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00:00:13,945 --> 00:00:15,320
GERBRAND CEDER:
So we'll do a lab

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00:00:15,320 --> 00:00:19,490
that essentially covers
modeling things with potentials.

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00:00:19,490 --> 00:00:22,700
You do such things like
calculate a vacancy formation

4
00:00:22,700 --> 00:00:25,620
energy, a surface energy.

5
00:00:25,620 --> 00:00:30,080
So the idea will be that
working with a simple energy

6
00:00:30,080 --> 00:00:32,090
model, first of all,
you learn the mechanics

7
00:00:32,090 --> 00:00:34,040
of these kind of
calculations, building

8
00:00:34,040 --> 00:00:38,850
supercells with a defect,
studying convergence of them.

9
00:00:38,850 --> 00:00:40,380
So that's one thing.

10
00:00:40,380 --> 00:00:43,910
You'll find that the hard
part is often figuring out

11
00:00:43,910 --> 00:00:46,910
exactly what calculation
to do and how

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00:00:46,910 --> 00:00:51,290
to put the numbers together
to get a physical quantity.

13
00:00:51,290 --> 00:00:53,450
And then you'll
also learn something

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00:00:53,450 --> 00:00:56,030
about the things we talked
about here, what kind

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00:00:56,030 --> 00:00:57,925
of results potentials give.

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00:00:57,925 --> 00:00:59,300
For example, you
[INAUDIBLE] tell

17
00:00:59,300 --> 00:01:03,050
you will find that the vacancy
formation energy is very

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00:01:03,050 --> 00:01:06,900
close to the cohesive energy,
something we discussed,

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00:01:06,900 --> 00:01:08,840
we talked about.

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00:01:08,840 --> 00:01:11,820
So those are the
things for Thursday.

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00:01:11,820 --> 00:01:14,233
And I think you'll have
about two weeks to finish

22
00:01:14,233 --> 00:01:16,400
the assignment, although
the first one is really not

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00:01:16,400 --> 00:01:17,120
that intensive.

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00:01:17,120 --> 00:01:21,163
You'll probably be able to
do it considerably faster.

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00:01:21,163 --> 00:01:22,580
So what I want to
do today is just

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00:01:22,580 --> 00:01:27,290
finish up empirical
energy models

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00:01:27,290 --> 00:01:29,870
by focusing on organic
materials and oxides.

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And I'll only do about
half of the lecture,

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00:01:31,940 --> 00:01:34,580
and then Professor Marzari
will give an introduction

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00:01:34,580 --> 00:01:38,610
to the next part, which
is the ab initio method.

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00:01:38,610 --> 00:01:40,610
So he'll start with some
work, some introduction

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00:01:40,610 --> 00:01:43,280
to quantum mechanics, and then
lead that for the next two

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00:01:43,280 --> 00:01:48,240
lectures into an initio methods
for energy calculations.

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00:01:48,240 --> 00:01:48,740
OK.

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00:01:51,890 --> 00:01:54,930
If you sort of
remember this diagram,

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00:01:54,930 --> 00:01:59,780
so what we did last time, we
talked about pair potentials,

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00:01:59,780 --> 00:02:02,940
discussed sort of the
formal way why they fail

38
00:02:02,940 --> 00:02:05,840
when things are very covalent.

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00:02:05,840 --> 00:02:07,790
And then I spent most
of the lecture talking

40
00:02:07,790 --> 00:02:12,200
about pair functionals,
things like the embedded atom

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00:02:12,200 --> 00:02:14,000
method, the blue model.

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00:02:14,000 --> 00:02:17,130
And we talked a little bit
about cluster potentials.

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00:02:17,130 --> 00:02:17,690
OK.

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00:02:17,690 --> 00:02:20,300
So I showed you the example
of a three-body potential

45
00:02:20,300 --> 00:02:23,480
for silicon.

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00:02:23,480 --> 00:02:26,120
Today, I want to talk
about organic systems

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00:02:26,120 --> 00:02:28,280
and Oxides Let me start
with organic systems,

48
00:02:28,280 --> 00:02:33,050
where in organic systems
potential modeling,

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00:02:33,050 --> 00:02:37,620
or call it empirical function
modeling, works extremely well.

50
00:02:37,620 --> 00:02:40,380
And there's a very
good reason for that.

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00:02:40,380 --> 00:02:42,530
And I'll come to that.

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00:02:42,530 --> 00:02:44,180
You tend to distinguish
between what

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00:02:44,180 --> 00:02:48,060
are called bonded and
non-bonded interactions.

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00:02:48,060 --> 00:02:51,410
So the bonded ones are the
one along covalent bonds.

55
00:02:51,410 --> 00:02:55,340
So let's say you had
two methane molecules

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00:02:55,340 --> 00:03:02,030
interacting, so CH4 and CH4.

57
00:03:07,810 --> 00:03:10,120
So the bonded interactions
would, for example,

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00:03:10,120 --> 00:03:11,290
be the ones here.

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00:03:11,290 --> 00:03:12,680
That's the only ones.

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00:03:12,680 --> 00:03:16,390
Say, the carbon-hydrogen bond
would be a bonded interaction.

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00:03:16,390 --> 00:03:19,100
The non-bonded interaction
are than the other ones,

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00:03:19,100 --> 00:03:21,650
the ones that are not
between covalent bonds.

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00:03:21,650 --> 00:03:24,760
So for example, this
hydrogen-hydrogen

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00:03:24,760 --> 00:03:26,860
would interact through
a non-bonded term.

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00:03:32,360 --> 00:03:34,925
And this becomes important
when you make interactions,

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00:03:34,925 --> 00:03:37,910
say, between long chain
molecules or polymers.

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00:03:37,910 --> 00:03:42,710
If you did sort of one
polymer and another one,

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00:03:42,710 --> 00:03:45,540
you would have bonded
interactions along the chain.

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00:03:45,540 --> 00:03:47,420
And you would have
non-bonded interactions

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00:03:47,420 --> 00:03:50,660
between different chain parts,
between different chains,

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00:03:50,660 --> 00:03:53,060
or between one part of
the chain and another

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00:03:53,060 --> 00:03:56,630
if it sort of were
to fold on itself.

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00:03:56,630 --> 00:03:59,300
So keep that in mind, bonded
and non-bonded interactions.

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00:03:59,300 --> 00:04:04,490
So I wanted to do a
short exercise in class

75
00:04:04,490 --> 00:04:08,000
making you think through
the relevant potential terms

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00:04:08,000 --> 00:04:12,680
you would need if you
wanted to, say, study water.

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00:04:12,680 --> 00:04:17,360
So here's water, H2O, one
oxygen, two hydrogens.

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00:04:17,360 --> 00:04:21,200
So I also want to make
you think through,

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00:04:21,200 --> 00:04:22,880
so that you realize
that it's really not

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00:04:22,880 --> 00:04:27,440
that hard to make a sort of
reasonable potential model

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00:04:27,440 --> 00:04:28,490
for an organic.

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00:04:28,490 --> 00:04:29,910
So let me make a wide slide.

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00:04:29,910 --> 00:04:32,980
So I got to erase that.

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00:04:37,230 --> 00:04:38,250
How do we erase here?

85
00:04:40,948 --> 00:04:43,130
There we go.

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00:04:43,130 --> 00:04:47,360
It's the very tedious
way of erasing.

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00:04:47,360 --> 00:04:47,860
OK.

88
00:04:51,170 --> 00:04:53,400
So who wants to start?

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00:04:53,400 --> 00:04:56,300
What should go in?

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00:04:56,300 --> 00:04:57,190
So we have water.

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00:05:01,970 --> 00:05:03,410
Think about it.

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00:05:03,410 --> 00:05:06,290
Let's say I was doing
water in the liquid state.

93
00:05:06,290 --> 00:05:07,940
So I have a bunch
of water molecules.

94
00:05:07,940 --> 00:05:09,850
So what kind of
interactions would I need?

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00:05:12,880 --> 00:05:13,380
Yes.

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00:05:13,380 --> 00:05:14,672
AUDIENCE: They all have dipole.

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00:05:14,672 --> 00:05:16,588
GERBRAND CEDER: Exactly,
they all have dipole.

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00:05:16,588 --> 00:05:17,890
It's a good one to start with.

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00:05:17,890 --> 00:05:21,930
So there's a small
amount of positive charge

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00:05:21,930 --> 00:05:23,580
on the hydrogens.

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00:05:23,580 --> 00:05:25,857
And there's some amount
of negative charge,

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00:05:25,857 --> 00:05:28,440
which is 2 times the amount of
positive charge on the protons.

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00:05:28,440 --> 00:05:29,315
So you have a dipole.

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00:05:33,350 --> 00:05:34,970
So you have a dipole on the two.

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00:05:34,970 --> 00:05:38,740
How would you model
that interaction?

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00:05:38,740 --> 00:05:39,650
AUDIENCE: [INAUDIBLE]

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00:05:39,650 --> 00:05:40,733
GERBRAND CEDER: I'm sorry?

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00:05:40,733 --> 00:05:42,107
AUDIENCE: [INAUDIBLE]

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00:05:42,107 --> 00:05:43,190
GERBRAND CEDER: I'm sorry.

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00:05:43,190 --> 00:05:43,732
I can't hear.

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AUDIENCE: Lennard-Jones.

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00:05:45,915 --> 00:05:48,040
GERBRAND CEDER: I wouldn't
do a Lennard-Jones model

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for the dipole
because the dipole is

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essentially electrostatic.

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00:05:52,603 --> 00:05:54,520
So there's sort of two
ways you could do this.

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You could set up directly a
dipole-dipole interaction,

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00:06:00,100 --> 00:06:05,460
where you centered the dipole
somewhere on the center of mass

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00:06:05,460 --> 00:06:06,970
of the molecule.

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00:06:06,970 --> 00:06:10,800
Or you could treat the charged
components of the dipole

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00:06:10,800 --> 00:06:13,050
explicitly.

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So you could say, you know, I
have a delta plus here, delta

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plus here.

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00:06:17,940 --> 00:06:19,530
I have a delta minus here.

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And that would interact
electrostatically.

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00:06:25,920 --> 00:06:30,720
And you could set up a direct
electrostatic interaction.

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00:06:30,720 --> 00:06:35,340
The dipole-dipole one is
probably easier to implement,

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00:06:35,340 --> 00:06:37,080
is also somewhat shorter range.

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But both would work.

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00:06:39,540 --> 00:06:43,920
If you treat the
molecule as rigid,

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then the dipole approximation
and the electrostatic one,

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where you take the
charge individually,

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is essentially the same.

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00:06:51,030 --> 00:06:54,030
But if the molecule,
of course, can flap,

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then the dipole moment is
changing on that molecule.

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So that's definitely one term.

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So what else would you use?

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00:07:03,154 --> 00:07:03,654
[INAUDIBLE]?

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AUDIENCE: OH bonding?

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00:07:06,050 --> 00:07:08,120
GERBRAND CEDER: OH bonding, OK.

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00:07:08,120 --> 00:07:13,070
You guys take all the hard
ones, all the hard ones first.

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00:07:13,070 --> 00:07:15,870
Yeah, so hydrogen bonding--

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00:07:15,870 --> 00:07:22,550
so if there's an oxygen here and
there's a water sort of here,

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00:07:22,550 --> 00:07:23,770
you have hydrogen bonding.

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What will you use for that?

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What kind of form?

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Any form is as good
as any other one.

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People sometimes just use
Lennard-Jones potentials

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00:07:42,990 --> 00:07:46,215
for that, but you can
use something else.

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So what else would you use?

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00:07:54,513 --> 00:07:55,680
What else should go in here?

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00:08:01,430 --> 00:08:05,330
Well, if you don't assume your
hydrogen molecule is rigid,

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00:08:05,330 --> 00:08:08,990
then you definitely need
stuff for the bonds there.

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00:08:08,990 --> 00:08:18,330
So you may need an OH bond
potential, which obviously

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00:08:18,330 --> 00:08:25,600
would have a bond stretching
part and then probably

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a bending part, so OH bending.

156
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What would you make these?

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00:08:37,875 --> 00:08:39,690
What kind of forms
would you use?

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00:08:47,190 --> 00:08:50,570
Too early in the
morning for this?

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00:08:50,570 --> 00:08:53,510
I mean, you know,
I would always say,

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00:08:53,510 --> 00:08:56,600
before you get into complicated
models, take the simplest form

161
00:08:56,600 --> 00:08:57,650
and see where you get.

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00:08:57,650 --> 00:08:59,330
For bending, you
could literally take

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00:08:59,330 --> 00:09:01,730
a quadratic in the bond angle.

164
00:09:04,580 --> 00:09:07,610
That allows you to do reasonable
stretch around the equilibrium

165
00:09:07,610 --> 00:09:08,420
geometry.

166
00:09:08,420 --> 00:09:11,750
And a water molecule
does not flop over.

167
00:09:11,750 --> 00:09:15,860
It's hydrogens don't really
flop over that easily.

168
00:09:15,860 --> 00:09:19,540
Stretch, you could also, again,
use just a spring constant,

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00:09:19,540 --> 00:09:23,160
so r minus r0 squared.

170
00:09:23,160 --> 00:09:27,300
And then if you feel you go too
far away out of the equilibrium

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00:09:27,300 --> 00:09:29,910
bond length, you could have
more complicated potentials.

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00:09:29,910 --> 00:09:35,790
But an OH bond is quite
stiff, so it's not

173
00:09:35,790 --> 00:09:40,047
going to vibrate with
an enormous amplitude.

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00:09:40,047 --> 00:09:41,130
So what else will we need?

175
00:09:48,320 --> 00:09:51,050
I think we're sort of
missing one essential term.

