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MICHELE: Welcome
to lecture three

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on quantum mechanical methods.

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00:00:30,000 --> 00:00:33,300
As you may have
guessed, I'm not Jeff,

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as much as I look
like him, I know.

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I'm actually a graduate
student of his.

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I'm Michele.

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So feel free to interrupt
me with any questions you

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00:00:41,580 --> 00:00:42,780
have during this.

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00:00:42,780 --> 00:00:45,270
And this is a really exciting
class today because today,

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we're actually going to
start talking about some

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of the quantum methods
that we've been doing

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in the background up till now.

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00:00:52,750 --> 00:00:54,960
So first of all, I
guess you guys know

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that you don't have
class on Thursday,

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but I guess for those
of you doing projects,

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you can go meet with
Professor Buehler then.

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00:01:04,209 --> 00:01:08,330
So on our outline, we are here.

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We're going to be talking a
little bit about modeling,

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the beginnings of
modeling today,

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and a little bit
about molecules.

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00:01:13,760 --> 00:01:18,350
So last time, we talked about
a single electron system.

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We talked about hydrogen, and
actually, that gave us a lot.

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00:01:21,200 --> 00:01:26,870
We were able to explain how
spectral lines came about, just

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from understanding the
basic structure of hydrogen.

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But today, we're going to
work on more-than-one electron

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00:01:34,150 --> 00:01:34,960
systems.

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00:01:34,960 --> 00:01:39,850
And you'll see that this is
where the computation really

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becomes necessary.

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So here's the outline.

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00:01:44,805 --> 00:01:46,180
So you can see
today, we're going

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to talk a little bit about
Hartree and Hartree-Fock

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00:01:48,305 --> 00:01:52,108
methods, and density
functional theory.

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00:01:52,108 --> 00:01:53,650
So I believe you
guys are going to be

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00:01:53,650 --> 00:01:58,690
using density functional theory
to your next problem set.

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00:02:01,360 --> 00:02:03,443
So just some review
from last time--

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00:02:03,443 --> 00:02:05,110
here's the Schrodinger
equation that I'm

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00:02:05,110 --> 00:02:08,949
sure you're all sick
of seeing by now.

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00:02:08,949 --> 00:02:12,280
In general, if our system
is time independent,

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00:02:12,280 --> 00:02:14,950
then our Hamiltonian is
going to be time independent.

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00:02:14,950 --> 00:02:17,020
And we can separate
the variables

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00:02:17,020 --> 00:02:19,720
into the spatial component
and the time component.

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00:02:19,720 --> 00:02:22,180
And then by dividing
out the time component,

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00:02:22,180 --> 00:02:24,610
we are left with this
stationary, time-independent

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00:02:24,610 --> 00:02:26,232
Schrodinger equation.

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00:02:26,232 --> 00:02:28,190
And that's the equation
that we've been solving

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and we will continue to solve.

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So as I said, last time we
talked about the hydrogen atom.

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We just solved its Hamiltonian.

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00:02:42,050 --> 00:02:45,380
The Hamiltonian itself is,
of course, the kinetic energy

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00:02:45,380 --> 00:02:47,490
and the potential energy.

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00:02:47,490 --> 00:02:49,790
The kinetic energy is this.

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00:02:49,790 --> 00:02:52,940
The potential energy is actually
just that Coulomb attraction

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between the electron and the
proton in that hydrogen atom.

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00:02:56,690 --> 00:03:01,490
And this, we saw, was
pretty simple to solve.

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00:03:01,490 --> 00:03:04,340
I guess we didn't actually do
it, but the way this is solved

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00:03:04,340 --> 00:03:06,380
is you switch into
spherical coordinates.

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00:03:06,380 --> 00:03:10,730
You solve for a radial
component of the wave function,

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00:03:10,730 --> 00:03:13,430
and an angular component
of the wave function.

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00:03:13,430 --> 00:03:16,760
So to any of you who have
had a lot of chemistry,

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00:03:16,760 --> 00:03:18,990
these probably
look very familiar.

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So this is the s, these three
are p, these five are d.

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00:03:28,390 --> 00:03:31,780
And these are basically
just the spherical harmonics

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that you get from
solving the theta and phi

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parts of the Hamiltonian.

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And when we combine these,
we get the total spatial wave

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00:03:44,540 --> 00:03:47,250
function for the hydrogen atom.

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So the spatial wave
function is described

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by these quantum numbers.

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And the principle quantum
number, l, the angular momentum

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component, and m l, which is
the angular momentum component,

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00:04:01,370 --> 00:04:03,240
projected along the z axis.

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00:04:03,240 --> 00:04:06,910
So you guys have probably
seen these a million times--

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00:04:06,910 --> 00:04:10,900
that leads to these
spatial wave functions.

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00:04:10,900 --> 00:04:13,780
So this is increasing
principal quantum number

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and this is increasing
angular momentum.

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All right, so we've all
seen this graph now a couple

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00:04:21,579 --> 00:04:22,240
of times.

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00:04:22,240 --> 00:04:27,502
Can somebody tell me what
this graph represents?

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AUDIENCE: [INAUDIBLE]

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00:04:35,185 --> 00:04:35,810
MICHELE: Right.

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00:04:35,810 --> 00:04:41,752
And why is it important that
there are these specific lines

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00:04:41,752 --> 00:04:42,710
that it can go between?

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00:04:42,710 --> 00:04:49,172
I mean, what phenomenon are
we capturing with this idea?

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00:04:49,172 --> 00:04:52,620
AUDIENCE: We can [INAUDIBLE]

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00:04:52,620 --> 00:04:54,140
MICHELE: Right.

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00:04:54,140 --> 00:05:00,440
So this is why people
observed spectral lines

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00:05:00,440 --> 00:05:02,600
instead of seeing a
continuous spectrum.

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00:05:02,600 --> 00:05:06,380
and so atoms, instead of
emitting a continuous spectrum,

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00:05:06,380 --> 00:05:08,390
emit specific lines.

97
00:05:08,390 --> 00:05:12,170
And that puzzle was solved by--

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00:05:12,170 --> 00:05:15,380
we actually already solved it
last time with the solution

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00:05:15,380 --> 00:05:18,620
to the hydrogen atom, because we
saw that the hydrogen atom had

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00:05:18,620 --> 00:05:20,690
these discrete energy levels.

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00:05:20,690 --> 00:05:23,780
And electrons could only
exist in those specific energy

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00:05:23,780 --> 00:05:24,570
levels.

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00:05:24,570 --> 00:05:26,990
And so when an electron went
between those two energy

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00:05:26,990 --> 00:05:31,400
levels, it would emit the
energy of that difference

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00:05:31,400 --> 00:05:34,250
in energy levels as a photon.

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00:05:34,250 --> 00:05:38,090
And that photon would
be a specific wavelength

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00:05:38,090 --> 00:05:39,650
corresponding to
a specific color,

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00:05:39,650 --> 00:05:41,900
and that's exactly
where this emission line

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00:05:41,900 --> 00:05:42,760
spectrum comes from.

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00:05:46,040 --> 00:05:50,240
So let's try to add
one more electron.

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00:05:50,240 --> 00:05:53,450
So this is our Hamiltonian
now for helium.

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00:05:53,450 --> 00:05:58,550
We still have the kinetic part,
the T for electrons 1 and 2.

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00:05:58,550 --> 00:06:02,540
We have the potential part
for electrons 1 and 2.

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00:06:02,540 --> 00:06:07,460
That's the interaction of
the electrons with the helium

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00:06:07,460 --> 00:06:08,880
nucleus.

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00:06:08,880 --> 00:06:11,660
But now we have this term
that's the Coulomb repulsion.

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00:06:11,660 --> 00:06:15,410
It's the two electrons
interacting with each other.

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00:06:15,410 --> 00:06:17,900
And this makes the
problem suddenly

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00:06:17,900 --> 00:06:20,420
unable to be solved
analytically,

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00:06:20,420 --> 00:06:23,030
just this addition
of one more electron.

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00:06:23,030 --> 00:06:26,230
I guess you could think of it as
sort of the three-body problem

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00:06:26,230 --> 00:06:30,530
in classical mechanics.

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00:06:30,530 --> 00:06:33,870
So last time, we also
talked about spin.

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00:06:33,870 --> 00:06:39,310
So we know that electrons are
either spin up or spin down.

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00:06:39,310 --> 00:06:42,265
Does anybody remember
the experiment

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00:06:42,265 --> 00:06:43,390
that was done to show this?

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00:06:46,126 --> 00:06:48,410
AUDIENCE: Stern-Gerlach.

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00:06:48,410 --> 00:06:51,390
MICHELE: Yeah, good job.

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00:06:51,390 --> 00:06:56,832
So the Stern-Gerlach
experiment, basically it

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00:06:56,832 --> 00:06:58,290
was a stream of
electrons that were

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00:06:58,290 --> 00:07:00,930
shot through this magnetic
field and then were

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00:07:00,930 --> 00:07:03,310
observed on a plate behind it.

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00:07:03,310 --> 00:07:06,430
So electrons being shot
through this magnetic field

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00:07:06,430 --> 00:07:10,720
is kind of like the spin
component of the wave function

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00:07:10,720 --> 00:07:12,140
being observed.

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00:07:12,140 --> 00:07:15,010
So now that we're observing
this wave function,

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00:07:15,010 --> 00:07:18,660
it can only be in
its eigenstates.

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00:07:18,660 --> 00:07:23,030
So those eigenstates
are here and here.

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00:07:23,030 --> 00:07:25,940
So you think if you
have an electron that

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00:07:25,940 --> 00:07:28,580
can interact with
a magnetic field,

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00:07:28,580 --> 00:07:32,370
its magnetic moment could be
pointing sort of any direction.

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00:07:32,370 --> 00:07:35,660
So if it's pointing
perpendicular

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00:07:35,660 --> 00:07:37,438
to the magnetic
field, why would you

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00:07:37,438 --> 00:07:39,230
expect there to be any
kind of interaction?

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00:07:39,230 --> 00:07:42,200
You just expect it to end
up in the middle here.

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00:07:42,200 --> 00:07:47,740
And similarly, it could be
anywhere between up or down.

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00:07:47,740 --> 00:07:50,060
But this is quantum mechanics.

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00:07:50,060 --> 00:07:53,030
This magnetic field is
making a measurement.

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00:07:53,030 --> 00:07:56,017
And so we all know that
once a measurement is

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00:07:56,017 --> 00:07:57,850
made in quantum mechanics,
the wave function

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00:07:57,850 --> 00:08:01,820
collapses to one of its
observable eigenstates.

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00:08:01,820 --> 00:08:07,970
So it may seem strange to you
that if you know something

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00:08:07,970 --> 00:08:11,645
about where magnetic moments
usually come from, usually

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00:08:11,645 --> 00:08:15,140
we talk about magnetic fields
being produced by electrons

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00:08:15,140 --> 00:08:17,060
physically spinning in space.

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00:08:17,060 --> 00:08:19,430
But this is just some
property of an electron

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00:08:19,430 --> 00:08:23,660
that creates a magnetic moment,
without any kind of movement.

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00:08:23,660 --> 00:08:25,970
So that's still something
that we don't really

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00:08:25,970 --> 00:08:29,180
understand today, what
exactly that spin means.

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00:08:29,180 --> 00:08:32,900
But we observe it and
it seems necessary,

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00:08:32,900 --> 00:08:37,669
so right now, all we can say is
that we know electrons either

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00:08:37,669 --> 00:08:39,020
spin up or they're spin down.

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00:08:41,780 --> 00:08:44,960
And Jeff showed you
last time this letter

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00:08:44,960 --> 00:08:49,010
to Pauli about
how even Pauli was

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00:08:49,010 --> 00:08:53,340
a little skeptical of
this spin quantum number,

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00:08:53,340 --> 00:08:57,440
but it's actually crucial
for his exclusion principle.

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00:08:57,440 --> 00:09:00,956
So does anybody remember the
Pauli exclusion principle?

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00:09:05,670 --> 00:09:06,170
Yeah.

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00:09:06,170 --> 00:09:09,510
AUDIENCE: [INAUDIBLE]

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00:09:09,510 --> 00:09:11,730
MICHELE: Exactly.

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00:09:11,730 --> 00:09:17,270
So all the electrons
have to have--

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00:09:17,270 --> 00:09:18,740
all four quantum
numbers have to be

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00:09:18,740 --> 00:09:20,972
different between any
pair of electrons.

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00:09:20,972 --> 00:09:22,430
So the four quantum
numbers, again,

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00:09:22,430 --> 00:09:23,690
are the three
spatial components,

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00:09:23,690 --> 00:09:25,523
and now we've added one
more, which is spin.

