waves all the time.

563
00:29:06,950 --> 00:29:10,340
So really, in our
computer, our unknowns

564
00:29:10,340 --> 00:29:14,720
are these coefficients of
this basis set expansion.

565
00:29:14,720 --> 00:29:20,150
And so our eigenstate equation,
once we multiply on the left

566
00:29:20,150 --> 00:29:26,550
by phi and star, and integrate,
is really a matrix problem.

567
00:29:26,550 --> 00:29:30,530
So I've written it here as
just the same eigenfunction

568
00:29:30,530 --> 00:29:32,520
equation written over there.

569
00:29:32,520 --> 00:29:37,190
And if we call hmn at the matrix
element of the Hamiltonian

570
00:29:37,190 --> 00:29:40,920
between, say, two plane waves
of different wavelengths,

571
00:29:40,920 --> 00:29:42,380
this is what our problem means.

572
00:29:42,380 --> 00:29:44,510
It's just a linear
algebra problem.

573
00:29:44,510 --> 00:29:47,990
We need to find the
eigenvalues for which there

574
00:29:47,990 --> 00:29:50,810
is a possible solution,
and the possible solution

575
00:29:50,810 --> 00:29:51,980
will be eigenstates.

576
00:29:51,980 --> 00:29:54,050
And an eigenstate
is nothing else

577
00:29:54,050 --> 00:29:56,930
than an appropriate
set of coefficients

578
00:29:56,930 --> 00:30:01,190
that satisfy this equation,
and those coefficients put back

579
00:30:01,190 --> 00:30:03,830
in here will give
us actually what

580
00:30:03,830 --> 00:30:06,935
are the full eigenstates
of our problem.

581
00:30:13,780 --> 00:30:16,190
So this was the sort
of self-consistent

582
00:30:16,190 --> 00:30:17,140
diagonalization.

583
00:30:17,140 --> 00:30:20,110
As I just said a
moment ago, we can also

584
00:30:20,110 --> 00:30:24,220
look at the problem as a
nonlinear minimization problem.

585
00:30:24,220 --> 00:30:28,990
Once we have decided on an
approximation for our exchange

586
00:30:28,990 --> 00:30:32,500
correlation functional, could be
a local density approximation,

587
00:30:32,500 --> 00:30:35,240
could be a generalized
gradient approximation.

588
00:30:35,240 --> 00:30:37,450
This is a well-defined
quantity in which,

589
00:30:37,450 --> 00:30:40,090
again, the external
potential is given

590
00:30:40,090 --> 00:30:43,120
by this array of
non-local pseudopotential.

591
00:30:43,120 --> 00:30:46,510
And the Hartree energy is just
the functional of the charge

592
00:30:46,510 --> 00:30:47,320
density.

593
00:30:47,320 --> 00:30:49,630
And the charge
density itself is thus

594
00:30:49,630 --> 00:30:52,600
the sum of the square
modulus of the orbitals.

595
00:30:52,600 --> 00:30:57,280
So in reality, this energy
is a function of that psi,

596
00:30:57,280 --> 00:31:02,380
or in other terms, is nothing
else than a very complex

597
00:31:02,380 --> 00:31:07,540
function of those c1
to cn coefficients

598
00:31:07,540 --> 00:31:09,320
of each eigenvector.

599
00:31:09,320 --> 00:31:13,780
So this is nothing else than
a minimization problem, again,

600
00:31:13,780 --> 00:31:18,160
on a number of variables
that can be 1,000 if you are

601
00:31:18,160 --> 00:31:20,110
studying two atoms of silicon.

602
00:31:20,110 --> 00:31:23,060
And it can be in the tens
of hundreds of thousands

603
00:31:23,060 --> 00:31:25,640
if you start to do really
serious calculations.

604
00:31:25,640 --> 00:31:28,300
So again, it's a
fairly complex problem.

605
00:31:28,300 --> 00:31:32,080
A huge number of variables
that you need to deal with,

606
00:31:32,080 --> 00:31:37,450
and a nonlinear
expression for the energy.

607
00:31:37,450 --> 00:31:41,950
But again, in principle, if we
have this explicit expression

608
00:31:41,950 --> 00:31:48,160
for the energy, E of psi,
where the psi that we consider

609
00:31:48,160 --> 00:31:51,850
around only the occupied
orbitals, what we can do

610
00:31:51,850 --> 00:31:56,350
is nothing else than take
the functional derivative

611
00:31:56,350 --> 00:31:57,395
with respect to that psi.

612
00:31:57,395 --> 00:31:59,020
I'll consider them
real here just

613
00:31:59,020 --> 00:32:02,200
to avoid sort of complex
conjugate numbers.

614
00:32:02,200 --> 00:32:05,350
And at the end,
this is nothing else

615
00:32:05,350 --> 00:32:10,990
than calculates the derivative
of the energy with respect

616
00:32:10,990 --> 00:32:16,720
to all the coefficients,
say, for i that goes 1 up

617
00:32:16,720 --> 00:32:21,740
to the cutoff, all the
coefficients of all

618
00:32:21,740 --> 00:32:23,110
the occupied orbitals.

619
00:32:26,920 --> 00:32:29,890
So you see, the larger
your system becomes,

620
00:32:29,890 --> 00:32:34,630
the more basis elements
you'll need to use.

621
00:32:34,630 --> 00:32:38,350
I mean, if you double
the size of your system,

622
00:32:38,350 --> 00:32:40,180
if you look at the
math, you'll actually

623
00:32:40,180 --> 00:32:42,460
need the twice the
number of plane waves,

624
00:32:42,460 --> 00:32:45,580
as it makes sense,
to describe a charge

625
00:32:45,580 --> 00:32:48,350
density or a wave function
with the same resolution.

626
00:32:48,350 --> 00:32:50,530
So you double the
size of your system.

627
00:32:50,530 --> 00:32:55,000
The number of plane waves to
describe a single particle

628
00:32:55,000 --> 00:32:58,630
orbital becomes twice
as large, but now you

629
00:32:58,630 --> 00:33:02,440
will also have twice as
many occupied orbitals.

630
00:33:02,440 --> 00:33:04,420
So just the number
of this coefficient

631
00:33:04,420 --> 00:33:07,100
has become four times as large.

632
00:33:07,100 --> 00:33:09,970
So again, this calculation
becomes very expensive

633
00:33:09,970 --> 00:33:10,910
very quickly.

634
00:33:10,910 --> 00:33:12,340
But again, this is well-defined.

635
00:33:12,340 --> 00:33:16,180
I mean, we can just
actually write explicitly

636
00:33:16,180 --> 00:33:18,700
the nonlinear function
of the previous slide

637
00:33:18,700 --> 00:33:22,190
in terms of the coefficients
of the plane waves.

638
00:33:22,190 --> 00:33:26,110
This is actually done in
one of the article posted.

639
00:33:26,110 --> 00:33:28,870
There is a review of modern
physics by Mike Payne

640
00:33:28,870 --> 00:33:33,580
and coworkers-- among others,
John [? Gianopulos ?] at MIT.

641
00:33:33,580 --> 00:33:35,710
And they actually
work out the algebra

642
00:33:35,710 --> 00:33:37,360
of all these derivatives.

643
00:33:37,360 --> 00:33:39,430
And then once you
have the derivatives,

644
00:33:39,430 --> 00:33:41,350
you have the
gradients, and you know

645
00:33:41,350 --> 00:33:45,340
how to move along and go
to the minimum following

646
00:33:45,340 --> 00:33:46,510
the gradients.

647
00:33:46,510 --> 00:33:50,530
The only difference with a sort
of regular minimization problem

648
00:33:50,530 --> 00:33:53,560
is that this is a
constraint problem.

649
00:33:53,560 --> 00:33:57,970
That is what we have because
these are really electrons,

650
00:33:57,970 --> 00:34:00,970
and not just sort of
arbitrary functions.

651
00:34:00,970 --> 00:34:06,080
The electrons need to be
meaningful quantum states,

652
00:34:06,080 --> 00:34:08,929
so they need to be orthonormal.

653
00:34:08,929 --> 00:34:12,880
So this derivatives
with respect to the c

654
00:34:12,880 --> 00:34:17,770
need actually to take
place on the hyper surface

655
00:34:17,770 --> 00:34:21,100
where these conditions
are satisfied.

656
00:34:21,100 --> 00:34:24,310
That is, if you were to evolve
the coefficients of the plane

657
00:34:24,310 --> 00:34:26,889
waves, what you would
find is that as soon

658
00:34:26,889 --> 00:34:29,530
as you have sort of changed
them by a little amount,

659
00:34:29,530 --> 00:34:32,260
your orbitals per
se are not going

660
00:34:32,260 --> 00:34:34,520
to be orthonormal anymore.

661
00:34:34,520 --> 00:34:38,560
So this constraint is
sort of fundamental,

662
00:34:38,560 --> 00:34:43,389
and this is what
ultimately limits sort of--

663
00:34:43,389 --> 00:34:47,710
or determines the computational
costs of our calculation.

664
00:34:47,710 --> 00:34:51,580
Because again, if we double
the size of the system,

665
00:34:51,580 --> 00:34:54,909
we'll have twice as
many plane waves,

666
00:34:54,909 --> 00:34:58,610
and we'll have twice as
many occupied orbitals.

667
00:34:58,610 --> 00:35:00,910
So we have already
a cost of fours.

668
00:35:00,910 --> 00:35:03,370
But those occupied
orbitals will need

669
00:35:03,370 --> 00:35:06,410
to be orthogonal to each other.

670
00:35:06,410 --> 00:35:09,100
And so the number of
these matrix elements

671
00:35:09,100 --> 00:35:12,340
that you need to calculate
has become also twice--

672
00:35:12,340 --> 00:35:15,000
or the number of
orbitals in this matrix

673
00:35:15,000 --> 00:35:17,260
have become twice, so the
number of matrix elements

674
00:35:17,260 --> 00:35:19,300
has become four times as large.

675
00:35:19,300 --> 00:35:21,790
You see, we double the
size of the system,

676
00:35:21,790 --> 00:35:23,920
we'll have twice as
many orbitals here,

677
00:35:23,920 --> 00:35:25,780
twice as many orbitals here.

678
00:35:25,780 --> 00:35:28,840
And this integral is
going to take place

679
00:35:28,840 --> 00:35:32,030
on a region in space
that's twice as large.

680
00:35:32,030 --> 00:35:36,530
So 2 by 2, by 2 gives
us a factor of 8,

681
00:35:36,530 --> 00:35:40,090
and so gives us the,
ultimately, cubic scaling

682
00:35:40,090 --> 00:35:43,180
of cost of density
functional calculation.

683
00:35:43,180 --> 00:35:47,260
You go from two atoms of silicon
to four atoms of silicon,

684
00:35:47,260 --> 00:35:51,700
your calculation has become
eight times more expensive.

685
00:35:51,700 --> 00:35:55,930
Hartree-Fock in its
original formulation scales

686
00:35:55,930 --> 00:35:57,400
as the fourth power.

687
00:35:57,400 --> 00:36:00,880
Other quantum chemistry approach
scales as the fifth, sixth,

688
00:36:00,880 --> 00:36:02,330
or seventh power.

689
00:36:02,330 --> 00:36:06,550
So it becomes very easy
to sort of reach, really,

690
00:36:06,550 --> 00:36:08,500
the limit of
calculation that you

691
00:36:08,500 --> 00:36:11,140
can do on a regular
computer or even

692
00:36:11,140 --> 00:36:12,850
on a regular supercomputer.

693
00:36:12,850 --> 00:36:16,330
And there is a lot of effort
to develop what are called

694
00:36:16,330 --> 00:36:17,980
linear scaling approaches.

695
00:36:17,980 --> 00:36:21,190
That these electronic structure
algorithm that scale linearly

696
00:36:21,190 --> 00:36:22,960
as the size of the system.

697
00:36:22,960 --> 00:36:25,360
And somehow, they are
all based on the idea

698
00:36:25,360 --> 00:36:28,840
that sort of physics or
quantum mechanics is local.