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00:09:51,050 --> 00:09:53,510
We have very few
non-bonded terms

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00:09:53,510 --> 00:09:58,060
We only have the attractive
part of the non-bonded terms,

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00:09:58,060 --> 00:10:01,070
the dipole-dipole and
the electrostatic.

179
00:10:01,070 --> 00:10:04,070
But there has to be
some amount of repulsion

180
00:10:04,070 --> 00:10:07,340
between the hydrogens.

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00:10:07,340 --> 00:10:11,900
These cannot come infinitely
close together and the same

182
00:10:11,900 --> 00:10:14,900
for the oxygen. So you need
some non-bonded van der Waals

183
00:10:14,900 --> 00:10:15,710
interactions.

184
00:10:31,610 --> 00:10:32,900
OK.

185
00:10:32,900 --> 00:10:36,830
And those you could easily
give a Lennard-Jones for.

186
00:10:36,830 --> 00:10:38,480
It seems like,
when you have that,

187
00:10:38,480 --> 00:10:43,160
you have a somewhat reasonable
approximation for water.

188
00:10:43,160 --> 00:10:43,850
OK.

189
00:10:43,850 --> 00:10:47,055
If you do all this and
you parameterize it well,

190
00:10:47,055 --> 00:10:47,930
what are you missing?

191
00:10:53,840 --> 00:10:56,450
Yeah-- which is kind
of a vague terminology.

192
00:10:56,450 --> 00:10:59,840
But you know, I think in
particular what you're missing

193
00:10:59,840 --> 00:11:04,080
is charge transfer,
which can occur in water.

194
00:11:04,080 --> 00:11:09,080
You can also temporarily
form H3O+, which is, I think,

195
00:11:09,080 --> 00:11:11,180
what's called a hydronium
or something like that.

196
00:11:11,180 --> 00:11:12,020
Hydronium?

197
00:11:12,020 --> 00:11:14,180
Yeah, hydronium ion.

198
00:11:14,180 --> 00:11:18,110
So the sort of more
subtle electronic transfer

199
00:11:18,110 --> 00:11:19,430
effects you would be missing.

200
00:11:19,430 --> 00:11:21,150
And you'd have to do
quantum mechanics for that.

201
00:11:21,150 --> 00:11:22,310
And even quantum
mechanics, that wouldn't

202
00:11:22,310 --> 00:11:24,080
be all that easy to get right.

203
00:11:27,940 --> 00:11:28,440
OK.

204
00:11:40,650 --> 00:11:43,770
You know, I plotted here
the bending term for water,

205
00:11:43,770 --> 00:11:48,660
so you get some idea of how
reasonable quadratics are.

206
00:11:48,660 --> 00:11:54,230
Here's the exact form,
exact essentially

207
00:11:54,230 --> 00:11:58,010
as probably as determined
from quantum mechanics.

208
00:11:58,010 --> 00:12:01,340
If you fit a harmonic through,
so this is your quadratic.

209
00:12:01,340 --> 00:12:07,250
This is k times theta
minus theta 0 squared.

210
00:12:07,250 --> 00:12:14,050
If you see, you're pretty
well within the range

211
00:12:14,050 --> 00:12:18,350
of the exact potential
for at least 10,

212
00:12:18,350 --> 00:12:21,850
15 degrees on each end of
the equilibrium bond angle.

213
00:12:21,850 --> 00:12:23,980
And even then you're
not too far off.

214
00:12:23,980 --> 00:12:27,040
This is in
kilocalories per mole.

215
00:12:27,040 --> 00:12:29,680
I would say even at
this level you're

216
00:12:29,680 --> 00:12:34,990
kind of less than a
kilocalorie per mole off.

217
00:12:34,990 --> 00:12:38,060
Notice that a quadratic,
by its nature,

218
00:12:38,060 --> 00:12:42,520
it's always your potential
is too weak on one side

219
00:12:42,520 --> 00:12:44,560
and it's too stiff
on the other side.

220
00:12:44,560 --> 00:12:48,340
Because if the real
potential is asymmetric,

221
00:12:48,340 --> 00:12:50,800
you'll be too stiff on one
side and too weak on the other.

222
00:12:50,800 --> 00:12:53,000
But with a third
order polynomial,

223
00:12:53,000 --> 00:12:56,140
which is the green line here,
you're already doing very good.

224
00:12:56,140 --> 00:13:00,580
So you often don't need, per
se, really complicated functions

225
00:13:00,580 --> 00:13:01,840
of these bond angles.

226
00:13:01,840 --> 00:13:03,340
Because usually the
more complicated

227
00:13:03,340 --> 00:13:06,780
your function, the more can
go wrong when you fit as well.

228
00:13:21,200 --> 00:13:24,050
OK.

229
00:13:24,050 --> 00:13:28,910
One other contribution that
you often use in long molecules

230
00:13:28,910 --> 00:13:32,880
is portion potentials.

231
00:13:32,880 --> 00:13:34,760
And this is actually
not a three-body effect.

232
00:13:34,760 --> 00:13:38,420
It's actually a
four-body effect.

233
00:13:38,420 --> 00:13:42,170
If you think of
it ethane, if you

234
00:13:42,170 --> 00:13:44,300
have carbons here
and hydrogens here--

235
00:13:44,300 --> 00:13:46,430
it's a little hard to see.

236
00:13:46,430 --> 00:13:48,570
But if you look at
it from the side,

237
00:13:48,570 --> 00:13:52,210
so there'd be three
hydrogens on this end.

238
00:13:55,480 --> 00:13:57,020
The hydrogens can line up.

239
00:13:57,020 --> 00:13:59,110
So if you look at it
from this direction,

240
00:13:59,110 --> 00:14:01,280
the carbons are on
top of each other.

241
00:14:01,280 --> 00:14:07,820
So the hydrogen on one
side are 120 degrees apart.

242
00:14:07,820 --> 00:14:11,300
Now, the hydrogens
from the higher carbon

243
00:14:11,300 --> 00:14:15,222
can either be in the
same configuration--

244
00:14:17,780 --> 00:14:20,164
that's called the
eclipsed configuration--

245
00:14:25,270 --> 00:14:26,410
or they could be here.

246
00:14:30,300 --> 00:14:32,760
And that's called a
staggered configuration.

247
00:14:36,360 --> 00:14:39,940
And the staggered one is
somewhat lower in energy.

248
00:14:39,940 --> 00:14:42,570
So to get that energy
difference right,

249
00:14:42,570 --> 00:14:44,280
you need, essentially,
a potential

250
00:14:44,280 --> 00:14:47,320
that describes the torsion
around the carbon-carbon bond.

251
00:14:47,320 --> 00:14:48,960
If you think of
the three hydrogens

252
00:14:48,960 --> 00:14:52,410
here in sort of fixed position
with respect to each other

253
00:14:52,410 --> 00:14:54,750
at all under 120
degrees and the ones

254
00:14:54,750 --> 00:14:57,810
here, to describe
that the energy

255
00:14:57,810 --> 00:15:00,300
difference between the
different rotational states

256
00:15:00,300 --> 00:15:01,980
you need a torsion potential.

257
00:15:01,980 --> 00:15:07,050
And a torsion potential,
you need at least four atoms

258
00:15:07,050 --> 00:15:10,770
to describe it, four sets of
coordinates to describe it.

259
00:15:10,770 --> 00:15:16,560
Because the two carbons,
say, define the bond along,

260
00:15:16,560 --> 00:15:18,240
which are doing the torsion.

261
00:15:18,240 --> 00:15:23,070
And then these vectors, say, one
of the hydrogens on each side

262
00:15:23,070 --> 00:15:26,380
describe the state
of torsion you're in.

263
00:15:26,380 --> 00:15:26,970
OK.

264
00:15:26,970 --> 00:15:32,820
So a torsion potential is, by
definition, a four-body effect.

265
00:15:32,820 --> 00:15:34,590
These potentials
are often written

266
00:15:34,590 --> 00:15:38,080
as cosines of some angle.

267
00:15:38,080 --> 00:15:40,530
And the reason is that
a torsion potential

268
00:15:40,530 --> 00:15:42,750
needs rotational symmetry.

269
00:15:42,750 --> 00:15:44,280
If you think of the
ethane molecule,

270
00:15:44,280 --> 00:15:46,620
for example, every
120 degrees you're

271
00:15:46,620 --> 00:15:49,350
in the same configuration
because the phase you're

272
00:15:49,350 --> 00:15:52,150
rotating has 120
degrees symmetry.

273
00:15:52,150 --> 00:15:53,190
Does everybody see that?

274
00:15:53,190 --> 00:15:58,630
So you need periodicity
in the angle.

275
00:15:58,630 --> 00:16:01,560
So that's why a cosine
is very convenient.

276
00:16:01,560 --> 00:16:08,200
And here's the ethane
rotation energy plot.

277
00:16:08,200 --> 00:16:12,670
So what you see is that
it's minimal at 60 degrees.

278
00:16:12,670 --> 00:16:14,560
This is the staggered
configuration.

279
00:16:18,660 --> 00:16:24,810
And this is 120 and 0 and 240.

280
00:16:24,810 --> 00:16:26,730
These are the eclipse
configurations.

281
00:16:29,820 --> 00:16:31,620
And so these are easy
potentials to make

282
00:16:31,620 --> 00:16:34,800
because all you have to look
at is what the symmetry is

283
00:16:34,800 --> 00:16:37,830
of your torsional states.

284
00:16:37,830 --> 00:16:42,120
So if, for some reason, we're
90 degrees, which is unusual,

285
00:16:42,120 --> 00:16:44,790
you'd have a cosine of 4 omega.

286
00:16:44,790 --> 00:16:47,700
If it's 180 degrees, you'd
have a cosine of 2 omega.

287
00:16:53,400 --> 00:16:55,800
The last one is maybe
slightly less important,

288
00:16:55,800 --> 00:16:58,440
but there's something called
out-of-plane or improper

289
00:16:58,440 --> 00:17:01,530
torsion, which is essentially
also a torsion angle.

290
00:17:01,530 --> 00:17:04,890
But it's not as obvious.

291
00:17:04,890 --> 00:17:09,720
In the previous one, in
ethane, the four atoms

292
00:17:09,720 --> 00:17:11,730
along which I
described the torsion

293
00:17:11,730 --> 00:17:17,190
were sort of in sequence along a
line, along the molecular axis.

294
00:17:17,190 --> 00:17:19,800
In improper torsion,
they're not.

295
00:17:19,800 --> 00:17:22,470
Improper torsion
describes, essentially,

296
00:17:22,470 --> 00:17:26,130
let's say you have a plane
defined by these three

297
00:17:26,130 --> 00:17:30,450
atoms, a, b, and c.

298
00:17:30,450 --> 00:17:34,140
How much A comes
out of that plane,

299
00:17:34,140 --> 00:17:36,780
that's essentially what
improper torsion describes.

300
00:17:36,780 --> 00:17:39,270
For example, that may be
important for something

301
00:17:39,270 --> 00:17:42,510
like ammonia.

302
00:17:42,510 --> 00:17:51,750
Let's say you have N, H, H,
H. So the ammonia molecule can

303
00:17:51,750 --> 00:17:52,620
flop.

304
00:17:52,620 --> 00:17:56,580
Sort of you can have the
nitrogen and three hydrogens

305
00:17:56,580 --> 00:18:00,840
sort of sticking out in a
tetrahedral configuration.

306
00:18:00,840 --> 00:18:03,330
And the nitrogen can sort
of go through and come out

307
00:18:03,330 --> 00:18:04,440
on the other end.

308
00:18:04,440 --> 00:18:06,570
And so it's in the
opposite state.

309
00:18:06,570 --> 00:18:08,550
And so it's going through
an improper torsion

310
00:18:08,550 --> 00:18:10,470
when it does that.

311
00:18:10,470 --> 00:18:12,480
So improper torsion can
be important in things

312
00:18:12,480 --> 00:18:15,420
that are, say, sp2
hybridized because then they

313
00:18:15,420 --> 00:18:16,360
want to be flat.

314
00:18:16,360 --> 00:18:19,050
And if one atom comes
out of the plane,

315
00:18:19,050 --> 00:18:20,520
you have an improper torsion.

316
00:18:20,520 --> 00:18:23,520
Or an sp3
hybridization-- where it

317
00:18:23,520 --> 00:18:26,010
wants to be tetrahedral
from a central atom.

318
00:18:26,010 --> 00:18:29,070
You have three bonds going
towards the tetrahedron.

319
00:18:29,070 --> 00:18:31,320
And as you push that in
and you squeeze that flat,

320
00:18:31,320 --> 00:18:33,570
you do an improper
torsion on it.

321
00:18:33,570 --> 00:18:35,900
And there's many ways you
can measure improper torsion.

322
00:18:35,900 --> 00:18:38,250
But it's essentially
defined by the angle

323
00:18:38,250 --> 00:18:41,260
between the plane,
defined by these three

324
00:18:41,260 --> 00:18:44,380
atoms and this and
the distance that

325
00:18:44,380 --> 00:18:45,770
describes the improper torsion.

326
00:18:51,210 --> 00:18:54,750
So I thought I'd show
you a real example.

327
00:18:54,750 --> 00:18:57,630
I stole this from somebody's
PhD thesis at MIT,

328
00:18:57,630 --> 00:18:58,500
and I forgot who.

329
00:18:58,500 --> 00:19:01,590
So I can't give you the
reference, unfortunately,

330
00:19:01,590 --> 00:19:04,350
but it was somebody
in our department.

331
00:19:04,350 --> 00:19:06,090
OK.

332
00:19:06,090 --> 00:19:09,330
Here's a potential model
that the person put together

333
00:19:09,330 --> 00:19:12,300
for poly-hydroxybenzoic
acid, which I think

334
00:19:12,300 --> 00:19:14,490
is a component of
a liquid crystal,

335
00:19:14,490 --> 00:19:17,030
but I don't really remember.