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00:09:29,410 --> 00:09:32,620
Pauli himself said, "Already
in my original paper

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00:09:32,620 --> 00:09:34,270
I stressed the
circumstance that I

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00:09:34,270 --> 00:09:36,910
was unable to give a logical
reason for the exclusion

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00:09:36,910 --> 00:09:40,870
principle or to deduce it
from more general assumptions.

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00:09:40,870 --> 00:09:43,120
I had always the feeling,
and I still have it today,

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00:09:43,120 --> 00:09:44,810
that this is a deficiency."

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00:09:44,810 --> 00:09:47,110
So Pauli came up with this.

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00:09:47,110 --> 00:09:49,200
He used it.

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00:09:49,200 --> 00:09:54,140
It explained the periodic
table, but he had no idea

186
00:09:54,140 --> 00:09:55,340
why this had to be true.

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00:09:55,340 --> 00:09:58,670
And doesn't it bother you
guys that there's just

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00:09:58,670 --> 00:10:00,860
this arbitrary rule
that we're saying

189
00:10:00,860 --> 00:10:04,010
no two electrons can have the
same four quantum numbers?

190
00:10:04,010 --> 00:10:06,620
Well, today, we're actually
going to explain this.

191
00:10:06,620 --> 00:10:08,540
We're going to explain
what Pauli could not.

192
00:10:11,990 --> 00:10:14,560
So let's move on to
new stuff, unless there

193
00:10:14,560 --> 00:10:15,870
are any questions so far.

194
00:10:18,720 --> 00:10:21,030
OK, so today, we're
going to be talking

195
00:10:21,030 --> 00:10:23,100
about what happens
when we have more

196
00:10:23,100 --> 00:10:24,790
than one electron in a system.

197
00:10:24,790 --> 00:10:27,750
So we could be
talking about helium

198
00:10:27,750 --> 00:10:31,680
with two electrons,
iron, 26 electrons.

199
00:10:31,680 --> 00:10:35,370
We'll move up to molecules,
solids, things with 100s,

200
00:10:35,370 --> 00:10:39,310
1,000s of electrons.

201
00:10:39,310 --> 00:10:43,790
Dirac said in 1929, "The
underlying physical laws

202
00:10:43,790 --> 00:10:45,620
necessary for the
mathematical theory

203
00:10:45,620 --> 00:10:47,960
of a large part of physics
and the whole of chemistry

204
00:10:47,960 --> 00:10:50,480
are thus completely
known, and the difficulty

205
00:10:50,480 --> 00:10:56,990
is only that the exact
application of these laws

206
00:10:56,990 --> 00:10:59,580
leads to equations much too
complicated to be soluble."

207
00:10:59,580 --> 00:11:02,120
So basically, he was
feeling pretty cocky here.

208
00:11:02,120 --> 00:11:04,615
It's like we understand
the theory of everything.

209
00:11:04,615 --> 00:11:05,990
We understand the
periodic table.

210
00:11:05,990 --> 00:11:07,760
There's nothing left
to do in chemistry.

211
00:11:07,760 --> 00:11:09,830
We understand now.

212
00:11:09,830 --> 00:11:12,080
We can write the equations
for how all molecules work,

213
00:11:12,080 --> 00:11:14,000
how all solids work.

214
00:11:14,000 --> 00:11:19,460
And it's the complication of
these mathematical formulas

215
00:11:19,460 --> 00:11:21,390
that turned out
to be the problem.

216
00:11:21,390 --> 00:11:26,270
So many years later, he still
was struggling with these.

217
00:11:26,270 --> 00:11:28,700
And he says, "If there's
no complete agreement

218
00:11:28,700 --> 00:11:31,310
between the results of one's
work and the experiment,

219
00:11:31,310 --> 00:11:34,170
one should not allow himself
to be too discouraged."

220
00:11:34,170 --> 00:11:37,180
So even after all
this time struggling

221
00:11:37,180 --> 00:11:38,930
to solve this equation,
he still could not

222
00:11:38,930 --> 00:11:40,110
agree with experiment.

223
00:11:40,110 --> 00:11:43,610
And even today, we still cannot
always agree with experiment,

224
00:11:43,610 --> 00:11:45,870
but that's no reason not to try.

225
00:11:45,870 --> 00:11:50,030
So this was our
basic Hamiltonian

226
00:11:50,030 --> 00:11:54,620
that we had for just
the hydrogen system.

227
00:11:54,620 --> 00:11:59,440
It was just the kinetic
and potential energy.

228
00:11:59,440 --> 00:12:00,880
It's as simple as that.

229
00:12:00,880 --> 00:12:06,640
Well, now our equation looks
a little bit more like this.

230
00:12:06,640 --> 00:12:09,990
So I don't want you to be too
intimidated by this equation.

231
00:12:09,990 --> 00:12:11,800
It seems to have a
lot of terms in it,

232
00:12:11,800 --> 00:12:15,700
so let's go through
them for a simple case.

233
00:12:15,700 --> 00:12:21,190
So let's say we'd
have the simplest

234
00:12:21,190 --> 00:12:24,330
molecule you can think of, H2.

235
00:12:24,330 --> 00:12:29,170
So let's say you have a
nucleus here, nucleus here,

236
00:12:29,170 --> 00:12:32,530
and an electron that
originally was associated

237
00:12:32,530 --> 00:12:35,920
with this one, and
another electron

238
00:12:35,920 --> 00:12:38,740
originally associated with this.

239
00:12:38,740 --> 00:12:40,570
So what are these terms
in the Hamiltonian?

240
00:12:40,570 --> 00:12:42,110
So what do we need to consider?

241
00:12:42,110 --> 00:12:47,480
So this first term here
is the kinetic energy

242
00:12:47,480 --> 00:12:49,280
of the nuclei themselves.

243
00:12:49,280 --> 00:12:52,290
So it's how this nucleus and
this nucleus are moving around.

244
00:12:52,290 --> 00:12:54,980
So in this very
simple case, we'd

245
00:12:54,980 --> 00:12:58,040
have two terms in this sum.

246
00:12:58,040 --> 00:13:03,410
This next is the interaction
[? C-C ?] of Z i and a Z j.

247
00:13:03,410 --> 00:13:06,650
So it's the interaction of
the nuclei with each other.

248
00:13:06,650 --> 00:13:11,010
It's the Coulomb repulsion
of nucleus 1 and nucleus 2.

249
00:13:11,010 --> 00:13:15,150
So in this case, that's
actually just one term.

250
00:13:15,150 --> 00:13:20,800
This next term is the kinetic
energy of the electrons.

251
00:13:20,800 --> 00:13:23,130
So it's, in this
case, two terms.

252
00:13:23,130 --> 00:13:26,490
We have electron 1 and electron
2 that are zooming around,

253
00:13:26,490 --> 00:13:29,710
and that's their kinetic energy.

254
00:13:29,710 --> 00:13:32,310
The next term is
there's only one Z,

255
00:13:32,310 --> 00:13:36,640
so it's the interaction of
the nucleus and the electrons.

256
00:13:36,640 --> 00:13:39,660
So here, we have four
of those interactions.

257
00:13:39,660 --> 00:13:42,810
We have electron 1 interacting
with nucleus 1 and 2,

258
00:13:42,810 --> 00:13:46,870
electron 2 interacting
with nucleus 1 and 2.

259
00:13:46,870 --> 00:13:50,780
And finally, we have electrons
interacting with themselves,

260
00:13:50,780 --> 00:13:54,980
so the Coulomb
repulsion of electrons.

261
00:13:54,980 --> 00:13:56,920
So in this case, again,
there's just one term

262
00:13:56,920 --> 00:13:58,660
because there are
only two electrons.

263
00:14:02,260 --> 00:14:08,100
So I think all of these
terms are labeled properly.

264
00:14:08,100 --> 00:14:11,910
So our equation now has
become rather complicated.

265
00:14:11,910 --> 00:14:13,980
Instead of just
having one coordinate,

266
00:14:13,980 --> 00:14:17,520
which was the relative
coordinate between the nucleus

267
00:14:17,520 --> 00:14:20,640
and the electron that we
had in the hydrogen case,

268
00:14:20,640 --> 00:14:23,280
now we have this
Hamiltonian that

269
00:14:23,280 --> 00:14:25,470
depends on the position
of all the nuclei

270
00:14:25,470 --> 00:14:27,120
and the position of
all the electrons,

271
00:14:27,120 --> 00:14:28,470
and our wave
function that depends

272
00:14:28,470 --> 00:14:30,803
on the position of all the
nuclei and all the electrons.

273
00:14:30,803 --> 00:14:34,450
So this has become a massively
complicated equation.

274
00:14:34,450 --> 00:14:40,420
So at this point, what
are we going to do?

275
00:14:40,420 --> 00:14:42,790
You might think to
yourself that you're stuck,

276
00:14:42,790 --> 00:14:45,430
that this equation
is too complicated.

277
00:14:45,430 --> 00:14:46,840
You'll never get anywhere.

278
00:14:46,840 --> 00:14:51,580
And just as you're
despairing, this man walks in,

279
00:14:51,580 --> 00:14:53,003
pulls down the
back of his shirt.

280
00:14:53,003 --> 00:14:55,420
Might take you a little while
to stare at these equations,

281
00:14:55,420 --> 00:14:59,650
because you'll recognize this as
being the equation we just had.

282
00:14:59,650 --> 00:15:04,420
And he's giving you the
Born-Oppenheimer approximation.

283
00:15:04,420 --> 00:15:07,470
So let's look at
this approximation,

284
00:15:07,470 --> 00:15:10,140
maybe not on somebody's back.

285
00:15:10,140 --> 00:15:17,610
So Born was a pretty
foundational guy

286
00:15:17,610 --> 00:15:18,530
in quantum mechanics.

287
00:15:18,530 --> 00:15:20,370
He did a lot of work
in quantum mechanics.

288
00:15:20,370 --> 00:15:26,010
And I'm sure you all recognize
the name Oppenheimer.

289
00:15:26,010 --> 00:15:29,350
So these two guys, what
they decided to do,

290
00:15:29,350 --> 00:15:34,280
they looked at this picture,
and they thought about the fact

291
00:15:34,280 --> 00:15:41,240
that a proton is 2,000 times
the mass of an electron.

292
00:15:41,240 --> 00:15:44,280
And so because this
electron is so heavy,

293
00:15:44,280 --> 00:15:46,370
it's just moving so
slowly with respect

294
00:15:46,370 --> 00:15:48,470
to these electrons that
are zipping around.

295
00:15:48,470 --> 00:15:51,660
And so basically, what
they decided to do

296
00:15:51,660 --> 00:15:54,790
is just ignore that
motion of the nucleons.

297
00:15:54,790 --> 00:15:59,528
So the nuclei are moving so
much slower than the electrons

298
00:15:59,528 --> 00:16:02,070
that basically, you can assume
that the electrons will figure

299
00:16:02,070 --> 00:16:04,445
out their ground state, figure
out where they need to be,

300
00:16:04,445 --> 00:16:07,590
by the time any kind
of nucleus has moved

301
00:16:07,590 --> 00:16:11,010
even a fraction of an Angstrom.

302
00:16:11,010 --> 00:16:14,790
So because we're now
neglecting the kinetic energy

303
00:16:14,790 --> 00:16:18,990
of the nucleons, we
can also just calculate

304
00:16:18,990 --> 00:16:22,590
the ion-ion interaction
classically.

305
00:16:22,590 --> 00:16:26,912
So that really simplifies
our Hamiltonian.

306
00:16:26,912 --> 00:16:28,870
So now, we're just left
with these three terms.

307
00:16:28,870 --> 00:16:31,950
So it's the kinetic
energy of the electrons

308
00:16:31,950 --> 00:16:39,450
only, the interactions of the
electrons with the background

309
00:16:39,450 --> 00:16:40,920
positive charge--

310
00:16:40,920 --> 00:16:42,630
so where the ions are--

311
00:16:42,630 --> 00:16:45,140
and the interaction of the
electrons with themselves.

312
00:16:48,640 --> 00:16:51,360
So we're from now
on, we're mostly

313
00:16:51,360 --> 00:16:55,350
going to call this term just
the "external potential," so

314
00:16:55,350 --> 00:16:58,500
"external" because we
don't care about the motion

315
00:16:58,500 --> 00:17:02,230
of these nuclei anymore.

316
00:17:02,230 --> 00:17:06,150
So this leads us to
actually starting

317
00:17:06,150 --> 00:17:09,030
to talk about how we do
these approximations.

