WEBVTT
00:00:00.000 --> 00:00:01.860
NICOLA MARZARI:
Welcome, everyone.
00:00:01.860 --> 00:00:06.450
Lecture 8-- hopefully this
will be a somehow simpler class
00:00:06.450 --> 00:00:07.890
to follow than last class.
00:00:07.890 --> 00:00:10.710
That was sort of
heavily theoretical.
00:00:10.710 --> 00:00:12.900
What we are going to
do today is really,
00:00:12.900 --> 00:00:15.690
after two or three slides of
recap of what we have seen
00:00:15.690 --> 00:00:18.240
in the past lecture,
we'll go into really sort
00:00:18.240 --> 00:00:21.300
of practical application of
density functional theory.
00:00:21.300 --> 00:00:24.990
We'll first discuss how
you set up calculations.
00:00:24.990 --> 00:00:26.795
That is, what are
the parameters,
00:00:26.795 --> 00:00:28.170
and what are the
objects that you
00:00:28.170 --> 00:00:29.840
manipulate when
you do an electron
00:00:29.840 --> 00:00:31.020
[INAUDIBLE] calculation?
00:00:31.020 --> 00:00:33.900
And then we start seeing
a number of examples
00:00:33.900 --> 00:00:35.730
to give you the feeling
for the properties
00:00:35.730 --> 00:00:37.680
that we can calculate
using density function
00:00:37.680 --> 00:00:40.710
theory for the accuracy
and for the pitfalls
00:00:40.710 --> 00:00:42.360
you need to be careful with.
00:00:42.360 --> 00:00:44.730
And in the next
class on Thursday,
00:00:44.730 --> 00:00:49.030
we'll finish up with
more case studies.
00:00:49.030 --> 00:00:52.110
So let me actually
spend three slides
00:00:52.110 --> 00:00:55.740
to recap what we have
seen in the last lecture.
00:00:55.740 --> 00:00:58.440
Starting somehow from
this, that, I would say,
00:00:58.440 --> 00:01:00.360
is the most fundamental one.
00:01:00.360 --> 00:01:04.120
And it's really rewriting
quantum mechanics.
00:01:04.120 --> 00:01:06.480
So this is really the
Schrodinger equation
00:01:06.480 --> 00:01:08.880
from the '60s, density
functional theory.
00:01:08.880 --> 00:01:11.100
And this is what it says--
that we don't really
00:01:11.100 --> 00:01:15.970
need to solve a many-body
differential equation to find
00:01:15.970 --> 00:01:19.260
that the ground state charge
density, and the ground state
00:01:19.260 --> 00:01:22.200
energy, and the ground state
properties for a system.
00:01:22.200 --> 00:01:26.910
But we can actually solve it
using a variational principle
00:01:26.910 --> 00:01:29.790
on the charge density,
and the functional
00:01:29.790 --> 00:01:32.220
that has to be minimized
is written here.
00:01:32.220 --> 00:01:35.760
That is, for any given
charged density and prime,
00:01:35.760 --> 00:01:39.540
we have, at least in principle,
a well defined functional,
00:01:39.540 --> 00:01:43.680
and we need to change to vary
the charge density and prime.
00:01:43.680 --> 00:01:46.650
And the minimum value that
is functional will take
00:01:46.650 --> 00:01:49.350
is actually the
ground state energy.
00:01:49.350 --> 00:01:52.920
Somehow, sort of the
weakness of all of this
00:01:52.920 --> 00:01:55.950
is that everything is
well defined in principle,
00:01:55.950 --> 00:01:57.930
but it doesn't work in practice.
00:01:57.930 --> 00:02:02.490
To remind you what were the
sort of conceptual steps, what
00:02:02.490 --> 00:02:06.120
Hohenberg and Kohn proved
from their first theorem
00:02:06.120 --> 00:02:09.900
is that, given any charge
density and prime--
00:02:09.900 --> 00:02:12.870
more or less any charge
density and prime,
00:02:12.870 --> 00:02:18.480
an external potential v
prime is well defined,
00:02:18.480 --> 00:02:22.560
for which that charge density
is going to be the ground state
00:02:22.560 --> 00:02:23.320
charge density.
00:02:23.320 --> 00:02:26.940
This was sort of the inverse
part of the first Hohenberg
00:02:26.940 --> 00:02:28.350
and Kohn theorem.
00:02:28.350 --> 00:02:32.900
And then, at least in
principle, the solution
00:02:32.900 --> 00:02:36.170
to the Schrodinger equation
corresponding to v prime
00:02:36.170 --> 00:02:37.880
is well defined.
00:02:37.880 --> 00:02:40.290
And we call that c prime.
00:02:40.290 --> 00:02:45.620
And so one can construct a
functional f that is just
00:02:45.620 --> 00:02:51.290
the expectation value
of p-- psi prime
00:02:51.290 --> 00:02:53.990
over the many-body
kinetic energy
00:02:53.990 --> 00:02:59.220
operator plus the many-body
electron-electron interaction.
00:02:59.220 --> 00:03:05.730
And so this here is the well
defined, at least in theory,
00:03:05.730 --> 00:03:08.720
density functional
term that is here.
00:03:08.720 --> 00:03:10.700
And the other term
is just the integral
00:03:10.700 --> 00:03:13.370
of the external potential
times the charge density.
00:03:13.370 --> 00:03:16.220
So everything is well
defined in principle,
00:03:16.220 --> 00:03:18.590
and so in this, it's a
perfect reformulation
00:03:18.590 --> 00:03:20.030
of the Schrodinger equation.
00:03:20.030 --> 00:03:22.460
The real problem
is that we don't
00:03:22.460 --> 00:03:27.860
have an exact representation
of this universal density
00:03:27.860 --> 00:03:28.890
functional.
00:03:28.890 --> 00:03:33.320
And so what Walter Kohn
and Lu Sham did was they
00:03:33.320 --> 00:03:36.110
tried to figure out
an approximation
00:03:36.110 --> 00:03:39.890
to get a universal functional.
00:03:39.890 --> 00:03:43.730
And so in the total
energy functional,
00:03:43.730 --> 00:03:48.080
these were-- this was their
choice for the approximation.
00:03:48.080 --> 00:03:49.820
That is what they
said-- is, well,
00:03:49.820 --> 00:03:53.690
let's try to figure out what
are the most relevant physical
00:03:53.690 --> 00:03:57.830
terms in this functional, and
then we'll sort of leave--
00:03:57.830 --> 00:04:01.250
sweep under the rug all
the many-body complexity
00:04:01.250 --> 00:04:02.270
of this problem.
00:04:02.270 --> 00:04:05.750
And we'll call data an
exchange correlation function.
00:04:05.750 --> 00:04:10.250
It is really where the fine
detail of your solutions are.
00:04:10.250 --> 00:04:13.940
And so what they said, in
this universal functional,
00:04:13.940 --> 00:04:19.250
we can extract terms like the
Hartree electrostatic energy,
00:04:19.250 --> 00:04:23.110
written here, that is a
very simple and well defined
00:04:23.110 --> 00:04:26.240
functional of the
charge density.
00:04:26.240 --> 00:04:28.850
The sort of second term
was the tricky one,
00:04:28.850 --> 00:04:30.620
because they wanted
to figure out
00:04:30.620 --> 00:04:33.140
what was a significant
contribution
00:04:33.140 --> 00:04:35.610
to the overall quantum
kinetic energy.
00:04:35.610 --> 00:04:37.580
And as we have seen,
there isn't really
00:04:37.580 --> 00:04:41.240
a good way to extract kinetic
energy that's basically
00:04:41.240 --> 00:04:43.910
a second derivative of the
wave function-- of curvature
00:04:43.910 --> 00:04:46.900
of the wave function
from a charged density.
00:04:46.900 --> 00:04:50.360
The sort of most remarkable
case is that of a plane wave.
00:04:50.360 --> 00:04:53.240
A plane wave of any wavelength
gives you a constant charge
00:04:53.240 --> 00:04:56.010
density, but gives you a very
different second derivative,
00:04:56.010 --> 00:04:58.560
very different kinetic energy.
00:04:58.560 --> 00:05:00.860
And so what Kohn and Sham did--
00:05:00.860 --> 00:05:08.240
they introduced a related system
of non-interacting electrons
00:05:08.240 --> 00:05:13.640
that, for a given charge density
n, they would have the same--
00:05:13.640 --> 00:05:18.170
sorry, they would have
the same ground state
00:05:18.170 --> 00:05:21.260
energy of our original problem.
00:05:21.260 --> 00:05:24.170
But because these are
non-interacting electrons,
00:05:24.170 --> 00:05:29.300
one can define exactly what
is the quantum kinetic energy
00:05:29.300 --> 00:05:32.310
of this set of
non-interacting electrons.
00:05:32.310 --> 00:05:34.910
So if you want, this
is just a definition,
00:05:34.910 --> 00:05:38.450
but it turns out to be a
definition of the second piece
00:05:38.450 --> 00:05:43.640
that makes our third piece
here very, very small.
00:05:43.640 --> 00:05:47.810
So by extracting, selecting
the green and the red term,
00:05:47.810 --> 00:05:50.480
we were left with an
exchange correlational term
00:05:50.480 --> 00:05:53.360
that, if you want, still
contain all the complexity
00:05:53.360 --> 00:05:56.090
of this problem that it's
well defined in principle,
00:05:56.090 --> 00:05:58.070
but we don't know how to solve.
00:05:58.070 --> 00:06:01.820
But for this, then, they
applied that the same idea
00:06:01.820 --> 00:06:03.230
of Thomas and Fermi.
00:06:03.230 --> 00:06:05.060
That is, they said,
well, maybe we
00:06:05.060 --> 00:06:09.050
could try to approximate
this exchange correlation
00:06:09.050 --> 00:06:13.130
term with a local
density approximation.
00:06:13.130 --> 00:06:17.960
That is, we try to construct
this energy term by integrating
00:06:17.960 --> 00:06:21.110
an infinitesimal volume
by infinitesimal volume,
00:06:21.110 --> 00:06:24.470
and each infinitesimal
volume will contribute
00:06:24.470 --> 00:06:27.800
to this many-body problem
and exchange correlation
00:06:27.800 --> 00:06:31.340
energy density that is the
exchange correlation energy
00:06:31.340 --> 00:06:35.360
density of the interacting
homogeneous electron
00:06:35.360 --> 00:06:36.890
gas at that density.
00:06:36.890 --> 00:06:39.500
So when, in 1980,
Ceperley and Alder
00:06:39.500 --> 00:06:43.235
first did very complex
calculation-- quantum Monte
00:06:43.235 --> 00:06:46.070
Carlo calculation
of the interacting,
00:06:46.070 --> 00:06:49.790
but homogeneous electron gas
as a function of the density,
00:06:49.790 --> 00:06:53.540
this functional, at
least in principle,
00:06:53.540 --> 00:06:56.540
was, for the first
time, parameterized
00:06:56.540 --> 00:06:58.220
over a whole set of density.
00:06:58.220 --> 00:07:01.340
Often we call that the
Perdew-Zunger parameterization,
00:07:01.340 --> 00:07:04.910
and so we had finally
a working expression.
00:07:04.910 --> 00:07:07.050
Once we have a
working expression,
00:07:07.050 --> 00:07:09.590
then the problem of
finding the ground state
00:07:09.590 --> 00:07:12.650
becomes our electronic
structure computational problem.
00:07:12.650 --> 00:07:17.000
That is how to find the
minimum of this functional.
00:07:17.000 --> 00:07:20.690
Note the very significant
difference with respect
00:07:20.690 --> 00:07:22.160
to Hartree-Fock.
00:07:22.160 --> 00:07:25.290
In Hartree-Fock, we had
a variational principle,
00:07:25.290 --> 00:07:27.680
and that led us
to our expression
00:07:27.680 --> 00:07:30.230
for the energy for the
Hartree-Fock equation,
00:07:30.230 --> 00:07:34.190
in which there was a well
defined exchange term.
00:07:34.190 --> 00:07:37.430
It was written somehow
complicated-- complex, but well
00:07:37.430 --> 00:07:38.150
defined.
00:07:38.150 --> 00:07:40.160
But because it was
basically coming
00:07:40.160 --> 00:07:42.890
from a variational principle,
we had the possibility
00:07:42.890 --> 00:07:46.040
of making Hartree-Fock
better and better
00:07:46.040 --> 00:07:49.400
by extending our class
of search function
00:07:49.400 --> 00:07:53.790
from Slater determinant
to more flexible classes.
00:07:53.790 --> 00:07:55.650
We could just take,
say, combination
00:07:55.650 --> 00:07:57.060
of Slater determinant.
00:07:57.060 --> 00:08:01.200
So Hartree-Fock, in principle,
can be made better and better
00:08:01.200 --> 00:08:03.090
in a systematic way.
00:08:03.090 --> 00:08:06.870
And the computational cost that
you will pay is horrendous,
00:08:06.870 --> 00:08:08.700
but it gives you an avenue.
00:08:08.700 --> 00:08:11.760
Density functional theory
doesn't give you an avenue.
00:08:11.760 --> 00:08:15.270
It sort of monolithically states
that there is going to be,
00:08:15.270 --> 00:08:19.020
in principle, a well defined
universal functional,
00:08:19.020 --> 00:08:21.900
and in the Kohn and Sham
sort of the composition that
00:08:21.900 --> 00:08:25.470
is going to be in particular
one single given exchange
00:08:25.470 --> 00:08:29.580
correlation functional,
but it doesn't give us
00:08:29.580 --> 00:08:32.940
any sort of systematic
route to find
00:08:32.940 --> 00:08:34.590
better and better functional.
00:08:34.590 --> 00:08:37.679
And that's why for many
years there wasn't really
00:08:37.679 --> 00:08:41.700
much applied work using
density functional theory.
00:08:41.700 --> 00:08:43.890
Sort of in the
early '70s, people
00:08:43.890 --> 00:08:46.110
started studying
very simple system,
00:08:46.110 --> 00:08:50.250
like the surface of a
metal sort of represented
00:08:50.250 --> 00:08:53.280
as a step in a free
electron gas solution.
00:08:53.280 --> 00:08:56.880
And then it's only sort of the
beginning of the '80s, when
00:08:56.880 --> 00:08:59.940
we had the
parameterization of LDA
00:08:59.940 --> 00:09:02.160
from the calculation
of Ceperely and Alder,
00:09:02.160 --> 00:09:05.970
that people could sort of put
together an overall working
00:09:05.970 --> 00:09:06.780
algorithm.