699
00:36:28,840 --> 00:36:32,320
So if your orbital at
the end is ultimately

700
00:36:32,320 --> 00:36:35,510
localized in a certain
region of space,

701
00:36:35,510 --> 00:36:38,800
it will be automatically
orthogonal to orbitals

702
00:36:38,800 --> 00:36:40,240
that are very far away.

703
00:36:40,240 --> 00:36:42,580
Because if this is
localized somewhere here,

704
00:36:42,580 --> 00:36:45,310
and this psi i is
localized somewhere there,

705
00:36:45,310 --> 00:36:48,410
their overlap will
be 0 by definition,

706
00:36:48,410 --> 00:36:51,550
and so we don't need to
worry about orthogonality.

707
00:36:51,550 --> 00:36:54,250
So somehow, locality
of physics, locality

708
00:36:54,250 --> 00:36:56,320
of quantum mechanics
in principle

709
00:36:56,320 --> 00:37:01,870
tells us that there are
linear scaling approaches that

710
00:37:01,870 --> 00:37:04,300
could work, although,
none of them

711
00:37:04,300 --> 00:37:08,140
have really made into sort of
production electronic structure

712
00:37:08,140 --> 00:37:12,010
at this stage, although there
is a lot of ongoing effort

713
00:37:12,010 --> 00:37:14,130
in many groups.

714
00:37:17,360 --> 00:37:20,720
So with this [INAUDIBLE]
we conclude also

715
00:37:20,720 --> 00:37:22,760
all the technicalities.

716
00:37:22,760 --> 00:37:24,860
And what we'll do in
the rest of the class

717
00:37:24,860 --> 00:37:27,540
will give you a sort
of panorama of what

718
00:37:27,540 --> 00:37:31,610
typical applications of density
function of theory calculation

719
00:37:31,610 --> 00:37:33,920
are going to do.

720
00:37:33,920 --> 00:37:42,230
And I'll go very
quickly over this.

721
00:37:42,230 --> 00:37:45,020
I have here sort
of cases in which

722
00:37:45,020 --> 00:37:49,580
we could be interested in
structural excitations.

723
00:37:49,580 --> 00:37:55,370
So when you start warming up a
system, a molecule, or a solid,

724
00:37:55,370 --> 00:37:59,990
you start exciting the different
normal modes of your molecule

725
00:37:59,990 --> 00:38:00,980
or of a solid.

726
00:38:00,980 --> 00:38:04,190
I've shown you some of the
possibilities for something

727
00:38:04,190 --> 00:38:07,310
like a carbon nanotube,
because that band--

728
00:38:07,310 --> 00:38:09,110
we have a banding mode up there.

729
00:38:09,110 --> 00:38:11,830
We have a pinching mode, and
we have a breathing mode.

730
00:38:11,830 --> 00:38:14,030
And so these are all
the possible structural

731
00:38:14,030 --> 00:38:16,010
excitations, and
you can actually

732
00:38:16,010 --> 00:38:20,170
calculate this structural
excitations using

733
00:38:20,170 --> 00:38:21,660
density functional theory.

734
00:38:21,660 --> 00:38:28,160
And I've given here a comparison
between the case of diamond--

735
00:38:28,160 --> 00:38:31,790
what is calculated with the
sort of state of the art

736
00:38:31,790 --> 00:38:33,590
[INAUDIBLE] code
like you are going

737
00:38:33,590 --> 00:38:36,800
to see in your
laboratory, and what

738
00:38:36,800 --> 00:38:39,230
is measured actually
with the neutron

739
00:38:39,230 --> 00:38:40,970
scattering, the red dots.

740
00:38:40,970 --> 00:38:45,140
So you see without really
any input parameters.

741
00:38:45,140 --> 00:38:48,110
And once you have really
phonon dispersion,

742
00:38:48,110 --> 00:38:51,620
you can calculate all the
thermodynamics of solids.

743
00:38:51,620 --> 00:38:55,550
That is really basically
based on the statistics

744
00:38:55,550 --> 00:38:59,250
of excitation of this
vibrational degrees of freedom.

745
00:38:59,250 --> 00:39:02,480
So you could calculate, say,
how your elastic constant

746
00:39:02,480 --> 00:39:05,790
for your bulk models
changes with temperature,

747
00:39:05,790 --> 00:39:08,210
and this is the
calculated black line,

748
00:39:08,210 --> 00:39:11,240
and you could compare
it with experiments.

749
00:39:11,240 --> 00:39:13,370
Or you could take
one of your slabs

750
00:39:13,370 --> 00:39:16,910
like you have seen in the first
laboratory in which you were

751
00:39:16,910 --> 00:39:20,510
calculating the surface
energy, and you could actually

752
00:39:20,510 --> 00:39:21,650
put it in motion.

753
00:39:21,650 --> 00:39:24,410
You could follow at a
given time temperature

754
00:39:24,410 --> 00:39:26,540
the dynamics of the atoms.

755
00:39:26,540 --> 00:39:28,970
And you want to have
a slab thick enough

756
00:39:28,970 --> 00:39:33,110
so the atoms in the middle
really act as bulk atoms.

757
00:39:33,110 --> 00:39:35,180
They don't see the
presence of the surface,

758
00:39:35,180 --> 00:39:38,600
and then it becomes very
easy to investigate what's

759
00:39:38,600 --> 00:39:40,710
happening on the surface.

760
00:39:40,710 --> 00:39:44,990
And so you can have sort
of a snapshot of sort

761
00:39:44,990 --> 00:39:49,820
of how the atoms are moving
on the outer layers, what

762
00:39:49,820 --> 00:39:51,890
are the sort of
typical displacements

763
00:39:51,890 --> 00:39:53,840
of typical mean
square displacements.

764
00:39:53,840 --> 00:39:58,730
Or you can say study how the
distance between the layers

765
00:39:58,730 --> 00:40:00,360
evolved with temperature.

766
00:40:00,360 --> 00:40:02,900
So you can look at, say,
how, as you increase

767
00:40:02,900 --> 00:40:05,240
the temperature of
your slab, the distance

768
00:40:05,240 --> 00:40:07,370
between the surface layer
and the second layer

769
00:40:07,370 --> 00:40:08,960
changes with temperature.

770
00:40:08,960 --> 00:40:11,090
And this would be
the computation,

771
00:40:11,090 --> 00:40:14,090
and here we have the
experimental value.

772
00:40:14,090 --> 00:40:17,660
And there are sort of lots
of interesting physics

773
00:40:17,660 --> 00:40:20,840
that takes place,
again, if you look

774
00:40:20,840 --> 00:40:23,540
at the distance between
the second and third layer,

775
00:40:23,540 --> 00:40:26,630
if the red line above and
the system expands there.

776
00:40:26,630 --> 00:40:28,970
And sort of with
computation, you can actually

777
00:40:28,970 --> 00:40:33,920
probe your system deeper and
deeper, where experiment starts

778
00:40:33,920 --> 00:40:36,530
to become very difficult.
It's almost impossible

779
00:40:36,530 --> 00:40:39,590
to look at what the fourth
layer in a surface does,

780
00:40:39,590 --> 00:40:43,190
and what the fifth
layer of a surface does.

781
00:40:43,190 --> 00:40:45,530
I think in order to keep
the balance of the lecture,

782
00:40:45,530 --> 00:40:48,720
I'll actually switch on to
Professor [? Sidor's ?] part

783
00:40:48,720 --> 00:40:51,720
so he can show you some
of the other application.

784
00:40:51,720 --> 00:40:54,710
And if we have time either
in one of the next classes,

785
00:40:54,710 --> 00:40:59,480
I'll show you some of
the other applications

786
00:40:59,480 --> 00:41:00,630
that we have mentioned.

787
00:41:00,630 --> 00:41:04,370
So I think I'll pass over
the lecture to Professor

788
00:41:04,370 --> 00:41:07,610
[? Sidor. ?]

789
00:41:07,610 --> 00:41:11,180
PROFESSOR 2: So Professor
[? Mazari ?] already gave you

790
00:41:11,180 --> 00:41:12,788
some generic applications.

791
00:41:12,788 --> 00:41:14,580
What I want to do for
the rest of the class

792
00:41:14,580 --> 00:41:17,360
is actually give
you some numbers,

793
00:41:17,360 --> 00:41:22,670
and really look seriously at
what the typical accuracies are

794
00:41:22,670 --> 00:41:24,733
that you can expect
from what we'll

795
00:41:24,733 --> 00:41:27,150
call density functional theory,
but what we mean with that

796
00:41:27,150 --> 00:41:30,860
is sort of standard
density function theory

797
00:41:30,860 --> 00:41:33,500
as we kind of explain
it to you in the class

798
00:41:33,500 --> 00:41:36,080
in the local density
or generalized gradient

799
00:41:36,080 --> 00:41:37,710
approximation.

800
00:41:37,710 --> 00:41:40,370
So these are sort of staples
of electronic structure methods

801
00:41:40,370 --> 00:41:40,870
now.

802
00:41:40,870 --> 00:41:44,030
But that doesn't mean that
there are, in some cases, not

803
00:41:44,030 --> 00:41:48,020
already better forms out
there, but they're often

804
00:41:48,020 --> 00:41:50,780
very much in the research stage,
and there wouldn't be things

805
00:41:50,780 --> 00:41:53,660
that you would either
easily do on real problems.

806
00:41:53,660 --> 00:41:55,280
You wouldn't get
your hands on them,

807
00:41:55,280 --> 00:41:57,663
and you wouldn't necessarily
easily learn from them.

808
00:41:57,663 --> 00:41:59,330
So again, what we're
going to talk about

809
00:41:59,330 --> 00:42:06,602
is the kind of standard staple
of electronic structure.

810
00:42:06,602 --> 00:42:09,060
Before I did that, I want to
say a little bit about a topic

811
00:42:09,060 --> 00:42:11,350
we haven't touched
about, which is

812
00:42:11,350 --> 00:42:14,730
what's called the spin
polarized version of density

813
00:42:14,730 --> 00:42:16,920
functional theory.

814
00:42:16,920 --> 00:42:19,990
If you remember, the
Hohenberg-Kohn theorem,

815
00:42:19,990 --> 00:42:22,890
everything is, in
essence, expressed

816
00:42:22,890 --> 00:42:25,710
in terms of the charge density.

817
00:42:25,710 --> 00:42:27,930
Everything is a function
of the charge density.

818
00:42:27,930 --> 00:42:33,780
And the electron spin never
explicitly appears in there.

819
00:42:33,780 --> 00:42:35,880
But of course,
electrons have spin.

820
00:42:35,880 --> 00:42:38,250
If you don't consider any
coupling with the angular

821
00:42:38,250 --> 00:42:40,650
momentum, it's really
just up or down

822
00:42:40,650 --> 00:42:46,050
spin, so plus or minus
a half [INAUDIBLE]..

823
00:42:46,050 --> 00:42:47,940
And we'll treat
here, in the lecture,

824
00:42:47,940 --> 00:42:50,260
spin as a scalar quantity.

825
00:42:50,260 --> 00:42:53,070
So really, spin is
just up or down--

826
00:42:53,070 --> 00:42:55,860
plus 1 or minus 1, or
plus a 1/2, or minus a 1/2

827
00:42:55,860 --> 00:42:57,540
in the appropriate units.

828
00:42:57,540 --> 00:43:00,390
And in reality,
spin, as soon as it

829
00:43:00,390 --> 00:43:05,010
applies to the angular momentum,
becomes a vector quantity.

830
00:43:05,010 --> 00:43:08,190
Also as soon as it couples
to a magnetic field

831
00:43:08,190 --> 00:43:10,350
from the environment, it
becomes a vector quantity.

832
00:43:10,350 --> 00:43:12,840
And people do that
now already, treating

833
00:43:12,840 --> 00:43:17,880
spin as a vector quantity, but
most codes that you will use,

834
00:43:17,880 --> 00:43:20,760
spin will simply be
treated as a scalar, which

835
00:43:20,760 --> 00:43:24,360
is fine for most purposes.