318
00:17:09,030 --> 00:17:12,510
So traditionally, there
have been two pathways

319
00:17:12,510 --> 00:17:13,750
that have been followed.

320
00:17:13,750 --> 00:17:16,980
There's what a lot of
quantum chemists have taken,

321
00:17:16,980 --> 00:17:20,670
and we'll talk about that
first, but there's also

322
00:17:20,670 --> 00:17:24,180
density functional theory, which
is what a lot of physicists

323
00:17:24,180 --> 00:17:26,017
have traditionally used.

324
00:17:26,017 --> 00:17:28,600
And I'm not going to talk about
either of these two in detail.

325
00:17:28,600 --> 00:17:32,670
But I'll just mention, the
Moller-Plesset perturbation

326
00:17:32,670 --> 00:17:35,220
theory is a perturbation
theory. so it's

327
00:17:35,220 --> 00:17:39,650
based on changing
the Hamiltonian,

328
00:17:39,650 --> 00:17:42,240
so assuming that the Hamiltonian
is pretty much something

329
00:17:42,240 --> 00:17:43,830
we can solve, and
then just adding

330
00:17:43,830 --> 00:17:46,110
a small correction to it.

331
00:17:46,110 --> 00:17:48,300
And the coupled
cluster approach is

332
00:17:48,300 --> 00:17:51,570
more of a traditional quantum
chemistry approach, where

333
00:17:51,570 --> 00:17:53,670
instead of modifying
the Hamiltonian,

334
00:17:53,670 --> 00:17:55,410
we just modify the
wave functions.

335
00:17:55,410 --> 00:17:58,330
And we play around
with the wave functions

336
00:17:58,330 --> 00:18:02,680
until they become something
that's more close to something

337
00:18:02,680 --> 00:18:05,230
that would exist in reality.

338
00:18:05,230 --> 00:18:10,250
So let's talk about the basis of
the quantum chemistry approach.

339
00:18:10,250 --> 00:18:12,370
So this is Hartree.

340
00:18:12,370 --> 00:18:15,280
He was working right
after World War

341
00:18:15,280 --> 00:18:18,730
I. I think this was
actually his dissertation,

342
00:18:18,730 --> 00:18:24,230
and he got his PhD for doing
this very important work.

343
00:18:24,230 --> 00:18:28,113
So what he decided is
we have this system

344
00:18:28,113 --> 00:18:29,030
that we want to solve.

345
00:18:29,030 --> 00:18:30,700
We have a wave
function that's made up

346
00:18:30,700 --> 00:18:35,000
of all of these electrons that
are zooming around in space.

347
00:18:35,000 --> 00:18:37,990
Well, what if that kind
of acted like a bunch

348
00:18:37,990 --> 00:18:39,800
of one-electron wave functions?

349
00:18:39,800 --> 00:18:41,770
What if we could
just assume that we

350
00:18:41,770 --> 00:18:44,710
could know where one electron
was, and another electron was,

351
00:18:44,710 --> 00:18:47,500
and we just could just multiply
all those together to give us

352
00:18:47,500 --> 00:18:49,430
the total wave function?

353
00:18:49,430 --> 00:18:53,480
So all of a sudden, this really
complicated wave function

354
00:18:53,480 --> 00:18:57,150
becomes something that's
pretty easy to separate.

355
00:18:57,150 --> 00:19:02,100
So because we can
separate them like this,

356
00:19:02,100 --> 00:19:05,880
we can separate them into a set
of-- so if we have electrons 1

357
00:19:05,880 --> 00:19:10,260
through n, we'll have
a set of Schrodinger

358
00:19:10,260 --> 00:19:16,060
equations 1 through n, for
each electron by itself.

359
00:19:16,060 --> 00:19:17,290
So this is great.

360
00:19:17,290 --> 00:19:19,770
We've separated our Hamiltonian.

361
00:19:19,770 --> 00:19:22,490
We now can solve
the system, right?

362
00:19:22,490 --> 00:19:25,060
So there is a problem here.

363
00:19:28,150 --> 00:19:34,720
This is the density
of electron j.

364
00:19:34,720 --> 00:19:38,830
We're doing a sum over j
from electron 1 through n,

365
00:19:38,830 --> 00:19:40,780
and just skipping the
electron that we're

366
00:19:40,780 --> 00:19:42,420
considering in this case.

367
00:19:42,420 --> 00:19:44,830
So all of us is saying
this electron that we'd

368
00:19:44,830 --> 00:19:47,050
like to solve depends
on the positions

369
00:19:47,050 --> 00:19:49,670
of every single other
electron in that system.

370
00:19:49,670 --> 00:19:53,800
So even in this most simplified
version of the Schrodinger

371
00:19:53,800 --> 00:19:57,010
equation, we still
have the one electron

372
00:19:57,010 --> 00:19:59,240
depending on all of
the other electrons.

373
00:19:59,240 --> 00:20:02,862
So how do we solve
something like this?

374
00:20:02,862 --> 00:20:03,570
Do you guys know?

375
00:20:06,288 --> 00:20:08,830
I think this is even a little
more complicated than something

376
00:20:08,830 --> 00:20:11,658
Mathematica can solve.

377
00:20:11,658 --> 00:20:13,060
AUDIENCE: [INAUDIBLE].

378
00:20:13,060 --> 00:20:15,070
MICHELE: Pretty much.

379
00:20:15,070 --> 00:20:28,010
Well-- what we do is we try
a self-consistent approach.

380
00:20:28,010 --> 00:20:31,940
We basically just guess what
these wave functions look like.

381
00:20:31,940 --> 00:20:35,180
We say, well in
this case, we know

382
00:20:35,180 --> 00:20:38,750
that it's a sigma bond that's
forming between these two,

383
00:20:38,750 --> 00:20:43,380
so probably the electron
is somewhere in here.

384
00:20:43,380 --> 00:20:46,100
And so if I guess some
structure that looks something

385
00:20:46,100 --> 00:20:50,590
like this is my wave
function, that's

386
00:20:50,590 --> 00:20:52,110
maybe a good starting point.

387
00:20:52,110 --> 00:20:56,870
And so we put input
that into this equation.

388
00:20:56,870 --> 00:20:59,048
We take out the first
electron, and we're

389
00:20:59,048 --> 00:21:00,840
going to now solve the
Schrodinger equation

390
00:21:00,840 --> 00:21:02,200
for the first electron.

391
00:21:02,200 --> 00:21:07,950
So we plug in what
we've guessed as being

392
00:21:07,950 --> 00:21:11,370
the wave functions for all the
other electrons into this term.

393
00:21:11,370 --> 00:21:14,670
And now we solve for
this one electron term,

394
00:21:14,670 --> 00:21:16,140
for electron number 1.

395
00:21:16,140 --> 00:21:18,450
And then we do that
again for electron 2,

396
00:21:18,450 --> 00:21:21,065
and so on until we've gone
through all of the electrons.

397
00:21:21,065 --> 00:21:22,440
And then when
we're done, we have

398
00:21:22,440 --> 00:21:24,720
a new set of wave functions.

399
00:21:24,720 --> 00:21:27,570
So maybe my guess was wrong
and it looks something more

400
00:21:27,570 --> 00:21:30,020
like this.

401
00:21:30,020 --> 00:21:32,220
And so we look at
that, and say, wow,

402
00:21:32,220 --> 00:21:34,980
those two really don't look
anything like each other.

403
00:21:34,980 --> 00:21:36,610
I need to try this again.

404
00:21:36,610 --> 00:21:38,460
And so we go
through this process

405
00:21:38,460 --> 00:21:41,290
over and over again
until finally,

406
00:21:41,290 --> 00:21:47,820
the solutions to these wave
functions prior to doing this

407
00:21:47,820 --> 00:21:49,950
and after going
through and solving

408
00:21:49,950 --> 00:21:54,960
all of these eigenfunctions
look pretty much the same.

409
00:21:54,960 --> 00:21:57,690
And once we've
done that, we know

410
00:21:57,690 --> 00:22:00,750
that we've probably gotten
something at least relatively

411
00:22:00,750 --> 00:22:03,330
close to the ground-state
solution, or the solution

412
00:22:03,330 --> 00:22:05,530
that we're looking for.

413
00:22:05,530 --> 00:22:07,440
So does that make
sense to you guys?

414
00:22:07,440 --> 00:22:08,773
Are there any questions?

415
00:22:12,720 --> 00:22:17,010
OK, because this is actually
pretty close to how DFT works,

416
00:22:17,010 --> 00:22:17,510
too.

417
00:22:17,510 --> 00:22:19,370
This is the
self-consistent method

418
00:22:19,370 --> 00:22:24,960
that is pretty common in
solving these problems.

419
00:22:24,960 --> 00:22:26,120
So this is great.

420
00:22:26,120 --> 00:22:27,740
We have a way of doing this.

421
00:22:27,740 --> 00:22:30,500
Given that we can come up
with some reasonable guess

422
00:22:30,500 --> 00:22:33,630
for our input wave
functions, we now

423
00:22:33,630 --> 00:22:35,380
can solve the Schrodinger
equation, right?

424
00:22:38,130 --> 00:22:39,660
Yes and no.

425
00:22:39,660 --> 00:22:43,410
This picture is nice
and simple and soluble,

426
00:22:43,410 --> 00:22:47,520
but it takes out all
of the interactions

427
00:22:47,520 --> 00:22:49,020
that you get, or
all of the effects

428
00:22:49,020 --> 00:22:50,770
that you get from
interactions beyond just

429
00:22:50,770 --> 00:22:52,980
that coulomb repulsion, of
electron-electron Coulomb

430
00:22:52,980 --> 00:22:53,710
repulsion.

431
00:22:53,710 --> 00:22:56,088
Because remember, this
is a quantum system.

432
00:22:56,088 --> 00:22:57,630
These electrons
don't behave like you

433
00:22:57,630 --> 00:23:02,880
expect they would in some kind
of classical Newtonian system.

434
00:23:02,880 --> 00:23:05,730
These electrons are weird.

435
00:23:05,730 --> 00:23:12,870
We'll see later on some of the
symmetries that we can look at.

436
00:23:12,870 --> 00:23:15,570
Actually, I think that's next.

437
00:23:15,570 --> 00:23:17,280
But these electrons--
you wouldn't

438
00:23:17,280 --> 00:23:22,475
expect to just be able
to just average over all

439
00:23:22,475 --> 00:23:23,100
of the effects.

440
00:23:23,100 --> 00:23:27,600
Because this truly was a
multi-electron function.

441
00:23:27,600 --> 00:23:29,610
We're missing some
critical physics here

442
00:23:29,610 --> 00:23:33,630
by just separating it out
into single electron functions

443
00:23:33,630 --> 00:23:36,220
and just multiplying
those back together.

444
00:23:36,220 --> 00:23:41,672
So we're missing
two important terms.

445
00:23:41,672 --> 00:23:43,380
They're going to be
called the "exchange"

446
00:23:43,380 --> 00:23:44,670
and the "correlation" term.

447
00:23:44,670 --> 00:23:47,790
And the fix to at least
part of this problem

448
00:23:47,790 --> 00:23:50,860
brings us back to spin.

449
00:23:50,860 --> 00:23:55,030
So as I said, in
quantum mechanics,

450
00:23:55,030 --> 00:23:57,490
symmetry is really,
really important.

451
00:23:57,490 --> 00:24:02,890
Symmetry tells us a lot
about the way nature works.

452
00:24:02,890 --> 00:24:04,570
So even in classical
mechanics, you guys

453
00:24:04,570 --> 00:24:10,630
have probably encountered
the symmetry of real space,

454
00:24:10,630 --> 00:24:13,880
the symmetry of time,
things like that.

455
00:24:13,880 --> 00:24:18,490
And that's what gives you the
conservation laws in mechanics.

456
00:24:18,490 --> 00:24:21,480
So just like that's
true in mechanics,

457
00:24:21,480 --> 00:24:23,650
it's also true in
quantum mechanics.

458
00:24:23,650 --> 00:24:26,850
So we're going to be looking
at one particular symmetry,

459
00:24:26,850 --> 00:24:29,520
and talking about how
that might give us

460
00:24:29,520 --> 00:24:32,050
some insight into this problem.

461
00:24:32,050 --> 00:24:35,120
So that symmetry is called
"exchange symmetry."

462
00:24:35,120 --> 00:24:38,520
So that's based on the
fact that electrons

463
00:24:38,520 --> 00:24:40,650
are indistinguishable.

464
00:24:40,650 --> 00:24:46,250
So what does it mean to be
really indistinguishable?