00:09:06.780 --> 00:09:09.870
And I've shown you in the
last class sort of the phase
00:09:09.870 --> 00:09:12.090
diagram of silicon,
the first time
00:09:12.090 --> 00:09:15.690
that we see somehow that this
theory is going to give us
00:09:15.690 --> 00:09:19.440
a very accurate practical result
and will be able to give us
00:09:19.440 --> 00:09:22.320
the lattice parameters,
the bulk models of silicon,
00:09:22.320 --> 00:09:26.250
just by plotting the energy
as a function of the lattice
00:09:26.250 --> 00:09:29.670
parameter, or it will give
us even the phase transition
00:09:29.670 --> 00:09:31.845
to high pressure
phases of silicon.
00:09:37.110 --> 00:09:41.790
What I had written in the
previous slide was the total
00:09:41.790 --> 00:09:44.760
energy in this density
functional paradigm,
00:09:44.760 --> 00:09:47.580
and we had, as a
computational goal now--
00:09:47.580 --> 00:09:49.620
and you see how
we address this--
00:09:49.620 --> 00:09:54.930
the task of minimizing
that total energy.
00:09:54.930 --> 00:09:58.500
And there are two main
routes that we can take.
00:09:58.500 --> 00:10:01.890
Once the exchange
correlation potential
00:10:01.890 --> 00:10:04.710
is written in an
explicit form, this
00:10:04.710 --> 00:10:09.330
is a well defined, even
if nonlinear, functional
00:10:09.330 --> 00:10:12.660
of this independent
single particle
00:10:12.660 --> 00:10:15.000
orbitals-- the Kohn
and Sham orbitals
00:10:15.000 --> 00:10:17.250
of which the charge
density is just
00:10:17.250 --> 00:10:19.660
the sum of the square moduli.
00:10:19.660 --> 00:10:24.180
And so we can look at this as a
nonlinear minimization problem.
00:10:24.180 --> 00:10:28.590
We need to find the orbitals
that minimize this expression,
00:10:28.590 --> 00:10:32.520
or as it often happens, we
can write the associated
00:10:32.520 --> 00:10:34.080
Euler-Lagrange equation.
00:10:34.080 --> 00:10:35.700
We have a variational
principle so we
00:10:35.700 --> 00:10:38.190
can take the functional--
the functional
00:10:38.190 --> 00:10:40.380
differential of that,
and that gives us
00:10:40.380 --> 00:10:44.205
a set of differential equation
that is what we call the Kohn
00:10:44.205 --> 00:10:45.480
and Sham equation.
00:10:45.480 --> 00:10:47.250
And again, they are
written here-- sort
00:10:47.250 --> 00:10:49.170
of not very different.
00:10:49.170 --> 00:10:52.770
If you look at it, from
the Hartree-Fock equations,
00:10:52.770 --> 00:10:54.420
there are really the same terms.
00:10:54.420 --> 00:10:57.385
There is a quantum
kinetic energy.
00:10:57.385 --> 00:11:00.220
And now these are single
particle orbitals.
00:11:00.220 --> 00:11:01.860
There is a Hartree term.
00:11:01.860 --> 00:11:04.020
There is an external
potential term,
00:11:04.020 --> 00:11:06.720
and the only difference
is in the way
00:11:06.720 --> 00:11:10.320
the exchange or exchange
correlation term is calculated.
00:11:10.320 --> 00:11:14.730
In Hartree-Fock, it was
an explicit integral
00:11:14.730 --> 00:11:15.520
of the orbital.
00:11:15.520 --> 00:11:18.060
In density functional
theory, it's
00:11:18.060 --> 00:11:22.950
going to be some kind of complex
function of the charge density.
00:11:22.950 --> 00:11:25.950
And we are going
to try and find out
00:11:25.950 --> 00:11:31.590
what are the eigenstates of this
Hamiltonian, of which the most
00:11:31.590 --> 00:11:35.670
important part that
you need to remember
00:11:35.670 --> 00:11:40.560
is that this Hamiltonian is
what we call self-consistent.
00:11:40.560 --> 00:11:44.790
That is, it's an operator
that actually depends
00:11:44.790 --> 00:11:48.450
on its own solution, because
you see, what we have is
00:11:48.450 --> 00:11:51.540
that the charge density
is given by the sum
00:11:51.540 --> 00:11:53.010
of the square moduli.
00:11:53.010 --> 00:11:58.370
And the charge density goes into
the expression of the Hartree
00:11:58.370 --> 00:12:01.560
operator, and goes into the
expression of the exchange
00:12:01.560 --> 00:12:03.250
correlation operator.
00:12:03.250 --> 00:12:07.410
So the Hamiltonian, acting on
this single particle orbitals,
00:12:07.410 --> 00:12:09.330
depend on the charge density.
00:12:09.330 --> 00:12:11.850
And the charge
density is a function
00:12:11.850 --> 00:12:13.810
of the orbital themselves.
00:12:13.810 --> 00:12:16.530
And so the problem has
become slightly different
00:12:16.530 --> 00:12:17.813
from your usual problem.
00:12:17.813 --> 00:12:19.230
The Schrodinger
equation gives you
00:12:19.230 --> 00:12:22.500
an operator for which you
need to find the eigenvectors.
00:12:22.500 --> 00:12:26.670
Here, you have an operator for
which you find eigenvectors,
00:12:26.670 --> 00:12:28.890
but then these
eigenvectors give you
00:12:28.890 --> 00:12:32.010
a charge density that,
put back in here,
00:12:32.010 --> 00:12:33.990
gives you a different operator.
00:12:33.990 --> 00:12:36.780
And so exactly like
in Hartree-Fock,
00:12:36.780 --> 00:12:39.960
you have found your
ground state solution
00:12:39.960 --> 00:12:42.390
only once you have
become self-consistent.
00:12:42.390 --> 00:12:45.450
You have Hamiltonian
whose eigenstates give you
00:12:45.450 --> 00:12:49.560
a charge density that gives you
the same Hamiltonian you had
00:12:49.560 --> 00:12:51.930
calculated the eigenstates for.
00:12:51.930 --> 00:12:55.140
And so all the
computational approaches
00:12:55.140 --> 00:12:57.390
to solve the density
functional problem,
00:12:57.390 --> 00:13:00.810
as those that solve the
Hartree-Fock problem
00:13:00.810 --> 00:13:03.780
are iterative approaches.
00:13:03.780 --> 00:13:07.170
We can't just find a
solution to an Hamiltonian,
00:13:07.170 --> 00:13:12.360
but we really need to make
that problem self-consistent.
00:13:12.360 --> 00:13:16.500
So this is where we
concluded in last class.
00:13:16.500 --> 00:13:20.130
And so now we'll actually
go into practical feature
00:13:20.130 --> 00:13:21.970
of density function theory.
00:13:21.970 --> 00:13:27.150
So starting from reminding
you that all the quality
00:13:27.150 --> 00:13:30.150
of your calculation
depend ultimately
00:13:30.150 --> 00:13:32.400
on the quality of
your functional.
00:13:32.400 --> 00:13:35.130
And for many years, the only
functional that was used
00:13:35.130 --> 00:13:38.820
was really the local density
approximation functional.
00:13:38.820 --> 00:13:43.500
In the late '80s and
early '90s, people
00:13:43.500 --> 00:13:46.890
started to develop what are
called generalized gradient
00:13:46.890 --> 00:13:48.060
approximation.
00:13:48.060 --> 00:13:51.960
That is, they constructed
functionals of the charge
00:13:51.960 --> 00:13:55.320
density that didn't only depend
on the charge density itself,
00:13:55.320 --> 00:13:57.900
but also on its gradient.
00:13:57.900 --> 00:14:00.000
Not in the sense of
a Taylor expansion,
00:14:00.000 --> 00:14:02.370
because a Taylor expansion
wouldn't actually
00:14:02.370 --> 00:14:05.370
satisfy a number of
symmetry properties
00:14:05.370 --> 00:14:08.310
that we know that the
exact exchange correlation
00:14:08.310 --> 00:14:09.850
functional would do.
00:14:09.850 --> 00:14:12.690
And so in this sense, all
these new approximation
00:14:12.690 --> 00:14:15.750
are called generalized
gradient approximation.
00:14:15.750 --> 00:14:19.500
And there is a little menagerie
of acronyms and symbols
00:14:19.500 --> 00:14:22.260
that really are sort of
build up upon the names
00:14:22.260 --> 00:14:25.050
of the few people that have
really done a lot of work
00:14:25.050 --> 00:14:26.230
in this field.
00:14:26.230 --> 00:14:29.100
And so you'll see a lot
John Perdew in this,
00:14:29.100 --> 00:14:33.570
represented by a P, or
Axel Becke in Canada
00:14:33.570 --> 00:14:37.680
by a B, or Kieron Burke
in Rutgers by another B.
00:14:37.680 --> 00:14:40.450
So these are some of the
most popular functionals.
00:14:40.450 --> 00:14:44.420
I would say that, by now,
sort of almost every one
00:14:44.420 --> 00:14:49.090
is standardized on PBE as
being the most reasonable
00:14:49.090 --> 00:14:52.060
and the most accurate
advanced approximation
00:14:52.060 --> 00:14:53.470
beyond local density.
00:14:53.470 --> 00:14:56.020
And so this is what we'll use.
00:14:56.020 --> 00:15:00.430
There are classes of
more complex functionals.
00:15:00.430 --> 00:15:02.770
The quantum chemistry
has done-- community
00:15:02.770 --> 00:15:05.020
has done a lot of
work in looking
00:15:05.020 --> 00:15:09.510
at hybrid functionals,
functional in which
00:15:09.510 --> 00:15:14.050
a certain percentage of the
exchange correlation functional
00:15:14.050 --> 00:15:16.810
comes from the density
function approximation.
00:15:16.810 --> 00:15:18.880
It could be LDA,
or in most cases,
00:15:18.880 --> 00:15:20.920
could be something like PB.
00:15:20.920 --> 00:15:25.400
And some amount of Hartree-Fock
exchange is mixed in.
00:15:25.400 --> 00:15:27.670
So they really
take Kohn and Sham
00:15:27.670 --> 00:15:30.760
equation, in which they
exchange correlation term
00:15:30.760 --> 00:15:34.060
as both a component that is
density functional theory-like
00:15:34.060 --> 00:15:37.210
and a component that
is Hartree-Fock-like.
00:15:37.210 --> 00:15:39.490
And some of them are,
like these two mention
00:15:39.490 --> 00:15:43.510
here, can work very well,
especially for molecules.
00:15:43.510 --> 00:15:48.460
Somehow, again, Hartree-Fock
comes from atoms and molecules,
00:15:48.460 --> 00:15:51.880
and it tends to work
better in that limit.
00:15:51.880 --> 00:15:53.650
Density functional
theory is built
00:15:53.650 --> 00:15:56.230
on an approximation
like the LDA that
00:15:56.230 --> 00:15:59.650
ultimately comes from the
homogeneous electron gas.
00:15:59.650 --> 00:16:03.250
So it tends to work better
for solids, and as usual,
00:16:03.250 --> 00:16:05.320
the most difficult
cases in which you
00:16:05.320 --> 00:16:06.660
have a combination of the two.
00:16:06.660 --> 00:16:09.310
If you want to study a
molecule on a solid surface,
00:16:09.310 --> 00:16:11.680
then none of these
two approaches
00:16:11.680 --> 00:16:14.290
work really exceedingly well.
00:16:14.290 --> 00:16:18.310
There is a lot of
work in developing
00:16:18.310 --> 00:16:21.790
more complex
functionals that tend
00:16:21.790 --> 00:16:25.300
to work better than any of these
that I have mentioned here.
00:16:25.300 --> 00:16:28.210
They tend to be
exceedingly complex.
00:16:28.210 --> 00:16:30.550
And so again, there
are sort of meta-GGA
00:16:30.550 --> 00:16:34.150
functional-- there is a lot
of more complex functional
00:16:34.150 --> 00:16:36.850
that starts to depend
not only on the density
00:16:36.850 --> 00:16:40.180
and on the gradients, but
then maybe on the Laplacian.
00:16:40.180 --> 00:16:43.270
They might depend on
the orbital themselves.
00:16:43.270 --> 00:16:47.620
So you find terms like exact
exchange functional and so on.
00:16:47.620 --> 00:16:49.330
There is a lot of
current work, and there
00:16:49.330 --> 00:16:50.710
isn't a unique solution.
00:16:50.710 --> 00:16:55.030
And also they tend to be so much
more computationally expensive
00:16:55.030 --> 00:16:57.460
that, at this stage, I
would say they are still
00:16:57.460 --> 00:17:02.320
limited to a development
effort more than an application
00:17:02.320 --> 00:17:03.280
effort.
00:17:03.280 --> 00:17:06.130
But you see, once we
have a functional--
00:17:06.130 --> 00:17:10.210
and this is even done sort
of with standard LDA or TGA,
00:17:10.210 --> 00:17:13.810
we can actually describe
with remarkable accuracy
00:17:13.810 --> 00:17:16.609
a number of property for
very different materials.
00:17:16.609 --> 00:17:19.540
This is a table that
Chris Pickard in Cambridge
00:17:19.540 --> 00:17:24.460
had given me, and you see we
have a combination of metals,
00:17:24.460 --> 00:17:28.300
semiconductors, oxides, alloys.
00:17:28.300 --> 00:17:30.970
And what we have
here, say, is a list
00:17:30.970 --> 00:17:35.060
of what is the experimental
lattice parameter,
00:17:35.060 --> 00:17:37.490
and what is the
theoretical prediction.
00:17:37.490 --> 00:17:39.490
And you see we are sort
of in the range in which
00:17:39.490 --> 00:17:42.760
the errors are around 1%.
00:17:42.760 --> 00:17:45.310
And I would say,
for most materials,
00:17:45.310 --> 00:17:48.370
we are really in this ballpark.
00:17:48.370 --> 00:17:51.730
The error on a lattice
parameter of a material that
00:17:51.730 --> 00:17:55.630
doesn't have exotic
electronic properties is not
00:17:55.630 --> 00:17:59.110
a high-Tc superconductor,
is not as strongly
00:17:59.110 --> 00:18:01.600
correlated electronic
material can
00:18:01.600 --> 00:18:04.930
be expected to be in the
range of an error that
00:18:04.930 --> 00:18:07.960
is between 1% and 2%.
00:18:07.960 --> 00:18:09.370
So very, very good.
00:18:09.370 --> 00:18:11.080
You see predictive power.