336
00:19:17,030 --> 00:19:22,470
And so essentially,
there are bonded terms,

337
00:19:22,470 --> 00:19:26,310
which are really these here.

338
00:19:26,310 --> 00:19:35,670
what they call valence
is the bond stretch--

339
00:19:35,670 --> 00:19:38,320
sorry, the bond bending term.

340
00:19:38,320 --> 00:19:42,570
So it has a form, angle minus
reference angle squared.

341
00:19:42,570 --> 00:19:46,170
So it's harmonic in
the bond bending.

342
00:19:46,170 --> 00:19:50,130
There's a torsion
potential along the chain.

343
00:19:50,130 --> 00:19:53,260
And there's an improper
torsion potential.

344
00:19:53,260 --> 00:19:58,150
And then the rest is there
are non-bonded terms.

345
00:19:58,150 --> 00:19:59,680
These are the ones
between, say, you

346
00:19:59,680 --> 00:20:03,600
know, this piece of the
chain kind of interacting

347
00:20:03,600 --> 00:20:06,870
with that piece as they
approach each other.

348
00:20:06,870 --> 00:20:09,330
And that's a van der Waals
term, which is simply model

349
00:20:09,330 --> 00:20:13,470
as a Lennard-Jones here
and a Coulombic term

350
00:20:13,470 --> 00:20:16,407
because there are charged
species on this chain.

351
00:20:16,407 --> 00:20:18,990
So everything that goes in the
model this sort of stuff you've

352
00:20:18,990 --> 00:20:19,490
seen.

353
00:20:19,490 --> 00:20:22,099
And it's all fairly
straightforward.

354
00:20:26,790 --> 00:20:33,390
Potentials work remarkably
well in organic systems.

355
00:20:33,390 --> 00:20:35,010
And that seems to
be at odds with what

356
00:20:35,010 --> 00:20:38,730
I said in the beginning, that
pair potentials in particular

357
00:20:38,730 --> 00:20:42,850
are bad for covalent materials.

358
00:20:42,850 --> 00:20:44,500
But you have to
remember why they

359
00:20:44,500 --> 00:20:46,510
are bad because it's
the same reason why

360
00:20:46,510 --> 00:20:48,100
they're so good in organics.

361
00:20:48,100 --> 00:20:53,470
The reason pair potentials are
bad in metals is because you

362
00:20:53,470 --> 00:20:58,360
cannot capture the energy
dependence on coordination.

363
00:20:58,360 --> 00:20:59,320
OK.

364
00:20:59,320 --> 00:21:01,630
Remember, if your
coordination stays the same,

365
00:21:01,630 --> 00:21:03,880
if essentially your
average density around you

366
00:21:03,880 --> 00:21:06,370
stays the same, you're
actually doing very well

367
00:21:06,370 --> 00:21:08,470
with pair potentials in metals.

368
00:21:08,470 --> 00:21:11,980
If you remember when I showed
you that in the embedded atom

369
00:21:11,980 --> 00:21:14,950
method you can transfer
pieces of the energy

370
00:21:14,950 --> 00:21:17,900
between the potential and
the embedding function,

371
00:21:17,900 --> 00:21:19,330
if you go back to
that, you'll see

372
00:21:19,330 --> 00:21:22,670
that, if the embedding
density is constant,

373
00:21:22,670 --> 00:21:26,032
then all you have in the
energy is a pair energy.

374
00:21:26,032 --> 00:21:27,490
So when you're at
constant density,

375
00:21:27,490 --> 00:21:29,150
pair potentials work fine.

376
00:21:29,150 --> 00:21:32,410
So why do potentials
work so well in organics?

377
00:21:32,410 --> 00:21:34,600
Well, the reason
is that they simply

378
00:21:34,600 --> 00:21:40,270
take different potentials
for different coordinations.

379
00:21:40,270 --> 00:21:42,760
Different coordinations in
sort of organic chemistry

380
00:21:42,760 --> 00:21:45,430
are different hybridizations.

381
00:21:45,430 --> 00:21:52,060
And so rather than, say,
have a generic carbon-carbon

382
00:21:52,060 --> 00:21:56,650
potential, they have a different
one for sp carbon, sp2 carbon,

383
00:21:56,650 --> 00:21:57,642
sp3 carbon.

384
00:21:57,642 --> 00:21:59,350
So essentially, they're
saying, you know,

385
00:21:59,350 --> 00:22:02,020
we're going to take every
coordination as different.

386
00:22:02,020 --> 00:22:04,420
And since there's
only a finite set,

387
00:22:04,420 --> 00:22:07,000
there's essentially
only a very finite set

388
00:22:07,000 --> 00:22:11,800
of chemical environments that
your atoms see in organics.

389
00:22:11,800 --> 00:22:13,630
You can simply just
parameterize them all,

390
00:22:13,630 --> 00:22:16,480
and you have a different
potential for all of them.

391
00:22:16,480 --> 00:22:19,990
So if your carbon is,
say, in polyethylene,

392
00:22:19,990 --> 00:22:25,050
so it's sp3 hybridized, you'll
take a very different potential

393
00:22:25,050 --> 00:22:28,890
than if you had an
sp2 hybridized carbon.

394
00:22:28,890 --> 00:22:32,550
Because in one case, you want
to impose 120 degree angles,

395
00:22:32,550 --> 00:22:34,950
in other case you want to
impose 109 degree angles.

396
00:22:34,950 --> 00:22:39,810
So you never, ever deal with
the change of coordination

397
00:22:39,810 --> 00:22:43,780
of the bonded system.

398
00:22:43,780 --> 00:22:46,120
In the non-bonded system,
you have to deal with it,

399
00:22:46,120 --> 00:22:48,538
but there you don't have
covalent bonds anyway.

400
00:22:48,538 --> 00:22:49,830
So that's why it works so well.

401
00:22:55,730 --> 00:23:00,040
So you can describe
very well things

402
00:23:00,040 --> 00:23:04,630
like bending of chains, polymer
chains, with potentials.

403
00:23:04,630 --> 00:23:07,330
And the reason is it's
essentially a local effect.

404
00:23:07,330 --> 00:23:09,490
You're always bending
the same bond.

405
00:23:09,490 --> 00:23:14,430
And whether that's
in polyethylene

406
00:23:14,430 --> 00:23:16,780
or whether that's an
sp3 carbon in something

407
00:23:16,780 --> 00:23:18,672
much more complicated,
it's essentially

408
00:23:18,672 --> 00:23:19,880
the same bond you're bending.

409
00:23:19,880 --> 00:23:21,940
So you can parameterize
it very well,

410
00:23:21,940 --> 00:23:25,300
and it's pretty much always
the same, independent of what's

411
00:23:25,300 --> 00:23:27,700
far away from you.

412
00:23:27,700 --> 00:23:30,100
But this also means
that there's stuff

413
00:23:30,100 --> 00:23:31,540
that it will not work for.

414
00:23:34,120 --> 00:23:37,450
You cannot do chemistry,
reaction chemistry,

415
00:23:37,450 --> 00:23:39,190
with potentials.

416
00:23:39,190 --> 00:23:42,550
Because the reason is,
in reaction chemistry,

417
00:23:42,550 --> 00:23:44,900
you will change
your coordination.

418
00:23:44,900 --> 00:23:47,980
So you could not study the
polymerization reaction

419
00:23:47,980 --> 00:23:50,650
of ethylene to polyethylene--

420
00:23:50,650 --> 00:23:53,950
of ethene to polyethylene.

421
00:23:53,950 --> 00:23:56,830
Because there you are
actually changing.

422
00:23:56,830 --> 00:24:00,970
Your carbon is changing
from sp2 to sp3.

423
00:24:00,970 --> 00:24:02,500
And that would
literally-- you would

424
00:24:02,500 --> 00:24:05,410
have to sort of, in your
simulation, midstream

425
00:24:05,410 --> 00:24:08,560
change potential.

426
00:24:08,560 --> 00:24:11,350
Because your potential is tied
to a particular hybridization.

427
00:24:11,350 --> 00:24:16,570
You can't really do
bond breaking reactions.

428
00:24:16,570 --> 00:24:18,640
And the other thing,
which is not fundamental--

429
00:24:18,640 --> 00:24:22,765
that a lot of these
potentials lack polarization.

430
00:24:22,765 --> 00:24:28,660
In the end, why does on
sp3 carbon and another one

431
00:24:28,660 --> 00:24:30,520
don't always behave the same?

432
00:24:30,520 --> 00:24:32,170
It's because there
can be charges

433
00:24:32,170 --> 00:24:35,330
that polarize their
electron density,

434
00:24:35,330 --> 00:24:38,095
but these are fairly
minor effects.

435
00:24:38,095 --> 00:24:39,470
But that's something
you will not

436
00:24:39,470 --> 00:24:42,200
get in these kind of schemes.

437
00:24:45,800 --> 00:24:50,840
You know, because there's
only a sort of finite set

438
00:24:50,840 --> 00:24:54,140
of chemistries in organic
chemistry or bonding

439
00:24:54,140 --> 00:24:58,280
environments, making
potentials for organic system

440
00:24:58,280 --> 00:25:01,280
is a cottage industry.

441
00:25:01,280 --> 00:25:05,210
They tend to call them
force fields in organics,

442
00:25:05,210 --> 00:25:07,160
but there really are potentials.

443
00:25:07,160 --> 00:25:11,360
And so there are a whole
sets of parameterizations

444
00:25:11,360 --> 00:25:13,190
that are available.

445
00:25:13,190 --> 00:25:16,220
And a lot of sort of
good codes will have

446
00:25:16,220 --> 00:25:17,977
many of these built in already.

447
00:25:17,977 --> 00:25:20,060
For example, there's
somewhere about [INAUDIBLE],,

448
00:25:20,060 --> 00:25:23,735
then there's the AMBER
force field, which will--

449
00:25:23,735 --> 00:25:26,150
you know, what they mean
with a force field is

450
00:25:26,150 --> 00:25:29,420
they will have potentials
for the most common bonds

451
00:25:29,420 --> 00:25:32,730
that you find, in this case,
in biology, but in general

452
00:25:32,730 --> 00:25:35,000
in organic chemistry.

453
00:25:35,000 --> 00:25:41,450
The CHARMM [INAUDIBLE] very
well, very often used one.

454
00:25:41,450 --> 00:25:43,340
CHARMM has
essentially potentials

455
00:25:43,340 --> 00:25:48,650
for almost all elements,
almost all light elements.

456
00:25:48,650 --> 00:25:53,155
So all these have been
parameterized for you.

457
00:25:53,155 --> 00:25:54,530
There's an
interesting discussion

458
00:25:54,530 --> 00:25:56,960
about all these force
fields on this website

459
00:25:56,960 --> 00:25:59,660
if you care to look
at it at some point.

460
00:26:02,240 --> 00:26:05,150
OK.

461
00:26:05,150 --> 00:26:09,110
So another field where
potentials have worked

462
00:26:09,110 --> 00:26:12,230
reasonably well as in oxides.

463
00:26:12,230 --> 00:26:18,080
Actually, oxides is probably
one of the earliest material

464
00:26:18,080 --> 00:26:23,030
classes in which there was very
extensive atomistic modeling

465
00:26:23,030 --> 00:26:25,760
going on.

466
00:26:25,760 --> 00:26:30,110
This goes way back to the
'60s before there were barely

467
00:26:30,110 --> 00:26:31,070
any usable computers.

468
00:26:31,070 --> 00:26:34,220
People were modeling oxides.

469
00:26:34,220 --> 00:26:37,130
And of course, people
weren't even thinking

470
00:26:37,130 --> 00:26:39,650
of doing quantum mechanics
computationally at the time.

471
00:26:39,650 --> 00:26:41,900
So they were using
empirical potential models.

472
00:26:41,900 --> 00:26:45,080
And the drive to work
on oxides was actually

473
00:26:45,080 --> 00:26:46,820
from the nuclear industry.

474
00:26:46,820 --> 00:26:49,520
There was one oxide in
particular they cared about.

475
00:26:49,520 --> 00:26:52,460
And that was uranium
oxide because it's

476
00:26:52,460 --> 00:26:55,880
an essential component
of your nuclear fuel.

477
00:26:55,880 --> 00:26:58,820
And this was one of
these early realizations

478
00:26:58,820 --> 00:27:01,580
that there's some stuff that was
much easier to do with modeling

479
00:27:01,580 --> 00:27:02,960
than to do experimentally.

480
00:27:02,960 --> 00:27:05,300
Doing experiments
on nuclear fuel rods

481
00:27:05,300 --> 00:27:08,670
is really kind of
painful and expensive.

482
00:27:08,670 --> 00:27:11,360
And so that's actually how
the modeling industry on oxide

483
00:27:11,360 --> 00:27:13,970
was born in the '60s
is to actually model

484
00:27:13,970 --> 00:27:16,430
point defects in uranium oxide.

485
00:27:16,430 --> 00:27:18,860
So it's a
well-established field.

486
00:27:18,860 --> 00:27:20,840
It's probably getting
superseded largely

487
00:27:20,840 --> 00:27:23,270
by doing quantum mechanics
now, but it's still

488
00:27:23,270 --> 00:27:26,840
sort of interesting to see
how reasonably well it works

489
00:27:26,840 --> 00:27:28,610
and what it doesn't work for.

490
00:27:28,610 --> 00:27:31,970
The standard, by all
means, is that you use

491
00:27:31,970 --> 00:27:34,280
a repulsive Buckingham term.

492
00:27:34,280 --> 00:27:38,690
So you have a potential that--

493
00:27:38,690 --> 00:27:42,020
so you have the
repulsive exponential.