465
00:24:46,250 --> 00:24:52,390
So if I have, maybe, two
pieces of chalk that look--

466
00:24:52,390 --> 00:24:53,950
I guess they're not
exactly the same,

467
00:24:53,950 --> 00:24:56,710
but they look pretty similar.

468
00:24:56,710 --> 00:24:59,560
And let's say I want to follow--

469
00:24:59,560 --> 00:25:02,080
I'm going to call this one
Fred and this one George.

470
00:25:02,080 --> 00:25:04,810
And I'm going to follow their
movement throughout space.

471
00:25:04,810 --> 00:25:07,240
So if I give them
an initial position

472
00:25:07,240 --> 00:25:10,950
and some initial velocity,
I know throughout all time

473
00:25:10,950 --> 00:25:12,700
which one is Fred and
which one is George.

474
00:25:12,700 --> 00:25:14,980
Because I knew their
exact initial position.

475
00:25:14,980 --> 00:25:16,960
I knew their exact
initial velocity.

476
00:25:16,960 --> 00:25:18,430
I can calculate
their trajectories.

477
00:25:18,430 --> 00:25:20,600
I know which one is which.

478
00:25:20,600 --> 00:25:23,710
But that's not
true for electrons.

479
00:25:23,710 --> 00:25:27,160
So does anybody know the
uncertainty principle?

480
00:25:27,160 --> 00:25:31,030
Can somebody tell me what
the uncertainty principle is?

481
00:25:31,030 --> 00:25:31,530
Yeah.

482
00:25:35,710 --> 00:25:37,634
Did you volunteer?

483
00:25:37,634 --> 00:25:39,580
AUDIENCE: No, I didn't.

484
00:25:39,580 --> 00:25:43,840
I [INAUDIBLE]
algebraic [INAUDIBLE]..

485
00:25:43,840 --> 00:25:46,080
MICHELE: Right.

486
00:25:46,080 --> 00:25:47,080
AUDIENCE: [INAUDIBLE].

487
00:25:50,060 --> 00:25:53,250
MICHELE: Right, so the exact
mathematical formulation

488
00:25:53,250 --> 00:25:57,090
is something like the
uncertainty in the momentum

489
00:25:57,090 --> 00:25:58,920
times the uncertainty
in the position

490
00:25:58,920 --> 00:26:02,252
has to be always greater
than or equal to basically

491
00:26:02,252 --> 00:26:02,835
some constant.

492
00:26:06,300 --> 00:26:11,570
So if I have my two
starting electrons,

493
00:26:11,570 --> 00:26:13,940
and I have some idea
of where it starts,

494
00:26:13,940 --> 00:26:17,390
and with some idea of what
velocity it starts with,

495
00:26:17,390 --> 00:26:22,190
and some other electron-- with
similarly not an exact position

496
00:26:22,190 --> 00:26:24,440
but some kind of position
and some kind of knowledge

497
00:26:24,440 --> 00:26:26,960
of the velocity--

498
00:26:26,960 --> 00:26:29,150
it's clear that over
time, I'm going to have

499
00:26:29,150 --> 00:26:31,040
no idea which one was which.

500
00:26:31,040 --> 00:26:33,380
They'll switch and I won't
be able to tell that they've

501
00:26:33,380 --> 00:26:36,050
switched because I can't follow
their trajectories the same way

502
00:26:36,050 --> 00:26:39,460
I can with classical objects.

503
00:26:39,460 --> 00:26:43,128
And furthermore, electrons don't
have any identifying marks.

504
00:26:43,128 --> 00:26:45,670
They don't have like a beauty
mark somewhere that can tell us

505
00:26:45,670 --> 00:26:47,470
which one was the
electron that we really

506
00:26:47,470 --> 00:26:49,510
fell in love with first.

507
00:26:49,510 --> 00:26:56,078
And so electrons act
completely indistinguishable.

508
00:26:56,078 --> 00:26:58,120
And so this is a symmetry
we're going to exploit.

509
00:27:01,170 --> 00:27:07,050
So let's say we have a
system full of electrons,

510
00:27:07,050 --> 00:27:08,580
and I pull a curtain
in front of you

511
00:27:08,580 --> 00:27:10,230
so you can't see
the system anymore.

512
00:27:10,230 --> 00:27:12,420
I take one electron, and
I take another electron,

513
00:27:12,420 --> 00:27:13,770
and I swap them exactly.

514
00:27:13,770 --> 00:27:16,140
I swap their positions,
I swap their momentum,

515
00:27:16,140 --> 00:27:18,660
I swap everything about them.

516
00:27:18,660 --> 00:27:21,420
And I open that
curtain again, you

517
00:27:21,420 --> 00:27:24,060
won't be able to tell at all
that anything's happened,

518
00:27:24,060 --> 00:27:26,290
because these electrons act
exactly like each other.

519
00:27:26,290 --> 00:27:29,130
So it doesn't matter if I
swapped one with another.

520
00:27:29,130 --> 00:27:32,820
So this might sound
a little bit dumb.

521
00:27:32,820 --> 00:27:34,750
I mean, you might be
sitting there thinking,

522
00:27:34,750 --> 00:27:35,670
OK, this is obvious.

523
00:27:35,670 --> 00:27:41,100
Why is she going on so long
about something so obvious?

524
00:27:41,100 --> 00:27:44,070
And the reason is
when we formalize

525
00:27:44,070 --> 00:27:47,310
this in terms of
mathematics, suddenly

526
00:27:47,310 --> 00:27:50,760
things become
really interesting.

527
00:27:50,760 --> 00:27:53,900
So we're going to define
an exchange operator, which

528
00:27:53,900 --> 00:27:56,660
is this chi 1 2.

529
00:27:56,660 --> 00:27:59,720
So chi 1 2 acting
on a system that

530
00:27:59,720 --> 00:28:04,820
has an electron 1 and electron
2, all it's going to do

531
00:28:04,820 --> 00:28:08,150
is swap electron
1 with electron 2.

532
00:28:08,150 --> 00:28:11,750
Just so this
formalism makes sense,

533
00:28:11,750 --> 00:28:15,980
this 1 refers to what
particular wave function it is.

534
00:28:15,980 --> 00:28:20,750
So this psi 1 refers to that
set of four quantum numbers,

535
00:28:20,750 --> 00:28:23,990
or set of whatever quantum
numbers describes that system.

536
00:28:23,990 --> 00:28:27,050
And similarly, psi 2
describes a different set

537
00:28:27,050 --> 00:28:31,340
of quantum numbers, and r1 is
the position of electron 1, r2

538
00:28:31,340 --> 00:28:32,870
is the position of electron 2.

539
00:28:37,070 --> 00:28:41,700
So when we use this exchange
operator on our system,

540
00:28:41,700 --> 00:28:45,440
we get something that
should be indistinguishable.

541
00:28:45,440 --> 00:28:48,210
And you can see that
if we act twice--

542
00:28:48,210 --> 00:28:51,450
so if we have electron 1
and 2, and we swap them,

543
00:28:51,450 --> 00:28:53,450
and then we act again
with the exchange operator

544
00:28:53,450 --> 00:28:56,240
and we swap 1 and 2,
we'll switch back.

545
00:28:56,240 --> 00:29:04,070
So we know mathematically that
this exchange operator, acted

546
00:29:04,070 --> 00:29:06,717
twice on a system,
leaves you exactly

547
00:29:06,717 --> 00:29:08,300
with the original
system that you had.

548
00:29:11,360 --> 00:29:16,940
So we have to assume
here that this wave

549
00:29:16,940 --> 00:29:20,060
function is going to be an
eigenstate of the exchange

550
00:29:20,060 --> 00:29:20,900
operator.

551
00:29:20,900 --> 00:29:25,430
And there is a quantum
mechanical reason for that.

552
00:29:25,430 --> 00:29:29,930
But what this tells us,
assuming that we've now

553
00:29:29,930 --> 00:29:32,090
acted with this
exchange operator twice,

554
00:29:32,090 --> 00:29:36,590
we know that the value of the
eigenvalue of the exchange

555
00:29:36,590 --> 00:29:40,180
operator squared is
equal to 1, which

556
00:29:40,180 --> 00:29:42,520
gives us that the
eigenvalue itself

557
00:29:42,520 --> 00:29:43,840
has to be plus or minus 1.

558
00:29:47,782 --> 00:29:49,240
So just to be clear,
what I've said

559
00:29:49,240 --> 00:29:56,050
is that when I act
on this wave function

560
00:29:56,050 --> 00:29:58,310
with this exchange operator--

561
00:29:58,310 --> 00:30:03,370
so I have my total
wave function, whatever

562
00:30:03,370 --> 00:30:07,790
it is, which is a function
of multiple electrons,

563
00:30:07,790 --> 00:30:12,310
when I act with this
exchange operator on it,

564
00:30:12,310 --> 00:30:18,760
I get something where
I've switched r1 and r2,

565
00:30:18,760 --> 00:30:21,410
and all the other electrons
are left the same.

566
00:30:21,410 --> 00:30:23,200
And what I said in
the previous slide

567
00:30:23,200 --> 00:30:24,680
is that these two states--

568
00:30:24,680 --> 00:30:29,540
this state and this state--
have to act exactly the same.

569
00:30:29,540 --> 00:30:35,500
But now I've also
told you that this

570
00:30:35,500 --> 00:30:47,290
is equal to some eigenfunction
times the original wave

571
00:30:47,290 --> 00:30:48,190
function.

572
00:30:48,190 --> 00:30:51,460
And this eigenfunction, I've
said, can be negative 1.

573
00:30:51,460 --> 00:30:58,590
So can anybody
reconcile this for me?

574
00:30:58,590 --> 00:31:02,850
How can this eigenfunction
be negative and yet

575
00:31:02,850 --> 00:31:05,610
have this original
eigenstate not be changed,

576
00:31:05,610 --> 00:31:12,660
or have no measurable
difference between the electrons

577
00:31:12,660 --> 00:31:15,810
being in the original positions
or 1 and 2 being switched?

578
00:31:37,760 --> 00:31:43,840
So the key to the question
is actually the term

579
00:31:43,840 --> 00:31:46,780
"measurable difference."

580
00:31:46,780 --> 00:31:51,370
So remember, Jeff has said
that in quantum mechanics,

581
00:31:51,370 --> 00:31:52,890
we have this wave function.

582
00:31:52,890 --> 00:31:55,140
We have no idea what the
wave function actually means.

583
00:31:55,140 --> 00:31:57,598
It's the square, the absolute
value square of the function,

584
00:31:57,598 --> 00:31:59,170
that actually is meaningful.

585
00:31:59,170 --> 00:32:02,598
So you can see, even if
this is a negative 1,

586
00:32:02,598 --> 00:32:04,890
and we have a negative 1 in
front of the wave function,

587
00:32:04,890 --> 00:32:08,050
when we square that, the
negative is going to disappear.

588
00:32:08,050 --> 00:32:12,690
So we still have this symmetry
because the thing that's

589
00:32:12,690 --> 00:32:16,230
measurable is the absolute value
squared of the wave function,

590
00:32:16,230 --> 00:32:19,140
and not the wave
function itself.

591
00:32:19,140 --> 00:32:24,070
But this is actually a pretty
key piece of quantum mechanics.

592
00:32:24,070 --> 00:32:29,130
So we have two possible
values for this wave function.

593
00:32:29,130 --> 00:32:32,610
So this wave function
can be positive 1

594
00:32:32,610 --> 00:32:33,720
or it can be negative 1.

595
00:32:42,740 --> 00:32:46,180
So any quantum
mechanical system is

596
00:32:46,180 --> 00:32:48,820
going to have a wave
function that either

597
00:32:48,820 --> 00:32:52,947
has the eigenvalue for
this equation positive 1

598
00:32:52,947 --> 00:32:53,530
or negative 1.

599
00:32:53,530 --> 00:32:57,790
When it's positive 1,
we call them "bosons."

600
00:32:57,790 --> 00:33:01,150
So this leads to a lot
of interesting physics,

601
00:33:01,150 --> 00:33:03,020
like the Bose-Einstein
condensate,

602
00:33:03,020 --> 00:33:05,020
which I don't know if you
guys have heard about.

603
00:33:05,020 --> 00:33:07,480
There's a lot of
cool experiments

604
00:33:07,480 --> 00:33:09,760
that have been done
on helium 4, where

605
00:33:09,760 --> 00:33:13,180
at a low enough temperature,
all of the helium atoms

606
00:33:13,180 --> 00:33:16,210
suddenly find themselves in
exactly the same ground state.

607
00:33:16,210 --> 00:33:22,220
And there's some really
bizarre consequences of that.