836
00:43:24,360 --> 00:43:29,360
We tend to refer to them
as just up and down spin.

837
00:43:29,360 --> 00:43:33,900
I'll often write that with
the either up and down arrow.

838
00:43:33,900 --> 00:43:41,140
Now why do you actually need
to treat the electron spin?

839
00:43:41,140 --> 00:43:44,890
Let me just sort of give
you a refresher of why

840
00:43:44,890 --> 00:43:46,930
you need to deal with the spin.

841
00:43:46,930 --> 00:43:50,230
Well, the reason is the Pauli
Exclusion Principle really.

842
00:43:50,230 --> 00:43:53,410
Is that the Pauli Exclusion
Principle tells you that two

843
00:43:53,410 --> 00:43:56,290
electrons cannot be in exactly
the same quantum state.

844
00:43:56,290 --> 00:43:57,740
Remember that?

845
00:43:57,740 --> 00:44:03,690
So that means that if you have
two up electrons approaching

846
00:44:03,690 --> 00:44:10,650
each other, versus say an
up and a down electron,

847
00:44:10,650 --> 00:44:13,050
these will approach
each other differently.

848
00:44:13,050 --> 00:44:18,790
Because these two are
in the same spin state,

849
00:44:18,790 --> 00:44:21,030
so if you bring them
very close together,

850
00:44:21,030 --> 00:44:24,880
they essentially now get
the same coordinate as well.

851
00:44:24,880 --> 00:44:27,930
So they almost have the
same quantum numbers now,

852
00:44:27,930 --> 00:44:31,180
and the Pauli exclusion
principle prevents that.

853
00:44:31,180 --> 00:44:34,410
So the Pauli Exclusion
Principle keeps electrons

854
00:44:34,410 --> 00:44:39,390
with parallel spin essentially
away from each other.

855
00:44:39,390 --> 00:44:43,170
Whereas if you have electrons
with empty parallel spin,

856
00:44:43,170 --> 00:44:44,917
even if these are at
the same position,

857
00:44:44,917 --> 00:44:46,500
they don't have the
same spin, so they

858
00:44:46,500 --> 00:44:48,400
don't have the same
set of quantum numbers,

859
00:44:48,400 --> 00:44:52,020
so the Pauli Exclusion
Principle doesn't act on them.

860
00:44:52,020 --> 00:44:53,730
And the Pauli
Exclusion Principle

861
00:44:53,730 --> 00:44:57,870
is essentially something
that keeps the electrons away

862
00:44:57,870 --> 00:45:01,060
without an explicit term
for it in the Hamiltonian.

863
00:45:01,060 --> 00:45:03,820
It's not like-- the coulombic
interaction, of course,

864
00:45:03,820 --> 00:45:05,362
keeps electrons away
from each other.

865
00:45:05,362 --> 00:45:06,737
But the Pauli
Exclusion Principle

866
00:45:06,737 --> 00:45:08,520
is essentially
something on top of that

867
00:45:08,520 --> 00:45:11,490
that comes from
anti-symmeterizing the wave.

868
00:45:11,490 --> 00:45:14,250
You don't see it
directly in the form--

869
00:45:14,250 --> 00:45:16,710
as a term in the form
of the Hamiltonian.

870
00:45:16,710 --> 00:45:18,780
And if you remember,
consequences of--

871
00:45:22,730 --> 00:45:24,380
it's going to take forever--

872
00:45:24,380 --> 00:45:26,390
the Pauli Exclusion
Principle is Hund's rule.

873
00:45:26,390 --> 00:45:29,570
That if you remember
atomic d level,

874
00:45:29,570 --> 00:45:35,040
say, for example, if you
add electrons to, say,

875
00:45:35,040 --> 00:45:39,810
five d levels, you're going
to add them with parallel spin

876
00:45:39,810 --> 00:45:43,530
first, because again, then
the Pauli Exclusion Principle

877
00:45:43,530 --> 00:45:45,060
is satisfied.

878
00:45:45,060 --> 00:45:50,010
And then you start filling them
with the anti-parallel levels.

879
00:45:50,010 --> 00:45:53,580
So this is going to
carry over in atoms--

880
00:45:53,580 --> 00:45:55,830
in solids-- I'm sorry.

881
00:45:55,830 --> 00:45:58,440
Basically, if those five, say--

882
00:45:58,440 --> 00:46:00,150
let's focus on d levels.

883
00:46:00,150 --> 00:46:02,730
If they remain degenerate--

884
00:46:02,730 --> 00:46:04,620
so they remain roughly
at the same level,

885
00:46:04,620 --> 00:46:07,290
you're going to fill them
according to Hund's rule.

886
00:46:07,290 --> 00:46:08,860
And that's what I've shown here.

887
00:46:08,860 --> 00:46:11,550
So in solids,
these d levels will

888
00:46:11,550 --> 00:46:14,130
split a little, which
is what I've shown here,

889
00:46:14,130 --> 00:46:17,280
but they don't split a lot.

890
00:46:17,280 --> 00:46:21,070
Then you're actually going to
fill them with parallel spin.

891
00:46:21,070 --> 00:46:24,600
And so if that's the
case, you have a lot

892
00:46:24,600 --> 00:46:27,420
of magnetic moment on your ion.

893
00:46:27,420 --> 00:46:30,170
You have five
electron spins here.

894
00:46:30,170 --> 00:46:32,670
You have no down spin, so you
have a strong magnetic moment.

895
00:46:32,670 --> 00:46:35,920
And I'll show in a second
where that becomes important.

896
00:46:35,920 --> 00:46:37,380
On the other hand,
let's say you're

897
00:46:37,380 --> 00:46:41,190
in an environment that
splits off two of these,

898
00:46:41,190 --> 00:46:43,350
but so much higher.

899
00:46:43,350 --> 00:46:47,340
At some point, you won't
satisfy Hund's rule anymore,

900
00:46:47,340 --> 00:46:49,770
because the energy
cost of-- let's say

901
00:46:49,770 --> 00:46:52,930
after you put in these
three green electrons, now

902
00:46:52,930 --> 00:46:54,310
you have to add two more.

903
00:46:54,310 --> 00:46:55,930
You have five electrons.

904
00:46:55,930 --> 00:46:57,430
To fill them with
parallel spin, you

905
00:46:57,430 --> 00:46:59,930
would have to put them here.

906
00:46:59,930 --> 00:47:02,700
But since those levers
are so much higher,

907
00:47:02,700 --> 00:47:06,620
you basically want to pay
the exchange penalty--

908
00:47:06,620 --> 00:47:08,810
they call it the
Hund's rule penalty--

909
00:47:08,810 --> 00:47:10,760
to put them in a lower
orbit when you put them

910
00:47:10,760 --> 00:47:12,300
in with anti-parallel.

911
00:47:12,300 --> 00:47:16,880
So in some sense, whether you
get a lot of magnetic spin--

912
00:47:16,880 --> 00:47:18,500
a lot of magnetic
moment left over

913
00:47:18,500 --> 00:47:22,400
depends on how much your
orbital will split in the end.

914
00:47:22,400 --> 00:47:23,900
But I'm going to
show you in the end

915
00:47:23,900 --> 00:47:26,810
that it can actually have
significant consequences

916
00:47:26,810 --> 00:47:29,840
on the physical properties
of your material.

917
00:47:29,840 --> 00:47:35,320
So in your density
functional calculation,

918
00:47:35,320 --> 00:47:38,290
you will carry a
magnetic moment locally

919
00:47:38,290 --> 00:47:41,930
when the up density and the
down density are not the same.

920
00:47:41,930 --> 00:47:45,100
And so I've sort of
already given away here

921
00:47:45,100 --> 00:47:48,220
how this problem is dealt with
in density functional theory.

922
00:47:50,750 --> 00:47:54,290
The interesting thing is that
if you think about it very hard,

923
00:47:54,290 --> 00:47:56,600
you shouldn't have
to deal with spin

924
00:47:56,600 --> 00:47:58,010
in density functional theory.

925
00:48:07,260 --> 00:48:09,510
Somehow, it put me
back at my old picture.

926
00:48:17,015 --> 00:48:20,000
In principle, remember
where we told you that--

927
00:48:20,000 --> 00:48:22,160
Professor [? Mazari ?]
told you that the energy

928
00:48:22,160 --> 00:48:27,290
and the potential is a
function of the charge density?

929
00:48:27,290 --> 00:48:30,380
So the charge itself
should actually

930
00:48:30,380 --> 00:48:33,710
also contain the information
about electron spin

931
00:48:33,710 --> 00:48:36,140
and magnetic moment, even
though it doesn't explicitly

932
00:48:36,140 --> 00:48:38,060
contain that.

933
00:48:38,060 --> 00:48:39,980
But for a given
charge density, there

934
00:48:39,980 --> 00:48:42,590
would probably be a certain
amount of spin polarization.

935
00:48:42,590 --> 00:48:45,980
So it should all be in that
functional, but remember,

936
00:48:45,980 --> 00:48:48,620
that's the functional
that we don't know.

937
00:48:48,620 --> 00:48:52,825
And so in practice, that
doesn't work very well.

938
00:48:52,825 --> 00:48:54,200
So what we do is
we really helped

939
00:48:54,200 --> 00:48:57,387
density functional theory
along by treating the up

940
00:48:57,387 --> 00:48:59,743
and the down density separately.

941
00:48:59,743 --> 00:49:01,160
But again, you
should keep in mind

942
00:49:01,160 --> 00:49:02,660
that in principle,
in the formalism

943
00:49:02,660 --> 00:49:05,430
of density functional theory,
you wouldn't have to do that.

944
00:49:05,430 --> 00:49:10,370
So if you do what's called,
say, the local spin density

945
00:49:10,370 --> 00:49:13,910
approximation, which is the spin
polarized version of the Local

946
00:49:13,910 --> 00:49:17,750
Density Approximation-- and
so goes under the name LSD--

947
00:49:17,750 --> 00:49:21,290
Lucy in the Sky with
Diamonds, or LSDA.

948
00:49:21,290 --> 00:49:24,440
And there's a version of that
for the GGA, which nobody--

949
00:49:24,440 --> 00:49:27,380
often people will just
call it LDA or GGA,

950
00:49:27,380 --> 00:49:30,710
but they will call it sometimes
spin polarized LDA or GGA.

951
00:49:30,710 --> 00:49:35,590
What you do there is that
you have a separate density

952
00:49:35,590 --> 00:49:37,750
for the up electrons
and a separate density

953
00:49:37,750 --> 00:49:42,680
for the down electrons, and the
two will interact differently.

954
00:49:42,680 --> 00:49:44,470
So the up/up will--

955
00:49:44,470 --> 00:49:46,950
up will interact
differently with up,

956
00:49:46,950 --> 00:49:49,300
with up, then up with down.

957
00:49:49,300 --> 00:49:51,670
And that comes from the way
the exchange correlation

958
00:49:51,670 --> 00:49:54,590
potentials are defined.

959
00:49:54,590 --> 00:49:58,600
So I think Professor
[? Mazari ?] mentioned

960
00:49:58,600 --> 00:50:01,690
restricted and unrestricted
to Hartree-Fock before

961
00:50:01,690 --> 00:50:04,135
very briefly, and this is
essentially the same idea.

962
00:50:11,090 --> 00:50:12,890
Sort of one quick tip
I want to give you

963
00:50:12,890 --> 00:50:17,150
is that if you have spin
polarized materials,

964
00:50:17,150 --> 00:50:20,510
it's often much more useful
to look at spin densities

965
00:50:20,510 --> 00:50:22,700
than at charge densities.

966
00:50:22,700 --> 00:50:25,220
One of the really cool things
about doing quantum mechanics

967
00:50:25,220 --> 00:50:27,470
is that you can actually
look at the charge densities,

968
00:50:27,470 --> 00:50:30,140
and look at the electrons,
which most people get

969
00:50:30,140 --> 00:50:32,720
very excited about the first
time they do quantum mechanics.