494
00:27:42,020 --> 00:27:44,750
You have the van der Waals term.

495
00:27:44,750 --> 00:27:46,830
That's your
short-range potential.

496
00:27:46,830 --> 00:27:49,430
And then you do electrostatics
because, of course,

497
00:27:49,430 --> 00:27:51,230
you have charged
ions when doing that.

498
00:27:51,230 --> 00:27:54,500
And so the electrostatics
give you your cohesion.

499
00:27:54,500 --> 00:27:56,840
That's the one that wants
to bring the ions together.

500
00:27:56,840 --> 00:27:59,705
And the Buckingham term
gives you your repulsion.

501
00:28:03,080 --> 00:28:06,710
To sum the long range
electrostatic part,

502
00:28:06,710 --> 00:28:08,570
people can sum in real space.

503
00:28:08,570 --> 00:28:10,130
If you sum this in
real space, it's

504
00:28:10,130 --> 00:28:11,540
only conditionally convergent.

505
00:28:11,540 --> 00:28:14,240
And people use what's
called the Ewald method.

506
00:28:14,240 --> 00:28:16,940
It's something you'll
see when you use codes.

507
00:28:16,940 --> 00:28:18,680
There actually are
faster techniques now,

508
00:28:18,680 --> 00:28:21,132
but this is still
definitely a standard.

509
00:28:21,132 --> 00:28:22,590
We're not going to
go into details,

510
00:28:22,590 --> 00:28:24,680
but essentially the
Ewald way is a way

511
00:28:24,680 --> 00:28:29,990
of partitioning your
energy sum in real space

512
00:28:29,990 --> 00:28:33,800
and in reciprocal space,
so that both converge.

513
00:28:33,800 --> 00:28:36,110
You can actually show
that, in either space,

514
00:28:36,110 --> 00:28:38,180
it doesn't converge very well.

515
00:28:38,180 --> 00:28:40,580
In real space, 1 over
r doesn't converge.

516
00:28:40,580 --> 00:28:42,680
In reciprocal space, it
really doesn't converge

517
00:28:42,680 --> 00:28:44,210
when you Fourier transform it.

518
00:28:44,210 --> 00:28:46,250
But you can partition
it, so that it

519
00:28:46,250 --> 00:28:47,510
converges in both space.

520
00:28:47,510 --> 00:28:50,330
And you basically
have to sum it up.

521
00:28:54,790 --> 00:28:59,810
And an essential piece that
was added in the late '60,

522
00:28:59,810 --> 00:29:03,610
early '70s was polarization.

523
00:29:03,610 --> 00:29:06,400
An important thing in
oxides is that you have

524
00:29:06,400 --> 00:29:08,900
strong electrostatic fields.

525
00:29:08,900 --> 00:29:11,860
So when you put an electron
density in a field,

526
00:29:11,860 --> 00:29:13,360
it can polarize.

527
00:29:13,360 --> 00:29:16,090
And this is particularly
important for the oxygen ion.

528
00:29:16,090 --> 00:29:19,300
If you put the oxygen ion in a
non-symmetric position, so one

529
00:29:19,300 --> 00:29:23,920
where the field doesn't
vanish, an oxygen ion

530
00:29:23,920 --> 00:29:25,330
is highly negatively charged.

531
00:29:25,330 --> 00:29:27,230
It means it has a
lot of electrons.

532
00:29:27,230 --> 00:29:31,400
So the electron density
cloud will polarize.

533
00:29:31,400 --> 00:29:35,780
And the oxygen will
actually carry a dipole.

534
00:29:35,780 --> 00:29:39,440
And that was actually modeled
in a sort of very ingenious way,

535
00:29:39,440 --> 00:29:40,730
I thought.

536
00:29:40,730 --> 00:29:42,920
The way polarization
is included is

537
00:29:42,920 --> 00:29:45,740
by what's called a
shell and spring model.

538
00:29:45,740 --> 00:29:49,670
Essentially, you treat
the polarizable ion

539
00:29:49,670 --> 00:29:51,660
by two entities.

540
00:29:51,660 --> 00:29:54,620
It has a nucleus
with a charge on it,

541
00:29:54,620 --> 00:29:57,410
and it has a shell
with a charge on it.

542
00:29:57,410 --> 00:30:00,440
And the two charges sum
to the charge of the ion.

543
00:30:00,440 --> 00:30:04,760
So if this were oxygen,
this would be 2 minus then.

544
00:30:04,760 --> 00:30:08,930
But what you do is the two
can move independently,

545
00:30:08,930 --> 00:30:12,280
but they are
connected by a spring.

546
00:30:12,280 --> 00:30:15,430
So they can't completely wander
off, so they are connected.

547
00:30:15,430 --> 00:30:17,830
But what happens now
if you put-- say,

548
00:30:17,830 --> 00:30:22,570
if you put a field
on this, the positive

549
00:30:22,570 --> 00:30:25,550
and the negative charge will
move in opposite direction.

550
00:30:25,550 --> 00:30:28,720
So you may go to a state
where the center, qn,

551
00:30:28,720 --> 00:30:30,730
the center of the
positive charge is here.

552
00:30:30,730 --> 00:30:33,470
And that's the shell.

553
00:30:33,470 --> 00:30:37,750
And so, now, this would be the
center of the positive charge.

554
00:30:37,750 --> 00:30:41,110
And the center of the negative
charge would be shifted.

555
00:30:41,110 --> 00:30:42,415
And you would have a dipole.

556
00:30:44,938 --> 00:30:46,480
So does everybody
see how this works?

557
00:30:49,430 --> 00:30:53,440
So the polarizability of
the ion is essentially

558
00:30:53,440 --> 00:30:56,080
determined by the
charge difference

559
00:30:56,080 --> 00:30:59,140
between the shell and the core
and the spring between them.

560
00:30:59,140 --> 00:31:01,180
If the spring is
infinitely stiff,

561
00:31:01,180 --> 00:31:03,490
then you cannot
polarize the ion.

562
00:31:03,490 --> 00:31:07,930
If the spring is soft, then you
can polarize the ion very well.

563
00:31:07,930 --> 00:31:15,020
And that's actually how the
spring constant is fitted.

564
00:31:15,020 --> 00:31:16,900
It's fitted to the
polarizability, often,

565
00:31:16,900 --> 00:31:19,180
of the free ion.

566
00:31:19,180 --> 00:31:24,130
And I think this is
actually the polarization.

567
00:31:24,130 --> 00:31:27,100
Typically, what interacts
with the other ions

568
00:31:27,100 --> 00:31:29,290
through the short range
term is the shell model.

569
00:31:32,710 --> 00:31:35,500
Usually, when you model
oxides with potentials,

570
00:31:35,500 --> 00:31:40,600
you tend to put a shell and
spring only on the anions

571
00:31:40,600 --> 00:31:43,630
because it's obviously the
anion that's most polarizable.

572
00:31:43,630 --> 00:31:45,940
It has the biggest
electron density cloud.

573
00:31:45,940 --> 00:31:48,580
It has the electrons.

574
00:31:48,580 --> 00:31:50,560
And cations are much
less polarizable

575
00:31:50,560 --> 00:31:52,720
because they're cations.

576
00:31:52,720 --> 00:31:55,048
They have lost most of
their valence electrons.

577
00:31:55,048 --> 00:31:56,590
Especially something
like-- let's say

578
00:31:56,590 --> 00:31:59,440
you do MgO, magnesium oxide.

579
00:31:59,440 --> 00:32:04,850
Magnesium is 2+, so
magnesium is an outlier.

580
00:32:04,850 --> 00:32:06,640
So it's a bare core.

581
00:32:06,640 --> 00:32:08,380
If you strip off two
electrons, there's

582
00:32:08,380 --> 00:32:11,410
nothing left to polarize
unless you think you're

583
00:32:11,410 --> 00:32:12,622
going to polarize the core.

584
00:32:12,622 --> 00:32:14,080
So you would there
just put a shell

585
00:32:14,080 --> 00:32:17,470
on a spring on the oxygen.
This becomes very important

586
00:32:17,470 --> 00:32:20,320
when you work in low
symmetry environments.

587
00:32:20,320 --> 00:32:24,310
Because whether your
ion polarizes or not

588
00:32:24,310 --> 00:32:26,980
depends on whether the
symmetry allows it to polarize.

589
00:32:29,620 --> 00:32:33,370
If you are sitting in a
center of point symmetry,

590
00:32:33,370 --> 00:32:36,460
let's say you're sitting in an
inversion center of a crystal,

591
00:32:36,460 --> 00:32:38,350
I don't care how
polarizable you are.

592
00:32:38,350 --> 00:32:42,160
The symmetry doesn't
allow you to polarize.

593
00:32:42,160 --> 00:32:45,490
But so it's when you do low
symmetry calculation, defects,

594
00:32:45,490 --> 00:32:50,140
surfaces, that polarizability
becomes quite important.

595
00:32:54,770 --> 00:32:56,830
So I wanted to show
you a real example.

596
00:32:56,830 --> 00:32:58,900
This is something we kind
of quickly calculated

597
00:32:58,900 --> 00:33:00,760
ourselves a few years ago.

598
00:33:00,760 --> 00:33:03,190
That's why it looks
like a crummy graph.

599
00:33:03,190 --> 00:33:05,890
Because this is the phonon
density of state for magnesium

600
00:33:05,890 --> 00:33:08,560
oxide with and
without polarization.

601
00:33:08,560 --> 00:33:10,720
So what this is is
really this axis

602
00:33:10,720 --> 00:33:13,010
is the frequency of the phonons.

603
00:33:13,010 --> 00:33:16,700
And this is the
density of states,

604
00:33:16,700 --> 00:33:19,910
how many phonons there are
with that frequency of energy.

605
00:33:19,910 --> 00:33:21,530
And essentially,
what you see is that,

606
00:33:21,530 --> 00:33:25,720
when you have no
polarization, you

607
00:33:25,720 --> 00:33:29,410
have a high frequency
tail of phonons.

608
00:33:29,410 --> 00:33:31,720
And remember, phonons--
even though MgO

609
00:33:31,720 --> 00:33:35,680
is a highly symmetric
crystal, the energy of phonons

610
00:33:35,680 --> 00:33:37,510
is determined by
low symmetry things.

611
00:33:37,510 --> 00:33:40,180
Because, you know, think of
a displacement of a phonon.

612
00:33:40,180 --> 00:33:44,010
You're now breaking the
translational symmetry.

613
00:33:44,010 --> 00:33:46,860
So you're not into
high symmetry 0 Kelvin

614
00:33:46,860 --> 00:33:49,440
crystalline environment
anymore, so polarization

615
00:33:49,440 --> 00:33:50,370
becomes relevant.

616
00:33:50,370 --> 00:33:53,340
When you turn on
polarization of the anions,

617
00:33:53,340 --> 00:33:56,290
you essentially dampen away
those high frequencies.

618
00:33:56,290 --> 00:33:56,790
OK.

619
00:34:05,040 --> 00:34:07,410
Let me actually, just
before I go here, come back

620
00:34:07,410 --> 00:34:14,489
to the issue of polarization is
important to take into account

621
00:34:14,489 --> 00:34:17,150
when you fit.

622
00:34:17,150 --> 00:34:20,330
There are people who
will show you potentials

623
00:34:20,330 --> 00:34:22,352
fit to the equation of state.

624
00:34:22,352 --> 00:34:23,810
So does everybody
know what I mean?

625
00:34:23,810 --> 00:34:25,820
The equation of state, I
mean literally the energy

626
00:34:25,820 --> 00:34:27,653
of the system as a
function of state lattice

627
00:34:27,653 --> 00:34:28,739
parameter or volume.

628
00:34:28,739 --> 00:34:32,000
So let's say I took MgO,
magnesium oxides or rock salt.

629
00:34:32,000 --> 00:34:35,120
I could really, let's say,
have initial calculations

630
00:34:35,120 --> 00:34:36,530
of different lattice parameters.

631
00:34:36,530 --> 00:34:38,596
And that would give
me a curve, energy

632
00:34:38,596 --> 00:34:39,679
versus lattice parameters.

633
00:34:39,679 --> 00:34:42,139
I could fit my
potential to that.

634
00:34:42,139 --> 00:34:45,409
If I do that, I have never
included any polarization

635
00:34:45,409 --> 00:34:47,280
effects.

636
00:34:47,280 --> 00:34:49,320
Because at every
lattice parameter,

637
00:34:49,320 --> 00:34:53,340
that is still a
high symmetry state.

638
00:34:53,340 --> 00:34:55,320
At none of these
lattice parameters

639
00:34:55,320 --> 00:34:58,110
is any of the ions
ever polarized,

640
00:34:58,110 --> 00:35:01,480
ever have a dipole moment on it.

641
00:35:01,480 --> 00:35:03,870
And so what you see
then, when you go and do

642
00:35:03,870 --> 00:35:06,330
defect calculations
with that potential,

643
00:35:06,330 --> 00:35:09,680
you will almost always
overestimate their energy.

644
00:35:09,680 --> 00:35:11,330
And the reason is
you've taken away

645
00:35:11,330 --> 00:35:14,390
a degree of freedom
of the system.

646
00:35:14,390 --> 00:35:17,010
If you actually then
calculate a defect,

647
00:35:17,010 --> 00:35:19,310
the system wants to polarize,
but you haven't given it

648
00:35:19,310 --> 00:35:21,750
the option because you haven't
included it in your model.

649
00:35:21,750 --> 00:35:23,570
So I would say be
aware of people who

650
00:35:23,570 --> 00:35:25,250
show you equations of state.

651
00:35:25,250 --> 00:35:28,790
You know, everybody can fit an
equation of state, everybody.