608
00:33:22,220 --> 00:33:24,370
However, what
we're interested in

609
00:33:24,370 --> 00:33:27,760
are the particles with
an eigenvalue of minus 1.

610
00:33:27,760 --> 00:33:30,130
Those are called "fermions."

611
00:33:30,130 --> 00:33:31,750
Electrons are fermions.

612
00:33:31,750 --> 00:33:33,790
So electrons,
therefore, need to have

613
00:33:33,790 --> 00:33:36,040
an eigenvalue of negative 1.

614
00:33:36,040 --> 00:33:39,950
And this is actually the key to
the Pauli exclusion principle.

615
00:33:39,950 --> 00:33:44,120
So this is why the Pauli
exclusion principle is true.

616
00:33:44,120 --> 00:33:46,630
So can somebody
explain to me that?

617
00:33:46,630 --> 00:33:49,570
It's kind of a subtle point.

618
00:33:49,570 --> 00:33:50,620
Does anybody see that?

619
00:34:05,120 --> 00:34:12,340
So let's look at a
wave function that

620
00:34:12,340 --> 00:34:14,260
doesn't follow the Pauli
exclusion principle

621
00:34:14,260 --> 00:34:15,469
and see what happens.

622
00:34:15,469 --> 00:34:19,570
So let's say I
have two electrons.

623
00:34:19,570 --> 00:34:24,929
I'm going to put them both
in the 1s with spin up.

624
00:34:24,929 --> 00:34:32,989
So what happens when I
operate with this exchange

625
00:34:32,989 --> 00:34:34,340
operator on this?

626
00:34:48,670 --> 00:34:52,630
So these two are
exactly the same.

627
00:34:52,630 --> 00:34:57,480
So I needed a negative 1
here, but I have a positive 1.

628
00:34:57,480 --> 00:35:04,860
So because this exchange
gives me the wrong sign,

629
00:35:04,860 --> 00:35:08,070
I know anything that has
some system like this, where

630
00:35:08,070 --> 00:35:11,520
I have the same quantum numbers
on two different particles,

631
00:35:11,520 --> 00:35:16,110
has to be a boson
and not a fermion.

632
00:35:16,110 --> 00:35:19,770
So let me just give you an
example of a wave function

633
00:35:19,770 --> 00:35:26,530
that does satisfy the
proper eigenstate for this.

634
00:35:26,530 --> 00:35:29,670
So if we have something
like 1 over root

635
00:35:29,670 --> 00:35:39,860
2 1s up for particle 1,
1s down for particle 2,

636
00:35:39,860 --> 00:35:47,810
minus 1s down for particle 1
times 1s up for particle 2.

637
00:35:54,230 --> 00:35:55,340
So can everybody see?

638
00:35:55,340 --> 00:35:58,760
If I switch 1 and 2, I'm going
to get the negative of this.

639
00:35:58,760 --> 00:36:02,450
Because these two are going to
switch and become this term,

640
00:36:02,450 --> 00:36:06,830
these two are going to
switch and become this term.

641
00:36:06,830 --> 00:36:07,550
Is that clear?

642
00:36:11,090 --> 00:36:13,720
I'm seeing a lot of blank faces.

643
00:36:13,720 --> 00:36:17,200
I hope that means that
people are way past this

644
00:36:17,200 --> 00:36:21,010
and have mastered this long ago.

645
00:36:21,010 --> 00:36:22,440
Are there any questions on this?

646
00:36:34,100 --> 00:36:35,430
To me, this is really exciting.

647
00:36:35,430 --> 00:36:38,180
This is great to be able to
finally explain something

648
00:36:38,180 --> 00:36:40,270
that Paul himself
couldn't explain.

649
00:36:40,270 --> 00:36:44,770
We actually can understand
quantum mechanically

650
00:36:44,770 --> 00:36:47,840
why electrons behave
the way they do,

651
00:36:47,840 --> 00:36:49,590
why we have the Pauli
exclusion principle,

652
00:36:49,590 --> 00:36:51,540
and therefore, why we
have the periodic table.

653
00:36:51,540 --> 00:36:56,500
I mean, this is a
pretty big consequence.

654
00:36:56,500 --> 00:36:59,350
[DOOR CLOSING]

655
00:36:59,350 --> 00:37:03,310
I guess it was so big,
he couldn't take it.

656
00:37:03,310 --> 00:37:12,360
All right, let's move on to
what the implication of this

657
00:37:12,360 --> 00:37:15,160
is in the Hartree method.

658
00:37:15,160 --> 00:37:17,730
So a guy named Fock
came along, and he

659
00:37:17,730 --> 00:37:20,520
has either a very fortunate
or very unfortunate name,

660
00:37:20,520 --> 00:37:23,280
depending on how
you feel about him.

661
00:37:23,280 --> 00:37:27,570
And he basically just took an
anti-symmetrized wave function.

662
00:37:27,570 --> 00:37:30,600
So anti-symmetrized just
means this characteristic,

663
00:37:30,600 --> 00:37:34,050
that when you act upon
it with this operator,

664
00:37:34,050 --> 00:37:37,080
you get a negative 1.

665
00:37:37,080 --> 00:37:40,980
So he just put in an
anti-symmetrizied wave function

666
00:37:40,980 --> 00:37:44,745
into Hartree's original
equation, which was just,

667
00:37:44,745 --> 00:37:45,870
if you remember, just this.

668
00:37:45,870 --> 00:37:51,900
It's the kinetic part
and then from the ions,

669
00:37:51,900 --> 00:37:54,370
the potential part from
the other electrons.

670
00:37:54,370 --> 00:37:56,700
So this was the
original equation.

671
00:37:56,700 --> 00:37:59,730
And when he plugged in an
anti-symmetrized wave function,

672
00:37:59,730 --> 00:38:03,600
suddenly, this term appeared.

673
00:38:03,600 --> 00:38:07,590
So this is the term we
call the "exchange term."

674
00:38:07,590 --> 00:38:09,870
If you look very
closely at it, you'll

675
00:38:09,870 --> 00:38:15,310
see that what it does
is quantum mechanically,

676
00:38:15,310 --> 00:38:17,160
two electrons with the
same spin will never

677
00:38:17,160 --> 00:38:18,160
be in the same position.

678
00:38:18,160 --> 00:38:22,060
And it basically subtracts
out that possibility,

679
00:38:22,060 --> 00:38:24,950
of two electrons with the same
spin being in the same place.

680
00:38:24,950 --> 00:38:29,590
So this actually reduces
the energy of the system

681
00:38:29,590 --> 00:38:33,160
because suddenly, all electrons
with the same spin have

682
00:38:33,160 --> 00:38:35,630
to be just a little bit
farther apart from each other.

683
00:38:35,630 --> 00:38:38,320
So that minimizes the Coulomb
interaction just a little bit.

684
00:38:41,340 --> 00:38:44,410
So actually, with
this correction,

685
00:38:44,410 --> 00:38:47,830
the results seem to come and
match up with experiments

686
00:38:47,830 --> 00:38:49,150
a lot closer.

687
00:38:49,150 --> 00:38:51,280
And this is actually
the foundation

688
00:38:51,280 --> 00:38:53,170
for molecular orbital theory.

689
00:38:53,170 --> 00:38:58,375
So just by adding this spin
consideration into the Hartree

690
00:38:58,375 --> 00:39:00,250
equation, we suddenly
have an equation that's

691
00:39:00,250 --> 00:39:03,610
actually pretty functional.

692
00:39:03,610 --> 00:39:05,680
So Jeff thinks this is
quite an emotional moment.

693
00:39:08,430 --> 00:39:15,600
However, as I said, there
were two energy terms

694
00:39:15,600 --> 00:39:19,360
that we were missing in that
original Hartree equation.

695
00:39:19,360 --> 00:39:21,660
So we found the exchange
term, but we're still

696
00:39:21,660 --> 00:39:25,080
missing the correlation term.

697
00:39:25,080 --> 00:39:27,145
So there are some ways
to deal with this,

698
00:39:27,145 --> 00:39:29,520
but I think the way we're
going to deal with this is just

699
00:39:29,520 --> 00:39:31,920
by moving on to DFT.

700
00:39:31,920 --> 00:39:34,950
So now, we're going to start
talking about DFT, which

701
00:39:34,950 --> 00:39:37,500
I think is what you
guys are mostly using,

702
00:39:37,500 --> 00:39:39,885
so you might find
this more interesting.

703
00:39:44,700 --> 00:39:48,990
So the Schrodinger equation
is just really hard

704
00:39:48,990 --> 00:39:51,420
to solve with all
of these electrons.

705
00:39:51,420 --> 00:39:55,570
As we've said, it
takes a lot of time.

706
00:39:55,570 --> 00:39:58,890
It takes a lot of
computational expense.

707
00:39:58,890 --> 00:40:03,633
So let's divide space into the
worst grid you could imagine.

708
00:40:03,633 --> 00:40:05,175
I guess the worst
would be one point,

709
00:40:05,175 --> 00:40:08,760
but the worst grid you can
imagine is 2 by 2 by 2.

710
00:40:08,760 --> 00:40:14,710
And let's think about
the number of points

711
00:40:14,710 --> 00:40:17,920
that we need to keep track of
if we're calculating something

712
00:40:17,920 --> 00:40:21,070
with little n electrons.

713
00:40:23,740 --> 00:40:26,210
So if you only have one
electron, it's only eight,

714
00:40:26,210 --> 00:40:29,030
and it's not too bad
to keep track of.

715
00:40:29,030 --> 00:40:30,910
If we go up all the
way to 1,000 electrons,

716
00:40:30,910 --> 00:40:32,890
suddenly we have this
huge number of points

717
00:40:32,890 --> 00:40:34,180
that we have to keep track of.

718
00:40:34,180 --> 00:40:38,180
And that's just inconceivable,
even with today's computers.

719
00:40:38,180 --> 00:40:39,970
Why try to do
something this big?

720
00:40:39,970 --> 00:40:42,670
That would take forever.

721
00:40:42,670 --> 00:40:44,700
So it'd be really
nice if we could

722
00:40:44,700 --> 00:40:48,810
think about some other
aspect of the system

723
00:40:48,810 --> 00:40:51,660
that we could calculate,
instead of keeping

724
00:40:51,660 --> 00:40:54,527
track of every single one of
those electrons separately.

725
00:40:58,510 --> 00:41:02,230
And a term like that is density.

726
00:41:02,230 --> 00:41:04,800
So if we consider
density instead,

727
00:41:04,800 --> 00:41:07,417
suddenly, it's pretty
much the same scaling

728
00:41:07,417 --> 00:41:09,000
no matter how many
electrons you have.

729
00:41:12,460 --> 00:41:16,850
So the electron
density definitely

730
00:41:16,850 --> 00:41:18,770
seems to be a lot more
manageable than dealing

731
00:41:18,770 --> 00:41:21,150
with each electron separately.

732
00:41:21,150 --> 00:41:25,400
And so at some point,
someone's, like, well,

733
00:41:25,400 --> 00:41:26,900
wouldn't it be nice
if we could just

734
00:41:26,900 --> 00:41:29,510
calculate with density instead
of with wave functions?

735
00:41:29,510 --> 00:41:31,460
Because think about it--
these wave functions

736
00:41:31,460 --> 00:41:33,418
don't mean anything anyway
until you've squared

737
00:41:33,418 --> 00:41:34,758
them to get the density.

738
00:41:34,758 --> 00:41:36,800
That's the only thing that
we can measure anyway.

739
00:41:36,800 --> 00:41:39,680
Why don't we just
look at the density?

740
00:41:39,680 --> 00:41:43,270
So what if I could come up
with a function of energy that

741
00:41:43,270 --> 00:41:46,440
was dependent on the
density of electrons?

742
00:41:46,440 --> 00:41:52,680
And Walter Kohn actually won
the Nobel Prize in Chemistry

743
00:41:52,680 --> 00:41:56,490
in 1998 with somebody else
for coming up with this idea.

744
00:41:59,250 --> 00:42:04,070
So as I said, one reason
that we want to go to DFT--

745
00:42:04,070 --> 00:42:08,270
it scales so much better than
the quantum chemistry methods.

746
00:42:08,270 --> 00:42:11,480
And we're going to have to spend
so much less time, especially

747
00:42:11,480 --> 00:42:13,030
as the systems get larger.

748
00:42:13,030 --> 00:42:15,840
So let's look at an example.

749
00:42:15,840 --> 00:42:19,520
So if we have two
atoms of silicon,

750
00:42:19,520 --> 00:42:23,870
DFT and the two sample
quantum chemistry

751
00:42:23,870 --> 00:42:26,300
codes all take about
the same amount of time.