00:18:11.080 --> 00:18:15.370
When we calculate with density
functional theory the lattice
00:18:15.370 --> 00:18:17.830
parameter of this, we
are really not trying
00:18:17.830 --> 00:18:20.500
to fit any potential at all.
00:18:20.500 --> 00:18:24.190
When we calculate the
lattice parameter of silver,
00:18:24.190 --> 00:18:25.960
we are just saying
there is going
00:18:25.960 --> 00:18:30.310
to be an array of Coulombic
potentials attracting
00:18:30.310 --> 00:18:33.280
the electrons with the
atomic number of silver.
00:18:33.280 --> 00:18:37.390
And we are looking at how the
total energy of this system
00:18:37.390 --> 00:18:39.460
varies with lattice parameters.
00:18:39.460 --> 00:18:43.090
And that's why, if you want,
electronic structure approaches
00:18:43.090 --> 00:18:46.360
tend to be extremely powerful,
because they are not fitted.
00:18:46.360 --> 00:18:52.180
We are just trying to find the
electronic structure solution.
00:18:52.180 --> 00:18:57.310
So how does it work in practice?
00:18:57.310 --> 00:19:00.010
And what I'm going
to describe here
00:19:00.010 --> 00:19:03.130
is what's become known by
now as the total-energy
00:19:03.130 --> 00:19:04.840
pseudopotential approach.
00:19:04.840 --> 00:19:07.930
This is really the approach
that has been developed
00:19:07.930 --> 00:19:11.668
to study solid systems.
00:19:11.668 --> 00:19:13.960
Again, you have to keep in
mind these two communities--
00:19:13.960 --> 00:19:18.460
one of sort of solid-state
extended studies of matter,
00:19:18.460 --> 00:19:20.890
and one of atoms and molecules.
00:19:20.890 --> 00:19:22.630
And really density
functional theory
00:19:22.630 --> 00:19:26.980
comes from these solid
state approaches.
00:19:26.980 --> 00:19:30.100
And for reasons that
we'll see in a moment that
00:19:30.100 --> 00:19:33.280
are really related
to what we use
00:19:33.280 --> 00:19:37.180
as a basis set to
describe electrons
00:19:37.180 --> 00:19:41.170
in a solid, what become
apparent very early on
00:19:41.170 --> 00:19:47.980
is that describing accurately
the core electron in an atom
00:19:47.980 --> 00:19:51.800
or in a solid would have
been exceedingly complex.
00:19:51.800 --> 00:19:54.430
If you think at it, whenever
you have an atom that
00:19:54.430 --> 00:19:57.550
forms a molecule or
a solid, you have
00:19:57.550 --> 00:19:59.770
a lot of electrons
that are going
00:19:59.770 --> 00:20:04.570
to be basically unaffected
by the chemical environment.
00:20:04.570 --> 00:20:10.060
If you take an iron atom, and
you put it in iron as a metal,
00:20:10.060 --> 00:20:13.420
you put as iron in a
transition metal oxide,
00:20:13.420 --> 00:20:16.990
or you put as iron as a center,
say, in a heme grouping--
00:20:16.990 --> 00:20:18.940
hemoglobin or myoglobin--
00:20:18.940 --> 00:20:21.880
what happens is that the
valence electrons or iron
00:20:21.880 --> 00:20:24.070
will redistribute themselves.
00:20:24.070 --> 00:20:27.550
But really, the core
electron of irons
00:20:27.550 --> 00:20:32.440
are so tightly bound to the
nucleus by order of magnitude
00:20:32.440 --> 00:20:34.120
in energy with
respect to the sort
00:20:34.120 --> 00:20:36.130
of typical energy
of valence electrons
00:20:36.130 --> 00:20:38.600
that they are
basically unaffected.
00:20:38.600 --> 00:20:41.530
And so in some ways,
we really don't
00:20:41.530 --> 00:20:45.970
want to carry all the
computational expense
00:20:45.970 --> 00:20:49.480
of describing core
electrons when
00:20:49.480 --> 00:20:52.030
we know that their
contribution is
00:20:52.030 --> 00:20:55.870
going to be a rigid contribution
that doesn't change depending
00:20:55.870 --> 00:20:57.700
on the chemical environment.
00:20:57.700 --> 00:20:59.950
The core electrons are
certainly important.
00:20:59.950 --> 00:21:00.850
They are there.
00:21:00.850 --> 00:21:02.980
They are bound to
the nucleus, and they
00:21:02.980 --> 00:21:04.225
screen the nucleus charge.
00:21:04.225 --> 00:21:08.500
In iron, the 1s electrons--
the two 1s electrons
00:21:08.500 --> 00:21:10.840
will be so tightly
bound to the nucleus
00:21:10.840 --> 00:21:13.150
that the nucleus
doesn't look to the 2s,
00:21:13.150 --> 00:21:17.500
or 2p, or 3s, 3p electrons
as saving 26 protons.
00:21:17.500 --> 00:21:20.260
But it really looks
like having 24 protons,
00:21:20.260 --> 00:21:23.960
because those two 1s
electrons screen completely.
00:21:23.960 --> 00:21:28.210
And so the 2s and 2p really
will also spin almost completely
00:21:28.210 --> 00:21:31.780
the nucleus from the point
of view of the 3s and the 3p.
00:21:31.780 --> 00:21:35.890
So we want to find out a way
of somehow taking into account
00:21:35.890 --> 00:21:37.900
the presence of
that core electrons,
00:21:37.900 --> 00:21:41.650
but we don't want to carry
that on in our calculation,
00:21:41.650 --> 00:21:43.750
because it's very expensive.
00:21:43.750 --> 00:21:49.060
Not to mention that the spatial
variation of core electrons
00:21:49.060 --> 00:21:51.010
is going to be extremely sharp.
00:21:51.010 --> 00:21:53.510
I'll show you an
example in a moment.
00:21:53.510 --> 00:21:56.710
And so we need a lot of
computational information
00:21:56.710 --> 00:22:00.820
to describe all the sharp
wiggles that core electrons
00:22:00.820 --> 00:22:02.570
will do around the nucleus.
00:22:02.570 --> 00:22:06.400
And this problem has been
solved that again from the late
00:22:06.400 --> 00:22:11.140
'70s to the early '80s by what
are called pseudopotential
00:22:11.140 --> 00:22:13.640
approaches-- in particular,
something that is called--
00:22:13.640 --> 00:22:15.400
and you'll see this term often--
00:22:15.400 --> 00:22:18.850
norm conserving pseudopotentials
that, in many ways,
00:22:18.850 --> 00:22:21.430
are some of the most
complex part of all
00:22:21.430 --> 00:22:23.390
these electron
structure approaches.
00:22:23.390 --> 00:22:27.520
And I'll just give you a
sort generic flavor of how
00:22:27.520 --> 00:22:29.410
they approach this problem.
00:22:34.800 --> 00:22:38.340
Once we have sort of
removed the core electrons
00:22:38.340 --> 00:22:41.310
from our problems, we
still need to find out
00:22:41.310 --> 00:22:44.640
what are the Kohn and Sham
orbitals that minimize
00:22:44.640 --> 00:22:46.470
our density functional problem.
00:22:46.470 --> 00:22:49.710
And as always-- and again, we'll
see this in the next slide--
00:22:49.710 --> 00:22:53.370
we need to find an appropriate
computational representation
00:22:53.370 --> 00:22:54.750
of these orbitals.
00:22:54.750 --> 00:22:58.920
And that is in particular
will expand those orbitals
00:22:58.920 --> 00:22:59.550
on a basis.
00:22:59.550 --> 00:23:04.830
That is, we'll represent any
possible orbital in a problem
00:23:04.830 --> 00:23:08.140
as a linear combination
of simple function.
00:23:08.140 --> 00:23:10.530
This is what is
called a basis set.
00:23:10.530 --> 00:23:13.650
So in our one dimensional
analysis problem,
00:23:13.650 --> 00:23:16.560
when you have got an arbitrary
function, you can expand--
00:23:16.560 --> 00:23:18.270
you can do, say,
Fourier analysis.
00:23:18.270 --> 00:23:21.810
You can expand it in a
series of sines and cosines.
00:23:21.810 --> 00:23:24.310
So the same general
problem applies here.
00:23:24.310 --> 00:23:27.090
We have arbitrary functions
in three dimensions that
00:23:27.090 --> 00:23:31.360
are our orbitals, and we want
to find an appropriate basis
00:23:31.360 --> 00:23:32.670
set that describe them.
00:23:32.670 --> 00:23:35.070
So this basis set
needs to be accurate--
00:23:35.070 --> 00:23:37.260
that this, needs to
be flexible enough
00:23:37.260 --> 00:23:40.800
to describe all the possible
wiggles of our orbitals,
00:23:40.800 --> 00:23:43.860
but also need to be
computationally convenient.
00:23:43.860 --> 00:23:47.200
And we'll go in
that in a moment.
00:23:47.200 --> 00:23:51.630
So once we have sort of put
together these elements,
00:23:51.630 --> 00:23:56.490
we have sort of our external
potential being represented
00:23:56.490 --> 00:23:59.700
by this set of
nuclei, and somehow,
00:23:59.700 --> 00:24:01.680
via the pseudopotential
approach,
00:24:01.680 --> 00:24:02.970
also the core electrons.
00:24:02.970 --> 00:24:06.240
And once we have decided
what is our basic set,
00:24:06.240 --> 00:24:10.830
we really need to go and
solve self-consistently
00:24:10.830 --> 00:24:12.780
either the Kohn
and Sham equation
00:24:12.780 --> 00:24:14.760
that you have seen
before, or we want
00:24:14.760 --> 00:24:18.430
to minimize the nonlinear
energy functional.
00:24:18.430 --> 00:24:21.450
And so there are sort
of several steps.
00:24:21.450 --> 00:24:27.040
Usually, what we'll do is we'll
start from a trial solution.
00:24:27.040 --> 00:24:29.730
Remember that, since
the Hamiltonian depends
00:24:29.730 --> 00:24:32.070
on the charge density,
we need to have
00:24:32.070 --> 00:24:34.860
a guess to our
initial charge density
00:24:34.860 --> 00:24:38.940
to even construct our operator,
because our operator depends
00:24:38.940 --> 00:24:41.650
on the ground state
charge density.
00:24:41.650 --> 00:24:43.800
So since that we don't have
the ground state charge
00:24:43.800 --> 00:24:45.510
density when we
start our problem,
00:24:45.510 --> 00:24:48.120
we need to start with
a trial solution.
00:24:48.120 --> 00:24:50.040
It could be a trial
charge density.
00:24:50.040 --> 00:24:51.720
Could be trial orbitals.
00:24:51.720 --> 00:24:57.220
But once we have that, all
our operator is well defined.
00:24:57.220 --> 00:25:00.090
So we can calculate all
the different terms--
00:25:00.090 --> 00:25:03.220
the quantum kinetic
energy, the Hartree energy,
00:25:03.220 --> 00:25:05.730
the exchange correlation terms.
00:25:05.730 --> 00:25:08.880
And we can try to
solve that Hamiltonian
00:25:08.880 --> 00:25:10.200
from that Hamiltonian.
00:25:10.200 --> 00:25:14.880
We'll find new orbitals,
and with those new orbitals,
00:25:14.880 --> 00:25:18.540
we'll calculate the ground
state charge density,
00:25:18.540 --> 00:25:21.120
and we'll obtain a
new Hamiltonian that
00:25:21.120 --> 00:25:23.910
then will sort of
keep iterating,
00:25:23.910 --> 00:25:27.990
finding new orbitals, new
charge density, new Hamiltonian,
00:25:27.990 --> 00:25:30.810
until it reaches a
fixed point, until we
00:25:30.810 --> 00:25:32.490
reach self-consistency.
00:25:32.490 --> 00:25:36.150
Or we can just take the approach
of minimizing the total energy
00:25:36.150 --> 00:25:38.140
functional to self-consistency.
00:25:38.140 --> 00:25:43.690
I've written in a more
compact way the problem here.
00:25:43.690 --> 00:25:48.610
Well, it'll come in a moment.
00:25:48.610 --> 00:25:55.510
So let's describe first
our first computational
00:25:55.510 --> 00:25:57.460
approximation in this problem--
00:25:57.460 --> 00:26:01.030
that is, the introduction
of pseudopotentials.
00:26:01.030 --> 00:26:03.850
And as I said,
what we want to do
00:26:03.850 --> 00:26:08.920
is we want to get rid of the
electrons in the inner shells,
00:26:08.920 --> 00:26:11.770
in the core of the atoms,
because they really
00:26:11.770 --> 00:26:16.270
don't have any contribution
to the valence chemical bonds.
00:26:16.270 --> 00:26:17.215
They are just there.
00:26:17.215 --> 00:26:19.030
They're important
because they screen
00:26:19.030 --> 00:26:22.690
the nucleus in a very specific
quantum mechanical term,
00:26:22.690 --> 00:26:26.200
but they really don't
change much when
00:26:26.200 --> 00:26:29.620
the valence
coordination changes.
00:26:29.620 --> 00:26:33.490
We call that, actually, a
frozen core approximation.
00:26:33.490 --> 00:26:38.020
That is, we take an atom--
again, we take an iron atom.
00:26:38.020 --> 00:26:42.100
We solve the density functional
problem for the iron atom.
00:26:42.100 --> 00:26:46.020
That is, we find a density
functional orbital for the 1s,
00:26:46.020 --> 00:26:50.740
2s, 2p, 3s, 3p,
and so on orbitals.
00:26:50.740 --> 00:26:55.510
For an atom, it's reasonable to
do that [INAUDIBLE] solution,
00:26:55.510 --> 00:26:57.280
even for the core
electrons, because you
00:26:57.280 --> 00:26:58.730
have spherical symmetry.
00:26:58.730 --> 00:27:01.540
And the problem is still
sort of fairly simple.
00:27:01.540 --> 00:27:04.810
And at this point, we say,
well, from now on, we are not
00:27:04.810 --> 00:27:08.710
going to describe
the full iron atom,
00:27:08.710 --> 00:27:13.060
but we are going always
to deal with a pseudo iron
00:27:13.060 --> 00:27:21.100
atom in which the nucleus and
the 1s, 2s, and 2p electrons
00:27:21.100 --> 00:27:23.290
have been frozen.
00:27:23.290 --> 00:27:27.400
And so what really the
valence electron need to see
00:27:27.400 --> 00:27:30.400
is a pseudo nucleus
that is not given
00:27:30.400 --> 00:27:32.770
just by the bare
Coulombic potential,
00:27:32.770 --> 00:27:36.640
but by the bare Coulombic
potential screened
00:27:36.640 --> 00:27:41.410
by this 1s, and 2s,
and 2p electron frozen
00:27:41.410 --> 00:27:43.570
in their atomic configuration.