652
00:35:28,790 --> 00:35:33,030
You know, any potential that has
more than one fitting parameter

653
00:35:33,030 --> 00:35:35,250
can fit an equation of state.

654
00:35:35,250 --> 00:35:38,490
It's the low symmetry details,
surface energies, defects,

655
00:35:38,490 --> 00:35:39,900
that are hard to reproduce.

656
00:35:43,470 --> 00:35:45,300
These days, just
like in organics,

657
00:35:45,300 --> 00:35:47,850
there are very good sources
of potentials for oxides.

658
00:35:47,850 --> 00:35:50,680
You don't really have
to go and fit your own.

659
00:35:50,680 --> 00:35:54,120
This is one paper that sort
of collected a bunch of them,

660
00:35:54,120 --> 00:35:55,697
which is a paper by Bush.

661
00:35:55,697 --> 00:35:57,030
They're actually on the web now.

662
00:35:57,030 --> 00:35:59,740
You can literally download
them in the format

663
00:35:59,740 --> 00:36:02,460
right away that's relevant
for a particular code,

664
00:36:02,460 --> 00:36:04,650
the set of standardized codes.

665
00:36:04,650 --> 00:36:07,680
The one we'll use in
the lab is called GULP,

666
00:36:07,680 --> 00:36:11,010
which is free for academics.

667
00:36:11,010 --> 00:36:15,120
But even other codes you can
pretty much get the form often

668
00:36:15,120 --> 00:36:17,670
right off the web.

669
00:36:17,670 --> 00:36:18,862
OK.

670
00:36:18,862 --> 00:36:22,590
Oh, this is actually one of
the tables in that paper.

671
00:36:22,590 --> 00:36:25,957
All I'm showing you this is that
you can see that they actually

672
00:36:25,957 --> 00:36:26,790
have the parameters.

673
00:36:26,790 --> 00:36:28,230
So this is the
parameter in front

674
00:36:28,230 --> 00:36:31,440
of the exponential between
the cation and the anion.

675
00:36:31,440 --> 00:36:33,485
This is the one in
the exponential.

676
00:36:33,485 --> 00:36:38,160
Remember, it's a
exponential minus r overall.

677
00:36:38,160 --> 00:36:41,400
This would be the
polarization, the core,

678
00:36:41,400 --> 00:36:42,900
and the shell polarization.

679
00:36:42,900 --> 00:36:45,850
In this case, on
some of the cations,

680
00:36:45,850 --> 00:36:47,550
they put it on as well.

681
00:36:47,550 --> 00:36:50,550
On the alkalis, they
didn't fit a polarization,

682
00:36:50,550 --> 00:36:53,640
since they're really not
very polarizable if you

683
00:36:53,640 --> 00:36:55,792
see lithium, sodium, potassium.

684
00:36:55,792 --> 00:36:57,250
But on some of the
heavier cations,

685
00:36:57,250 --> 00:37:00,510
they did fit polarization terms.

686
00:37:00,510 --> 00:37:02,010
But the main
important polarization

687
00:37:02,010 --> 00:37:03,552
is the one on the
oxygen. So you have

688
00:37:03,552 --> 00:37:08,010
a core charge and a shell
charge and a spring constant.

689
00:37:08,010 --> 00:37:10,950
This is actually the
anion spring constant.

690
00:37:10,950 --> 00:37:12,510
The nice thing about
these potentials

691
00:37:12,510 --> 00:37:15,210
is that they were
fit all together.

692
00:37:15,210 --> 00:37:20,890
They actually fitted all these
potentials, the cation, anion,

693
00:37:20,890 --> 00:37:24,258
and the anion-ion terms
all in one massive fitting.

694
00:37:24,258 --> 00:37:25,800
And what that means
is that they have

695
00:37:25,800 --> 00:37:28,860
a consistent
anion-anion potential

696
00:37:28,860 --> 00:37:30,960
across multiple oxides.

697
00:37:30,960 --> 00:37:32,820
The problem with fitting
potentials in oxide

698
00:37:32,820 --> 00:37:36,450
has always been the
oxygen-oxygen one.

699
00:37:36,450 --> 00:37:39,700
Because in an oxide,
the oxygens are big.

700
00:37:39,700 --> 00:37:41,520
They're about 1.4 angstrom.

701
00:37:41,520 --> 00:37:43,060
A cation is small.

702
00:37:43,060 --> 00:37:45,600
Like, magnesium is,
say, 0.7 angstrom.

703
00:37:45,600 --> 00:37:49,680
So really an oxide is big
oxygens touching each other

704
00:37:49,680 --> 00:37:52,390
and the cations sitting in
the spaces between them.

705
00:37:52,390 --> 00:37:54,180
So the potential
between the oxygens

706
00:37:54,180 --> 00:37:57,600
is always the one
that kind of sets

707
00:37:57,600 --> 00:38:01,770
a lot of your dynamical effects,
a lot of your lattice parameter

708
00:38:01,770 --> 00:38:02,290
effects.

709
00:38:02,290 --> 00:38:05,640
So it's the one that's
hardest to deal with.

710
00:38:05,640 --> 00:38:07,080
It's also the problem.

711
00:38:07,080 --> 00:38:09,630
If there's a reason that
this sometimes fails,

712
00:38:09,630 --> 00:38:12,960
it's often because of
the anion potential.

713
00:38:12,960 --> 00:38:15,870
Even though we think
of oxygen as a 2- ion,

714
00:38:15,870 --> 00:38:19,380
it turns out it's actually
a very fluffy ion.

715
00:38:19,380 --> 00:38:24,510
That last p electron you
put on it to make it 2-

716
00:38:24,510 --> 00:38:27,540
is actually not stably
bound in a vacuum.

717
00:38:27,540 --> 00:38:30,450
It's bound largely by
the electrostatic field

718
00:38:30,450 --> 00:38:31,980
in the crystal.

719
00:38:31,980 --> 00:38:35,310
Remember, from the cations
comes a positive field.

720
00:38:35,310 --> 00:38:37,680
So the electron really wants
to be there on the oxygen.

721
00:38:37,680 --> 00:38:40,360
And that's what binds
the last p electron.

722
00:38:40,360 --> 00:38:41,790
So the problem
with the oxygen is

723
00:38:41,790 --> 00:38:45,240
that it relaxes
its wave functions

724
00:38:45,240 --> 00:38:46,300
depending on environment.

725
00:38:46,300 --> 00:38:48,160
So it's not a hard ion at all.

726
00:38:48,160 --> 00:38:50,340
And that's often
referred to as breeding.

727
00:38:50,340 --> 00:38:51,125
The oxygen breeds.

728
00:38:51,125 --> 00:38:53,250
So what that really means--
that the oxygen doesn't

729
00:38:53,250 --> 00:38:56,550
have the same size in
different environments.

730
00:38:56,550 --> 00:38:59,540
And this is where it gets
problematic for potentials

731
00:38:59,540 --> 00:39:01,540
because you model your
potential with, remember,

732
00:39:01,540 --> 00:39:04,740
this Buckingham potential, which
has a well-defined minimum.

733
00:39:04,740 --> 00:39:07,960
It sort of treats the oxygen
as the same every time,

734
00:39:07,960 --> 00:39:10,650
but this breeding basically
messes that up somewhat,

735
00:39:10,650 --> 00:39:13,770
that your oxygen has
somewhat different sides

736
00:39:13,770 --> 00:39:14,910
in different environments.

737
00:39:14,910 --> 00:39:19,350
If you're doing monoxides,
you pick that up

738
00:39:19,350 --> 00:39:22,950
in that difference in the
cation anion potential.

739
00:39:22,950 --> 00:39:26,445
If you're just comparing, say,
MgO and CaO, magnesium oxide

740
00:39:26,445 --> 00:39:29,520
and calcium oxide,
you can pick up

741
00:39:29,520 --> 00:39:32,640
the inconsistency between
the oxygen-oxygen potential

742
00:39:32,640 --> 00:39:36,330
in the two by modifying
the metal oxygen

743
00:39:36,330 --> 00:39:37,680
potential somewhat.

744
00:39:37,680 --> 00:39:40,530
But it's going to do complicated
oxides with a lot of metals.

745
00:39:40,530 --> 00:39:45,600
And at the same time you quite
can't play that game anymore.

746
00:39:45,600 --> 00:39:49,710
This is the charge density of
the oxygen in calcium oxide.

747
00:39:49,710 --> 00:39:51,990
And on the same scale,
this is the charge density

748
00:39:51,990 --> 00:39:53,610
in magnesium oxide.

749
00:39:53,610 --> 00:39:57,300
If you actually see,
it's not quite the same.

750
00:39:57,300 --> 00:40:00,060
It's actually not nearly as
spherical in calcium oxide

751
00:40:00,060 --> 00:40:02,070
as in magnesium oxide.

752
00:40:02,070 --> 00:40:04,900
And calcium is bigger.

753
00:40:04,900 --> 00:40:10,780
And these are in exactly
the same structure.

754
00:40:10,780 --> 00:40:14,710
The last one is really
the more problematic one,

755
00:40:14,710 --> 00:40:19,450
is that most oxides are
actually-- more and more,

756
00:40:19,450 --> 00:40:22,090
we find out that they
are less hard in valence

757
00:40:22,090 --> 00:40:23,650
than we thought they were.

758
00:40:23,650 --> 00:40:26,950
We always think of oxides as
there's 2- on the oxygens.

759
00:40:26,950 --> 00:40:30,340
And then I figured out
what my cation charge is.

760
00:40:30,340 --> 00:40:32,590
And for lithium, it's 1+.

761
00:40:32,590 --> 00:40:34,750
For magnesium, it's 2+.

762
00:40:34,750 --> 00:40:37,660
What we find more and more
is that the ions really don't

763
00:40:37,660 --> 00:40:40,480
have full charge in oxides.

764
00:40:40,480 --> 00:40:45,670
So what I showed you here, this
is mixtures of zirconium oxide

765
00:40:45,670 --> 00:40:47,570
and calcium oxide.

766
00:40:47,570 --> 00:40:51,010
And what is shown, the negative
numbers that are shown,

767
00:40:51,010 --> 00:40:55,152
are the charges on the
oxygen. And what you see

768
00:40:55,152 --> 00:40:56,860
is that they're kind
of all over the map.

769
00:40:56,860 --> 00:41:01,290
They depend, essentially, on
what cations are around them.

770
00:41:01,290 --> 00:41:02,910
Zirconium is 4+.

771
00:41:02,910 --> 00:41:04,770
Calcium is 2+.

772
00:41:04,770 --> 00:41:07,230
So if you actually have a
lot of zirconium around,

773
00:41:07,230 --> 00:41:10,620
you tend to make the
oxygen more negative than

774
00:41:10,620 --> 00:41:13,050
if you have calcium around you.

775
00:41:13,050 --> 00:41:16,260
And so what this
essentially means

776
00:41:16,260 --> 00:41:19,500
is that, when you do
empirical potential modeling

777
00:41:19,500 --> 00:41:23,530
and you use full
charges, in some sense,

778
00:41:23,530 --> 00:41:25,470
your energetics is too harsh.

779
00:41:25,470 --> 00:41:28,320
You tend to
overestimate energies

780
00:41:28,320 --> 00:41:31,560
because you have too large an
electrostatic contribution.

781
00:41:31,560 --> 00:41:34,650
In reality, when you treat
the full electron density

782
00:41:34,650 --> 00:41:37,080
in quantum mechanics,
the electron density

783
00:41:37,080 --> 00:41:40,200
will relax away a little
from these full charges

784
00:41:40,200 --> 00:41:45,500
and, thereby, sort of moderate
the electrostatic effect.

785
00:41:45,500 --> 00:41:48,950
So this is probably
the hardest problem

786
00:41:48,950 --> 00:41:54,760
in modeling oxides of
empirical potentials.

787
00:41:54,760 --> 00:41:56,680
This becomes
extremely problematic

788
00:41:56,680 --> 00:42:00,310
when you do transition metal
oxides because transition metal

789
00:42:00,310 --> 00:42:02,260
have variable valence.

790
00:42:02,260 --> 00:42:04,570
Magnesium, at least, we
always think of as 2+.

791
00:42:04,570 --> 00:42:06,910
But what if you're
doing iron oxides?

792
00:42:06,910 --> 00:42:09,550
Iron can be 2+, 3+, or 4+.

793
00:42:09,550 --> 00:42:12,940
And at some point, you will
do conformation changes

794
00:42:12,940 --> 00:42:15,640
in your material, and
your valence will change.

795
00:42:15,640 --> 00:42:17,950
And you can't deal with
that in a pair potential

796
00:42:17,950 --> 00:42:20,170
because it's the
same as in organics.

797
00:42:20,170 --> 00:42:24,370
A 2+ and a 3+ iron have a
different potential with oxygen

798
00:42:24,370 --> 00:42:26,480
because they have a
different repulsion.

799
00:42:26,480 --> 00:42:29,500
And so there are people who
have tried to make models

800
00:42:29,500 --> 00:42:34,500
with variable charge where you
have literally the short range

801
00:42:34,500 --> 00:42:36,840
repulsion, the
electrostatic term,

802
00:42:36,840 --> 00:42:39,660
and then a kind of
ionization term.

803
00:42:39,660 --> 00:42:44,340
So you would actually minimize
not just over atomic positions,

804
00:42:44,340 --> 00:42:46,440
but also over
charges on the ions.

805
00:42:46,440 --> 00:42:47,940
So there would be
some function that

806
00:42:47,940 --> 00:42:50,610
tells you how hard it is to
pull an electron off and put it

807
00:42:50,610 --> 00:42:51,540
on another.

808
00:42:51,540 --> 00:42:53,040
It's just that after
a while you run

809
00:42:53,040 --> 00:42:54,908
into so many fitting problems.