752
00:42:26,300 --> 00:42:28,940
If we try to calculate
100 atoms of silicon,

753
00:42:28,940 --> 00:42:31,980
suddenly DFT is 5 hours,
which might sound long.

754
00:42:31,980 --> 00:42:36,020
But the Moller-Plesset
is a year,

755
00:42:36,020 --> 00:42:39,180
and the coupled cluster
approach is 2,000 years.

756
00:42:39,180 --> 00:42:40,800
I don't think
anybody's tested that,

757
00:42:40,800 --> 00:42:43,700
but I wouldn't recommend it.

758
00:42:43,700 --> 00:42:48,010
But suddenly, you can see why
if you want to do anything

759
00:42:48,010 --> 00:42:52,150
more than just two atoms,
maybe DFT would be a good way

760
00:42:52,150 --> 00:42:54,750
to go about it.

761
00:42:54,750 --> 00:42:59,540
So this is a graph just showing,
again, that DFT scales better

762
00:42:59,540 --> 00:43:01,190
than all the other approaches.

763
00:43:04,450 --> 00:43:07,930
So again, the basic
tenet of DFT is

764
00:43:07,930 --> 00:43:09,850
we've taken this
really complicated,

765
00:43:09,850 --> 00:43:13,000
multi-electron wave
function, which we already

766
00:43:13,000 --> 00:43:15,440
had trouble enough calculating
in the quantum chemistry

767
00:43:15,440 --> 00:43:15,940
method.

768
00:43:15,940 --> 00:43:18,190
I mean, we already
found problems

769
00:43:18,190 --> 00:43:20,470
that if we tried
to simplify this

770
00:43:20,470 --> 00:43:25,240
by making this a function
of each individual electron,

771
00:43:25,240 --> 00:43:29,080
we suddenly lost all
the electron correlation

772
00:43:29,080 --> 00:43:31,840
and exchange, which is
pretty crucial to getting

773
00:43:31,840 --> 00:43:35,150
these calculations right.

774
00:43:35,150 --> 00:43:37,550
And we know that the square
of this wave function

775
00:43:37,550 --> 00:43:39,830
is the electron density anyway.

776
00:43:39,830 --> 00:43:42,665
So let's just think
about the density.

777
00:43:45,850 --> 00:43:50,440
So does anybody know what a
functional is, any math people?

778
00:43:50,440 --> 00:43:51,620
AUDIENCE: [INAUDIBLE].

779
00:43:51,620 --> 00:43:52,370
MICHELE: Great.

780
00:43:52,370 --> 00:43:53,090
Great.

781
00:43:53,090 --> 00:43:58,700
So generally a function, you
input a number and output

782
00:43:58,700 --> 00:43:59,630
a number.

783
00:43:59,630 --> 00:44:03,330
A functional, you input a
function and output a number.

784
00:44:03,330 --> 00:44:05,810
So this is called "density
functional theory,"

785
00:44:05,810 --> 00:44:10,460
because we are writing a
functional of the function

786
00:44:10,460 --> 00:44:12,030
density.

787
00:44:12,030 --> 00:44:14,580
Density is a function.

788
00:44:14,580 --> 00:44:17,190
At every point in space,
there is a particular value

789
00:44:17,190 --> 00:44:19,690
of the density.

790
00:44:19,690 --> 00:44:23,330
And so we would like
to be able to solve

791
00:44:23,330 --> 00:44:28,310
this energy as a
function of the density

792
00:44:28,310 --> 00:44:31,920
or I guess a functional
of the density.

793
00:44:31,920 --> 00:44:34,950
So of course, we're going to
see all the familiar terms

794
00:44:34,950 --> 00:44:37,460
showing up-- the
kinetic energy, the ion

795
00:44:37,460 --> 00:44:40,470
potential, the
ion-electron interaction,

796
00:44:40,470 --> 00:44:41,720
electron-electron interaction.

797
00:44:41,720 --> 00:44:43,762
We're going to see all
the same terms showing up.

798
00:44:43,762 --> 00:44:46,602
And we'll just have to deal
with them slightly differently.

799
00:44:49,440 --> 00:44:52,490
So the way we actually
go about solving this

800
00:44:52,490 --> 00:44:57,560
is we write some equation of
all of the different terms,

801
00:44:57,560 --> 00:45:01,160
as far as we know
what they look like.

802
00:45:01,160 --> 00:45:07,540
We write out the density in
terms of its wave functions,

803
00:45:07,540 --> 00:45:09,250
and then do a
functional derivative

804
00:45:09,250 --> 00:45:11,630
of one of these wave functions.

805
00:45:11,630 --> 00:45:13,620
And so the functional
derivative works--

806
00:45:13,620 --> 00:45:16,940
it's basically just a
variational principle.

807
00:45:16,940 --> 00:45:19,340
Are you guys familiar with
the variational principle

808
00:45:19,340 --> 00:45:20,848
in quantum mechanics?

809
00:45:25,160 --> 00:45:30,150
The lowest energy possible for
a system is the ground state.

810
00:45:30,150 --> 00:45:34,190
And so any time that you can
find any system that gives you

811
00:45:34,190 --> 00:45:37,880
a lower energy, you're getting
closer to the ground state.

812
00:45:37,880 --> 00:45:42,560
So I have my basic
Schrodinger equation.

813
00:45:45,440 --> 00:45:49,040
If I can change something
about this wave function

814
00:45:49,040 --> 00:45:51,950
such that when I solve this
equation I get a lower energy,

815
00:45:51,950 --> 00:45:55,197
I probably have something that's
closer to the true ground state

816
00:45:55,197 --> 00:45:56,030
than I used to have.

817
00:45:56,030 --> 00:45:58,808
Because the smallest
possible value

818
00:45:58,808 --> 00:46:00,350
for this energy that
I can get out of

819
00:46:00,350 --> 00:46:03,020
this equation is going
to be the ground state.

820
00:46:05,940 --> 00:46:09,800
So that's actually used a
lot in quantum mechanics.

821
00:46:09,800 --> 00:46:12,860
So maybe I don't know exactly
what this function looks like,

822
00:46:12,860 --> 00:46:15,320
but I can kind of guess what
this function looks like.

823
00:46:15,320 --> 00:46:21,890
And I can minimize this energy
with respect to what exactly

824
00:46:21,890 --> 00:46:23,900
this wave function looks like.

825
00:46:23,900 --> 00:46:26,320
And that's exactly
what we're doing here.

826
00:46:26,320 --> 00:46:30,182
We're minimizing this
energy the best that we can,

827
00:46:30,182 --> 00:46:32,640
to try to get something that's
as close to the ground state

828
00:46:32,640 --> 00:46:34,620
as we can.

829
00:46:34,620 --> 00:46:38,530
So that leads to the
Kohn-Sham equations.

830
00:46:38,530 --> 00:46:40,610
Again, this is pretty
much the same Hamiltonian

831
00:46:40,610 --> 00:46:44,330
you've been looking at all
class, all last week, too.

832
00:46:44,330 --> 00:46:45,460
We have the kinetic term.

833
00:46:45,460 --> 00:46:49,120
We have the potential term, and
that acts on a wave function,

834
00:46:49,120 --> 00:46:52,780
and gives us some eigenvalue
times the wave function.

835
00:46:52,780 --> 00:46:55,595
So in this case, our
potential is broken up

836
00:46:55,595 --> 00:46:56,970
into a couple of
different terms.

837
00:46:56,970 --> 00:47:00,770
So we've already made the
Born-Oppenheimer approximation.

838
00:47:00,770 --> 00:47:03,610
So we're ignoring the
ion-ion potential.

839
00:47:03,610 --> 00:47:04,870
We can add that in whenever.

840
00:47:04,870 --> 00:47:08,020
That doesn't change.

841
00:47:08,020 --> 00:47:11,430
We have the ion interaction
with the electrons.

842
00:47:11,430 --> 00:47:15,070
We have the electrons
interacting with themselves.

843
00:47:15,070 --> 00:47:17,620
And now we have this new term
that's called the "exchange

844
00:47:17,620 --> 00:47:19,055
correlation potential."

845
00:47:21,730 --> 00:47:25,920
So this is the energy right
here that we were missing before

846
00:47:25,920 --> 00:47:27,780
in the quantum
chemistry approach.

847
00:47:32,932 --> 00:47:34,390
But there's a
problem-- we actually

848
00:47:34,390 --> 00:47:36,520
don't know what this
exchange correlation

849
00:47:36,520 --> 00:47:39,050
function looks like.

850
00:47:39,050 --> 00:47:43,510
So people use a bunch of
different approximations.

851
00:47:43,510 --> 00:47:47,410
The two most commonly used are
the local density approximation

852
00:47:47,410 --> 00:47:50,830
and the general
gradient approximation.

853
00:47:50,830 --> 00:47:54,460
And they're pretty
self-explanatory.

854
00:47:54,460 --> 00:47:56,710
The local density
approximation, you just

855
00:47:56,710 --> 00:48:02,980
approximate the local density
as being a flat in a local area.

856
00:48:02,980 --> 00:48:04,630
And the general
gradient approximation,

857
00:48:04,630 --> 00:48:06,800
you assume a first derivative.

858
00:48:06,800 --> 00:48:17,590
So if you have some potential
that looks like this,

859
00:48:17,590 --> 00:48:22,270
the local density approximation
might approximate it

860
00:48:22,270 --> 00:48:26,920
as looking something like that,
whereas the general gradient

861
00:48:26,920 --> 00:48:30,340
approximation would get
the slopes for each.

862
00:48:33,190 --> 00:48:35,710
So generally, the general
gradient approximation

863
00:48:35,710 --> 00:48:37,557
ends up being a little
bit more accurate,

864
00:48:37,557 --> 00:48:39,640
but they both have their
strengths and weaknesses.

865
00:48:39,640 --> 00:48:41,640
And depending on what
problem you want to solve,

866
00:48:41,640 --> 00:48:44,230
you really need to pay
attention to what you

867
00:48:44,230 --> 00:48:46,910
should expect from either one.

868
00:48:46,910 --> 00:48:52,030
So how does this
process actually work?

869
00:48:52,030 --> 00:48:55,110
So this is a lot like the
solution to the Hartree

870
00:48:55,110 --> 00:48:55,890
equation.

871
00:48:55,890 --> 00:49:02,430
We start with our guess at the
position of the ions, which

872
00:49:02,430 --> 00:49:04,290
may or may not change.

873
00:49:04,290 --> 00:49:06,810
From there, we
also need to start

874
00:49:06,810 --> 00:49:10,080
with a couple initial parameters
that we didn't talk about

875
00:49:10,080 --> 00:49:11,570
for the Hartree case.

876
00:49:11,570 --> 00:49:14,070
So in this case, we're going
to pick a cut-off for the plane

877
00:49:14,070 --> 00:49:14,620
wave basis.

878
00:49:14,620 --> 00:49:16,953
So that's something I'm going
to talk about in a minute.

879
00:49:16,953 --> 00:49:18,480
I haven't talked about that yet.

880
00:49:18,480 --> 00:49:22,100
Then we guess what our electrons
look like in the system.

881
00:49:22,100 --> 00:49:24,440
From that guess
of the electrons,

882
00:49:24,440 --> 00:49:28,730
we calculate the terms in the
Hamiltonian that we're missing.

883
00:49:28,730 --> 00:49:29,900
We solve the Hamiltonian.

884
00:49:29,900 --> 00:49:32,080
We just diagonalize it.

885
00:49:32,080 --> 00:49:37,150
And that diagonalization
gives us a new density.

886
00:49:37,150 --> 00:49:40,270
And then again, we ask,
is this density close

887
00:49:40,270 --> 00:49:42,860
enough to our input density
or is it pretty far away?

888
00:49:42,860 --> 00:49:46,750
And if it's pretty far
away, we go back up,

889
00:49:46,750 --> 00:49:50,140
use that new calculated density,
again to calculate the terms

890
00:49:50,140 --> 00:49:53,250
in the Hamiltonian
that we were missing,

891
00:49:53,250 --> 00:49:56,820
diagonalize the Hamiltonian
again, get a new density.

892
00:49:56,820 --> 00:50:01,020
And we just keep doing this
again and again until we've

893
00:50:01,020 --> 00:50:02,957
found something
that we say is close

894
00:50:02,957 --> 00:50:05,040
enough to the original
density that we've probably

895
00:50:05,040 --> 00:50:07,360
found something that's
a good approximation.

896
00:50:07,360 --> 00:50:09,810
So you'll notice
that this is actually

897
00:50:09,810 --> 00:50:11,160
different from the Hartree.