00:27:43.570 --> 00:27:46.420
And there is a theorem that
we won't dwell into by-- from
00:27:46.420 --> 00:27:48.280
Barth and Gelatt from the '80s--
00:27:48.280 --> 00:27:52.210
that says that this freezing
of the core electrons
00:27:52.210 --> 00:27:54.310
is actually a very
good approximation.
00:27:54.310 --> 00:27:55.390
It doesn't really matter.
00:27:55.390 --> 00:27:57.790
It's been verified
over and over again.
00:27:57.790 --> 00:28:00.820
It Is the last thing
we have to worry about.
00:28:00.820 --> 00:28:03.220
To be precise, the
only thing that we
00:28:03.220 --> 00:28:05.680
have to worry about in
density functional theory,
00:28:05.680 --> 00:28:09.940
apart from making sure that
our technical approximations--
00:28:09.940 --> 00:28:13.150
computation approximation
are all accurate
00:28:13.150 --> 00:28:15.100
is the exchange
correlation functional.
00:28:15.100 --> 00:28:18.200
That is truly the
only source of error.
00:28:18.200 --> 00:28:20.710
And so here, if you
want, I've represented
00:28:20.710 --> 00:28:24.460
the idea of a pseudo potential.
00:28:24.460 --> 00:28:27.790
This would have been the
standard solution for an atom.
00:28:27.790 --> 00:28:30.940
Actually, here I've chosen
something simpler than iron.
00:28:30.940 --> 00:28:32.860
I've chosen aluminum.
00:28:32.860 --> 00:28:38.590
And so we would have had a
Kohn and Sham set of equations
00:28:38.590 --> 00:28:42.670
for aluminum in which,
again, this term
00:28:42.670 --> 00:28:46.990
here contains the Hartree
potential, the exchange
00:28:46.990 --> 00:28:51.220
correlation potential,
and external potential
00:28:51.220 --> 00:28:53.620
that would be just the
bare nuclear potential.
00:28:53.620 --> 00:28:54.940
We solve this problem.
00:28:54.940 --> 00:28:55.800
What we find?
00:28:55.800 --> 00:28:58.570
Well, we find this
hierarchy of states.
00:28:58.570 --> 00:29:02.890
You see, we find two
electrons in the 1s state.
00:29:02.890 --> 00:29:06.850
We find two electrons in the
2s, two electrons in the 2p.
00:29:06.850 --> 00:29:09.880
But really, the electrons
that do all the chemistry
00:29:09.880 --> 00:29:12.910
are the 3s and 3p electrons.
00:29:12.910 --> 00:29:17.320
And you see the enormous
difference in energy scales.
00:29:17.320 --> 00:29:21.610
So the binding energy of the
1s electrons to the nucleus
00:29:21.610 --> 00:29:24.880
is 1,500 electron volts.
00:29:24.880 --> 00:29:27.100
There is no way,
unless you are sort
00:29:27.100 --> 00:29:31.360
of throwing X-rays of very
high energy to those electrons,
00:29:31.360 --> 00:29:36.040
that you are going to affect or
perturb in any significant way
00:29:36.040 --> 00:29:37.540
these 1s electrons.
00:29:37.540 --> 00:29:41.560
Remember, the energy
of a hydrogen bond
00:29:41.560 --> 00:29:45.920
is 0.29 electrons-- is a
fraction of an electronvolt.
00:29:45.920 --> 00:29:49.490
So this is four orders
of magnitude larger.
00:29:49.490 --> 00:29:53.320
So what we want is we want
to say this set of electrons
00:29:53.320 --> 00:29:56.200
are so tightly
bound that we don't
00:29:56.200 --> 00:29:59.770
need to consider how they
change during the formation
00:29:59.770 --> 00:30:01.250
of a chemical bond.
00:30:01.250 --> 00:30:06.550
And so what we want to
find is a new potential
00:30:06.550 --> 00:30:10.240
that we call our
pseudopotential, such
00:30:10.240 --> 00:30:14.950
that the eigenstates
in the presence
00:30:14.950 --> 00:30:22.270
of this pseudopotential give us
solutions that really reproduce
00:30:22.270 --> 00:30:27.290
exactly the valence electron
solution of the original atom.
00:30:27.290 --> 00:30:32.680
So you see, what we want is
we want to go from 13 over r,
00:30:32.680 --> 00:30:37.240
the Coulombic potential
that is contained in here
00:30:37.240 --> 00:30:43.900
for the bare aluminum nucleus,
to a new potential that somehow
00:30:43.900 --> 00:30:48.250
contains both the bare
potential of the nucleus,
00:30:48.250 --> 00:30:51.290
the screening from
the core electrons,
00:30:51.290 --> 00:30:56.560
and it is constructed according
to a well-defined prescription
00:30:56.560 --> 00:31:00.130
so that the eigenstates of
this new set of Kohn and Sham
00:31:00.130 --> 00:31:03.070
equation are in order.
00:31:03.070 --> 00:31:07.120
The lowest eigenstates
are actually
00:31:07.120 --> 00:31:11.620
the valence eigenstates
of our regional problem.
00:31:11.620 --> 00:31:15.460
We get the same eigenvalues,
and we get, in a way
00:31:15.460 --> 00:31:21.340
that I'll specify in a moment,
the same eigenfunctions.
00:31:21.340 --> 00:31:25.270
And so here, it's how we would
actually look at this problem.
00:31:25.270 --> 00:31:30.070
And this is a figure courtesy
of Chris [INAUDIBLE]..
00:31:30.070 --> 00:31:33.760
So you see, what we
usually ever in an atom
00:31:33.760 --> 00:31:37.730
is a Coulombic potential,
is this thin red line here.
00:31:37.730 --> 00:31:40.420
This is the potential
as a function
00:31:40.420 --> 00:31:42.190
of the radial distance.
00:31:42.190 --> 00:31:43.840
So it diverges.
00:31:43.840 --> 00:31:46.660
It goes to minus
infinity Coulombically.
00:31:46.660 --> 00:31:53.610
For aluminum, it
would go as 13 over r.
00:31:53.610 --> 00:32:00.600
What are the solutions, say,
for the 1s state in this--
00:32:00.600 --> 00:32:05.730
sorry, what would be the
solution for a valence electron
00:32:05.730 --> 00:32:07.110
in this potential?
00:32:07.110 --> 00:32:10.770
Well, there are going to be
all the sort of core electrons,
00:32:10.770 --> 00:32:15.280
but then a valence electron
would look something like this.
00:32:15.280 --> 00:32:19.740
It has-- you see this enormous
number of oscillations.
00:32:19.740 --> 00:32:22.620
The reason why those
oscillations are there
00:32:22.620 --> 00:32:26.940
is that eigenfunction of
the Kohn and Sham equation
00:32:26.940 --> 00:32:29.880
need to be orthogonal
to each other.
00:32:29.880 --> 00:32:31.320
This is another
one of these sort
00:32:31.320 --> 00:32:34.770
of fundamental quantum
mechanical rules.
00:32:34.770 --> 00:32:36.620
It's basically the
Pauli principles.
00:32:36.620 --> 00:32:40.530
You can't have two electrons
in the same quantum states.
00:32:40.530 --> 00:32:43.380
And in particular,
electrons corresponding
00:32:43.380 --> 00:32:47.110
to different quantum states need
to be orthogonal to each other.
00:32:47.110 --> 00:32:52.060
So the integral of the psi
star of theta 3s electron
00:32:52.060 --> 00:32:56.070
in aluminum times the psi of
the 1s electron in aluminum
00:32:56.070 --> 00:32:57.360
needs to be 0.
00:32:57.360 --> 00:33:02.440
And so what it means is that the
higher you go, the more wiggles
00:33:02.440 --> 00:33:02.940
you have.
00:33:02.940 --> 00:33:05.910
And these wiggles-- that
is, these changes of signs
00:33:05.910 --> 00:33:09.000
is what allows
you orthogonality.
00:33:09.000 --> 00:33:16.575
So if you have a 1s electron
that looks like this,
00:33:16.575 --> 00:33:19.260
a 1s electron will
have actually sort
00:33:19.260 --> 00:33:23.940
of exponential to the minus
r decaying wave function.
00:33:23.940 --> 00:33:28.290
The wave function say-- already
the 2s wave function needs
00:33:28.290 --> 00:33:31.200
to be orthogonal
to this electron,
00:33:31.200 --> 00:33:36.720
and so it needs to change
sign to allow orthogonality
00:33:36.720 --> 00:33:43.630
when you take the product
of these two orbitals.
00:33:43.630 --> 00:33:48.930
So orthogonality creates
a lot of wiggles,
00:33:48.930 --> 00:33:55.680
and these wiggles also
make the charge density
00:33:55.680 --> 00:33:58.950
coming from a valence
electron more spread
00:33:58.950 --> 00:34:01.830
towards the outside
of the nucleus.
00:34:01.830 --> 00:34:06.660
There is, if you want,
another quantum mechanical
00:34:06.660 --> 00:34:10.860
derived effect in which
valence electrons are
00:34:10.860 --> 00:34:14.820
moved outwards because of
this orthogonality constraint.
00:34:14.820 --> 00:34:20.100
And so you see this would be,
in light blue, the exact wave
00:34:20.100 --> 00:34:23.920
function for a valence electron.
00:34:23.920 --> 00:34:28.860
And what we want to find
is a pseudopotential that
00:34:28.860 --> 00:34:33.690
substitutes for the original
bare Coulombic potential--
00:34:33.690 --> 00:34:37.440
and it's written here in sort
of-- with a thick red line--
00:34:37.440 --> 00:34:42.690
that is constructed so that its
ground state wave function--
00:34:42.690 --> 00:34:45.449
that is here in a thick blue--
00:34:45.449 --> 00:34:50.340
has the same energy eigenvalue
of the valence electron.
00:34:50.340 --> 00:34:54.870
And in many ways, it's identical
to the original wave function.
00:34:54.870 --> 00:34:58.290
And the way we make it
identical is actually
00:34:58.290 --> 00:35:01.620
we only require
it to be identical
00:35:01.620 --> 00:35:05.760
to our original solution
outside the core,
00:35:05.760 --> 00:35:09.090
because outside of
the core of the atom
00:35:09.090 --> 00:35:11.820
is where chemistry takes place.
00:35:11.820 --> 00:35:18.750
Chemistry will be here when
atoms bind with other atoms.
00:35:18.750 --> 00:35:24.330
Inside here, this is the region
where core electron sits,
00:35:24.330 --> 00:35:29.790
and that's a region that will
never overlap with other atoms,
00:35:29.790 --> 00:35:32.520
again, because there are
sort of already these very
00:35:32.520 --> 00:35:34.710
strongly bound electrons.
00:35:34.710 --> 00:35:38.940
And to bring two atoms to
have their core regions to
00:35:38.940 --> 00:35:43.260
overlap requires
enormous pressure,
00:35:43.260 --> 00:35:44.640
hundreds of gigapascals.
00:35:44.640 --> 00:35:46.810
So we never really
have to worry.
00:35:46.810 --> 00:35:48.990
So what we are
trying to do here is
00:35:48.990 --> 00:35:53.730
we want to construct a
pseudopotential in thick red
00:35:53.730 --> 00:35:58.840
such that its ground state
wave functions in thick blue
00:35:58.840 --> 00:36:04.710
have the same eigenvalues of
my original eigenfunctions
00:36:04.710 --> 00:36:06.900
in thin blue lines.
00:36:06.900 --> 00:36:10.290
And in many ways, they
represent the same physics
00:36:10.290 --> 00:36:12.450
and the same
chemistry, because they
00:36:12.450 --> 00:36:16.950
are sort of identical
outside the core.
00:36:16.950 --> 00:36:21.840
So the thick red line is
how we describe an atom.
00:36:21.840 --> 00:36:23.970
The bare Coulombic potential--
00:36:23.970 --> 00:36:28.110
but we also built all
the screening action
00:36:28.110 --> 00:36:30.540
from the core
electrons, and you see
00:36:30.540 --> 00:36:32.670
the central feature
of the pseudopotential
00:36:32.670 --> 00:36:37.320
is that, if you want, it becomes
repulsive around the center
00:36:37.320 --> 00:36:39.180
of the atom because
we really need
00:36:39.180 --> 00:36:45.180
to keep the valence electron
to be mostly delocalized
00:36:45.180 --> 00:36:48.060
in the valence region.
00:36:48.060 --> 00:36:50.130
How to build this
pseudopotential
00:36:50.130 --> 00:36:53.400
is actually a complex
and somehow dark art.
00:36:53.400 --> 00:36:58.090
By now, it's been perfected,
and that-- there are basically
00:36:58.090 --> 00:37:00.700
tables of pseudopotential.
00:37:00.700 --> 00:37:05.340
So it's a problem you almost
never have to worry anymore.
00:37:05.340 --> 00:37:07.230
An electronic
structure code will
00:37:07.230 --> 00:37:09.510
have a set of pseudopotential.
00:37:09.510 --> 00:37:11.970
And in the course
of the years, there
00:37:11.970 --> 00:37:15.180
have been two flavors
that have been developed.
00:37:15.180 --> 00:37:19.050
The first one that was
actually, I would say,
00:37:19.050 --> 00:37:23.390
invented at Bell Labs at the
end of the '70s by Hamann,
00:37:23.390 --> 00:37:25.710
Schluter, and Chiang
is what are called
00:37:25.710 --> 00:37:28.620
norm conserving
pseudopotential, of which I'll
00:37:28.620 --> 00:37:31.650
describe this plot in a moment.
00:37:31.650 --> 00:37:35.780
There was a sort of improvement
of this norm conserving
00:37:35.780 --> 00:37:39.300
pseudopotential that are called
ultrasoft pseudopotential,
00:37:39.300 --> 00:37:42.870
developed by David Vanderbilt
at Rutgers University.
00:37:42.870 --> 00:37:48.630
They tend to be much
less expensive to be--
00:37:48.630 --> 00:37:51.930
to use in practical
calculation, basically
00:37:51.930 --> 00:37:54.540
because they are
smoother and smoother,
00:37:54.540 --> 00:37:59.970
and so they can be described
with sort of a smaller
00:37:59.970 --> 00:38:02.340
basis set, and they
tend to be more
00:38:02.340 --> 00:38:08.040
accurate over a broader range
of coordination and energies.