810
00:42:54,908 --> 00:42:56,700
The more parameters
you have in your model,

811
00:42:56,700 --> 00:42:59,010
the more things you have to
fit that you have to wonder

812
00:42:59,010 --> 00:43:01,980
whether it's worth doing that.

813
00:43:01,980 --> 00:43:04,572
And these models have not
sort of caught on in general.

814
00:43:04,572 --> 00:43:06,780
If you're interested, I can
give you some references,

815
00:43:06,780 --> 00:43:12,880
but they're not quite
in widespread use.

816
00:43:12,880 --> 00:43:14,380
So I kind of talked
about this here,

817
00:43:14,380 --> 00:43:15,850
the limitations
of pair potentials

818
00:43:15,850 --> 00:43:18,900
in oxides, the oxygen
change, the charge effect,

819
00:43:18,900 --> 00:43:20,670
and so many body effects.

820
00:43:20,670 --> 00:43:22,470
So let me sort of
summarize this,

821
00:43:22,470 --> 00:43:27,470
and then I'll hand over
Professor Marzari what

822
00:43:27,470 --> 00:43:29,700
I think we've seen.

823
00:43:29,700 --> 00:43:33,020
So in metals, you have
the issue of coordination.

824
00:43:33,020 --> 00:43:34,670
If you should
remember one thing,

825
00:43:34,670 --> 00:43:38,960
it's bond strength
depends on coordination.

826
00:43:38,960 --> 00:43:41,083
And you should remember
how that showed up

827
00:43:41,083 --> 00:43:43,250
in things like vacancy
formation energies and things

828
00:43:43,250 --> 00:43:44,292
like surface contraction.

829
00:43:44,292 --> 00:43:47,030
When you make a surface, the
layers tend to contract in.

830
00:43:47,030 --> 00:43:50,090
So if you have less
bonds, they're stronger.

831
00:43:50,090 --> 00:43:53,750
And that's the problem
for pair potentials.

832
00:43:53,750 --> 00:43:55,610
And methods like
embedded atom method

833
00:43:55,610 --> 00:43:58,160
do a great job of
fixing that problem.

834
00:43:58,160 --> 00:44:00,290
They're still missing
a lot of things.

835
00:44:00,290 --> 00:44:02,870
They don't have, in the end,
subtle electronic hybridization

836
00:44:02,870 --> 00:44:03,230
effects.

837
00:44:03,230 --> 00:44:05,180
So you still won't get
everything out of them,

838
00:44:05,180 --> 00:44:06,950
but they fixed
that main problem.

839
00:44:06,950 --> 00:44:09,140
In organic molecules,
the reason you're saved

840
00:44:09,140 --> 00:44:12,650
is because essentially you have
a limited set of environments.

841
00:44:12,650 --> 00:44:14,930
And you really parameterize
for each of them.

842
00:44:14,930 --> 00:44:18,140
And people have essentially
done that for you.

843
00:44:18,140 --> 00:44:20,000
The work is done.

844
00:44:20,000 --> 00:44:24,650
In oxides, I would say
they're sort of in between.

845
00:44:24,650 --> 00:44:27,260
You do reasonably
well for things

846
00:44:27,260 --> 00:44:30,490
that are sort of gross
topological changes.

847
00:44:30,490 --> 00:44:33,100
But because of the variable
charge effect, the oxygen

848
00:44:33,100 --> 00:44:36,220
breeding, and sometimes
charge transfer,

849
00:44:36,220 --> 00:44:39,560
you will miss that
subtle energetic effect.

850
00:44:39,560 --> 00:44:41,800
The other thing is that,
in transition metals,

851
00:44:41,800 --> 00:44:43,420
you often have magnetic moments.

852
00:44:43,420 --> 00:44:46,780
In transition metals, you have
a lot of unpaired electrons

853
00:44:46,780 --> 00:44:49,600
because the orbitals
are very localized.

854
00:44:49,600 --> 00:44:52,700
So you will tend to populate
them following Hund's rules.

855
00:44:52,700 --> 00:44:55,240
So you will tend to build up
magnetic moments on the ions.

856
00:44:55,240 --> 00:44:57,160
And so you will have
magnetic interactions.

857
00:44:57,160 --> 00:45:01,510
And these are almost never
dealt with in pair potentials.

858
00:45:01,510 --> 00:45:05,800
So I'm going to hand off here to
Professor Marzari, who's going

859
00:45:05,800 --> 00:45:07,987
to teach you quantum mechanics.

860
00:45:07,987 --> 00:45:09,070
NICOLA MARZARI: Excellent.

861
00:45:09,070 --> 00:45:13,100
So welcome to the second
part of the lecture.

862
00:45:13,100 --> 00:45:16,240
What we are going to see
in the next few classes

863
00:45:16,240 --> 00:45:18,670
are actually electronic
structure methods

864
00:45:18,670 --> 00:45:20,380
for modeling materials.

865
00:45:20,380 --> 00:45:22,540
And what I've
decided to do today

866
00:45:22,540 --> 00:45:26,170
is really just a brief
introduction and a refresher

867
00:45:26,170 --> 00:45:28,480
of quantum mechanics
or just an introduction

868
00:45:28,480 --> 00:45:29,860
if you haven't seen before.

869
00:45:29,860 --> 00:45:31,900
And this is mostly
because, at the end,

870
00:45:31,900 --> 00:45:34,480
electronic structure
methods involves

871
00:45:34,480 --> 00:45:37,420
the solution of a
Schrodinger equation

872
00:45:37,420 --> 00:45:39,770
or a relative of the
Schrodinger equation.

873
00:45:39,770 --> 00:45:42,460
And so I wanted to
remind you about that.

874
00:45:42,460 --> 00:45:45,970
This is sort of what we can do
more or less easily nowadays.

875
00:45:45,970 --> 00:45:48,520
This is a system with
a few hundred atoms.

876
00:45:48,520 --> 00:45:51,400
It's actually myoglobin
with a carbon monoxide

877
00:45:51,400 --> 00:45:53,660
molecules attached to it.

878
00:45:53,660 --> 00:45:55,990
And it's the kind
of calculation that

879
00:45:55,990 --> 00:46:01,690
would take a day or so on
a smaller research cluster.

880
00:46:01,690 --> 00:46:05,410
It's also a calculation that
basically predicts that we all

881
00:46:05,410 --> 00:46:07,840
would be dead in a
nanosecond or so,

882
00:46:07,840 --> 00:46:11,740
just because the error in the
binding energy of that molecule

883
00:46:11,740 --> 00:46:14,170
that you get with the standard
quantum mechanical method

884
00:46:14,170 --> 00:46:19,220
is actually not enough to sort
of compare with experiments.

885
00:46:19,220 --> 00:46:23,530
So let's see a few cases
in sort of which we

886
00:46:23,530 --> 00:46:28,060
think quantum mechanical
simulations are very relevant.

887
00:46:28,060 --> 00:46:32,380
The first case is one of
structure and bonding.

888
00:46:32,380 --> 00:46:36,310
Basically, a solid or a
molecule stays together

889
00:46:36,310 --> 00:46:39,370
because there is an
electronic glue that

890
00:46:39,370 --> 00:46:42,910
compensate the repulsion
between the nuclei.

891
00:46:42,910 --> 00:46:46,060
And the structure and stability
of different structures

892
00:46:46,060 --> 00:46:48,130
is determined by
the compensation

893
00:46:48,130 --> 00:46:49,950
of repulsive effects.

894
00:46:49,950 --> 00:46:51,940
So all nuclear repel each other.

895
00:46:51,940 --> 00:46:55,660
All electrons repel each
other, but nuclei and electrons

896
00:46:55,660 --> 00:46:57,310
attract each other.

897
00:46:57,310 --> 00:47:01,120
And say, here we are seeing
two different structures.

898
00:47:01,120 --> 00:47:03,700
On the left is the
high temperature

899
00:47:03,700 --> 00:47:06,040
cubic phase of lead titanite.

900
00:47:06,040 --> 00:47:08,440
And on the right, we
have the low temperature

901
00:47:08,440 --> 00:47:11,140
ferroelectric phase
of lead titanite.

902
00:47:11,140 --> 00:47:13,720
And you see it has
very subtle differences

903
00:47:13,720 --> 00:47:17,680
and very subtle differences
in the charge density.

904
00:47:17,680 --> 00:47:20,050
At the end, a solid
is really made

905
00:47:20,050 --> 00:47:23,800
by spherical atoms
that sort of interfere

906
00:47:23,800 --> 00:47:27,430
very lightly, destructively,
or constructively.

907
00:47:27,430 --> 00:47:30,190
And here you can see
sort of affirmation

908
00:47:30,190 --> 00:47:34,030
of some kind of stronger bond
between the white oxygen,

909
00:47:34,030 --> 00:47:37,510
in this case, and
the green lead atoms.

910
00:47:37,510 --> 00:47:39,400
And we are at the
stage in which sort

911
00:47:39,400 --> 00:47:43,390
of state of the art electronic
structure approaches

912
00:47:43,390 --> 00:47:47,230
are actually able to calculate
accurately these energy

913
00:47:47,230 --> 00:47:48,130
differences.

914
00:47:48,130 --> 00:47:51,160
And so they can give you
the zero temperature phase

915
00:47:51,160 --> 00:47:52,190
stability.

916
00:47:52,190 --> 00:47:54,770
And as you learn in
the rest of the class,

917
00:47:54,770 --> 00:47:57,310
once you have a good
energetic model,

918
00:47:57,310 --> 00:47:59,500
you can also calculate
the thermodynamics.

919
00:47:59,500 --> 00:48:03,070
Basically, thermodynamics
is the statistical mechanics

920
00:48:03,070 --> 00:48:05,710
of the thermal excitation
for such a system.

921
00:48:09,660 --> 00:48:12,230
Another case in which
electronic structure methods

922
00:48:12,230 --> 00:48:15,080
are obviously important
is when you are actually

923
00:48:15,080 --> 00:48:18,380
interested in electronic
structure properties.

924
00:48:18,380 --> 00:48:20,870
So if you want to
calculate or predict

925
00:48:20,870 --> 00:48:23,300
electronic properties,
optical properties,

926
00:48:23,300 --> 00:48:26,150
and, to a large extent,
magnetic properties,

927
00:48:26,150 --> 00:48:27,735
you really can't
use a potential.

928
00:48:27,735 --> 00:48:30,200
A potential, if anything,
gives you a structure,

929
00:48:30,200 --> 00:48:32,990
but will never give you
the electronic excitation.

930
00:48:32,990 --> 00:48:35,570
And this is actually a beautiful
case from recent research.

931
00:48:35,570 --> 00:48:39,200
This is from Professor Bawendi
in the chemistry department.

932
00:48:39,200 --> 00:48:41,540
And what you are
seeing here is actually

933
00:48:41,540 --> 00:48:46,130
a suspension of cadmium
selenides nanoparticles.

934
00:48:46,130 --> 00:48:49,250
And to a large extent,
the optical excitation

935
00:48:49,250 --> 00:48:52,010
can actually be modeled
just by the excitation

936
00:48:52,010 --> 00:48:55,010
of an electron confined
in a spherical potential.

937
00:48:55,010 --> 00:48:57,283
Also, one can do sort
of better than that.

938
00:48:57,283 --> 00:48:58,700
But what's happening
here is that,

939
00:48:58,700 --> 00:49:02,780
depending on the size, the
spacing between the excited

940
00:49:02,780 --> 00:49:03,770
level changes.

941
00:49:03,770 --> 00:49:06,080
And basically,
this system absorbs

942
00:49:06,080 --> 00:49:08,220
light in different ways.

943
00:49:08,220 --> 00:49:12,120
And so they can actually make
for great chemical sensors.

944
00:49:12,120 --> 00:49:14,240
And this is, again,
electronic properties

945
00:49:14,240 --> 00:49:16,850
is what we predict
with quantum mechanics.

946
00:49:16,850 --> 00:49:22,550
And finally, we can follow
chemical bonding, chemical

947
00:49:22,550 --> 00:49:25,560
breaking and chemical
forming reactions.

948
00:49:25,560 --> 00:49:28,350
So say you want to study
a chemical reaction.

949
00:49:28,350 --> 00:49:31,320
And this is a sort of classic
case for organic chemistry.

950
00:49:31,320 --> 00:49:33,980
This is a cycloaddition
reaction or what

951
00:49:33,980 --> 00:49:37,160
is called a Diels-Alder
reaction in which, basically,

952
00:49:37,160 --> 00:49:39,800
an ethylene molecule
break apart,

953
00:49:39,800 --> 00:49:42,357
and we form an aromatic bond.

954
00:49:42,357 --> 00:49:43,940
Well, this is something
that you could

955
00:49:43,940 --> 00:49:45,660
do with quantum mechanics.

956
00:49:45,660 --> 00:49:48,650
This is a picture that comes
from Bill Goddard's site in Cal

957
00:49:48,650 --> 00:49:49,310
Tech.

958
00:49:49,310 --> 00:49:51,590
And you see we are
actually following

959
00:49:51,590 --> 00:49:55,730
the highest occupied molecular
orbital as the reaction

960
00:49:55,730 --> 00:49:56,690
proceeds.

961
00:49:56,690 --> 00:49:59,750
And this is really the reason
why a potential wouldn't

962
00:49:59,750 --> 00:50:03,020
be able to describe
this to a large extent

963
00:50:03,020 --> 00:50:05,690
because what is
taking place is really

964
00:50:05,690 --> 00:50:09,080
a reorganization of
the electronic state.

965
00:50:09,080 --> 00:50:13,280
You see, ultimately, it's really
constructive and destructive

966
00:50:13,280 --> 00:50:16,520
interference of
electrons as standing

967
00:50:16,520 --> 00:50:20,420
waves in this potential that
determine the stability.