898
00:50:11,160 --> 00:50:14,370
So this looks very similar to
the solution of the Hartree

899
00:50:14,370 --> 00:50:20,520
equation, but you'll notice that
instead of solving one electron

900
00:50:20,520 --> 00:50:23,550
wave function at
a time, every time

901
00:50:23,550 --> 00:50:25,680
we have to do the
self-consistent cycle.

902
00:50:25,680 --> 00:50:28,800
All we have to do here is just
diagonalize this Hamiltonian.

903
00:50:28,800 --> 00:50:31,230
So this is why the
process speeds up so much.

904
00:50:36,600 --> 00:50:39,090
There are a bunch of
software that people

905
00:50:39,090 --> 00:50:42,030
use for DFT calculations.

906
00:50:42,030 --> 00:50:43,090
Some of them are free.

907
00:50:43,090 --> 00:50:45,030
Some of them are not.

908
00:50:45,030 --> 00:50:47,100
I think you guys
are using Siesta,

909
00:50:47,100 --> 00:50:50,230
which is not on this table.

910
00:50:50,230 --> 00:50:53,580
But all of these
different software

911
00:50:53,580 --> 00:50:56,610
have their strengths
and weaknesses.

912
00:50:56,610 --> 00:50:58,500
And depending on what
you want to calculate,

913
00:50:58,500 --> 00:51:00,480
you should probably
think about which

914
00:51:00,480 --> 00:51:03,160
model is the best for you.

915
00:51:03,160 --> 00:51:09,870
But we're going to be looking
at a PWscf input file,

916
00:51:09,870 --> 00:51:12,300
just to give you a general
idea of what these input files

917
00:51:12,300 --> 00:51:15,673
look like. and I guess pretty
soon you'll find out anyway,

918
00:51:15,673 --> 00:51:17,340
since you'll have to
write some of them.

919
00:51:20,330 --> 00:51:22,830
So I promised you I was going
to talk about basis functions.

920
00:51:22,830 --> 00:51:25,500
That was that one parameter
in the self-consistent cycle

921
00:51:25,500 --> 00:51:28,860
that you have to pick.

922
00:51:28,860 --> 00:51:35,340
So a basis function is
basically just whatever function

923
00:51:35,340 --> 00:51:39,240
you like, what function you want
to deal with mathematically,

924
00:51:39,240 --> 00:51:43,260
that you can write your
wave functions in terms of.

925
00:51:43,260 --> 00:51:47,370
So all we do is we take our wave
functions and we expand them.

926
00:51:47,370 --> 00:51:50,550
So we get some linear
combination of whatever basis

927
00:51:50,550 --> 00:51:51,600
we like.

928
00:51:51,600 --> 00:51:55,710
So usually, people pick
Gaussians, or plane waves,

929
00:51:55,710 --> 00:51:57,690
or things that are
really easy to calculate,

930
00:51:57,690 --> 00:52:03,300
because you can write
any wave function

931
00:52:03,300 --> 00:52:05,760
in terms of a linear
combination of these,

932
00:52:05,760 --> 00:52:08,910
as long as you have
a complete basis.

933
00:52:11,640 --> 00:52:17,310
So we pick these spaces such
that it's orthonormalized.

934
00:52:17,310 --> 00:52:22,950
What we can do from
that is when you

935
00:52:22,950 --> 00:52:25,663
multiply one of
these by another one,

936
00:52:25,663 --> 00:52:26,830
you'll get a delta function.

937
00:52:26,830 --> 00:52:30,750
So basically, if I
multiply through by--

938
00:52:30,750 --> 00:52:33,510
so I have to integrate
over all space

939
00:52:33,510 --> 00:52:36,490
because these functions
exist over all space.

940
00:52:36,490 --> 00:52:37,950
So I integrate over all space.

941
00:52:37,950 --> 00:52:42,690
I pick one of these
particular basic functions

942
00:52:42,690 --> 00:52:44,280
that I multiply by.

943
00:52:44,280 --> 00:52:49,810
And that picks out exactly
the j basis function here.

944
00:52:49,810 --> 00:52:51,720
So this whole term,
when I integrate out,

945
00:52:51,720 --> 00:52:54,870
because it's orthogonal, I
only get the j term here.

946
00:52:54,870 --> 00:52:57,270
Because it's
normalized, I get 1.

947
00:52:57,270 --> 00:53:01,470
So I'm just left with this
E and c j on this side.

948
00:53:01,470 --> 00:53:05,970
On this side, when I do that
same integration over all space

949
00:53:05,970 --> 00:53:11,212
and multiplication by
this j basis function,

950
00:53:11,212 --> 00:53:13,170
I get something more
complicated because I have

951
00:53:13,170 --> 00:53:17,130
this Hamiltonian in between.

952
00:53:17,130 --> 00:53:20,130
So this Hamilton,
this H i j function,

953
00:53:20,130 --> 00:53:26,920
is actually not quite
as simple as it looks.

954
00:53:26,920 --> 00:53:34,820
It's an integration over all
space of the Hamiltonian acting

955
00:53:34,820 --> 00:53:42,280
on the basis set i, and then
the inner product of that

956
00:53:42,280 --> 00:53:47,612
with basis set j, the
complex conjugate.

957
00:53:51,280 --> 00:53:54,310
But this is something
we can calculate.

958
00:53:54,310 --> 00:53:56,290
This actually ends up
being the Hamiltonian

959
00:53:56,290 --> 00:53:58,940
that we diagonalize in the end.

960
00:53:58,940 --> 00:54:02,260
So you can see that the choice
of these basis functions,

961
00:54:02,260 --> 00:54:05,110
to make it something that's
easy to calculate this term,

962
00:54:05,110 --> 00:54:08,650
is going to be really crucial
for getting the shortest

963
00:54:08,650 --> 00:54:10,450
possible computation time.

964
00:54:16,570 --> 00:54:22,310
One that's commonly
used is plane waves.

965
00:54:22,310 --> 00:54:25,220
This is a plane wave.

966
00:54:25,220 --> 00:54:31,670
The exact form of the ones
we use, e to the i G j dot r.

967
00:54:31,670 --> 00:54:41,660
You guys know the e to
the i x is equal to cosine

968
00:54:41,660 --> 00:54:43,820
x plus i sine x.

969
00:54:51,867 --> 00:54:53,950
So you can see why these
are called "plane waves,"

970
00:54:53,950 --> 00:54:59,470
because in real space, they're
basically just some wave.

971
00:55:06,550 --> 00:55:09,010
So this r is the
real space vector.

972
00:55:09,010 --> 00:55:11,770
What we have to do
is pick what j's

973
00:55:11,770 --> 00:55:13,700
we're going to use
in our simulation.

974
00:55:13,700 --> 00:55:17,830
So really, if we're going to
expand any function in terms

975
00:55:17,830 --> 00:55:21,170
of this basis, we need
the spaces to be complete,

976
00:55:21,170 --> 00:55:25,010
which means basically
what it says.

977
00:55:25,010 --> 00:55:30,340
It's "complete," meaning you can
expand anything in terms of it.

978
00:55:30,340 --> 00:55:33,610
But we don't really want
a complete basis set,

979
00:55:33,610 --> 00:55:35,250
because for this to
be complete, this G

980
00:55:35,250 --> 00:55:36,580
would have to go to infinity.

981
00:55:36,580 --> 00:55:41,230
We'd have to go for every number
for G between 0 and infinity,

982
00:55:41,230 --> 00:55:45,280
and we don't have the
computational time to do that,

983
00:55:45,280 --> 00:55:48,580
because the size of our
Hamiltonian that we diagonalize

984
00:55:48,580 --> 00:55:51,910
depends on the number of these
functions that we put in.

985
00:55:51,910 --> 00:55:56,020
So what we hope to do
is pick a maximum G

986
00:55:56,020 --> 00:56:00,900
that's sufficient to capture
all of the physics that happens,

987
00:56:00,900 --> 00:56:03,360
but not so big that our
computational time takes

988
00:56:03,360 --> 00:56:04,210
forever.

989
00:56:04,210 --> 00:56:09,150
So as I increase G, I
get shorter and shorter

990
00:56:09,150 --> 00:56:10,590
wavelengths.

991
00:56:10,590 --> 00:56:13,380
And at some point,
this wavelength

992
00:56:13,380 --> 00:56:14,970
is going to be so
short that it's

993
00:56:14,970 --> 00:56:19,350
going to represent a really
high kinetic energy system.

994
00:56:19,350 --> 00:56:21,270
And it's going to
be just un-physical.

995
00:56:21,270 --> 00:56:27,030
I mean, if I have an atom here,
an atom here, an atom here,

996
00:56:27,030 --> 00:56:29,040
probably the variation
in the electron density

997
00:56:29,040 --> 00:56:33,040
is not this great right
around that nucleus.

998
00:56:33,040 --> 00:56:36,630
And so at some point,
we can just cut off G,

999
00:56:36,630 --> 00:56:38,665
and say we have enough.

1000
00:56:38,665 --> 00:56:40,290
We can probably
capture all the physics

1001
00:56:40,290 --> 00:56:47,330
we need just going up
to the G that we have.

1002
00:56:47,330 --> 00:56:50,740
So you can see actually how this
plane wave basis works really

1003
00:56:50,740 --> 00:56:53,020
well for periodic crystals.

1004
00:56:53,020 --> 00:56:57,680
I just drew nuclei here.

1005
00:56:57,680 --> 00:57:02,780
So when you have an
infinite crystal,

1006
00:57:02,780 --> 00:57:09,920
you have a nucleus here, a
nucleus here, a nucleus here.

1007
00:57:09,920 --> 00:57:11,930
And because they're
all the same,

1008
00:57:11,930 --> 00:57:14,180
you don't expect that the
wave function or the density

1009
00:57:14,180 --> 00:57:16,370
should be different
at this nucleus

1010
00:57:16,370 --> 00:57:18,380
than it is at this nucleus.

1011
00:57:18,380 --> 00:57:22,870
So this plane wave is actually
a great representation of that

1012
00:57:22,870 --> 00:57:26,050
because the density is the
same here as it is here,

1013
00:57:26,050 --> 00:57:27,130
as it'll be here.

1014
00:57:27,130 --> 00:57:30,520
And so these plane
waves are actually

1015
00:57:30,520 --> 00:57:34,440
ideal for calculating
periodic crystals.

1016
00:57:34,440 --> 00:57:36,690
So how are we going
to attack molecules?

1017
00:57:42,240 --> 00:57:45,700
What we're going to do is
put these molecules in a box.

1018
00:57:45,700 --> 00:57:48,900
We're going to put them in a
box with some amount of vacuum

1019
00:57:48,900 --> 00:57:54,630
on either side, and then repeat
that box in all directions.

1020
00:57:54,630 --> 00:58:01,010
In x, in y, in z, we're going
to infinitely repeat this box.

1021
00:58:01,010 --> 00:58:04,810
And so now instead of having
an atom here, an atom here,

1022
00:58:04,810 --> 00:58:06,210
an atom here,
we're going to have

1023
00:58:06,210 --> 00:58:11,240
a molecule here, and a molecule
here, and a molecule here.

1024
00:58:11,240 --> 00:58:13,300
So you can see how
maybe with this,

1025
00:58:13,300 --> 00:58:15,460
these plane waves might
represent that system

1026
00:58:15,460 --> 00:58:16,470
a little bit better.

1027
00:58:16,470 --> 00:58:18,220
But there are some
things you have to take

1028
00:58:18,220 --> 00:58:19,510
into consideration with this.

1029
00:58:22,630 --> 00:58:25,240
Since there's a molecule here
and some distance away there's

1030
00:58:25,240 --> 00:58:27,620
another molecule, they might
interact with each other.

1031
00:58:27,620 --> 00:58:29,890
And if you want just the
properties of that molecule

1032
00:58:29,890 --> 00:58:32,650
by itself in vacuum, you
don't want them interacting

1033
00:58:32,650 --> 00:58:34,590
with other molecules.

1034
00:58:34,590 --> 00:58:36,130
So you need to make
sure that you've

1035
00:58:36,130 --> 00:58:38,770
included enough vacuum
between these molecules,

1036
00:58:38,770 --> 00:58:42,370
between the molecule and
the edge of its unit cell,

1037
00:58:42,370 --> 00:58:46,280
that the interaction
will be negligible

1038
00:58:46,280 --> 00:58:47,920
or will be on the
order of the error

1039
00:58:47,920 --> 00:58:51,290
that you have from everything
else in your calculation.