00:38:08.040 --> 00:38:10.770
But so the general idea
of the pseudopotential
00:38:10.770 --> 00:38:13.050
from a different way--
00:38:13.050 --> 00:38:17.190
in the previous slide, I
showed you how it looks like.
00:38:17.190 --> 00:38:20.000
But the genetic idea
of a pseudopotential
00:38:20.000 --> 00:38:22.730
is that we want to
create something
00:38:22.730 --> 00:38:26.480
that contains both the nucleus
and these core electrons.
00:38:26.480 --> 00:38:30.140
And that basically will
act on the valence electron
00:38:30.140 --> 00:38:34.910
in the same way as the true
problem of core electron
00:38:34.910 --> 00:38:36.830
explicitly and nucleus.
00:38:36.830 --> 00:38:40.220
And so what we say is
that we want a pseudo
00:38:40.220 --> 00:38:44.990
potential to scatter
an incoming wave,
00:38:44.990 --> 00:38:48.440
an incoming electron
in the same way
00:38:48.440 --> 00:38:54.610
as our real set of core
electrons in an atom
00:38:54.610 --> 00:38:56.180
would do that.
00:38:56.180 --> 00:38:59.920
And in order to do
that very accurately,
00:38:59.920 --> 00:39:03.040
for this pseudopotential,
what we need to do
00:39:03.040 --> 00:39:06.460
is actually to have
the pseudopotential
00:39:06.460 --> 00:39:10.990
not just be a single local
form, as in the previous slide.
00:39:10.990 --> 00:39:15.160
It's shown you the sort of
red thick line that was just
00:39:15.160 --> 00:39:18.700
a unique form of the
potential as a function of r,
00:39:18.700 --> 00:39:22.510
but we actually want
our pseudopotential
00:39:22.510 --> 00:39:27.760
to be different, depending on
the angular momentum component
00:39:27.760 --> 00:39:29.750
of the electron coming.
00:39:29.750 --> 00:39:31.390
So if you have a
valence electron,
00:39:31.390 --> 00:39:33.530
and you have a
valence wave function,
00:39:33.530 --> 00:39:36.700
you can actually say that that
valence wave function will
00:39:36.700 --> 00:39:43.570
have a 20% angular symmetry of
the S type, 30% of the P type,
00:39:43.570 --> 00:39:47.000
30% of the D type, and
so on and so forth.
00:39:47.000 --> 00:39:51.460
And what you do is
you act differently
00:39:51.460 --> 00:39:54.070
on the S component
of the wave function,
00:39:54.070 --> 00:39:57.610
on the P component of the wave
function, on the D component,
00:39:57.610 --> 00:39:59.240
and so on and so forth.
00:39:59.240 --> 00:40:05.080
And so this would be the
different radial parts
00:40:05.080 --> 00:40:07.090
of the pseudopotential,
depending on which
00:40:07.090 --> 00:40:08.590
components they act on.
00:40:08.590 --> 00:40:12.070
They all have this
overall 1 over r--
00:40:12.070 --> 00:40:14.710
or z over r asymptotic trend.
00:40:14.710 --> 00:40:18.680
But as we get more close to the
nucleus, they start deferring.
00:40:18.680 --> 00:40:21.760
You see, from the
core region inwards,
00:40:21.760 --> 00:40:23.540
they are all different.
00:40:23.540 --> 00:40:27.760
And in particular, they are
all, or at least in part,
00:40:27.760 --> 00:40:33.360
repulsive to again represent
this frozen set of electrons.
00:40:33.360 --> 00:40:36.730
And this is how the
corresponding wave functions
00:40:36.730 --> 00:40:37.910
would look like.
00:40:37.910 --> 00:40:41.140
And I've made-- again,
this is for an indium atom,
00:40:41.140 --> 00:40:47.350
a comparison between, say, the
p all electron wave function
00:40:47.350 --> 00:40:51.340
and the pseudo wave function.
00:40:51.340 --> 00:40:55.600
And you see they are identical
in the valence region,
00:40:55.600 --> 00:40:59.740
but in the core region, the
true all electron valence wave
00:40:59.740 --> 00:41:01.780
function is a lot
of oscillation,
00:41:01.780 --> 00:41:04.990
while the pseudo wave
function is much smoother.
00:41:04.990 --> 00:41:08.290
And this smoothness
means that we
00:41:08.290 --> 00:41:14.200
can describe this pseudo wave
function in sines and cosines
00:41:14.200 --> 00:41:21.200
with a much smaller set of waves
at different wavelengths then
00:41:21.200 --> 00:41:23.750
we would have to
do if we were to--
00:41:23.750 --> 00:41:27.590
if we had the need to describe
all this high frequency
00:41:27.590 --> 00:41:28.640
oscillation.
00:41:28.640 --> 00:41:32.690
So at the end,
pseudopotential are just there
00:41:32.690 --> 00:41:37.760
to sort of give us a
chance to avoid describing
00:41:37.760 --> 00:41:41.540
all these high frequency
oscillation of the true valence
00:41:41.540 --> 00:41:43.760
wave function, and
also to give us
00:41:43.760 --> 00:41:48.320
the possibility of avoiding
the explicit description
00:41:48.320 --> 00:41:50.390
of the core
electrons that really
00:41:50.390 --> 00:41:54.200
don't change their shape
in going from the atom
00:41:54.200 --> 00:41:54.830
to the solid.
00:41:58.070 --> 00:42:02.360
So now with this, we have
an exchange correlation
00:42:02.360 --> 00:42:03.320
functional.
00:42:03.320 --> 00:42:07.190
We have this new approximation
to the external potential
00:42:07.190 --> 00:42:11.420
given by a sort of
array of pseudopotential
00:42:11.420 --> 00:42:13.250
centered on each nucleus.
00:42:13.250 --> 00:42:17.540
And so we are back to this
sort of self-consistent problem
00:42:17.540 --> 00:42:22.610
of having to solve these
Kohn and Sham differential
00:42:22.610 --> 00:42:23.870
equations.
00:42:23.870 --> 00:42:27.590
And again, the way we want
to attack this problem
00:42:27.590 --> 00:42:34.550
on a computer is by making this
problem into a numerical matrix
00:42:34.550 --> 00:42:36.020
algebra problem.
00:42:36.020 --> 00:42:41.180
And the way we go about this
is by choosing a busy set
00:42:41.180 --> 00:42:44.390
and expanding our
orbitals in a basis.
00:42:44.390 --> 00:42:48.980
That is, we say by saying that
any sort of generating function
00:42:48.980 --> 00:42:51.860
can be thought of as
a linear combination
00:42:51.860 --> 00:42:54.200
of a set of simple function.
00:42:54.200 --> 00:42:58.340
Simple function, for which we
can do analytical operation--
00:42:58.340 --> 00:43:04.100
that is, we are able, say, to
calculate the matrix elements,
00:43:04.100 --> 00:43:07.700
say, of the kinetic energy
between this simple function,
00:43:07.700 --> 00:43:10.340
or we are able to
calculate analytically
00:43:10.340 --> 00:43:14.660
the second derivative
of this basis function.
00:43:14.660 --> 00:43:18.890
Well, by making this ansatz,
by saying that a generic wave
00:43:18.890 --> 00:43:21.050
function is a linear
combination of this,
00:43:21.050 --> 00:43:25.790
our problem transforms from a
differential analysis problem
00:43:25.790 --> 00:43:28.960
to a problem of finding
these coefficients.
00:43:28.960 --> 00:43:33.620
And again, sort of the
basis set that we'll
00:43:33.620 --> 00:43:37.460
choose in all sort of our
practical computational lab
00:43:37.460 --> 00:43:40.730
will really be a basis
set of sines and cosines,
00:43:40.730 --> 00:43:42.620
or what we call plane waves.
00:43:42.620 --> 00:43:45.590
That is especially
adapted and especially
00:43:45.590 --> 00:43:48.110
useful for the case of solids.
00:43:48.110 --> 00:43:50.520
And I've given you
an example here.
00:43:50.520 --> 00:43:53.000
Suppose that what
we want to describe
00:43:53.000 --> 00:43:58.070
is a Gaussian charge
density, something
00:43:58.070 --> 00:44:03.350
that is a little bit similar to
the charge density in an atom.
00:44:03.350 --> 00:44:07.440
Well, this localized
charge density
00:44:07.440 --> 00:44:12.850
can be actually expressed
fairly accurately
00:44:12.850 --> 00:44:19.780
as a linear combination of
very few sines or cosines
00:44:19.780 --> 00:44:21.220
with different wavelengths.
00:44:21.220 --> 00:44:23.920
You see, I go from
this wavelength
00:44:23.920 --> 00:44:25.930
to smaller and
smaller wavelength.
00:44:25.930 --> 00:44:29.080
And if I choose my
coefficients such
00:44:29.080 --> 00:44:32.080
that I have constructive
interference here
00:44:32.080 --> 00:44:36.070
in the center and destructive
interference outside,
00:44:36.070 --> 00:44:39.940
I actually get just by
something nine terms here,
00:44:39.940 --> 00:44:43.040
something that basically
describes perfectly
00:44:43.040 --> 00:44:44.570
my charge density.
00:44:44.570 --> 00:44:47.680
So this will be our general
sort of approach to the problem.
00:44:47.680 --> 00:44:50.260
We choose a basis set.
00:44:50.260 --> 00:44:54.130
That is, we choose a set
of elementary functions
00:44:54.130 --> 00:44:58.330
that are particularly suited
to describe our problem,
00:44:58.330 --> 00:45:02.230
and for which we can do all
the analysis that we want.
00:45:02.230 --> 00:45:04.630
That is, we are able to
calculate second derivatives.
00:45:04.630 --> 00:45:08.710
We are able to calculate
matrix elements analytically.
00:45:08.710 --> 00:45:11.410
And then our
computational problem
00:45:11.410 --> 00:45:16.550
becomes really just a problem
of finding the coefficients.
00:45:16.550 --> 00:45:20.530
And this is one of those
approximations for which you
00:45:20.530 --> 00:45:22.370
need to test the accuracy.
00:45:22.370 --> 00:45:26.260
That is, if you use a
very small business set,
00:45:26.260 --> 00:45:30.220
if you use only 10 plain waves
with different wavelengths,
00:45:30.220 --> 00:45:32.920
your calculation will
be very inexpensive,
00:45:32.920 --> 00:45:36.070
because you have only 10
coefficients to work with.
00:45:36.070 --> 00:45:39.130
But it probably won't
be very accurate.
00:45:39.130 --> 00:45:41.620
If you use one
million plane waves,
00:45:41.620 --> 00:45:44.260
your calculation will
be extremely accurate,
00:45:44.260 --> 00:45:46.760
but it will be also
very expensive.
00:45:46.760 --> 00:45:50.320
So this is an approximation,
but an approximation-- and this
00:45:50.320 --> 00:45:53.680
is fundamental-- for which
you can control the accuracy.
00:45:53.680 --> 00:45:56.200
That is, you just need to
make sure that your business
00:45:56.200 --> 00:45:58.180
set is good enough.
00:45:58.180 --> 00:46:02.860
And that is something that
you can always check and test.
00:46:02.860 --> 00:46:06.640
You can't test, apart from
comparison with experiment,
00:46:06.640 --> 00:46:09.690
if your exchange correlation
functional is good enough.
00:46:09.690 --> 00:46:11.890
That's why we put
so much importance,
00:46:11.890 --> 00:46:15.760
and you should never forget that
really all the accuracy or all
00:46:15.760 --> 00:46:19.270
the errors that ultimately
come in your calculation
00:46:19.270 --> 00:46:21.550
come from the exchange
correlational function.
00:46:21.550 --> 00:46:25.480
All the other computational
numerical approximation
00:46:25.480 --> 00:46:29.680
using pseudopotentials,
using a finite basis set
00:46:29.680 --> 00:46:32.200
are all approximations
that can be tested.
00:46:32.200 --> 00:46:35.920
And we always assume that a
well done electronic structure
00:46:35.920 --> 00:46:38.650
calculation will have
those under control.
00:46:38.650 --> 00:46:41.050
You are using enough basis sets.
00:46:41.050 --> 00:46:44.020
You are using accurate
pseudopotential, and so on.
00:46:46.660 --> 00:46:49.120
As I said, in solids,
for a reason that you'll
00:46:49.120 --> 00:46:52.870
see in a moment, really the
universal basis of choice
00:46:52.870 --> 00:46:55.060
is that of plane
waves-- basically
00:46:55.060 --> 00:46:58.450
the e to the I
complex exponential,
00:46:58.450 --> 00:46:59.830
and you'll see why.
00:46:59.830 --> 00:47:02.290
But in a sort of--
00:47:02.290 --> 00:47:07.600
atoms and molecules-- their
almost universal basis
00:47:07.600 --> 00:47:11.980
choice is that a
combination of Gaussians.
00:47:11.980 --> 00:47:14.800
And the reason why
Gaussians have been chosen,
00:47:14.800 --> 00:47:17.590
and actually why the most
famous quantum chemistry
00:47:17.590 --> 00:47:20.950
code is called Gaussian
is that it's very easy
00:47:20.950 --> 00:47:24.460
to do the analytical
integrals that
00:47:24.460 --> 00:47:28.450
are present in the exchange
term for Hartree-Fock.
00:47:28.450 --> 00:47:31.320
So if you want an exchange
term for Hartree-Fock,
00:47:31.320 --> 00:47:34.270
the term means the
choice of Gaussian
00:47:34.270 --> 00:47:39.880
as a particularly
convenient basis set.
00:47:39.880 --> 00:47:43.120
And again, the product of
two Gaussians is a Gaussian,
00:47:43.120 --> 00:47:48.430
and that's why this is
ultimately a good choice.
00:47:48.430 --> 00:47:51.340
This is what has developed.
00:47:51.340 --> 00:47:56.230
At the end, you can still use
Gaussians to study solids,
00:47:56.230 --> 00:47:58.960
or you can use plane
waves to study molecules.
00:47:58.960 --> 00:48:03.310
And there is even a sort
of richer phenomenology.
00:48:03.310 --> 00:48:07.300
But generally speaking,
these two approaches,
00:48:07.300 --> 00:48:10.780
that of localized
functions in chemistry
00:48:10.780 --> 00:48:16.630
and of delocalized function
in extended systems,
00:48:16.630 --> 00:48:23.110
are the basic categories.
00:48:27.550 --> 00:48:32.050
We'll start from the
point of view of solids,
00:48:32.050 --> 00:48:33.700
and then you'll
see why we actually
00:48:33.700 --> 00:48:37.210
describe-- we use plain waves
to describe these orbitals.