968
00:50:20,420 --> 00:50:22,370
And again, this is
sort of done with sort

969
00:50:22,370 --> 00:50:26,850
of current state of the art
electronics structure methods.

970
00:50:26,850 --> 00:50:29,120
So this is what we are
going to learn, basically.

971
00:50:29,120 --> 00:50:33,020
And a picture that I always
want you to have in mind

972
00:50:33,020 --> 00:50:35,930
is what I call the
standard model of matter.

973
00:50:35,930 --> 00:50:39,420
That is everything you
look at, a beta solid,

974
00:50:39,420 --> 00:50:43,130
a gas, or a molecule,
ultimately can

975
00:50:43,130 --> 00:50:45,730
be described as sort of
the current condition,

976
00:50:45,730 --> 00:50:48,170
so at the sort of
average energy that

977
00:50:48,170 --> 00:50:52,040
is typical of room
temperature or sort of roughly

978
00:50:52,040 --> 00:50:55,940
orders of magnitude
above and below it.

979
00:50:55,940 --> 00:50:59,630
First of all, by
ionic nuclei, that

980
00:50:59,630 --> 00:51:03,200
is really where all the mass
of your system is concentrated.

981
00:51:03,200 --> 00:51:06,950
Remember, a sort of nucleus as
actually a typical size of 10

982
00:51:06,950 --> 00:51:10,820
to the minus 14, 10 to
the minus 15 meters.

983
00:51:10,820 --> 00:51:14,510
But it's really where
all the mass lies.

984
00:51:14,510 --> 00:51:15,740
And the amazing thing--

985
00:51:15,740 --> 00:51:17,150
I always find this amazing--

986
00:51:17,150 --> 00:51:21,560
is that to a large extent,
even for a nucleus,

987
00:51:21,560 --> 00:51:25,610
Newton's equation of
motion still apply.

988
00:51:25,610 --> 00:51:27,290
I mean, in reality,
there are sort

989
00:51:27,290 --> 00:51:29,960
of a number of subtle
quantum effects.

990
00:51:29,960 --> 00:51:33,530
We'll mention this sort of
in the course of this class.

991
00:51:33,530 --> 00:51:36,020
But for all practical
purposes, you

992
00:51:36,020 --> 00:51:38,510
should start thinking
as this nuclei

993
00:51:38,510 --> 00:51:41,960
as behaving as
classical particles

994
00:51:41,960 --> 00:51:46,200
and moving around following the
Newton's equation of motion.

995
00:51:46,200 --> 00:51:49,100
So they have a mass,
and they feel a force.

996
00:51:49,100 --> 00:51:51,650
And the force that
they feel is basically

997
00:51:51,650 --> 00:51:54,830
a repulsive force from
all the other nuclei

998
00:51:54,830 --> 00:52:02,390
and the attractive force
from this electronic glue.

999
00:52:02,390 --> 00:52:06,240
These nuclei are actually
surrounded by electrons.

1000
00:52:06,240 --> 00:52:07,690
When you study the
periodic table,

1001
00:52:07,690 --> 00:52:10,370
you have studied all
the orbital shells.

1002
00:52:10,370 --> 00:52:17,210
And really what happens is that
all the lowest core electrons

1003
00:52:17,210 --> 00:52:21,140
are so tightly bound
that they really do not

1004
00:52:21,140 --> 00:52:26,210
sort of change in the process of
the forming of a chemical bond

1005
00:52:26,210 --> 00:52:28,290
or the breaking of
a chemical bond.

1006
00:52:28,290 --> 00:52:32,630
So say, if you are thinking at
any sort of second row element,

1007
00:52:32,630 --> 00:52:35,390
carbon, nitrogen,
oxygen, well, we

1008
00:52:35,390 --> 00:52:39,140
know that there are 1s
electrons in there, sort

1009
00:52:39,140 --> 00:52:41,030
of the one that we
have first filled.

1010
00:52:41,030 --> 00:52:43,700
Those 1s electrons
are core electrons.

1011
00:52:43,700 --> 00:52:47,930
And they really do not change
in the evolution of this system,

1012
00:52:47,930 --> 00:52:52,582
unless we are really at energy
so high that we can start sort

1013
00:52:52,582 --> 00:52:54,290
of kicking those
around, but that doesn't

1014
00:52:54,290 --> 00:52:56,520
happen for practical purposes.

1015
00:52:56,520 --> 00:52:59,000
So again, in your
standard model of matter,

1016
00:52:59,000 --> 00:53:03,830
you need to think the nucleus
surrounded by very tightly

1017
00:53:03,830 --> 00:53:06,980
bound shell of electrons.

1018
00:53:06,980 --> 00:53:10,040
And it's really only
the outer electrons,

1019
00:53:10,040 --> 00:53:12,590
what we call the
valence electrons, that

1020
00:53:12,590 --> 00:53:18,560
are those are so lightly bound
to the nucleus that can start

1021
00:53:18,560 --> 00:53:22,790
creating constructive and
destructive interference when

1022
00:53:22,790 --> 00:53:24,770
they fill another nucleus.

1023
00:53:24,770 --> 00:53:27,980
And so they create
chemical bonds.

1024
00:53:27,980 --> 00:53:30,410
See, if you just look
at the, say, charge

1025
00:53:30,410 --> 00:53:33,170
density in something
like platinum,

1026
00:53:33,170 --> 00:53:35,600
this is sort of a
platinum surface

1027
00:53:35,600 --> 00:53:37,962
to which a methane molecule
is attached to sort

1028
00:53:37,962 --> 00:53:40,190
of the beginning of catalysis.

1029
00:53:40,190 --> 00:53:42,620
You see, at the
end, what we have

1030
00:53:42,620 --> 00:53:47,785
we have spherical atoms with
very, very tiny bonding.

1031
00:53:47,785 --> 00:53:49,910
And again, it's not very
different from the picture

1032
00:53:49,910 --> 00:53:54,830
that we have seen before of the
oxygen atom in calcium oxide.

1033
00:53:54,830 --> 00:53:58,490
Obviously, all the relevant
energy is in this bonding.

1034
00:53:58,490 --> 00:54:01,790
The difference between platinum
in the metal state or, say,

1035
00:54:01,790 --> 00:54:04,580
a platinum atom
solvated in water,

1036
00:54:04,580 --> 00:54:07,400
is the difference in
this chemical bonding.

1037
00:54:07,400 --> 00:54:10,940
So studying what happens
here is very important.

1038
00:54:10,940 --> 00:54:14,780
But to a large extent, this
sort of general picture

1039
00:54:14,780 --> 00:54:18,080
of the charge density of
the atom being spherical

1040
00:54:18,080 --> 00:54:18,710
is conserved.

1041
00:54:18,710 --> 00:54:21,470
It's even in a
covalently bonded system.

1042
00:54:21,470 --> 00:54:24,170
This is a metallic
bonded system, platinum.

1043
00:54:24,170 --> 00:54:27,290
Up here, we have a covalent
bonded system that is methane.

1044
00:54:27,290 --> 00:54:29,600
And again, to a
large extent, you

1045
00:54:29,600 --> 00:54:32,510
see the spheres
that are the atoms.

1046
00:54:32,510 --> 00:54:35,390
But all what we care
in understanding

1047
00:54:35,390 --> 00:54:37,340
the stability of
different structure

1048
00:54:37,340 --> 00:54:41,060
is actually what those
last valence electrons do.

1049
00:54:41,060 --> 00:54:43,310
And obviously, those are
the lastly bound one.

1050
00:54:43,310 --> 00:54:47,510
When you start sort of creating
an atom, you have a nucleus.

1051
00:54:47,510 --> 00:54:49,340
And you start putting electrons.

1052
00:54:49,340 --> 00:54:51,510
You fill up sort
of all the levels.

1053
00:54:51,510 --> 00:54:55,550
The first one are incredibly
tightly bound to the nucleus.

1054
00:54:55,550 --> 00:54:58,640
And then the more you
add, the more these

1055
00:54:58,640 --> 00:55:01,350
electrons screen the nucleus.

1056
00:55:01,350 --> 00:55:03,230
And so the next
electrons are going

1057
00:55:03,230 --> 00:55:06,080
to see less and less
attractive charge to the point

1058
00:55:06,080 --> 00:55:08,180
that the last electron
sees really very

1059
00:55:08,180 --> 00:55:09,500
little attractive charge.

1060
00:55:09,500 --> 00:55:11,930
And so it's a valence
electron ready to bind.

1061
00:55:11,930 --> 00:55:13,350
Yes.

1062
00:55:13,350 --> 00:55:17,850
AUDIENCE: [INAUDIBLE]

1063
00:55:17,850 --> 00:55:19,380
NICOLA MARZARI:
Very, very good--

1064
00:55:19,380 --> 00:55:22,230
this is the difference
between if you

1065
00:55:22,230 --> 00:55:24,870
want the classical
behavior of particle

1066
00:55:24,870 --> 00:55:28,020
and the quantum mechanical
behavior of particle.

1067
00:55:28,020 --> 00:55:31,800
Ultimately, I like to think
of it as the indetermination

1068
00:55:31,800 --> 00:55:32,520
principle.

1069
00:55:32,520 --> 00:55:36,810
That is, if the electron
were to fall on the nucleus,

1070
00:55:36,810 --> 00:55:38,820
we would be in a
situation in which

1071
00:55:38,820 --> 00:55:41,850
both its position
and its momentum

1072
00:55:41,850 --> 00:55:44,790
are known with
infinite precision.

1073
00:55:44,790 --> 00:55:46,500
And that is something
that doesn't

1074
00:55:46,500 --> 00:55:49,020
happen in quantum mechanics.

1075
00:55:49,020 --> 00:55:51,870
And we'll see it
in a different way.

1076
00:55:51,870 --> 00:55:56,490
Ultimately, electrons
behave as wave.

1077
00:55:56,490 --> 00:56:00,180
And so the ground
state of that wave

1078
00:56:00,180 --> 00:56:02,190
is something in
which the electron

1079
00:56:02,190 --> 00:56:05,200
can't live on the nucleus.

1080
00:56:05,200 --> 00:56:07,320
If you want another
sort of analogy,

1081
00:56:07,320 --> 00:56:10,110
could be one of an organ pipe.

1082
00:56:10,110 --> 00:56:13,770
If you have an organ pipe,
there is a fundamental harmonic

1083
00:56:13,770 --> 00:56:16,530
that you can play
and high harmonics.

1084
00:56:16,530 --> 00:56:19,920
But there is nothing
lower in energy or sort

1085
00:56:19,920 --> 00:56:24,000
of deeper in bass than
the fundamental harmonic.

1086
00:56:24,000 --> 00:56:27,090
So because, basically,
electrons behave

1087
00:56:27,090 --> 00:56:31,500
as quantum mechanical particles,
that is behave as wave,

1088
00:56:31,500 --> 00:56:34,830
their lowest energy
meaningful state

1089
00:56:34,830 --> 00:56:38,250
is one in which they do not
collapse to the nucleus,

1090
00:56:38,250 --> 00:56:39,840
but they sort of
create-- and we'll

1091
00:56:39,840 --> 00:56:44,820
see this in a moment-- sort
of an appropriate interference

1092
00:56:44,820 --> 00:56:48,280
a stable stationary
state around the nucleus.

1093
00:56:48,280 --> 00:56:50,520
And sort of, if you want,
this is the difference

1094
00:56:50,520 --> 00:56:53,850
between quantum mechanics
and classical mechanics.

1095
00:56:53,850 --> 00:56:57,690
If an electron was a
classical particle,

1096
00:56:57,690 --> 00:56:59,730
it would collapse
on the nucleus.

1097
00:56:59,730 --> 00:57:02,190
Because it's not, it doesn't.

1098
00:57:02,190 --> 00:57:06,180
And you know, quantum mechanics
has been largely understanding

1099
00:57:06,180 --> 00:57:09,120
what are these rules that
quantum object follows

1100
00:57:09,120 --> 00:57:10,380
that are just different.

1101
00:57:10,380 --> 00:57:12,840
And you know, sort of you'll
see some of these rules.

1102
00:57:12,840 --> 00:57:15,420
And there are sort of different
ways in explaining this.

1103
00:57:15,420 --> 00:57:18,660
I like to think that basically
the indetermination principle.

1104
00:57:18,660 --> 00:57:21,930
At the end, a quantum
particle has a certain amount

1105
00:57:21,930 --> 00:57:24,900
of indetermination that
can actually be quantified.

1106
00:57:24,900 --> 00:57:29,340
If that quantum particle were
to collapse on the nucleus,

1107
00:57:29,340 --> 00:57:32,820
the indetermination of the
two conjugate variables,

1108
00:57:32,820 --> 00:57:36,480
the momentum and the
position, would actually be 0.

1109
00:57:36,480 --> 00:57:39,010
And that can't happen.

1110
00:57:39,010 --> 00:57:42,930
You know, as much as you start,
say, in classical mechanics,

1111
00:57:42,930 --> 00:57:47,070
that a particle follows
Newton's equation of motion

1112
00:57:47,070 --> 00:57:50,320
and not something different,
in quantum mechanics,

1113
00:57:50,320 --> 00:57:53,880
you learn a set of rules that
these quantum particles follow

1114
00:57:53,880 --> 00:57:55,590
and that have been verified.

1115
00:57:55,590 --> 00:57:57,840
And one of these
rules basically says

1116
00:57:57,840 --> 00:58:01,470
that the lowest energy
state for that particle

1117
00:58:01,470 --> 00:58:05,190
is not the one in which it
collapses, but the one in which

1118
00:58:05,190 --> 00:58:07,860
it sort of forms an
appropriate interference

1119
00:58:07,860 --> 00:58:11,170
pattern around the nucleus,
but not collapse on it.