970
00:50:32,720 --> 00:50:35,090
It's kind of cool-- you can
look at where the electrons go.

971
00:50:35,090 --> 00:50:36,507
Well, the first
thing you learn is

972
00:50:36,507 --> 00:50:39,290
that you don't see much when
you look at charge densities.

973
00:50:39,290 --> 00:50:41,720
You typically see
big blobs of charge,

974
00:50:41,720 --> 00:50:44,930
and it's very hard to see
any fine structure of bonding

975
00:50:44,930 --> 00:50:46,700
in blobs of charge density.

976
00:50:46,700 --> 00:50:48,370
And I wanted to
show you an example.

977
00:50:48,370 --> 00:50:49,760
Here's lithium cobalt oxide.

978
00:50:49,760 --> 00:50:51,080
It's a transition metal oxide.

979
00:50:51,080 --> 00:50:54,370
It's a layered material--
layers of oxygen

980
00:50:54,370 --> 00:50:55,660
here-- the red things.

981
00:50:55,660 --> 00:50:58,270
And then layers of cobalt,
and layers of lithium.

982
00:50:58,270 --> 00:51:01,180
And if you plot the
charges in a plane--

983
00:51:01,180 --> 00:51:05,415
this is actually a plane
in the plane of the figure.

984
00:51:05,415 --> 00:51:06,790
If you look at
the charges-- this

985
00:51:06,790 --> 00:51:08,470
is a picture of
the charge density.

986
00:51:08,470 --> 00:51:11,200
You see big blobs, and you
see the oxygen layers here.

987
00:51:11,200 --> 00:51:13,510
And where the
oxygens are 2 minus,

988
00:51:13,510 --> 00:51:16,102
so remember that you're showing
only the valence electrons.

989
00:51:16,102 --> 00:51:17,810
So these have a lot
of valence electrons,

990
00:51:17,810 --> 00:51:20,530
and so you see a
lot of intensity.

991
00:51:20,530 --> 00:51:23,590
Cobalt has somewhat less
valence electrons on it,

992
00:51:23,590 --> 00:51:25,360
so you see less intensity here.

993
00:51:25,360 --> 00:51:27,770
And lithium is ionized
to lithium plus,

994
00:51:27,770 --> 00:51:29,920
so it has no valence
electrons on it,

995
00:51:29,920 --> 00:51:33,160
so you see almost nothing here.

996
00:51:33,160 --> 00:51:36,490
But in essence, this doesn't
give you a lot of detail.

997
00:51:36,490 --> 00:51:41,160
If you actually take a
material like this, and rather

998
00:51:41,160 --> 00:51:46,120
than plot the charges, you plot
the spin polarization density.

999
00:51:46,120 --> 00:51:50,100
So let's say up minus
down, or down minus up.

1000
00:51:50,100 --> 00:51:52,830
So it's how much magnetic
moment there is locally.

1001
00:51:52,830 --> 00:51:54,428
You get much cleaner pictures.

1002
00:51:54,428 --> 00:51:55,470
This is the same picture.

1003
00:51:55,470 --> 00:51:56,790
It's a slightly
different material

1004
00:51:56,790 --> 00:51:57,960
with different ions in it.

1005
00:51:57,960 --> 00:52:00,480
It's with nickel
and manganese in it.

1006
00:52:00,480 --> 00:52:02,850
But here's the
oxygen. By the way,

1007
00:52:02,850 --> 00:52:04,950
I should have told you
red is the neutral color

1008
00:52:04,950 --> 00:52:07,740
in this picture, so it's 0.

1009
00:52:07,740 --> 00:52:10,060
So now you don't see
the oxygen at all,

1010
00:52:10,060 --> 00:52:11,910
and the reason is oxygen
doesn't have spin.

1011
00:52:11,910 --> 00:52:12,888
It's a filled shell.

1012
00:52:12,888 --> 00:52:15,180
Every time you have filled
shells, you don't have spin.

1013
00:52:15,180 --> 00:52:18,390
So looking at spin density
often allows you to filter out

1014
00:52:18,390 --> 00:52:21,090
certain ions, and you
really-- the transition metals

1015
00:52:21,090 --> 00:52:24,390
tend to have spin on them,
and you see that very clearly.

1016
00:52:24,390 --> 00:52:27,300
So it's often a trick that I
just wanted to share with you.

1017
00:52:31,490 --> 00:52:34,190
So I want to go in sort
of the last half hour--

1018
00:52:34,190 --> 00:52:39,230
is go through the kind
of numerical accuracy,

1019
00:52:39,230 --> 00:52:43,340
and then slowly try to connect
that to the physical accuracy

1020
00:52:43,340 --> 00:52:44,840
that you get in properties.

1021
00:52:44,840 --> 00:52:47,780
So if you want to use--

1022
00:52:47,780 --> 00:52:50,120
have initial methods in
density functional theory

1023
00:52:50,120 --> 00:52:52,670
to get to engineering
properties,

1024
00:52:52,670 --> 00:52:55,280
a lot of steps you have to
make, because in the end,

1025
00:52:55,280 --> 00:52:56,408
we calculate simple things.

1026
00:52:56,408 --> 00:52:58,700
We've got to get charge
densities, and band structures,

1027
00:52:58,700 --> 00:52:59,720
and energies.

1028
00:52:59,720 --> 00:53:01,320
And you'll talk
later to somebody,

1029
00:53:01,320 --> 00:53:04,550
and they want to know what's the
corrosion resistance of this.

1030
00:53:04,550 --> 00:53:08,280
And corrosion resistance
is not a quantum operator,

1031
00:53:08,280 --> 00:53:10,010
so you need to
take a lot of steps

1032
00:53:10,010 --> 00:53:13,730
to go from the simple, what I
would call primitive output,

1033
00:53:13,730 --> 00:53:15,260
to engineering properties.

1034
00:53:15,260 --> 00:53:17,060
But before you even
take that step,

1035
00:53:17,060 --> 00:53:19,310
you need to understand
the kind of accuracy--

1036
00:53:19,310 --> 00:53:21,920
how reliable your output is.

1037
00:53:21,920 --> 00:53:25,698
And so I collected
a lot of results.

1038
00:53:25,698 --> 00:53:27,740
And I was going to start
with the simple things--

1039
00:53:27,740 --> 00:53:30,238
the energies of the atoms.

1040
00:53:30,238 --> 00:53:32,780
So here's a collection, and it
looks like a bunch of numbers,

1041
00:53:32,780 --> 00:53:34,980
but there's a very
systematic trend in it.

1042
00:53:34,980 --> 00:53:36,627
So what I show for
a bunch of atoms--

1043
00:53:36,627 --> 00:53:38,960
and these all should have a
minus sign in front of them,

1044
00:53:38,960 --> 00:53:42,230
because if you sum up all the
electronic states in the atom,

1045
00:53:42,230 --> 00:53:46,218
they're obviously binding,
so they should be negative.

1046
00:53:46,218 --> 00:53:48,260
So there's always the
experimental line, at least

1047
00:53:48,260 --> 00:53:50,670
for most of them.

1048
00:53:50,670 --> 00:53:54,540
The LDA number, and then
the GGA with a fairly recent

1049
00:53:54,540 --> 00:53:57,730
implementation of the
exchange correlation function.

1050
00:53:57,730 --> 00:54:01,770
And so if you look
carefully at the numbers,

1051
00:54:01,770 --> 00:54:05,260
let's take one here-- let's
take carbon for example.

1052
00:54:05,260 --> 00:54:09,330
So experiment is 275688.

1053
00:54:09,330 --> 00:54:11,970
Oops, that wasn't good.

1054
00:54:11,970 --> 00:54:17,130
LDA, the binding energy
is somewhat weaker--

1055
00:54:17,130 --> 00:54:19,380
almost is an
electron volt weaker.

1056
00:54:19,380 --> 00:54:24,420
And GGA is slightly
closer to the experiment.

1057
00:54:24,420 --> 00:54:26,460
That's typically
what you'll see.

1058
00:54:26,460 --> 00:54:28,020
If you look at all
the other atoms,

1059
00:54:28,020 --> 00:54:31,890
you'll see a very
systematic trend.

1060
00:54:31,890 --> 00:54:38,070
In the LDA, in the atoms, the
electrons are not bound enough.

1061
00:54:38,070 --> 00:54:43,838
In the GGA, They are somewhat
closer to the experiment.

1062
00:54:46,770 --> 00:54:50,040
It gets more interesting
when you look at molecules,

1063
00:54:50,040 --> 00:54:52,870
because now you can talk about
a physical binding energy.

1064
00:54:52,870 --> 00:54:56,340
And so the one we look at
here is for very simple

1065
00:54:56,340 --> 00:54:58,720
diatomic molecules.

1066
00:54:58,720 --> 00:55:00,010
What's their binding energy?

1067
00:55:00,010 --> 00:55:02,690
So what's their energy
to pull them apart?

1068
00:55:02,690 --> 00:55:06,900
So if you think about
it, you have a molecule--

1069
00:55:06,900 --> 00:55:10,980
has an ab.

1070
00:55:10,980 --> 00:55:13,830
This is the vector between them.

1071
00:55:13,830 --> 00:55:16,690
You'll have sort of something
that looks like this.

1072
00:55:16,690 --> 00:55:20,100
And so we're looking at
what's that well depth here--

1073
00:55:20,100 --> 00:55:21,050
the binding energy.

1074
00:55:23,670 --> 00:55:28,770
If you look at hydrogen, H2,
you have the experimental number

1075
00:55:28,770 --> 00:55:31,410
here, the LDA number.

1076
00:55:31,410 --> 00:55:33,450
You see now that
the LDA number--

1077
00:55:33,450 --> 00:55:35,610
that the binding is too strong.

1078
00:55:35,610 --> 00:55:38,310
The H2 molecule is
bound too strongly.

1079
00:55:38,310 --> 00:55:39,240
It's not so bad.

1080
00:55:39,240 --> 00:55:43,620
It's only about 5% in hydrogen.
And the GGA, the binding

1081
00:55:43,620 --> 00:55:45,720
is too weak.

1082
00:55:45,720 --> 00:55:49,455
For reference, I've also put
uncorrected Hartree-Fock here.

1083
00:55:49,455 --> 00:55:51,330
Professor [? Mazari ?]
showed you essentially

1084
00:55:51,330 --> 00:55:54,870
what Hartree-Fock is,
which is essentially

1085
00:55:54,870 --> 00:55:58,260
having the Hartree term for
the self-consistent Coulomb

1086
00:55:58,260 --> 00:56:00,420
interaction from
the other electrons,

1087
00:56:00,420 --> 00:56:03,870
and the exact exchange with
no correlation effects.

1088
00:56:06,476 --> 00:56:09,810
This is an interesting one that
you will actually often use--

1089
00:56:09,810 --> 00:56:13,920
O2-- any time you look
at oxidation reactions,

1090
00:56:13,920 --> 00:56:14,820
for example.

1091
00:56:14,820 --> 00:56:20,160
Experiment-- oxygen is only
about minus 5.2 ev binding.

1092
00:56:20,160 --> 00:56:24,590
LDA binds by a
whopping 7 and 1/2.

1093
00:56:24,590 --> 00:56:27,390
So you're more than
two electron volts off.

1094
00:56:27,390 --> 00:56:30,080
GGA gets you a little
closer in this case.

1095
00:56:34,270 --> 00:56:38,815
Uncorrected Hartree-Fock
is off the charts.

1096
00:56:38,815 --> 00:56:40,690
And this is something
you'll generically see.

1097
00:56:40,690 --> 00:56:43,185
Uncorrected Hartree-Fock,
very few people

1098
00:56:43,185 --> 00:56:44,560
would actually
use that any more.

1099
00:56:44,560 --> 00:56:46,990
It's sort of for binding
energies way off the charts.

1100
00:56:49,966 --> 00:56:51,545
These things are important.

1101
00:56:51,545 --> 00:56:53,170
If you now-- let's
say you want to look

1102
00:56:53,170 --> 00:56:55,460
at an oxidation reaction.