00:48:37.210 --> 00:48:39.640
And when you deal
with solid, you
00:48:39.640 --> 00:48:44.560
need to be aware of one of
the fundamental symmetries
00:48:44.560 --> 00:48:48.520
that the eigenstates
of our solution
00:48:48.520 --> 00:48:52.180
in our periodic potential,
as what happens in solids,
00:48:52.180 --> 00:48:53.470
satisfy.
00:48:53.470 --> 00:48:57.160
That is, in particular,
the Hamiltonian of a solid
00:48:57.160 --> 00:49:00.190
is periodically invariant.
00:49:00.190 --> 00:49:02.290
That's the characteristic
of a solid.
00:49:02.290 --> 00:49:07.570
It's a regular periodic array of
externals pseudopotential that
00:49:07.570 --> 00:49:08.770
[INAUDIBLE].
00:49:08.770 --> 00:49:11.440
So in the language
of quantum mechanics,
00:49:11.440 --> 00:49:15.340
we say that the
Hamiltonian commutes
00:49:15.340 --> 00:49:18.040
with the translational operator.
00:49:18.040 --> 00:49:21.280
Translation operator
by a vector--
00:49:21.280 --> 00:49:23.750
that is, a direct
lattice vector.
00:49:23.750 --> 00:49:30.100
So if you take out cubic system
and you displace, you translate
00:49:30.100 --> 00:49:34.750
your set of potentials by
1 cubic lattice vector,
00:49:34.750 --> 00:49:38.060
you have the same operator.
00:49:38.060 --> 00:49:39.800
That is, trivially
to say that, if you
00:49:39.800 --> 00:49:45.770
have a regular array of blue
dots that is infinite in space,
00:49:45.770 --> 00:49:50.660
and you displace them, and
the displaced one r crosses,
00:49:50.660 --> 00:49:55.190
and they have been displaced
by this lattice vector r here,
00:49:55.190 --> 00:49:57.470
and both the red dots
and the blue dots
00:49:57.470 --> 00:49:59.960
are infinite,
well, by displacing
00:49:59.960 --> 00:50:01.910
the blue dot into
the red dots, you
00:50:01.910 --> 00:50:05.120
have the same identical problem.
00:50:05.120 --> 00:50:09.170
When two operator commuting
quantum mechanics, what we know
00:50:09.170 --> 00:50:14.420
is that we can find eigenstates
that are common eigenstates.
00:50:14.420 --> 00:50:18.080
That is, they are eigenstates
both of the first operator,
00:50:18.080 --> 00:50:19.760
and then on the second operator.
00:50:19.760 --> 00:50:22.610
I won't dwell on
this in case you
00:50:22.610 --> 00:50:26.450
are not familiar with parts
of this quantum mechanical
00:50:26.450 --> 00:50:27.210
problem.
00:50:27.210 --> 00:50:31.310
But what I want to highlight
is that the net result,
00:50:31.310 --> 00:50:34.850
the fact that we can
choose eigenvectors that
00:50:34.850 --> 00:50:37.025
are common eigenvectors
of the Hamiltonians
00:50:37.025 --> 00:50:40.280
and of the translation
operator, tells us
00:50:40.280 --> 00:50:44.300
that a genetic eigenvector
for aperiodic potential
00:50:44.300 --> 00:50:49.610
needs to have a very well
defined symmetry and a very
00:50:49.610 --> 00:50:51.350
well defined symmetric form.
00:50:51.350 --> 00:50:55.040
And that's sort of summarized
in the Bloch theorem.
00:50:55.040 --> 00:50:59.330
And that tells us
that an eigenvector
00:50:59.330 --> 00:51:04.540
will be given by a term--
00:51:04.540 --> 00:51:10.420
a function that has the same
periodicity of the crystal
00:51:10.420 --> 00:51:16.510
times a plane wave--
00:51:16.510 --> 00:51:19.510
again, a complex
exponential that
00:51:19.510 --> 00:51:22.960
modulates these
periodic parts and that
00:51:22.960 --> 00:51:25.370
can have any wavelength.
00:51:25.370 --> 00:51:29.590
So in other words, what the
Bloch theorem here is telling
00:51:29.590 --> 00:51:33.640
us is that the solutions,
the eigenstates
00:51:33.640 --> 00:51:37.960
of aperiodic Hamiltonian can
have any possible periodicity.
00:51:37.960 --> 00:51:41.920
You don't have to think at
the electronic eigenstates
00:51:41.920 --> 00:51:46.480
of aperiodic Hamiltonian as
having the same periodicity
00:51:46.480 --> 00:51:48.010
of your lattice.
00:51:48.010 --> 00:51:51.790
But the shape that they
have is actually this.
00:51:51.790 --> 00:51:54.400
That is the product
of two function--
00:51:54.400 --> 00:52:00.250
one that has any arbitrary
wavelength, and another term--
00:52:00.250 --> 00:52:04.310
the so-called periodic part
of our orbitals-- that does
00:52:04.310 --> 00:52:09.260
have the same periodicity
of the lattice.
00:52:09.260 --> 00:52:12.100
This can be proven
in a variety of ways,
00:52:12.100 --> 00:52:16.390
and somehow I summarized
here why ultimately this
00:52:16.390 --> 00:52:18.940
is the shape that
the eigenfunction
00:52:18.940 --> 00:52:21.100
of the Hamiltonian
needs to have,
00:52:21.100 --> 00:52:24.820
basically because if
we have a translation
00:52:24.820 --> 00:52:26.650
and the Hamiltonian
is invariant,
00:52:26.650 --> 00:52:31.330
the charge density of your
solid must also be invariant.
00:52:31.330 --> 00:52:34.150
And this solution
satisfies that,
00:52:34.150 --> 00:52:38.440
because if you look at the
wave function-- not at r,
00:52:38.440 --> 00:52:41.410
but r plus a direct
lattice vector--
00:52:41.410 --> 00:52:44.050
this term will be
unchanged by definition
00:52:44.050 --> 00:52:46.330
because this term
has the periodicity.
00:52:46.330 --> 00:52:50.800
This term will become
e to the ik times
00:52:50.800 --> 00:52:54.597
scalar product r plus a
direct lattice vector.
00:52:54.597 --> 00:52:56.680
But when you look at the
charge density-- that is,
00:52:56.680 --> 00:52:58.100
the square modulus--
00:52:58.100 --> 00:52:59.580
this term disappears.
00:52:59.580 --> 00:53:03.700
So only this remains
in the charge density.
00:53:03.700 --> 00:53:05.950
And you also want
two translations
00:53:05.950 --> 00:53:08.590
being equivalent to the
sum of the other two.
00:53:08.590 --> 00:53:12.820
And again, if you translate
that by two consecutive lattice
00:53:12.820 --> 00:53:15.070
vectors, this term
doesn't change.
00:53:15.070 --> 00:53:19.210
And the exponential of a
sum of two lattice vectors
00:53:19.210 --> 00:53:22.150
is going to be just the
product of two exponentials
00:53:22.150 --> 00:53:23.920
for each lattice vector.
00:53:23.920 --> 00:53:26.920
But the fundamental
concept here is
00:53:26.920 --> 00:53:29.320
this-- that the
wave function are
00:53:29.320 --> 00:53:33.830
modulated by this wavelength
and we actually classify them.
00:53:33.830 --> 00:53:36.730
We do these two quantum numbers.
00:53:36.730 --> 00:53:40.150
As we classify orbitals
in the periodic table,
00:53:40.150 --> 00:53:41.830
we give them quantum numbers.
00:53:41.830 --> 00:53:46.120
We say this is going to be
a 3s electron with spin-up.
00:53:46.120 --> 00:53:49.960
We are saying what
the quantum number nlm
00:53:49.960 --> 00:53:52.030
and spin are for that electron.
00:53:52.030 --> 00:53:55.840
In a solid, these are
our quantum numbers.
00:53:55.840 --> 00:53:58.240
We have a bond index.
00:53:58.240 --> 00:54:02.290
That is, if you wanted solid
equivalent of your energy
00:54:02.290 --> 00:54:07.570
levels, and it's actually
a discrete integer index
00:54:07.570 --> 00:54:11.860
that somehow classifies
the different eigenvectors,
00:54:11.860 --> 00:54:18.620
but those eigenvectors can
have also any wavelength.
00:54:18.620 --> 00:54:21.730
And so the wavelength
of the block orbital
00:54:21.730 --> 00:54:26.270
is another quantum number
that characterize our system.
00:54:26.270 --> 00:54:28.990
So when you actually
look at a molecule,
00:54:28.990 --> 00:54:31.960
you think that in
terms of energy levels,
00:54:31.960 --> 00:54:34.900
and you draw an energy
diagram in which you will
00:54:34.900 --> 00:54:38.110
have a 1s, 2s, 3s energy level.
00:54:38.110 --> 00:54:41.110
When you think of the
possible energy levels
00:54:41.110 --> 00:54:44.050
from four electrons
in a solid, you'll
00:54:44.050 --> 00:54:48.850
draw a more complex band
diagram in which you
00:54:48.850 --> 00:54:52.600
will sort of classify
levels, not only in terms
00:54:52.600 --> 00:54:57.320
of discrete integers, but also
in terms of the wavelength.
00:54:57.320 --> 00:55:02.620
And so this is what is usually
band energy diagram, of which I
00:55:02.620 --> 00:55:03.880
have an example here.
00:55:15.030 --> 00:55:19.760
So this is-- would be the
solid equivalent of the energy
00:55:19.760 --> 00:55:20.490
levels.
00:55:20.490 --> 00:55:22.880
So for a molecule,
you would have
00:55:22.880 --> 00:55:26.360
a discrete set of energy level.
00:55:26.360 --> 00:55:31.100
For a solid, you don't have
only a set of discrete quantum
00:55:31.100 --> 00:55:36.140
numbers n, but you have also a
continuous set of wavelengths.
00:55:36.140 --> 00:55:40.580
So you have bands
that show you how
00:55:40.580 --> 00:55:43.280
your energy of that
eigenvector changes
00:55:43.280 --> 00:55:47.000
depending on the wavelength
of that eigenvector.
00:55:47.000 --> 00:55:50.810
And what I've plotted
here is actually
00:55:50.810 --> 00:55:56.060
the band energy diagram for
a free electron in an FCC
00:55:56.060 --> 00:55:57.290
lattice.
00:55:57.290 --> 00:56:00.050
What I'm saying is
that, suppose that I'm
00:56:00.050 --> 00:56:05.000
considering an electron that
doesn't feel any potential.
00:56:05.000 --> 00:56:08.130
What would be its band energy?
00:56:08.130 --> 00:56:10.070
Well, it's band
energy is actually
00:56:10.070 --> 00:56:12.000
going to be a parabola.
00:56:12.000 --> 00:56:17.800
So it's something that
we would represent
00:56:17.800 --> 00:56:21.860
as this sort of
energy, as being really
00:56:21.860 --> 00:56:24.740
proportional to its wavelength.
00:56:24.740 --> 00:56:30.470
That is, again, because a
free electron is represented
00:56:30.470 --> 00:56:35.600
by a plain wave e to the ikr.
00:56:35.600 --> 00:56:39.320
And so the second
derivative of a plain wave--
00:56:39.320 --> 00:56:41.510
that is, the quantum
kinetic energy--
00:56:41.510 --> 00:56:44.150
is just the k squared term.
00:56:44.150 --> 00:56:48.030
And if an electron is free,
there is no potential.
00:56:48.030 --> 00:56:53.690
So we can think of the energy
as a function of wavelength
00:56:53.690 --> 00:56:58.350
for a free electron to
be perfectly parabolic.
00:56:58.350 --> 00:57:01.080
But now what we can
think for a moment
00:57:01.080 --> 00:57:06.660
is also how to
represent this parabola.
00:57:06.660 --> 00:57:10.740
In principle, this parabola
is a single parabola infinite
00:57:10.740 --> 00:57:12.930
as a function of wavelength.
00:57:12.930 --> 00:57:20.550
But we can also, for a moment,
represent the data in an FCC
00:57:20.550 --> 00:57:24.090
or in any arbitrary
periodic lattice.
00:57:24.090 --> 00:57:28.110
So what we could be saying is
that suppose that now there
00:57:28.110 --> 00:57:32.430
is an imaginary geometric
structure that is,
00:57:32.430 --> 00:57:38.220
say, in real space Bravais
lattice with FCC periodicity,
00:57:38.220 --> 00:57:41.220
and that, in reciprocal
space, will correspond
00:57:41.220 --> 00:57:43.470
to a BCC reciprocal lattice.
00:57:43.470 --> 00:57:48.870
And what I want to do is I
want to take this parabola
00:57:48.870 --> 00:57:51.510
and fold it.
00:57:51.510 --> 00:57:57.600
So instead of having a
single parabolic branch that
00:57:57.600 --> 00:58:04.140
extends to infinity, every
time I hit in Brillouin zone
00:58:04.140 --> 00:58:07.050
in the reciprocal
space a boundary,
00:58:07.050 --> 00:58:10.650
I fold this parabola back.
00:58:10.650 --> 00:58:14.670
So what I've shown here is
really the band energy diagram
00:58:14.670 --> 00:58:18.720
that is a parabola,
supposing for a moment
00:58:18.720 --> 00:58:24.390
that my electron leaves in the
geometry of an FCC lattice.
00:58:24.390 --> 00:58:26.460
And this is just to
give you the feeling
00:58:26.460 --> 00:58:30.390
that sometimes something
that looks very complex
00:58:30.390 --> 00:58:35.550
is just a representation
of something very simple
00:58:35.550 --> 00:58:37.560
in a specific geometry.
00:58:37.560 --> 00:58:40.350
And now you see in a
moment what happens
00:58:40.350 --> 00:58:47.760
if we study the electron in a
system that truly has the FCC
00:58:47.760 --> 00:58:49.630
Bravais lattice in real space.
00:58:49.630 --> 00:58:54.250
And so it is a BCC Bravais
lattice in reciprocal space.
00:58:54.250 --> 00:58:55.830
And that is silicon.
00:58:55.830 --> 00:59:00.210
And you see that the
band energy dispersions
00:59:00.210 --> 00:59:05.160
for electrons in silicon
looked, actually very, very
00:59:05.160 --> 00:59:07.200
similar to free electrons.