1040
00:58:51,290 --> 00:58:53,080
So once we've done
this, we can then

1041
00:58:53,080 --> 00:58:55,400
start our self-consistent loop.

1042
00:58:55,400 --> 00:58:57,280
So what happens?

1043
00:58:57,280 --> 00:59:02,380
We put in where the ions are.

1044
00:59:02,380 --> 00:59:04,040
We calculate our potential.

1045
00:59:04,040 --> 00:59:08,637
We've guessed at some
original electron density.

1046
00:59:08,637 --> 00:59:10,720
We've calculated all the
terms in the Hamiltonian.

1047
00:59:10,720 --> 00:59:12,610
We've diagonalized
that Hamiltonian.

1048
00:59:12,610 --> 00:59:16,090
That diagonalization
gives us our energy.

1049
00:59:16,090 --> 00:59:20,590
And then the program said, well,
that new density wasn't really

1050
00:59:20,590 --> 00:59:21,650
close to the old density.

1051
00:59:21,650 --> 00:59:22,608
So let's do that again.

1052
00:59:22,608 --> 00:59:25,010
And it does that again,
and calculates energy.

1053
00:59:25,010 --> 00:59:29,090
And you can see how the
energy always goes down.

1054
00:59:29,090 --> 00:59:30,470
So this is a negative number.

1055
00:59:30,470 --> 00:59:31,798
The energy always goes down.

1056
00:59:31,798 --> 00:59:33,340
So that's the
variational principle--

1057
00:59:33,340 --> 00:59:35,500
as you approach the
correct ground state,

1058
00:59:35,500 --> 00:59:37,282
you get lower in energy.

1059
00:59:37,282 --> 00:59:38,740
And you can see
that the difference

1060
00:59:38,740 --> 00:59:41,550
between these successively
gets smaller and smaller.

1061
00:59:41,550 --> 00:59:44,050
And that's what you really want
to have happen, because that

1062
00:59:44,050 --> 00:59:46,180
means that your
energy is converging,

1063
00:59:46,180 --> 00:59:49,420
and you'll end up with a
pretty good approximation

1064
00:59:49,420 --> 00:59:51,980
of the ground state energy.

1065
00:59:51,980 --> 00:59:54,800
You've also ended up
with the charge density,

1066
00:59:54,800 --> 00:59:57,320
because that's what you've been
calculating this whole time

1067
00:59:57,320 --> 00:59:59,730
to calculate the energy.

1068
00:59:59,730 --> 01:00:01,890
So just with those
two properties,

1069
01:00:01,890 --> 01:00:04,050
and being kind of
clever, you can actually

1070
01:00:04,050 --> 01:00:08,070
calculate a huge range of
important molecular and solid

1071
01:00:08,070 --> 01:00:08,590
properties.

1072
01:00:08,590 --> 01:00:11,400
So you calculate at least
you can guess at a structure.

1073
01:00:11,400 --> 01:00:14,130
You can calculate bulk
modulus, shear modulus,

1074
01:00:14,130 --> 01:00:15,750
elastic constants.

1075
01:00:15,750 --> 01:00:19,710
You can calculate vibrational
properties and sound velocity.

1076
01:00:19,710 --> 01:00:23,550
Binding energy is
actually just probably

1077
01:00:23,550 --> 01:00:27,480
going to be this energy
that you've calculated.

1078
01:00:27,480 --> 01:00:29,430
You can kind of guess
at reaction pathways.

1079
01:00:29,430 --> 01:00:32,430
Those are a little bit harder
to do with this approach.

1080
01:00:32,430 --> 01:00:34,770
You can calculate
forces, pressure, stress.

1081
01:00:34,770 --> 01:00:36,990
You can calculate all
of these properties just

1082
01:00:36,990 --> 01:00:41,280
with this simple calculation
that we can now do.

1083
01:00:41,280 --> 01:00:44,350
But how do we know that these
calculations are meaningful?

1084
01:00:44,350 --> 01:00:47,130
How do we know that we've
actually said anything useful?

1085
01:00:47,130 --> 01:00:50,430
Dirac was complaining
that he found

1086
01:00:50,430 --> 01:00:52,680
that his calculations never
really matched experiment,

1087
01:00:52,680 --> 01:00:54,690
but don't lose heart.

1088
01:00:54,690 --> 01:00:58,350
But when should you lose heart?

1089
01:00:58,350 --> 01:01:01,830
Or when should you
maybe check to make sure

1090
01:01:01,830 --> 01:01:05,070
that what you're doing is
as accurate as you can get?

1091
01:01:05,070 --> 01:01:07,970
We need to talk
about convergence.

1092
01:01:07,970 --> 01:01:10,490
So there are a couple
of different properties

1093
01:01:10,490 --> 01:01:15,530
that you need to worry about.

1094
01:01:15,530 --> 01:01:17,660
We talked about is
the basis big enough.

1095
01:01:17,660 --> 01:01:20,660
So that's, did I
pick a high enough G

1096
01:01:20,660 --> 01:01:26,600
that I got enough wiggliness
of my wave functions, my input

1097
01:01:26,600 --> 01:01:29,480
basis wave functions,
that I've captured

1098
01:01:29,480 --> 01:01:30,930
all the relevant physics?

1099
01:01:30,930 --> 01:01:33,710
And so this is the trade-off.

1100
01:01:33,710 --> 01:01:37,460
It takes more time if you
have a larger basis set,

1101
01:01:37,460 --> 01:01:40,520
but you also lose some
accuracy by decreasing

1102
01:01:40,520 --> 01:01:42,020
the size of your basis set.

1103
01:01:42,020 --> 01:01:46,910
And at some point,
when you've taken out

1104
01:01:46,910 --> 01:01:48,942
too many basic
functions, you suddenly

1105
01:01:48,942 --> 01:01:50,900
get something that's not
going to be meaningful

1106
01:01:50,900 --> 01:01:53,540
at all because you just don't
have the right structure

1107
01:01:53,540 --> 01:01:54,560
of the potential.

1108
01:01:57,410 --> 01:01:59,750
We talked about is
the box big enough.

1109
01:01:59,750 --> 01:02:02,000
Did you include enough
vacuum so that you

1110
01:02:02,000 --> 01:02:06,110
don't get interactions between
molecules in neighboring unit

1111
01:02:06,110 --> 01:02:06,710
cells?

1112
01:02:06,710 --> 01:02:09,620
Because if you want the
properties of the molecule

1113
01:02:09,620 --> 01:02:11,540
by itself in vacuum,
you don't want

1114
01:02:11,540 --> 01:02:14,270
the interaction of those two
molecules to play a part.

1115
01:02:14,270 --> 01:02:18,480
But here we get, actually, a
trade-off with the basis set.

1116
01:02:18,480 --> 01:02:20,690
So the bigger you make
your box, the bigger

1117
01:02:20,690 --> 01:02:22,790
you need your basis set to be.

1118
01:02:22,790 --> 01:02:32,210
So you can see if this is
the size of my box here,

1119
01:02:32,210 --> 01:02:33,290
and this is my cut-off--

1120
01:02:33,290 --> 01:02:34,820
G has a wavelength of this--

1121
01:02:34,820 --> 01:02:37,910
if I suddenly make
my box much bigger,

1122
01:02:37,910 --> 01:02:39,740
this wavelength
is going to expand

1123
01:02:39,740 --> 01:02:45,300
and it's going to be a
much bigger wavelength.

1124
01:02:45,300 --> 01:02:47,570
And so maybe now
this is actually

1125
01:02:47,570 --> 01:02:49,640
an important wavelength
to my system.

1126
01:02:49,640 --> 01:02:52,040
And maybe even a
smaller wavelength

1127
01:02:52,040 --> 01:02:55,440
is an important
wavelength to my system.

1128
01:02:55,440 --> 01:02:57,360
Does that make sense?

1129
01:02:57,360 --> 01:02:59,880
OK, so making your
box bigger means

1130
01:02:59,880 --> 01:03:01,710
that you need a bigger
basis set, which

1131
01:03:01,710 --> 01:03:07,900
means more computational
time, but on the other hand,

1132
01:03:07,900 --> 01:03:10,200
you don't want to include
the effects of intermolecular

1133
01:03:10,200 --> 01:03:11,520
interaction.

1134
01:03:11,520 --> 01:03:16,350
Finally, you can ask if you
exited the self-consistent loop

1135
01:03:16,350 --> 01:03:18,360
at the right time.

1136
01:03:18,360 --> 01:03:21,932
So I keep saying, well,
we compare the old density

1137
01:03:21,932 --> 01:03:23,890
to the new density and
if they're close enough,

1138
01:03:23,890 --> 01:03:25,110
then we assume it's right.

1139
01:03:25,110 --> 01:03:28,170
But what's that definition
of "close enough?"

1140
01:03:28,170 --> 01:03:31,200
And how do you know that that
definition of "close enough"

1141
01:03:31,200 --> 01:03:32,907
is close enough?

1142
01:03:32,907 --> 01:03:34,740
So that's another
property that you're going

1143
01:03:34,740 --> 01:03:35,823
to have to converge along.

1144
01:03:38,550 --> 01:03:42,980
So let's just take a look
at an example input file,

1145
01:03:42,980 --> 01:03:45,410
and see how some of these
properties that you need

1146
01:03:45,410 --> 01:03:47,900
to converge along, and the
other properties you might care

1147
01:03:47,900 --> 01:03:51,750
about, show up in this system.

1148
01:03:51,750 --> 01:03:57,930
So we might care about
putting in the right atoms,

1149
01:03:57,930 --> 01:03:59,670
talk about how the
atoms are related

1150
01:03:59,670 --> 01:04:03,000
to each other spatially,
and then these three

1151
01:04:03,000 --> 01:04:08,030
parameters that we talked about
that we need to converge along.

1152
01:04:08,030 --> 01:04:13,430
So this is just a
sample of water.

1153
01:04:13,430 --> 01:04:16,880
And so here, this is
the size of the cell.

1154
01:04:16,880 --> 01:04:20,710
This is the number of atoms,
number of types of atoms.

1155
01:04:20,710 --> 01:04:25,790
This is ecut, this
is the energy cutoff,

1156
01:04:25,790 --> 01:04:29,636
which refers to
the size of your G,

1157
01:04:29,636 --> 01:04:33,630
and then your atoms and
the atomic positions.

1158
01:04:33,630 --> 01:04:36,790
And these are just
the pseudo potentials.

1159
01:04:36,790 --> 01:04:39,690
So we haven't talked at all
about pseudo potentials,

1160
01:04:39,690 --> 01:04:44,190
but just so you know,
we talked about the fact

1161
01:04:44,190 --> 01:04:50,240
that when we think
about atoms we only care

1162
01:04:50,240 --> 01:04:52,820
about the valence electrons.

1163
01:04:52,820 --> 01:04:55,070
So all those core
electrons still exist,

1164
01:04:55,070 --> 01:04:56,640
but we don't want
to calculate them.

1165
01:04:56,640 --> 01:04:59,600
So we use pseudo potentials
to sort of capture

1166
01:04:59,600 --> 01:05:01,370
those inner electrons
a little bit,

1167
01:05:01,370 --> 01:05:04,713
and tell us how those
inner electrons are

1168
01:05:04,713 --> 01:05:07,130
going to affect the valence
electrons and everything else.

1169
01:05:07,130 --> 01:05:13,100
Because when we
have an atom that

1170
01:05:13,100 --> 01:05:16,110
has some core of electrons--

1171
01:05:16,110 --> 01:05:20,510
so let's say this
has 27 protons in it.

1172
01:05:20,510 --> 01:05:22,220
An electron out
here is not going

1173
01:05:22,220 --> 01:05:25,970
to feel that 27 proton charge.

1174
01:05:25,970 --> 01:05:28,940
It's going to feel that
27 protons shielded

1175
01:05:28,940 --> 01:05:31,950
by some number of
core electrons.

1176
01:05:31,950 --> 01:05:38,300
And so to get the Coulomb
interaction right there,

1177
01:05:38,300 --> 01:05:41,150
we need to have some
kind of modified version

1178
01:05:41,150 --> 01:05:45,170
of what exactly our atoms'
potential looks like.

1179
01:05:45,170 --> 01:05:49,050
So that's what the pseudo
potentials are about.

1180
01:05:49,050 --> 01:05:53,010
So that's it.

1181
01:05:53,010 --> 01:05:57,420
We've covered the basis of
quantum chemistry and density

1182
01:05:57,420 --> 01:05:59,500
functional theory today.

1183
01:05:59,500 --> 01:06:01,040
Are there any questions?

1184
01:06:06,680 --> 01:06:08,890
All right, thank you.