1103
00:56:55,460 --> 00:56:57,050
So that means that
at some point,

1104
00:56:57,050 --> 00:57:00,320
you're going to calculate
the state of an oxide,

1105
00:57:00,320 --> 00:57:03,380
and compare the chemical
potential of the oxygen

1106
00:57:03,380 --> 00:57:07,550
there to that of oxygen gas.

1107
00:57:07,550 --> 00:57:11,215
So of course, you have a
big error in the oxygen gas.

1108
00:57:11,215 --> 00:57:12,840
The question is, how
much of that error

1109
00:57:12,840 --> 00:57:14,590
carries over to the solid?

1110
00:57:14,590 --> 00:57:16,620
If you make exactly the
same error in the solid,

1111
00:57:16,620 --> 00:57:19,280
then the reaction energy
is perfect, Because.

1112
00:57:19,280 --> 00:57:21,310
You're going to
subtract the two.

1113
00:57:21,310 --> 00:57:23,745
And a lot of
practical things you

1114
00:57:23,745 --> 00:57:25,120
do with density
functional theory

1115
00:57:25,120 --> 00:57:26,680
depend on error cancellation.

1116
00:57:29,410 --> 00:57:32,020
The thing is that you will
have more error cancellation

1117
00:57:32,020 --> 00:57:36,603
as the states that you subtract
are more physically similar.

1118
00:57:36,603 --> 00:57:38,020
But the problem
is, let's say, you

1119
00:57:38,020 --> 00:57:40,810
look at oxidation of a metal--
aluminum plus oxygen going

1120
00:57:40,810 --> 00:57:43,300
to aluminum oxide.

1121
00:57:43,300 --> 00:57:45,190
The oxygen in
aluminum oxide is very

1122
00:57:45,190 --> 00:57:47,660
different from the oxygen
in the O2 molecule,

1123
00:57:47,660 --> 00:57:50,800
so not all the
error will cancel.

1124
00:57:50,800 --> 00:57:56,047
Let's say it was so bad
that you kept 2 ev error.

1125
00:57:56,047 --> 00:58:00,160
So 2 ev error in the
molecule is one ev per oxygen

1126
00:58:00,160 --> 00:58:01,770
if you want to think
of it that way.

1127
00:58:01,770 --> 00:58:06,540
If you're not careful about
it, that's an enormous effect.

1128
00:58:06,540 --> 00:58:09,000
Think about it-- the
chemical potential which

1129
00:58:09,000 --> 00:58:11,040
relates directly
to the energy goes

1130
00:58:11,040 --> 00:58:15,500
like the logarithm of the
partial pressure of oxygen.

1131
00:58:15,500 --> 00:58:19,340
Remember that mu is
mu 0 plus rt log p?

1132
00:58:19,340 --> 00:58:21,470
So if now you invert
that, that means your--

1133
00:58:21,470 --> 00:58:24,470
if you wanted to calculate, say,
a partial pressure of oxygen

1134
00:58:24,470 --> 00:58:28,210
at which something oxidized,
your oxygen pressure

1135
00:58:28,210 --> 00:58:32,280
goes exponentially
with the energetics.

1136
00:58:32,280 --> 00:58:35,790
So if you have a 1 ev
error at room temperature,

1137
00:58:35,790 --> 00:58:40,620
your error in the
oxygen pressure

1138
00:58:40,620 --> 00:58:43,200
is the exponential
of one ev over kt,

1139
00:58:43,200 --> 00:58:46,440
which is off the charts.

1140
00:58:46,440 --> 00:58:49,670
So you have to be a little
careful with these kinds

1141
00:58:49,670 --> 00:58:51,830
of error calibrations.

1142
00:58:51,830 --> 00:58:54,680
Fortunately, we'll
see later when

1143
00:58:54,680 --> 00:58:57,140
we look at reaction between
solids, most of the error

1144
00:58:57,140 --> 00:59:00,560
tends to cancel, and we get
much, much better accuracy.

1145
00:59:00,560 --> 00:59:03,860
If all our reaction energies
were wrong by one ev,

1146
00:59:03,860 --> 00:59:05,150
we wouldn't be here.

1147
00:59:05,150 --> 00:59:07,230
We'd be out of business.

1148
00:59:07,230 --> 00:59:09,740
But you have to keep in
mind that you get less error

1149
00:59:09,740 --> 00:59:14,517
cancellation as the states
you're comparing are different.

1150
00:59:14,517 --> 00:59:16,850
The more they're different,
the less error cancellation.

1151
00:59:16,850 --> 00:59:18,380
That's sort of a rule of thumb.

1152
00:59:18,380 --> 00:59:20,240
And so going from
a gas to a solid

1153
00:59:20,240 --> 00:59:21,860
is a significant difference.

1154
00:59:21,860 --> 00:59:23,810
Often, what people
do is that if you

1155
00:59:23,810 --> 00:59:26,030
want to get practical
results, they'll

1156
00:59:26,030 --> 00:59:28,730
add a correction to this
which they fit at one point.

1157
00:59:32,470 --> 00:59:34,360
So these are the
small molecules.

1158
00:59:34,360 --> 00:59:35,380
Let's go to the solids.

1159
00:59:38,265 --> 00:59:39,890
Here I don't have
the binding energies,

1160
00:59:39,890 --> 00:59:41,350
but I have the
lattice parameters.

1161
00:59:41,350 --> 00:59:44,200
But you'll see
something that's very

1162
00:59:44,200 --> 00:59:46,697
consistent with the molecules.

1163
00:59:46,697 --> 00:59:48,280
If you look at the
lattice parameters,

1164
00:59:48,280 --> 00:59:50,620
you compare, say,
the experimental ones

1165
00:59:50,620 --> 00:59:52,360
versus the LDA ones.

1166
00:59:52,360 --> 00:59:55,510
What you'll see is that the
LDA ones almost always--

1167
00:59:55,510 --> 00:59:58,707
and I think always in
this case are smaller.

1168
00:59:58,707 --> 01:00:00,790
Actually, yeah, because
the difference is actually

1169
01:00:00,790 --> 01:00:02,230
negative.

1170
01:00:02,230 --> 01:00:04,450
The GGA results
are always bigger.

1171
01:00:07,160 --> 01:00:10,230
This is rather consistent
whatever material you do.

1172
01:00:10,230 --> 01:00:15,640
You will find almost always
that the LDA gives you

1173
01:00:15,640 --> 01:00:18,480
lattice parameters that are too
small by a factor of a percent,

1174
01:00:18,480 --> 01:00:20,720
sometimes, 2%.

1175
01:00:20,720 --> 01:00:24,560
And so people refer to that
as the over-binding of LDA.

1176
01:00:24,560 --> 01:00:26,660
LDA binds somewhat too strongly.

1177
01:00:26,660 --> 01:00:29,220
Remember you saw that in
the molecules as well.

1178
01:00:29,220 --> 01:00:31,340
Oxygen had a 7 ev
binding energy,

1179
01:00:31,340 --> 01:00:34,270
and it should only have
a 5 ev binding energy.

1180
01:00:34,270 --> 01:00:36,020
And so in solids, the
way that comes out

1181
01:00:36,020 --> 01:00:38,520
is that your equilibrium lattice
parameters are slightly too

1182
01:00:38,520 --> 01:00:40,020
small.

1183
01:00:40,020 --> 01:00:43,140
Actually, I'm not sure that
I know of a single result

1184
01:00:43,140 --> 01:00:46,800
where LDA gives a lattice
parameter that's too big.

1185
01:00:46,800 --> 01:00:49,957
I've seen that on some
occasions in papers,

1186
01:00:49,957 --> 01:00:52,290
but it's almost always an
indication that the people did

1187
01:00:52,290 --> 01:00:54,843
the calculation wrong.

1188
01:00:54,843 --> 01:00:56,260
Actually, an LDA
lattice parameter

1189
01:00:56,260 --> 01:00:58,840
that agrees with experiment
is usually wrong--

1190
01:00:58,840 --> 01:01:00,390
a wrong calculation.

1191
01:01:00,390 --> 01:01:04,032
GGA is much more unpredictable.

1192
01:01:04,032 --> 01:01:06,490
The ones that I've shown here
because they're simple metals

1193
01:01:06,490 --> 01:01:08,650
and semiconductors give
you a lattice parameter

1194
01:01:08,650 --> 01:01:10,120
that's too large.

1195
01:01:10,120 --> 01:01:12,190
In GGA, it's actually
also possible to get

1196
01:01:12,190 --> 01:01:14,630
a lattice parameter that's
too small, although it's rare.

1197
01:01:14,630 --> 01:01:17,260
Most of the time, you're
on the higher side,

1198
01:01:17,260 --> 01:01:20,200
but it's less predictable,
and that's sort of slightly

1199
01:01:20,200 --> 01:01:21,700
problematic with the GGA.

1200
01:01:21,700 --> 01:01:24,315
In LDA, a good
guess of the lattice

1201
01:01:24,315 --> 01:01:26,440
parameters-- you calculate
your lattice parameters,

1202
01:01:26,440 --> 01:01:28,120
and you know you're
on the low side.

1203
01:01:28,120 --> 01:01:29,560
You always know that the
real lattice parameter

1204
01:01:29,560 --> 01:01:30,550
is going to be bigger.

1205
01:01:30,550 --> 01:01:34,660
In GGA, it's slightly
more difficult to predict

1206
01:01:34,660 --> 01:01:36,370
on which side you are.

1207
01:01:36,370 --> 01:01:38,890
But in metals, you do tend
to be on the high side.

1208
01:01:41,410 --> 01:01:44,140
That actually has consequences
for other properties,

1209
01:01:44,140 --> 01:01:46,330
like the bulk modulus.

1210
01:01:46,330 --> 01:01:49,420
If you compare, say, the
experimental bulk modulus

1211
01:01:49,420 --> 01:01:53,810
to the LDA one, s you'll find
is that in almost all cases--

1212
01:01:53,810 --> 01:01:55,810
and I think in all cases
that I've shown-- well,

1213
01:01:55,810 --> 01:01:58,150
all cases except silicon--

1214
01:01:58,150 --> 01:02:00,650
the LDA bulk modulus
is too large,

1215
01:02:00,650 --> 01:02:02,890
so the material is
too stiff that means.

1216
01:02:02,890 --> 01:02:06,880
And that kind of goes together
with the over-binding.

1217
01:02:06,880 --> 01:02:09,760
Remember, the bonding
energy is too high,

1218
01:02:09,760 --> 01:02:11,698
the lattice parameter
is too small.

1219
01:02:11,698 --> 01:02:13,990
All that is kind of in
agreement with the material also

1220
01:02:13,990 --> 01:02:15,040
being too stiff.

1221
01:02:15,040 --> 01:02:18,220
As you compress the
material, it gets stiffer.

1222
01:02:18,220 --> 01:02:20,800
GGA, most of the time, if
you see from deviations,

1223
01:02:20,800 --> 01:02:23,110
has to be on the other side.

1224
01:02:23,110 --> 01:02:25,120
It tends to be too soft.

1225
01:02:25,120 --> 01:02:29,350
And bulk modulus
effects that will

1226
01:02:29,350 --> 01:02:34,150
transfer, for example, also
into vibrational frequencies.

1227
01:02:34,150 --> 01:02:37,360
In material, when you're
too hard, too stiff,

1228
01:02:37,360 --> 01:02:39,070
you'll have higher
vibrational frequency.

1229
01:02:39,070 --> 01:02:40,870
When you're too
soft, you'll have

1230
01:02:40,870 --> 01:02:42,580
lower vibrational frequencies.

1231
01:02:45,650 --> 01:02:46,795
Here's the same for oxides.

1232
01:02:46,795 --> 01:02:48,170
You're not exactly
learning a lot

1233
01:02:48,170 --> 01:02:51,050
new by looking at them, except
that in oxides, the errors just

1234
01:02:51,050 --> 01:02:52,175
tend to be slightly larger.