00:59:07.200 --> 00:59:09.210
You see now for a
moment something
00:59:09.210 --> 00:59:11.160
that you thought it
was always extremely
00:59:11.160 --> 00:59:14.340
complex, like the band
diagram for silicon,
00:59:14.340 --> 00:59:18.780
turns out to be just a
parabola, just a free electron
00:59:18.780 --> 00:59:20.760
slightly modified.
00:59:20.760 --> 00:59:23.880
And you can clearly see
all the same pieces,
00:59:23.880 --> 00:59:25.920
but obviously now
these electrons,
00:59:25.920 --> 00:59:28.080
these valence
electrons in silicon
00:59:28.080 --> 00:59:30.820
feel the periodic potential.
00:59:30.820 --> 00:59:33.120
And so the overall
solution is modified.
00:59:33.120 --> 00:59:36.480
And for those of you
familiar with solids,
00:59:36.480 --> 00:59:39.840
there is a band gap
between valence electrons
00:59:39.840 --> 00:59:41.320
and conduction electrons.
00:59:41.320 --> 00:59:43.650
And here, we would
be at a gamma point.
00:59:43.650 --> 00:59:45.630
This would be the
lowest valence band,
00:59:45.630 --> 00:59:50.490
and here we would have the
heavy and the light whole bands
00:59:50.490 --> 00:59:52.380
for the top valence--
00:59:52.380 --> 00:59:54.180
for the top valence bands.
00:59:54.180 --> 01:00:00.150
But what this is telling us
is that electrons in a solid
01:00:00.150 --> 01:00:04.050
really look like free
electrons with a little bit
01:00:04.050 --> 01:00:09.840
of a perturbation that is enough
to transform this parabola
01:00:09.840 --> 01:00:12.450
into something different.
01:00:12.450 --> 01:00:15.670
And again, because they
are almost free electrons,
01:00:15.670 --> 01:00:19.860
and because they have
the periodicity--
01:00:19.860 --> 01:00:21.930
the periodic part of
the Bloch orbitals
01:00:21.930 --> 01:00:24.810
need to have the periodicity
of the crystal lattice
01:00:24.810 --> 01:00:26.790
that we are studying--
it will actually
01:00:26.790 --> 01:00:32.640
be extremely appropriate
to use as a basis set
01:00:32.640 --> 01:00:37.860
to describe at each k
point in the Brillouin.
01:00:37.860 --> 01:00:39.990
The periodic part of
the Bloch orbitals
01:00:39.990 --> 01:00:43.800
will be extremely convenient
to use plane waves.
01:00:43.800 --> 01:00:45.600
And this is the
reason, ultimately,
01:00:45.600 --> 01:00:49.830
why I written here,
sort of again reminding
01:00:49.830 --> 01:00:52.890
some basics of crystallography.
01:00:52.890 --> 01:00:57.370
I've taken, say, the
zinc blende structure.
01:00:57.370 --> 01:01:04.230
So again, an FCC lattice here,
where these three here in green
01:01:04.230 --> 01:01:09.040
would be the primitive lattice
vectors of the FCC lattice--
01:01:09.040 --> 01:01:14.910
so what I am calling
here a1, a2 and a3.
01:01:14.910 --> 01:01:16.950
And these three lattices--
01:01:16.950 --> 01:01:23.400
three vector lattices in 1/2,
1/2, 0, 0, 1/2, 1/2, and 1/2,
01:01:23.400 --> 01:01:28.620
0, 1/2 really represents
my Bravais lattice
01:01:28.620 --> 01:01:34.260
by repeating blue atoms at any
linear combination of three--
01:01:34.260 --> 01:01:35.700
of these three vectors.
01:01:35.700 --> 01:01:39.720
I span all the infinite
blue sublattice
01:01:39.720 --> 01:01:41.850
of the cations in a zinc blende.
01:01:41.850 --> 01:01:46.840
And the zinc blende is just a
parallel compound of silicon.
01:01:46.840 --> 01:01:50.080
It's just that the second atom
in the unit cell is different.
01:01:50.080 --> 01:01:52.860
And again, by
applying translations
01:01:52.860 --> 01:01:55.170
that are linear combination
of these green vectors,
01:01:55.170 --> 01:01:57.560
I span all the blue lattice.
01:01:57.560 --> 01:01:59.570
And silicon would be
identical to this.
01:01:59.570 --> 01:02:02.580
Just the red and the
blue would be identical.
01:02:02.580 --> 01:02:07.700
So having defined
direct vector lattices,
01:02:07.700 --> 01:02:15.170
the a1, a2, and a3, what I can
define is a set of dual vectors
01:02:15.170 --> 01:02:21.380
that we call reciprocal lattice
vectors g1, g2, and g3 that
01:02:21.380 --> 01:02:26.780
are such that the scalar
product between g1 and g2
01:02:26.780 --> 01:02:29.000
is either 1 or 0.
01:02:29.000 --> 01:02:32.810
And this is the [INAUDIBLE]
delta times 2 pi.
01:02:32.810 --> 01:02:34.100
This is a definition.
01:02:34.100 --> 01:02:37.070
This is what we know
is the definition
01:02:37.070 --> 01:02:42.050
of the reciprocal
lattice vector,
01:02:42.050 --> 01:02:45.290
but what is very
important for us
01:02:45.290 --> 01:02:53.120
is that these primitive
reciprocal lattice vectors are
01:02:53.120 --> 01:02:57.980
the fundamental descriptor of
all the plane waves that I want
01:02:57.980 --> 01:03:03.410
to use in describing a solid
and the reason for that
01:03:03.410 --> 01:03:12.560
is that a plane wave written
like this exponential of iGr,
01:03:12.560 --> 01:03:17.180
where the G here, where
this vector is just
01:03:17.180 --> 01:03:19.880
a linear combination
with integer
01:03:19.880 --> 01:03:23.810
number of my primitive
lattice vectors--
01:03:23.810 --> 01:03:29.360
so it's going to be l
with l integer G1 times G2
01:03:29.360 --> 01:03:33.600
to times n G3.
01:03:33.600 --> 01:03:39.180
So well defined by the
triplet of numbers l, m, n.
01:03:39.180 --> 01:03:42.720
Well, these plane
waves, e to the iGr,
01:03:42.720 --> 01:03:46.440
for any linear combination
of g1, g2, and g3
01:03:46.440 --> 01:03:50.820
is a function,
defined in real space,
01:03:50.820 --> 01:03:56.130
that is a periodicity
compatible with the periodicity
01:03:56.130 --> 01:03:58.170
of my direct lattice.
01:03:58.170 --> 01:04:02.760
So suppose that we were in
one dimension for a moment,
01:04:02.760 --> 01:04:08.340
and I would have a system
that has this periodicity.
01:04:08.340 --> 01:04:12.030
And remember, the periodic
part of the Bloch orbitals
01:04:12.030 --> 01:04:15.460
will have the same
periodicity in real space.
01:04:15.460 --> 01:04:19.530
So maybe the
periodic part of that
01:04:19.530 --> 01:04:23.460
will look something like this,
with the same periodicity.
01:04:26.680 --> 01:04:32.890
Well, then what I want
to do is really expand
01:04:32.890 --> 01:04:37.060
this periodic function
as a linear combination
01:04:37.060 --> 01:04:40.270
with appropriate
coefficient of plane waves
01:04:40.270 --> 01:04:42.490
with different
wavelengths, but I
01:04:42.490 --> 01:04:45.730
need to choose these
plane waves to all
01:04:45.730 --> 01:04:50.860
have the same periodicity
of the green lattice.
01:04:50.860 --> 01:04:57.010
So my plane waves will
only look like this,
01:04:57.010 --> 01:05:00.430
and this will be the sort
of fundamental harmonic
01:05:00.430 --> 01:05:01.420
if you want.
01:05:01.420 --> 01:05:08.250
And then another plane wave
could have this periodicity,
01:05:08.250 --> 01:05:12.840
and so on and so forth, going
to higher and higher wavelength.
01:05:12.840 --> 01:05:17.400
But the compact mathematical
representation of this
01:05:17.400 --> 01:05:18.540
is in here.
01:05:18.540 --> 01:05:21.630
That is, given a
direct lattice defined
01:05:21.630 --> 01:05:24.780
by the three vectors
a1, a2, and a3,
01:05:24.780 --> 01:05:27.540
I can define a
reciprocal lattice
01:05:27.540 --> 01:05:31.650
that is represented by the
three vector G1, G1, and G3.
01:05:31.650 --> 01:05:36.300
And any linear combination with
integer numbers of this G1, G2,
01:05:36.300 --> 01:05:40.680
and G3 will give me a vector
G such that the complex
01:05:40.680 --> 01:05:44.430
exponential e to
the iGr is actually
01:05:44.430 --> 01:05:48.810
a periodicity that is compatible
with my direct lattice.
01:05:48.810 --> 01:05:53.580
That is, the moment I
create a direct lattice,
01:05:53.580 --> 01:05:56.340
there are an infinite
number of wavelengths
01:05:56.340 --> 01:05:58.980
that are not compatible
with the periodicity
01:05:58.980 --> 01:06:00.960
that I've thrown into my system.
01:06:00.960 --> 01:06:02.160
Those disappear.
01:06:02.160 --> 01:06:04.440
I don't want to have
anything to deal with.
01:06:04.440 --> 01:06:09.240
What I'm left is with
a numerable infinity
01:06:09.240 --> 01:06:13.590
of plane waves that are all
compatible with the direct
01:06:13.590 --> 01:06:15.700
lattice that I'm choosing from.
01:06:15.700 --> 01:06:18.480
And you see now there
is a natural way
01:06:18.480 --> 01:06:21.450
of both choosing plane waves--
01:06:21.450 --> 01:06:22.650
they will be here--
01:06:22.650 --> 01:06:26.580
and also a natural way to
choose the most important one,
01:06:26.580 --> 01:06:30.660
because when we choose our
basis set for our calculation,
01:06:30.660 --> 01:06:35.130
we'll start from the plane
waves of longer wavelength.
01:06:35.130 --> 01:06:38.430
That is, that the
blue wavelength
01:06:38.430 --> 01:06:39.810
that I've plotted here--
01:06:39.810 --> 01:06:42.480
and will decrease
the wavelength.
01:06:42.480 --> 01:06:45.720
That will include more
and more plane wave
01:06:45.720 --> 01:06:49.260
that sort of have finer
and finer wavelength, finer
01:06:49.260 --> 01:06:50.760
and finer resolution.
01:06:50.760 --> 01:06:53.790
And that can be
just naturally done
01:06:53.790 --> 01:06:58.350
by including plane waves
with G vectors that
01:06:58.350 --> 01:07:02.280
have a larger square modulus.
01:07:02.280 --> 01:07:07.470
That is the larger this G vector
here, the finer the resolution,
01:07:07.470 --> 01:07:11.020
the higher the wavelength
of this plane wave.
01:07:11.020 --> 01:07:13.560
So there is truly a
natural way to choose
01:07:13.560 --> 01:07:16.740
basis set now,
because, again, what
01:07:16.740 --> 01:07:19.660
we have is a reciprocal space.
01:07:19.660 --> 01:07:22.890
And so we'll have a
Brillouin zone in red.
01:07:22.890 --> 01:07:25.650
We'll have, if I'm
in two dimension,
01:07:25.650 --> 01:07:28.690
the two G1 and G2 vectors.
01:07:28.690 --> 01:07:32.640
And so the linear combination
G's that I've described
01:07:32.640 --> 01:07:40.710
will be given by the infinite,
but discrete set of G vectors
01:07:40.710 --> 01:07:43.060
represented here.
01:07:43.060 --> 01:07:48.060
And so only the G vectors
denoted here with a cross
01:07:48.060 --> 01:07:50.580
give rise to a
plane wave that has
01:07:50.580 --> 01:07:54.870
the compatible periodicity
with my periodic boundary
01:07:54.870 --> 01:07:56.200
conditions.
01:07:56.200 --> 01:08:00.900
And so in order to choose
a basis set, what I do,
01:08:00.900 --> 01:08:06.270
I just draw a circle, or in
three dimensions, a sphere,
01:08:06.270 --> 01:08:11.130
and I decide to include
all the vector that
01:08:11.130 --> 01:08:13.140
sit inside that sphere.
01:08:13.140 --> 01:08:16.410
And by making that
sphere larger and larger,
01:08:16.410 --> 01:08:18.930
I'm going to include
G vectors that
01:08:18.930 --> 01:08:22.200
have larger and larger modules
that is finer and finer
01:08:22.200 --> 01:08:23.130
wavelength.
01:08:23.130 --> 01:08:27.810
And that is what is called
your cutoff in your basis set.
01:08:27.810 --> 01:08:32.040
So one of the fundamental
parameters of your calculation
01:08:32.040 --> 01:08:36.210
will be choosing your cutoff
for your plane wave basis set--
01:08:36.210 --> 01:08:40.859
that is, choosing the radius of
that sphere that includes plane
01:08:40.859 --> 01:08:46.050
waves with wavelength compatible
to the periodic boundary
01:08:46.050 --> 01:08:49.920
conditions up to a
certain resolution.
01:08:49.920 --> 01:08:53.310
You make this cutoff
sphere larger and larger.
01:08:53.310 --> 01:08:57.510
You systematically include
finer and finer resolutions,
01:08:57.510 --> 01:09:00.750
so you become better and better
at describing your problem.
01:09:00.750 --> 01:09:04.260
And you'll always need to
check in your first calculation
01:09:04.260 --> 01:09:06.120
that your basis
set is good enough.
01:09:06.120 --> 01:09:10.240
That is, by increasing
your cutoff radius,
01:09:10.240 --> 01:09:15.210
you will not see any
significant physical change
01:09:15.210 --> 01:09:18.210
in the quantities that
you are calculating.
01:09:18.210 --> 01:09:21.510
You don't see a physical
change in the energy
01:09:21.510 --> 01:09:22.529
as a function of volume.
01:09:22.529 --> 01:09:26.430
You don't see a change in the
forces acting on atoms, and so
01:09:26.430 --> 01:09:27.300
on and so forth.
01:09:30.590 --> 01:09:32.300
This will take forever.
01:09:32.300 --> 01:09:36.500
Again, this is not the
only choice of basis set,
01:09:36.500 --> 01:09:40.160
but as you can see, it's
really the appropriate one
01:09:40.160 --> 01:09:42.350
for periodic systems.
01:09:42.350 --> 01:09:48.770
And it has sort of a set of nice
advantages, of which I would
01:09:48.770 --> 01:09:53.660
say the most important
is the fact that it's
01:09:53.660 --> 01:09:58.970
systematic in the sense that
you can continuously improve
01:09:58.970 --> 01:10:05.810
the resolution in your problem
by including more plane waves.
01:10:05.810 --> 01:10:15.310
There is a negative
side to this that
01:10:15.310 --> 01:10:20.110
says that it doesn't have
any information on how
01:10:20.110 --> 01:10:23.860
valence electrons should
look like close to a nucleus
01:10:23.860 --> 01:10:25.390
or close to a core.
01:10:25.390 --> 01:10:29.230
It requires a very large
number of basis elements.
01:10:29.230 --> 01:10:31.210
So plane wave
calculation tend to have
01:10:31.210 --> 01:10:33.580
a lot of elements in that.
01:10:33.580 --> 01:10:39.610
There are a number of
other important practical
01:10:39.610 --> 01:10:40.630
consequences.
01:10:40.630 --> 01:10:43.720
In particular, as
much as Gaussians,
01:10:43.720 --> 01:10:46.960
they allow for very
easy evaluation
01:10:46.960 --> 01:10:49.400
of some of the analytical
terms that we need.
01:10:49.400 --> 01:10:54.220
So it's very easy to take
the gradient of a plane wave
01:10:54.220 --> 01:10:56.470
or the Laplacian
of a plane wave,
01:10:56.470 --> 01:11:00.100
because basically the derivative
of a complex exponential,
01:11:00.100 --> 01:11:04.360
if you have e the iGr, the
derivative with respect
01:11:04.360 --> 01:11:08.020
to r of this is just
the function itself
01:11:08.020 --> 01:11:10.450
times the vector G. This
would be the gradient,
01:11:10.450 --> 01:11:13.360
and the Laplacian-- that is,
the second derivative-- is just
01:11:13.360 --> 01:11:16.220
G squared times
the wave function.
01:11:16.220 --> 01:11:19.000
And so all these terms
are easy to calculate,
01:11:19.000 --> 01:11:24.490
and there is a sort of more
subtle conclusion of this,
01:11:24.490 --> 01:11:29.290
that if you start to have a
calculation in which atoms
01:11:29.290 --> 01:11:34.360
move, like a molecular dynamic
calculation in which you need
01:11:34.360 --> 01:11:36.610
to calculate things
like forces, you
01:11:36.610 --> 01:11:39.520
need to calculate the derivative
of the energy with respect
01:11:39.520 --> 01:11:41.290
to the position of an atom.
01:11:41.290 --> 01:11:46.240
But now the energy in
itself is an expression
01:11:46.240 --> 01:11:49.270
that involves linear
combination of your basis set.
01:11:49.270 --> 01:11:52.660
Well, this basis
set does not depend
01:11:52.660 --> 01:11:54.830
on the position of the atom.
01:11:54.830 --> 01:11:58.780
So there is no term in the
force in the derivative
01:11:58.780 --> 01:12:01.090
of the energy with
respect to the position
01:12:01.090 --> 01:12:02.500
that comes from this.
01:12:02.500 --> 01:12:06.100
If, on the other hand, you are
using, say, a Gaussian that
01:12:06.100 --> 01:12:09.700
was centered on an atom, a
Gaussian centered on an atom
01:12:09.700 --> 01:12:13.010
would have in there the
position of the atom.
01:12:13.010 --> 01:12:15.340
And so in order to take
the derivative, the force,
01:12:15.340 --> 01:12:18.550
you would also need to take a
derivative of your basis set.
01:12:18.550 --> 01:12:21.310
And these are what are
called Pulay terms.
01:12:21.310 --> 01:12:24.430
And that just add another
layer of complexity
01:12:24.430 --> 01:12:25.760
to your calculation.
01:12:25.760 --> 01:12:27.530
And again, this is,
I think, ultimately,
01:12:27.530 --> 01:12:31.330
is one of the reasons why
in the solid-state community
01:12:31.330 --> 01:12:36.700
problems like molecular
dynamics and forced relaxation
01:12:36.700 --> 01:12:40.060
were developed earlier,
basically because it's much
01:12:40.060 --> 01:12:42.400
simpler to calculate forces.
01:12:42.400 --> 01:12:47.400
And I want to give
you an example on how
01:12:47.400 --> 01:12:50.340
certain parts of
your problem become
01:12:50.340 --> 01:12:53.370
very easy in your solution.
01:12:53.370 --> 01:12:56.640
In particular, remember
sort of in the construction
01:12:56.640 --> 01:12:59.730
of the self-consistent operator
from the charge density
01:12:59.730 --> 01:13:03.120
we needed to construct
the Hartree operator
01:13:03.120 --> 01:13:07.200
from the charge density integral
of n over r minus r prime.
01:13:07.200 --> 01:13:09.150
Well, that's
basically a solution
01:13:09.150 --> 01:13:11.910
of what is called
the Poisson equation.
01:13:11.910 --> 01:13:17.400
That is, given a charge
density, the Hartree potential
01:13:17.400 --> 01:13:23.620
is really coming from the
solution of this differential
01:13:23.620 --> 01:13:24.280
equation.
01:13:24.280 --> 01:13:26.710
That is, the Laplacian,
the second derivative
01:13:26.710 --> 01:13:29.890
of the Hartree potential
is equal to minus,
01:13:29.890 --> 01:13:35.320
in atomic units, 4 pi the
n electronic charge density
01:13:35.320 --> 01:13:38.110
taken as positive here
in particular, just
01:13:38.110 --> 01:13:40.210
to get the signs right.
01:13:40.210 --> 01:13:41.950
This is a differential equation.
01:13:41.950 --> 01:13:43.810
We need to solve
it in the course
01:13:43.810 --> 01:13:46.420
of our self-consistent
problem, because remember,
01:13:46.420 --> 01:13:48.670
we have sort of
diagonalized Hamiltonian.
01:13:48.670 --> 01:13:50.020
We have gotten some orbitals.
01:13:50.020 --> 01:13:52.515
From those orbitals, we have
calculated a charge density.
01:13:52.515 --> 01:13:53.890
But now, from the
charge density,
01:13:53.890 --> 01:13:56.500
we need to calculate the
electrostatic potential.
01:13:56.500 --> 01:13:58.330
And that's how it works.
01:13:58.330 --> 01:14:01.000
Well, this differential
equation is actually
01:14:01.000 --> 01:14:04.360
trivial to solve
if, for a moment,
01:14:04.360 --> 01:14:07.090
you think at a
plane wave solution.
01:14:07.090 --> 01:14:11.490
That is, you think
that your potential--
01:14:11.490 --> 01:14:14.220
that is, a function
of r is now being
01:14:14.220 --> 01:14:20.120
written as a linear combination
where the coefficients are
01:14:20.120 --> 01:14:25.520
called v of g of plane waves.
01:14:25.520 --> 01:14:28.070
So I'm taking my potential.
01:14:28.070 --> 01:14:29.990
It's going to have
the same periodicity
01:14:29.990 --> 01:14:32.120
of the reciprocal
lattice, and I'm
01:14:32.120 --> 01:14:35.570
writing it out as a linear
combination of waves.
01:14:35.570 --> 01:14:39.990
And I'm doing the same for
the charge density here.
01:14:39.990 --> 01:14:43.730
So the charge
density in itself is
01:14:43.730 --> 01:14:51.270
going to be given by
a linear combination
01:14:51.270 --> 01:14:55.410
with coefficient
G of plane waves.
01:14:55.410 --> 01:15:00.000
So this is my expansion in
plane waves of this real space
01:15:00.000 --> 01:15:01.110
functions.
01:15:01.110 --> 01:15:03.540
And then the algebra is
trivial, because to take
01:15:03.540 --> 01:15:07.050
the derivative of this red
term here, what I obtain
01:15:07.050 --> 01:15:11.970
is nothing else
than the sum over g.
01:15:11.970 --> 01:15:19.960
Second derivative will give me
r minus g squared v Hartree of G
01:15:19.960 --> 01:15:22.930
e to the iGr.
01:15:22.930 --> 01:15:30.070
And that is-- needs to be
equal to, well, minus 4 pi sum
01:15:30.070 --> 01:15:34.420
over G n of G e to the iGr.
01:15:34.420 --> 01:15:37.300
So I have done nothing
else than inserting
01:15:37.300 --> 01:15:39.580
the explicit expansion
in plane waves
01:15:39.580 --> 01:15:42.070
of my potential of
my charge density
01:15:42.070 --> 01:15:45.020
in the differential equation.
01:15:45.020 --> 01:15:49.330
And now, well, sort
of mathematically I
01:15:49.330 --> 01:15:53.290
can actually sort of multiply
the left and the right hand
01:15:53.290 --> 01:15:58.570
term for something like e to
the minus iG prime times r.
01:15:58.570 --> 01:16:01.810
And then I can
integrate in the r.
01:16:01.810 --> 01:16:04.150
And this is just the
mathematical operation
01:16:04.150 --> 01:16:06.910
that on an
orthonormal basis set,
01:16:06.910 --> 01:16:11.920
as this is, tells me that,
in order for this equality
01:16:11.920 --> 01:16:15.310
to be satisfied, what I
really need to have is
01:16:15.310 --> 01:16:19.690
that each coefficient
corresponding to the same G
01:16:19.690 --> 01:16:22.510
vector on the left hand side
and on the right hand side
01:16:22.510 --> 01:16:24.340
is separately equal.
01:16:24.340 --> 01:16:31.020
So the solution to the
Poisson-Boltzmann equation
01:16:31.020 --> 01:16:34.290
is trivial, because
what I need to have is
01:16:34.290 --> 01:16:41.520
that G square vG is
equal to 4 pi nG.
01:16:41.520 --> 01:16:46.350
That is, each coefficient
needs to be identical G by G.
01:16:46.350 --> 01:16:49.830
So if I have a charge
density in real space,
01:16:49.830 --> 01:16:53.010
I can expand it in plane waves.
01:16:53.010 --> 01:16:55.290
And this coefficient and
nothing else than a Fourier
01:16:55.290 --> 01:16:58.470
transform, so a computer
is very good, given
01:16:58.470 --> 01:17:00.240
a periodic function
in real space,
01:17:00.240 --> 01:17:02.310
to give me these coefficients.
01:17:02.310 --> 01:17:05.520
And once I have this
coefficient n of G,
01:17:05.520 --> 01:17:09.690
I will know instantly
what are the coefficients
01:17:09.690 --> 01:17:13.590
of the potential that is
the solution of this Poisson
01:17:13.590 --> 01:17:16.980
equation, because those
vG coefficients are really
01:17:16.980 --> 01:17:21.750
just my charge density
coefficient multiplied by 4 pi
01:17:21.750 --> 01:17:23.550
and divided by G square.
01:17:23.550 --> 01:17:25.680
So you see with
plane waves, a lot
01:17:25.680 --> 01:17:28.500
of the analytical work
in a transaction problem
01:17:28.500 --> 01:17:31.230
becomes trivial.
01:17:31.230 --> 01:17:37.440
And here, it's contained
one of the trickiest part
01:17:37.440 --> 01:17:39.420
of electronic
structure calculation.
01:17:39.420 --> 01:17:42.930
And the reason why electronic
structure calculation
01:17:42.930 --> 01:17:47.370
of large system becomes more
and more computationally
01:17:47.370 --> 01:17:50.700
ill-defined, and more
and more difficult to do
01:17:50.700 --> 01:17:54.390
as the size of the
system becomes larger,
01:17:54.390 --> 01:17:57.840
because as the size of
the system becomes larger,
01:17:57.840 --> 01:18:02.520
your cell in real space
will become larger.
01:18:02.520 --> 01:18:05.430
And your Brillouin zone
in reciprocal space
01:18:05.430 --> 01:18:07.410
will become smaller and smaller.
01:18:07.410 --> 01:18:10.350
As the three a1, a2,
a3 vectors becomes
01:18:10.350 --> 01:18:13.590
larger, the three dual
reciprocal space vector--
01:18:13.590 --> 01:18:16.360
G1, G1, and G3--
becomes smaller.
01:18:16.360 --> 01:18:21.330
So the smallest of the G vector
will become smaller and smaller
01:18:21.330 --> 01:18:23.490
as the supercell grows.
01:18:23.490 --> 01:18:26.700
And so then you see that
there is an instability,
01:18:26.700 --> 01:18:30.930
because this component
of the Hartree potential
01:18:30.930 --> 01:18:35.610
is given by this n coefficient
divided by G square.
01:18:35.610 --> 01:18:39.660
As your system becomes larger,
G square becomes smaller.
01:18:39.660 --> 01:18:43.260
And so as smaller
numerical instability
01:18:43.260 --> 01:18:47.580
in your charge density gives
you a large instability
01:18:47.580 --> 01:18:48.940
in your potential.
01:18:48.940 --> 01:18:51.300
And so you need to be
more and more careful.
01:18:51.300 --> 01:18:53.640
You need to become
more and more careful
01:18:53.640 --> 01:18:57.930
in your electrostatic solution
as the problem becomes
01:18:57.930 --> 01:18:58.875
larger and larger.
01:19:02.170 --> 01:19:04.380
This will take forever to say.
01:19:04.380 --> 01:19:07.620
I think that I need to
stop here for today,
01:19:07.620 --> 01:19:11.640
and what we'll see in the
next class on Thursday
01:19:11.640 --> 01:19:14.790
will sort of wrap up the
last few technical details
01:19:14.790 --> 01:19:18.780
that you need to go into the
computational lab on Tuesday,
01:19:18.780 --> 01:19:23.040
and we'll start discussing
case studies for our density
01:19:23.040 --> 01:19:24.780
functional problem.
01:19:24.780 --> 01:19:28.440
As a reminder-- and I've
emailed and hope all of you have
01:19:28.440 --> 01:19:30.030
received my email--
01:19:30.030 --> 01:19:32.940
in order to do the
computational lab on Tuesday,
01:19:32.940 --> 01:19:36.150
it is essential that
each and every one of you
01:19:36.150 --> 01:19:39.090
has an independent
personal computer
01:19:39.090 --> 01:19:42.780
account on the computer class,
because this calculation will
01:19:42.780 --> 01:19:44.100
become expensive.
01:19:44.100 --> 01:19:48.330
And we need to run them not
on the main a central node.
01:19:48.330 --> 01:19:52.440
That will collapse if more than
two or three of you run on it.
01:19:52.440 --> 01:19:55.260
But we need to spool
it via queuing system
01:19:55.260 --> 01:20:01.020
on the nodes of the cluster that
lie beneath the master nose.
01:20:01.020 --> 01:20:02.320
This is all for today.
01:20:02.320 --> 01:20:06.890
Enjoy the snow, and see
you on Thursday morning.