1235
01:02:58,090 --> 01:03:02,080
So here's the summary
for geometry prediction.

1236
01:03:02,080 --> 01:03:05,170
You almost always-- and I
would probably say always--

1237
01:03:05,170 --> 01:03:07,090
under-predict with LDA.

1238
01:03:07,090 --> 01:03:10,540
Less systematic errors with GGA.

1239
01:03:10,540 --> 01:03:13,450
For normal materials like
semiconductors and metals,

1240
01:03:13,450 --> 01:03:17,950
often, your errors are
confined to order 1% to 2%.

1241
01:03:17,950 --> 01:03:20,180
In transition metal oxides--
and if I have a chance,

1242
01:03:20,180 --> 01:03:21,722
I'll say a little
bit more about that

1243
01:03:21,722 --> 01:03:25,450
later because they have
electronic structures where

1244
01:03:25,450 --> 01:03:30,680
the LDA and GGA approximations
are particularly harsh on.

1245
01:03:30,680 --> 01:03:32,750
You tend to have
somewhat bigger errors.

1246
01:03:32,750 --> 01:03:36,340
But I may say a little more
about that if we get to it.

1247
01:03:36,340 --> 01:03:40,570
So I want to say something
about predicting structure,

1248
01:03:40,570 --> 01:03:44,300
and about the energy
scale that's required.

1249
01:03:44,300 --> 01:03:46,300
So this is often
something you want to do.

1250
01:03:46,300 --> 01:03:50,050
You want to know if I put my
energy in this arrangement,

1251
01:03:50,050 --> 01:03:52,398
is that lower in energy
than some other arrangement,

1252
01:03:52,398 --> 01:03:54,940
so I can kind of predict what
the most stable arrangement is.

1253
01:03:54,940 --> 01:03:56,898
So that could be [INAUDIBLE]
crystal structure,

1254
01:03:56,898 --> 01:03:59,750
but it's the same for if
you look, say, at a surface.

1255
01:03:59,750 --> 01:04:02,140
So I wanted to give you
a feeling for the scale

1256
01:04:02,140 --> 01:04:04,340
of energetic differences.

1257
01:04:04,340 --> 01:04:08,260
So for vanadium, I've listed
the atomic energy here

1258
01:04:08,260 --> 01:04:09,700
in [INAUDIBLE].

1259
01:04:09,700 --> 01:04:12,025
This is the energy
of all its electrons,

1260
01:04:12,025 --> 01:04:15,820
so not just the valence
electrons actually.

1261
01:04:15,820 --> 01:04:19,120
The energy for FCC
vanadium-- so remember,

1262
01:04:19,120 --> 01:04:25,160
the first line is the
atom, not in a solid.

1263
01:04:25,160 --> 01:04:31,550
The second line is the FCC iron,
and the third one is BCC iron.

1264
01:04:31,550 --> 01:04:32,850
So look at the differences.

1265
01:04:32,850 --> 01:04:34,790
First of all, if
you go from the atom

1266
01:04:34,790 --> 01:04:39,552
to the solid, your first four
digits don't even change.

1267
01:04:39,552 --> 01:04:41,010
And again, that's
a reflection of--

1268
01:04:41,010 --> 01:04:44,140
well, a lot of your deep
core states don't change.

1269
01:04:44,140 --> 01:04:45,870
But you would see
something similar even

1270
01:04:45,870 --> 01:04:47,995
with a pseudopotential
approximation where you only

1271
01:04:47,995 --> 01:04:49,990
deal with the valence electrons.

1272
01:04:49,990 --> 01:04:59,090
So the cohesive energy is only
0.03% of the total energy.

1273
01:04:59,090 --> 01:05:01,610
So if you're calculating--
the reason I'm saying this--

1274
01:05:01,610 --> 01:05:04,250
if you're calculating the
cohesive energy by first

1275
01:05:04,250 --> 01:05:07,160
calculating the total
energy of a solid,

1276
01:05:07,160 --> 01:05:08,780
and then calculating
atomic energy,

1277
01:05:08,780 --> 01:05:10,977
you'd better do these
things damn accurate,

1278
01:05:10,977 --> 01:05:12,560
because you're going
to subtract them,

1279
01:05:12,560 --> 01:05:15,170
and most of what you
subtract is the same.

1280
01:05:15,170 --> 01:05:18,140
So to get any significance
in your result,

1281
01:05:18,140 --> 01:05:22,540
you need to have high
numerical accuracy.

1282
01:05:22,540 --> 01:05:24,670
And that's not a big
problem with a lot of code,

1283
01:05:24,670 --> 01:05:27,560
but I want you to
keep that in mind.

1284
01:05:27,560 --> 01:05:30,040
But few people care about
the cohesive energy.

1285
01:05:30,040 --> 01:05:35,770
Let's say you want to know
whether vanadium is FCC or BCC.

1286
01:05:35,770 --> 01:05:38,110
So you could
calculate it as BCC.

1287
01:05:38,110 --> 01:05:43,750
Now the FCC/BCC
difference is only 0.001%

1288
01:05:43,750 --> 01:05:45,640
of the total energy.

1289
01:05:45,640 --> 01:05:48,530
And these are not
complicated structures.

1290
01:05:48,530 --> 01:05:51,760
So in many cases, we're going
to work with energy differences

1291
01:05:51,760 --> 01:05:53,170
that are fractions of--

1292
01:05:53,170 --> 01:05:56,590
so that are 10 to the
minus 6, 10 to the minus 7

1293
01:05:56,590 --> 01:05:58,318
times the total energy.

1294
01:05:58,318 --> 01:05:59,860
So it's sort of
telling you something

1295
01:05:59,860 --> 01:06:04,660
about how much numerical
accuracy you need.

1296
01:06:04,660 --> 01:06:06,820
If you want to look
at mixing energies--

1297
01:06:06,820 --> 01:06:09,310
let's say I mix vanadium
with something else--

1298
01:06:09,310 --> 01:06:10,960
platinum-- and you
want to know what's

1299
01:06:10,960 --> 01:06:14,620
the mixing enthalpy, because
that sets the whole temperature

1300
01:06:14,620 --> 01:06:18,460
scale for mixing, the whole
phase diagram topology.

1301
01:06:18,460 --> 01:06:19,810
That tends to be a fraction--

1302
01:06:19,810 --> 01:06:24,140
10 to minus 6, 10 to the
minus 7 of the total energy.

1303
01:06:24,140 --> 01:06:28,030
So my former advisor used to
compare this to-- let's say

1304
01:06:28,030 --> 01:06:31,480
you want to know the
weight of a captain that

1305
01:06:31,480 --> 01:06:33,490
sails a big supertanker.

1306
01:06:33,490 --> 01:06:35,680
It's like weighing the
tanker with the captain

1307
01:06:35,680 --> 01:06:38,210
and without the captain, and
looking at the difference,

1308
01:06:38,210 --> 01:06:40,510
and that's the weight
of the captain.

1309
01:06:40,510 --> 01:06:44,750
You're almost at a kind of
relative scale like that here.

1310
01:06:44,750 --> 01:06:47,710
So the [? cute ?]
thing really is

1311
01:06:47,710 --> 01:06:49,990
that all these
approximations we make

1312
01:06:49,990 --> 01:06:55,120
to density functional theory
are obviously not this accurate.

1313
01:06:55,120 --> 01:06:58,840
The total energy is not
accurate up to a fraction of 10

1314
01:06:58,840 --> 01:07:00,370
to the minus 6.

1315
01:07:00,370 --> 01:07:03,400
The only reason we're here and
we can get physical behavior

1316
01:07:03,400 --> 01:07:06,430
right is because
a lot of the error

1317
01:07:06,430 --> 01:07:08,200
the density functional
theory makes in LDA

1318
01:07:08,200 --> 01:07:09,980
and GGA is systematic.

1319
01:07:09,980 --> 01:07:12,070
And so a lot of it
cancels away when

1320
01:07:12,070 --> 01:07:14,290
you take energy differences.

1321
01:07:14,290 --> 01:07:18,790
When I do FCC and BCC vanadium,
yes, I may have an error of 10

1322
01:07:18,790 --> 01:07:21,640
to minus 4 in the
energy, but most of it

1323
01:07:21,640 --> 01:07:24,680
cancels away when I take
the energy difference.

1324
01:07:24,680 --> 01:07:26,343
And that's why we're lucky.

1325
01:07:26,343 --> 01:07:27,760
But you have to
keep that in mind,

1326
01:07:27,760 --> 01:07:30,070
because again, the less
cancellation you have,

1327
01:07:30,070 --> 01:07:40,410
the bigger your
error on the result.

1328
01:07:40,410 --> 01:07:44,640
So again, let me show you how
well or how badly it does.

1329
01:07:44,640 --> 01:07:46,950
So I did a very simple thing.

1330
01:07:46,950 --> 01:07:50,460
I looked at how well
does it predict, say,

1331
01:07:50,460 --> 01:07:53,010
the structure of the elements.

1332
01:07:53,010 --> 01:07:57,430
This is done in GGA, a standard
[? Trudel ?] potential method.

1333
01:07:57,430 --> 01:08:00,315
So this comes out-- you may
have to look at your hand

1334
01:08:00,315 --> 01:08:04,290
out because this is extremely
fuzzy on the screen.

1335
01:08:04,290 --> 01:08:07,860
In red, I did metals
that are experimentally

1336
01:08:07,860 --> 01:08:12,330
FCC, and green I did metals
that are experimentally BCC.

1337
01:08:12,330 --> 01:08:15,510
Now what I show you is
the calculated energy

1338
01:08:15,510 --> 01:08:18,630
difference between FCC
and BCC, and it's actually

1339
01:08:18,630 --> 01:08:25,880
the first line below every
element kind of like this.

1340
01:08:25,880 --> 01:08:29,920
And so when that's positive, the
BCC energy is higher than FCC,

1341
01:08:29,920 --> 01:08:30,670
so it's going to--

1342
01:08:30,670 --> 01:08:32,859
FCC is preferred over BCC.

1343
01:08:32,859 --> 01:08:40,120
If it's negative like here,
then BCC is preferred over FCC.

1344
01:08:40,120 --> 01:08:42,220
And so if you look--
so the color is

1345
01:08:42,220 --> 01:08:45,910
the experimental result,
the number is calculated.

1346
01:08:45,910 --> 01:08:49,830
So if you look at them
carefully, they're all correct.

1347
01:08:49,830 --> 01:08:51,569
It's negative when
we have green,

1348
01:08:51,569 --> 01:08:57,569
it's positive when we have red.

1349
01:08:57,569 --> 01:09:00,420
You can do a more
subtle comparison.

1350
01:09:00,420 --> 01:09:02,399
Look at the difference
between HCP--

1351
01:09:02,399 --> 01:09:04,537
Hexagonal Close Pack-- and FCC.

1352
01:09:04,537 --> 01:09:06,120
And the reason that
that's more subtle

1353
01:09:06,120 --> 01:09:09,450
is HCP FCC are much more alike.

1354
01:09:09,450 --> 01:09:10,428
They're all close pack.

1355
01:09:10,428 --> 01:09:11,970
It's just the
difference in stacking.

1356
01:09:11,970 --> 01:09:15,870
ab, ab, versus abc, abc.

1357
01:09:15,870 --> 01:09:18,030
And so again, you'll see
that they're all correct.

1358
01:09:20,640 --> 01:09:22,500
The red ones are the FCC ones.

1359
01:09:22,500 --> 01:09:24,870
They're the ones where that
first line is positive.

1360
01:09:24,870 --> 01:09:29,069
By the way, that number is
in kilojoules per mole here.

1361
01:09:29,069 --> 01:09:32,790
The yellow ones, that
number is negative.

1362
01:09:32,790 --> 01:09:40,840
So we get the structure of the
elements essentially correct.

1363
01:09:40,840 --> 01:09:43,720
There are notable exceptions.

1364
01:09:43,720 --> 01:09:45,520
In LDA, iron is wrong.

1365
01:09:45,520 --> 01:09:50,850
Iron is FCC in LDA, not BCC,
but in GGA, that's corrected.

1366
01:09:50,850 --> 01:09:53,020
And then there are the weirdos.

1367
01:09:53,020 --> 01:09:56,780
If you go deep down
in the periodic table,

1368
01:09:56,780 --> 01:10:01,120
especially f electron
metals have--

1369
01:10:01,120 --> 01:10:04,480
the f states are extremely
localized even in metals,

1370
01:10:04,480 --> 01:10:07,540
and so electron correlation
becomes very important here.

1371
01:10:07,540 --> 01:10:09,290
And I may say a little
more about that.

1372
01:10:09,290 --> 01:10:12,820
And so there you'll start
to see failures LDA and GGA.

1373
01:10:12,820 --> 01:10:15,730
An important one is plutonium.

1374
01:10:15,730 --> 01:10:18,650
Plutonium is kind of
important for obvious reasons,

1375
01:10:18,650 --> 01:10:22,090
especially if you work at
national labs these days.

1376
01:10:22,090 --> 01:10:26,770
And so people are building
more sophisticated methods

1377
01:10:26,770 --> 01:10:30,550
to deal with materials
such as plutonium.

1378
01:10:30,550 --> 01:10:33,357
Typically, when you work with
f electron metals, sometimes,

1379
01:10:33,357 --> 01:10:35,440
you'll get the answer
right, sometimes, you won't.

1380
01:10:35,440 --> 01:10:38,005
But you should be a
little more careful.

1381
01:10:42,590 --> 01:10:43,910
Let me skip this.

1382
01:10:43,910 --> 01:10:45,650
You can get the--

1383
01:10:45,650 --> 01:10:52,010
most of the time, you'll get the
structure of compounds right.

1384
01:10:52,010 --> 01:10:54,530
If you go to transition
metal oxides--

1385
01:10:54,530 --> 01:10:56,390
so I sort of went
from metallic elements

1386
01:10:56,390 --> 01:10:58,460
now to transition metal oxides--

1387
01:10:58,460 --> 01:11:00,890
most of the time, you also
get the structure right,

1388
01:11:00,890 --> 01:11:03,170
but things get more subtle.

1389
01:11:03,170 --> 01:11:05,990
In transition metal oxides,
the transition metal

1390
01:11:05,990 --> 01:11:08,120
has local d states.

1391
01:11:08,120 --> 01:11:11,480
And I showed you before they
often have significant spin

1392
01:11:11,480 --> 01:11:13,640
polarization, so the
first thing you need to do

1393
01:11:13,640 --> 01:11:16,430
is turn spin polarization
on, or you really

1394
01:11:16,430 --> 01:11:18,020
get the wrong answer.

1395
01:11:18,020 --> 01:11:19,980
But it often gets worse.

1396
01:11:19,980 --> 01:11:24,090
Remember that your spin is a
scalar, so it's up or down.

1397
01:11:24,090 --> 01:11:26,780
So now you have a
spatial degree of freedom

1398
01:11:26,780 --> 01:11:29,510
of how to organize
that spin on the ions.

1399
01:11:29,510 --> 01:11:31,520
If you have a bunch of
ions, you could put them

1400
01:11:31,520 --> 01:11:32,720
all with the same direction.

1401
01:11:32,720 --> 01:11:36,343
That's a ferromagnet,
or you could put them

1402
01:11:36,343 --> 01:11:37,760
with sort of
alternating direction

1403
01:11:37,760 --> 01:11:39,980
as a kind of a anti-ferromagnet.

1404
01:11:39,980 --> 01:11:43,280
And then there's many ways to
make them anti-ferromagnetic.

1405
01:11:43,280 --> 01:11:45,200
And unfortunately, in
transition metal oxide,

1406
01:11:45,200 --> 01:11:47,900
it often matters
because there's not only

1407
01:11:47,900 --> 01:11:51,590
a strong spin polarization
effect on the energy,

1408
01:11:51,590 --> 01:11:53,040
but there's a
fairly strong effect

1409
01:11:53,040 --> 01:11:58,940
of the interaction between
spins on different ions.

1410
01:11:58,940 --> 01:12:01,260
And I'm showing
you a result here.

1411
01:12:01,260 --> 01:12:02,720
This is a simple
crystal structure.

1412
01:12:02,720 --> 01:12:04,610
It's a structure of
lithium manganese oxide.

1413
01:12:04,610 --> 01:12:08,450
It's an ordered rock salt.
These are still very simple.

1414
01:12:08,450 --> 01:12:11,030
But the correct answer--

1415
01:12:11,030 --> 01:12:14,283
I'm showing the comparison
here between two structures

1416
01:12:14,283 --> 01:12:16,200
only labeled by their
symmetry, unfortunately.

1417
01:12:16,200 --> 01:12:19,730
One is C2/m, and one is pmmn.

1418
01:12:19,730 --> 01:12:23,060
There are similar structures,
but one is orthorhombic,

1419
01:12:23,060 --> 01:12:24,800
and one is monoclinic.

1420
01:12:24,800 --> 01:12:26,900
The correct answer is pmmn--

1421
01:12:26,900 --> 01:12:28,610
is the real crystal structure.

1422
01:12:28,610 --> 01:12:31,620
If you do a non-spin
polarized calculation--

1423
01:12:31,620 --> 01:12:34,700
so that's not even
allowing spin on the ions--

1424
01:12:34,700 --> 01:12:35,960
you get a whopping error.

1425
01:12:35,960 --> 01:12:41,720
I mean, C2/m is lower in energy
by 250 mill-electron volt

1426
01:12:41,720 --> 01:12:43,280
per formal unit.

1427
01:12:43,280 --> 01:12:44,240
That's very large.

1428
01:12:44,240 --> 01:12:49,020
In kilojoules, that'd be
25 kilojoules per mole.

1429
01:12:49,020 --> 01:12:50,420
It's a very large error.

1430
01:12:50,420 --> 01:12:51,890
If you turn on
spin polarization,

1431
01:12:51,890 --> 01:12:53,098
will make them ferromagnetic.

1432
01:12:53,098 --> 01:12:53,930
They're degenerate.

1433
01:12:53,930 --> 01:12:56,060
And if you make them
anti-ferromagnetic,

1434
01:12:56,060 --> 01:12:59,030
this one is the
lowest in energy.

1435
01:12:59,030 --> 01:13:04,100
Now here's a very common
mistake that people make.

1436
01:13:04,100 --> 01:13:06,350
If you take this material
at room temperature,

1437
01:13:06,350 --> 01:13:08,700
it's paramagnetic.

1438
01:13:08,700 --> 01:13:11,160
Some people say, well,
it's paramagnetic,

1439
01:13:11,160 --> 01:13:13,020
so I shouldn't have
any spin polarization--

1440
01:13:13,020 --> 01:13:14,160
there's no net moment.

1441
01:13:14,160 --> 01:13:17,910
That is so wrong, because
a paramagnet still

1442
01:13:17,910 --> 01:13:19,200
has a local moment.

1443
01:13:19,200 --> 01:13:21,390
The ions still have
a moment on them,

1444
01:13:21,390 --> 01:13:24,120
it's just randomly oriented.

1445
01:13:24,120 --> 01:13:27,710
So you still need to
represent that moment.

1446
01:13:27,710 --> 01:13:30,210
Because it turns out that's the
biggest effect on the energy

1447
01:13:30,210 --> 01:13:32,460
is the fact that you
have that local moment.

1448
01:13:32,460 --> 01:13:34,560
It's not necessarily
how they're arranged.

1449
01:13:34,560 --> 01:13:35,970
You can actually see that here.

1450
01:13:35,970 --> 01:13:39,630
How they're arranged makes
you go from this difference

1451
01:13:39,630 --> 01:13:41,100
to this difference.

1452
01:13:41,100 --> 01:13:42,810
But turning on the
local moment makes

1453
01:13:42,810 --> 01:13:46,750
you go from this difference
to that difference.

1454
01:13:46,750 --> 01:13:48,310
So never fall in that trap.

1455
01:13:48,310 --> 01:13:52,090
It's really only non-magnetic
materials or diamagnetic

1456
01:13:52,090 --> 01:13:57,770
materials for which you don't
really need spin polarization.

1457
01:13:57,770 --> 01:14:01,740
Now why is this
effect so important?

1458
01:14:01,740 --> 01:14:06,560
It's really because if
you spin polarize an ion,

1459
01:14:06,560 --> 01:14:08,920
you fill different orbitals.

1460
01:14:08,920 --> 01:14:12,470
I mean, I've shown that here
with a bunch of d orbitals.

1461
01:14:12,470 --> 01:14:14,840
And this is typically how
they split in most oxides.

1462
01:14:14,840 --> 01:14:16,850
Every time an ion
is octahedral--

1463
01:14:16,850 --> 01:14:21,140
five d orbitals tend to
split in pairs of 3, 2--

1464
01:14:21,140 --> 01:14:22,100
3 down, 2 up.

1465
01:14:22,100 --> 01:14:25,520
In some cases, 2 down, 3 up.

1466
01:14:25,520 --> 01:14:27,020
But let's say you
have to put four--

1467
01:14:27,020 --> 01:14:28,490
five electrons in there.

1468
01:14:28,490 --> 01:14:30,830
How many do I have?

1469
01:14:30,830 --> 01:14:33,290
I'm missing-- no,
four electrons.

1470
01:14:33,290 --> 01:14:34,280
If you put them--

1471
01:14:34,280 --> 01:14:35,390
that's called high spin.

1472
01:14:35,390 --> 01:14:38,120
So all parallel spin,
you put them like this.

1473
01:14:38,120 --> 01:14:40,980
If you put them low spin,
you put them like this.

1474
01:14:40,980 --> 01:14:42,630
So here we have no moment.

1475
01:14:42,630 --> 01:14:47,370
So these two ions have
different chemical properties,

1476
01:14:47,370 --> 01:14:50,580
because the electrons
occupy different orbitals.

1477
01:14:50,580 --> 01:14:57,120
These are different d orbitals,
and so this orbital points

1478
01:14:57,120 --> 01:15:01,540
in a different direction,
for example, than this one.

1479
01:15:01,540 --> 01:15:04,720
So by spin
polarizing, you create

1480
01:15:04,720 --> 01:15:05,980
essentially a different ion.

1481
01:15:05,980 --> 01:15:08,920
It's not really an
issue of magnetism,

1482
01:15:08,920 --> 01:15:11,470
because magnetic effects tend
to be small in materials.

1483
01:15:11,470 --> 01:15:14,230
But it's the fact that
you create chemically

1484
01:15:14,230 --> 01:15:18,870
a different ion because
you fill different levels.

1485
01:15:18,870 --> 01:15:21,680
That's really why these
energy differences are so big.

1486
01:15:26,533 --> 01:15:28,700
I sort of want to end with
showing you some reaction

1487
01:15:28,700 --> 01:15:30,620
energies very quickly.

1488
01:15:30,620 --> 01:15:33,200
And I'm going to sort make
it systematically harder.

1489
01:15:33,200 --> 01:15:34,580
So here's a simple one.

1490
01:15:34,580 --> 01:15:37,460
A metal, lithium, BCC,
with another metal,

1491
01:15:37,460 --> 01:15:40,640
aluminum forming a
lithium-aluminum compound.

1492
01:15:40,640 --> 01:15:44,480
Here's the experimental reaction
energy, here's the LDA one.

1493
01:15:44,480 --> 01:15:46,880
10% off.

1494
01:15:46,880 --> 01:15:48,020
That's classic.

1495
01:15:48,020 --> 01:15:50,840
Metallic reaction
energies, you're somewhere

1496
01:15:50,840 --> 01:15:54,020
in the range 5% to 15%.

1497
01:15:54,020 --> 01:15:55,640
The one I show
here, copper-gold,

1498
01:15:55,640 --> 01:15:58,740
is a notable exception
where you're over 50% off.

1499
01:15:58,740 --> 01:16:02,030
But most of them,
it's much simpler.

1500
01:16:06,360 --> 01:16:11,150
So in metals, you tend to get
very good reaction energies.

1501
01:16:11,150 --> 01:16:15,770
I want to show you the
case where things go wrong.

1502
01:16:15,770 --> 01:16:20,330
Where your errors become
bigger is in redox reactions.

1503
01:16:20,330 --> 01:16:22,670
And I've shown sort of
here three different ones.

1504
01:16:22,670 --> 01:16:23,630
They're all related.

1505
01:16:23,630 --> 01:16:25,670
They're essentially a
reaction between an oxide,

1506
01:16:25,670 --> 01:16:29,180
or in this case, a
phosphate with a metal

1507
01:16:29,180 --> 01:16:31,110
to react the two together.

1508
01:16:31,110 --> 01:16:33,530
And if you look at
the reaction energies,

1509
01:16:33,530 --> 01:16:34,950
you're not considerably off.

1510
01:16:34,950 --> 01:16:37,760
GGA gives you 2.8 electronvolts
for this reaction.

1511
01:16:37,760 --> 01:16:40,320
Experiment is 3.5.

1512
01:16:40,320 --> 01:16:44,150
This one, which is very
similar, the error is 30%.

1513
01:16:44,150 --> 01:16:46,190
You get 3.3 electronvolts.

1514
01:16:46,190 --> 01:16:48,260
Experiment is 4.6.

1515
01:16:48,260 --> 01:16:49,880
Why is that?

1516
01:16:49,880 --> 01:16:52,820
Well, it has to do with the
lack of error cancellation.

1517
01:16:52,820 --> 01:16:54,380
If you look in
detail what happens

1518
01:16:54,380 --> 01:16:56,450
to the electronic structure
in these materials,

1519
01:16:56,450 --> 01:16:58,710
these are redox reactions.

1520
01:16:58,710 --> 01:17:02,240
So if you do the math on the
valences-- and believe me,

1521
01:17:02,240 --> 01:17:06,220
this ion here is 3 plus.

1522
01:17:06,220 --> 01:17:07,780
Phosphorus, 5 plus.

1523
01:17:07,780 --> 01:17:09,760
Oxygen is 2 minus, so
you can do the math.

1524
01:17:09,760 --> 01:17:14,200
Iron here is 2 plus,
and lithium is 1 plus.

1525
01:17:14,200 --> 01:17:16,408
So what has happened
in this reaction?

1526
01:17:16,408 --> 01:17:18,700
Well, you've taken essentially
an electron from lithium

1527
01:17:18,700 --> 01:17:24,910
in it's metallic state, and
put it on the iron 3 plus

1528
01:17:24,910 --> 01:17:27,190
to make iron 2 plus.

1529
01:17:27,190 --> 01:17:29,980
So essentially, you've
transferred an electron

1530
01:17:29,980 --> 01:17:35,240
from metallic lithium and
ionized lithium to the iron

1531
01:17:35,240 --> 01:17:37,940
to reduce it from
3 plus to 2 plus.

1532
01:17:37,940 --> 01:17:39,800
But think about
what's that doing?

1533
01:17:39,800 --> 01:17:42,003
That electron in lithium--

1534
01:17:42,003 --> 01:17:43,670
lithium is there in
the alkaline metals.

1535
01:17:43,670 --> 01:17:45,210
That's an s electron.

1536
01:17:45,210 --> 01:17:47,840
So that's a wide
delocalized orbital,

1537
01:17:47,840 --> 01:17:49,250
and I think it's metallic--

1538
01:17:49,250 --> 01:17:54,440
and you're transferring it to a
localized d state on the iron.

1539
01:17:54,440 --> 01:17:56,900
So that electron is
essentially being transferred

1540
01:17:56,900 --> 01:17:59,750
between extremely
different states,

1541
01:17:59,750 --> 01:18:02,750
and this is what's killing you.

1542
01:18:02,750 --> 01:18:05,780
Because you transfer between
such different states,

1543
01:18:05,780 --> 01:18:08,900
you start losing a lot of the
sort of cancellation of errors

1544
01:18:08,900 --> 01:18:09,920
that you need.

1545
01:18:09,920 --> 01:18:12,650
That's the functional theory.

1546
01:18:12,650 --> 01:18:15,980
And in particular,
the error here

1547
01:18:15,980 --> 01:18:17,960
comes from something
quite particular.

1548
01:18:23,190 --> 01:18:28,030
It comes from what we call
the self-interaction error.

1549
01:18:28,030 --> 01:18:29,880
So I'm trying to sort
make you understand

1550
01:18:29,880 --> 01:18:31,590
where these errors
come from so that when

1551
01:18:31,590 --> 01:18:35,400
you work on your application,
you get a bit of a feeling for.

1552
01:18:35,400 --> 01:18:39,090
If you remember how we
solve all these quantum

1553
01:18:39,090 --> 01:18:41,040
mechanical equations,
we reduce them

1554
01:18:41,040 --> 01:18:43,575
to one electronic
equations, where

1555
01:18:43,575 --> 01:18:45,700
you have the kinetic energy,
the nuclear potential.

1556
01:18:45,700 --> 01:18:48,025
And then this
effective potential

1557
01:18:48,025 --> 01:18:49,650
which, remember what
all goes in there.

1558
01:18:49,650 --> 01:18:51,150
The effective the
potential-- that's

1559
01:18:51,150 --> 01:18:53,400
the one that has the
exchange correlation in it,

1560
01:18:53,400 --> 01:18:58,350
but it also has the Hartree
field, so the coulombic field

1561
01:18:58,350 --> 01:18:59,790
from all the electrons.

1562
01:18:59,790 --> 01:19:04,500
Well, that field includes
the electron itself.

1563
01:19:04,500 --> 01:19:06,330
That's the sort of
oddity in essence.

1564
01:19:06,330 --> 01:19:08,250
When you calculate
the charge density,

1565
01:19:08,250 --> 01:19:10,440
that's the charge density
of all the electrons.

1566
01:19:10,440 --> 01:19:12,390
We then calculate
the potential coming

1567
01:19:12,390 --> 01:19:16,020
from the charge density, that's
the potential coming from all

1568
01:19:16,020 --> 01:19:17,280
the electrons.

1569
01:19:17,280 --> 01:19:20,350
But you operate that now
on a single electron.

1570
01:19:20,350 --> 01:19:24,140
So the electron is
feeling its own potential.

1571
01:19:24,140 --> 01:19:27,410
Part of the exchange
correlation term

1572
01:19:27,410 --> 01:19:31,057
corrects for that,
but not all of it.

1573
01:19:31,057 --> 01:19:32,640
And the problem is
that the correction

1574
01:19:32,640 --> 01:19:35,670
doesn't operate as well on
different forms of the charge

1575
01:19:35,670 --> 01:19:36,520
density.

1576
01:19:36,520 --> 01:19:41,350
In a metal, you have a small
self-interaction error.

1577
01:19:41,350 --> 01:19:44,430
And the reason is if you
look at a state in a metal,

1578
01:19:44,430 --> 01:19:45,840
sort of very delocalized.

1579
01:19:45,840 --> 01:19:48,220
So very spread out
charge density.

1580
01:19:48,220 --> 01:19:51,360
So if you want to think of
it, the part of the electron

1581
01:19:51,360 --> 01:19:54,630
here doesn't feel much of
the charge density coming--

1582
01:19:54,630 --> 01:19:56,850
of the potential coming from
that piece of the charge

1583
01:19:56,850 --> 01:20:00,820
density, because
they're very far away.

1584
01:20:00,820 --> 01:20:05,080
Whereas if you do a very
localized state, in some sense,

1585
01:20:05,080 --> 01:20:08,230
then the potential
from the electron

1586
01:20:08,230 --> 01:20:11,320
is very high where the
electron itself is sitting,

1587
01:20:11,320 --> 01:20:13,090
because it's all very close.

1588
01:20:13,090 --> 01:20:16,840
If you put an electron in
a delta function if you--

1589
01:20:16,840 --> 01:20:19,202
if you didn't have an
uncertainty principle,

1590
01:20:19,202 --> 01:20:21,160
and you calculate its
potential, it's basically

1591
01:20:21,160 --> 01:20:22,810
sitting on top of itself then.

1592
01:20:22,810 --> 01:20:26,620
You'd have an infinite
self-interaction.

1593
01:20:26,620 --> 01:20:30,190
So the more local the state
is, the more self-interaction

1594
01:20:30,190 --> 01:20:31,180
you have.

1595
01:20:31,180 --> 01:20:32,980
And the exchange
correlation functional

1596
01:20:32,980 --> 01:20:37,000
can't quite correct these
two in the same way.

1597
01:20:37,000 --> 01:20:39,760
And remember that the exchange
correlation correction comes

1598
01:20:39,760 --> 01:20:43,420
from homogeneous
charge densities,

1599
01:20:43,420 --> 01:20:46,540
so it tends to correct
the metallic state better

1600
01:20:46,540 --> 01:20:48,400
than the localized state.

1601
01:20:48,400 --> 01:20:52,960
And so this is why that redox
reaction I showed you had a big

1602
01:20:52,960 --> 01:20:56,980
error, because we were
transferring from a state that

1603
01:20:56,980 --> 01:21:00,040
was metallic-- the electron
went from the lithium state--

1604
01:21:00,040 --> 01:21:02,590
to the transition metal state.

1605
01:21:02,590 --> 01:21:04,510
And somehow, the
self-interaction error

1606
01:21:04,510 --> 01:21:07,270
doesn't cancel.

1607
01:21:07,270 --> 01:21:10,030
And you will see
things like that

1608
01:21:10,030 --> 01:21:13,460
whenever you transfer electrons
between quite different states,

1609
01:21:13,460 --> 01:21:17,056
so that's something
to keep in mind.

1610
01:21:17,056 --> 01:21:19,720
I think I'm running out of
time here, so yeah, let me

1611
01:21:19,720 --> 01:21:20,230
stop here.

1612
01:21:23,060 --> 01:21:24,830
So I'm summarizing
here because this is

1613
01:21:24,830 --> 01:21:26,900
the stuff I went over before.

1614
01:21:26,900 --> 01:21:29,750
In general, you do pretty well.

1615
01:21:29,750 --> 01:21:33,960
I think if I'd given this
summary 10 years ago,

1616
01:21:33,960 --> 01:21:37,100
I would have been even more
optimistic, because most people

1617
01:21:37,100 --> 01:21:40,460
worked on metals and
semiconductors which

1618
01:21:40,460 --> 01:21:42,640
tend to be fairly
delocalized state,

1619
01:21:42,640 --> 01:21:45,140
so LDA and GGA do quite well.

1620
01:21:45,140 --> 01:21:49,310
I think as we dig into
more complicated materials,

1621
01:21:49,310 --> 01:21:54,140
we have learned more about
the errors of LDA and GGA.

1622
01:21:54,140 --> 01:21:55,580
But on sort of
classic metals, you

1623
01:21:55,580 --> 01:21:58,130
do pretty well with lattice
constants, reaction energies,

1624
01:21:58,130 --> 01:22:00,860
and cohesive energies.

1625
01:22:00,860 --> 01:22:03,500
But now there is a series of
methods under development--

1626
01:22:03,500 --> 01:22:06,230
and if we have some time,
we might sort of just broach

1627
01:22:06,230 --> 01:22:07,040
them--

1628
01:22:07,040 --> 01:22:09,200
to deal better with
the correlation energy,

1629
01:22:09,200 --> 01:22:12,020
with the self-interaction
energy to solve

1630
01:22:12,020 --> 01:22:13,760
these problems of
both energetics

1631
01:22:13,760 --> 01:22:17,090
and also electronic structure--
things such as the band gap.

1632
01:22:17,090 --> 01:22:18,050
So I'll end here.

1633
01:22:18,050 --> 01:22:20,450
And remember, on
Tuesday, we have lab.

1634
01:22:20,450 --> 01:22:24,970
So you meet in the lab, and then
Thursday, we'll be back here.