1120
00:58:11,170 --> 00:58:14,970
So if you want, it's truly
quantum mechanics at its heart.

1121
00:58:19,770 --> 00:58:23,730
And now, you see, if you
really want to sort of recover

1122
00:58:23,730 --> 00:58:27,900
some of the subtleties
that are somehow

1123
00:58:27,900 --> 00:58:30,490
invisible in this charge,
there's the picture.

1124
00:58:30,490 --> 00:58:32,580
If you want sort
of to recover what

1125
00:58:32,580 --> 00:58:37,835
are the tiny differences between
them sort of electronic levels,

1126
00:58:37,835 --> 00:58:39,210
well, what you
could do-- this is

1127
00:58:39,210 --> 00:58:41,580
sort of typical in a
transactional simulation.

1128
00:58:41,580 --> 00:58:44,790
You could take a
charge the difference

1129
00:58:44,790 --> 00:58:48,090
between a system in
which you have a methane

1130
00:58:48,090 --> 00:58:49,800
molecule attached
to the platinum,

1131
00:58:49,800 --> 00:58:51,900
or you don't have
a methane molecule.

1132
00:58:51,900 --> 00:58:54,450
And there, in this
difference, you

1133
00:58:54,450 --> 00:58:58,800
start sort of seeing patterns
of chemical bonding emerging.

1134
00:58:58,800 --> 00:59:00,690
And so you see sort
of places where

1135
00:59:00,690 --> 00:59:04,200
the charge density of your
system increases or decreases.

1136
00:59:04,200 --> 00:59:06,670
And you can start
seeing here actually

1137
00:59:06,670 --> 00:59:09,820
the d orbitals of the
platinum and the formation of

1138
00:59:09,820 --> 00:59:12,430
and the breaking of
the chemical bonds.

1139
00:59:12,430 --> 00:59:15,570
So there is this general
picture of atoms and spheres,

1140
00:59:15,570 --> 00:59:19,020
but then all what we care
are those subtle effects

1141
00:59:19,020 --> 00:59:20,980
that determine bonding.

1142
00:59:20,980 --> 00:59:24,620
And this is actually to put
some energy numbers on it.

1143
00:59:24,620 --> 00:59:27,330
Sort of this is
what relevant was.

1144
00:59:27,330 --> 00:59:30,840
We live at around 300
Kelvin, sort of give or take

1145
00:59:30,840 --> 00:59:32,610
1 order of magnitude.

1146
00:59:32,610 --> 00:59:35,730
And the kinetic
energy of an atom

1147
00:59:35,730 --> 00:59:39,000
in a sort of ideal
perfect gas is basically

1148
00:59:39,000 --> 00:59:41,950
0.04 electron volts.

1149
00:59:41,950 --> 00:59:45,000
So this is sort of
the average energy

1150
00:59:45,000 --> 00:59:48,480
that is available
because of temperature.

1151
00:59:48,480 --> 00:59:49,950
And so you can
understand that it's

1152
00:59:49,950 --> 00:59:56,220
sort of that is somehow a
button limit for chemical bonds.

1153
00:59:56,220 --> 00:59:59,580
If the chemical bonds
are, say, in our system,

1154
00:59:59,580 --> 01:00:02,850
we're going to be
comparable or smaller

1155
01:00:02,850 --> 01:00:05,310
than 0.04 electrons volts.

1156
01:00:05,310 --> 01:00:09,180
We would actually evaporate
right away and disappear.

1157
01:00:09,180 --> 01:00:15,930
So if we actually look at the
binding energy of the hydrogen

1158
01:00:15,930 --> 01:00:20,610
bond in a dimer between
two water molecule,

1159
01:00:20,610 --> 01:00:24,160
it's sort of roughly one
order of magnitude larger.

1160
01:00:24,160 --> 01:00:26,010
So really water at
room temperature

1161
01:00:26,010 --> 01:00:27,900
is sort of bound together.

1162
01:00:27,900 --> 01:00:31,710
It's a liquid at
0.29 electrons volts.

1163
01:00:31,710 --> 01:00:34,840
And now, you start understanding
the standard model of matter.

1164
01:00:34,840 --> 01:00:39,300
If you think at what is the
binding energy of an electron

1165
01:00:39,300 --> 01:00:40,300
just to a proton--

1166
01:00:40,300 --> 01:00:42,510
this is really the
hydrogen atom--

1167
01:00:42,510 --> 01:00:47,640
that binding energy is almost
2 orders of magnitude larger,

1168
01:00:47,640 --> 01:00:49,830
13 electron volts.

1169
01:00:49,830 --> 01:00:52,800
And so you can think
that, if we start

1170
01:00:52,800 --> 01:00:55,620
going from hydrogen to
the sort of other atoms

1171
01:00:55,620 --> 01:01:01,050
in the periodic table, the 1s
electron in the other atoms

1172
01:01:01,050 --> 01:01:03,990
are going to be even
more tightly bound.

1173
01:01:03,990 --> 01:01:06,930
And so they will
never be affected

1174
01:01:06,930 --> 01:01:13,260
by sort of the average energy
present at current temperature.

1175
01:01:13,260 --> 01:01:17,760
And actually, this number
scales as the square

1176
01:01:17,760 --> 01:01:19,390
of the atomic number.

1177
01:01:19,390 --> 01:01:21,210
So those core
electrons, in particular

1178
01:01:21,210 --> 01:01:24,570
those one core electrons, are
going to be terribly bound.

1179
01:01:24,570 --> 01:01:29,280
The only chemistry that we do
is with the highest electrons

1180
01:01:29,280 --> 01:01:33,480
whose binding energy is sort
of comparable, basically,

1181
01:01:33,480 --> 01:01:36,260
to this number.

1182
01:01:36,260 --> 01:01:42,500
OK, I think we are
particularly lucky in having

1183
01:01:42,500 --> 01:01:45,530
sort of a number of
books that are actually

1184
01:01:45,530 --> 01:01:50,000
excellent references for
electronic structure research

1185
01:01:50,000 --> 01:01:51,020
nowadays.

1186
01:01:51,020 --> 01:01:56,450
And I sort of mentioned here
three of the ones I like more.

1187
01:01:56,450 --> 01:01:58,220
They are all very recent.

1188
01:01:58,220 --> 01:02:03,200
There is the book from Richard
Martin at Urbana-Champaign

1189
01:02:03,200 --> 01:02:05,720
that I would really
say it's, by now,

1190
01:02:05,720 --> 01:02:08,690
the Bible of electronic
structure methods.

1191
01:02:08,690 --> 01:02:12,470
It's a very complete
and very detailed book.

1192
01:02:12,470 --> 01:02:14,090
So this is what
you would probably

1193
01:02:14,090 --> 01:02:17,570
want to have if you were
actually to do research

1194
01:02:17,570 --> 01:02:19,880
in electronic structure,
especially seen

1195
01:02:19,880 --> 01:02:23,340
from the point of view of
solid-state or condensed matter

1196
01:02:23,340 --> 01:02:25,430
and solids.

1197
01:02:25,430 --> 01:02:28,070
Another very beautiful
book that is, I think,

1198
01:02:28,070 --> 01:02:31,490
the closest in
spirit to this class

1199
01:02:31,490 --> 01:02:33,740
is the book by Mike
Finnis at the University

1200
01:02:33,740 --> 01:02:37,670
of Belfast, Interatomic
Forces in Condensed Matter.

1201
01:02:37,670 --> 01:02:41,390
It covers both quantum
mechanical approaches,

1202
01:02:41,390 --> 01:02:45,440
and it also shows how potential
approaches, as those described

1203
01:02:45,440 --> 01:02:47,990
by Professor Ceder in
the previous lectures,

1204
01:02:47,990 --> 01:02:50,270
actually emerge
very naturally from

1205
01:02:50,270 --> 01:02:52,100
quantum mechanical methods.

1206
01:02:52,100 --> 01:02:54,530
And we'll see a
little bit of this.

1207
01:02:54,530 --> 01:02:59,270
And the third book is by Tim
Kaxiras at Harvard University.

1208
01:02:59,270 --> 01:03:03,080
And again, it's an
electronic structure book

1209
01:03:03,080 --> 01:03:08,780
with a sort of very broad
perspective on solids

1210
01:03:08,780 --> 01:03:10,800
and condensed matter systems.

1211
01:03:10,800 --> 01:03:14,750
So any of this is actually
a very good reference.

1212
01:03:14,750 --> 01:03:17,030
And so what I want to do
after this introduction

1213
01:03:17,030 --> 01:03:19,100
in the next 15
minutes is give you

1214
01:03:19,100 --> 01:03:23,510
a refresher of quantum mechanics
in case you have never seen it.

1215
01:03:23,510 --> 01:03:26,180
And we can also discuss
actually some reading in case

1216
01:03:26,180 --> 01:03:27,080
you're interested.

1217
01:03:27,080 --> 01:03:29,310
And so this is how I do it.

1218
01:03:29,310 --> 01:03:32,090
If you go to London,
Piccadilly Circus, well, there

1219
01:03:32,090 --> 01:03:34,220
is the Reduced
Shakespeare Company.

1220
01:03:34,220 --> 01:03:37,460
They do in 90 minutes
all Shakespeare tragedies

1221
01:03:37,460 --> 01:03:38,870
and all Shakespeare comedies.

1222
01:03:38,870 --> 01:03:42,440
And so here the plan is to do
in 15 minutes all of quantum

1223
01:03:42,440 --> 01:03:46,160
mechanics sort of to give you
the flavor of what goes on.

1224
01:03:46,160 --> 01:03:49,790
And you see, reducing
expectation for over 20 years.

1225
01:03:49,790 --> 01:03:51,740
I actually really like them.

1226
01:03:51,740 --> 01:03:52,460
OK.

1227
01:03:52,460 --> 01:03:55,940
So this really goes
to answer the question

1228
01:03:55,940 --> 01:03:57,560
that we had before.

1229
01:03:57,560 --> 01:03:59,720
And that's sort
of, historically,

1230
01:03:59,720 --> 01:04:04,610
sort of between the 1900s and
sort of the mid-'20s, there was

1231
01:04:04,610 --> 01:04:09,500
a sort of emerging experimental
consensus on a number

1232
01:04:09,500 --> 01:04:11,580
of phenomenon.

1233
01:04:11,580 --> 01:04:15,140
The first one was sort of
studying the properties

1234
01:04:15,140 --> 01:04:17,030
of light in particular.

1235
01:04:17,030 --> 01:04:19,940
Light, ultimately, is
an electromagnetic wave.

1236
01:04:19,940 --> 01:04:22,730
And people discovered
the photoelectric effect.

1237
01:04:22,730 --> 01:04:26,450
That's actually Albert
Einstein paper 100 years ago.

1238
01:04:26,450 --> 01:04:30,230
This year is that the 100th
anniversary of the annus

1239
01:04:30,230 --> 01:04:34,370
mirabilis of Albert Einstein,
in which he was basically

1240
01:04:34,370 --> 01:04:37,630
giving an explanation of
the photoelectric effect.

1241
01:04:37,630 --> 01:04:42,080
There are illuminating material,
something like selenium.

1242
01:04:42,080 --> 01:04:46,340
And basically, your
electromagnetic radiation

1243
01:04:46,340 --> 01:04:50,990
excite discretely electrons
out of the material.

1244
01:04:50,990 --> 01:04:53,540
And that can sort
of be rationalized

1245
01:04:53,540 --> 01:04:58,430
in terms of quanta of energy,
discrete amounts of energy,

1246
01:04:58,430 --> 01:05:01,410
what we call photons
being exchanged.

1247
01:05:01,410 --> 01:05:03,980
So something that people
used to think as continuum,

1248
01:05:03,980 --> 01:05:07,730
as electromagnetic wave that can
have, say, any, in principle,

1249
01:05:07,730 --> 01:05:11,420
wavelength is actually
sort of made up

1250
01:05:11,420 --> 01:05:14,330
of massless particles
that have a sort

1251
01:05:14,330 --> 01:05:16,290
of discrete amount of energy.

1252
01:05:16,290 --> 01:05:17,850
So that was one thing.

1253
01:05:17,850 --> 01:05:20,480
The other thing that
sort of was fairly exotic

1254
01:05:20,480 --> 01:05:22,490
that Planck explained
and sort of, you know,

1255
01:05:22,490 --> 01:05:24,110
that's the birth
of quantum theory,

1256
01:05:24,110 --> 01:05:26,450
was actually looking
at the emission

1257
01:05:26,450 --> 01:05:28,490
of an incandescent body.

1258
01:05:28,490 --> 01:05:31,160
That's what is called
black body radiation.

1259
01:05:31,160 --> 01:05:34,040
And the spectrum
could be explained

1260
01:05:34,040 --> 01:05:38,690
in terms of excitation of a
gas-like of hot particles.

1261
01:05:38,690 --> 01:05:42,110
But again, spectrum is an
electromagnetic radiation.

1262
01:05:42,110 --> 01:05:44,960
And so there was this
sort of general idea

1263
01:05:44,960 --> 01:05:48,770
of electromagnetic waves have
particle-like properties.

1264
01:05:48,770 --> 01:05:51,800
What was really
sort of interesting

1265
01:05:51,800 --> 01:05:56,420
was sort of the discovery that
particles, like electrons,

1266
01:05:56,420 --> 01:05:58,640
can have wave-like properties.

1267
01:05:58,640 --> 01:06:00,320
And that actually
sort of determined,

1268
01:06:00,320 --> 01:06:04,670
ultimately, sort of the
development of wave mechanics

1269
01:06:04,670 --> 01:06:06,980
or quantum mechanics
and, in particular,

1270
01:06:06,980 --> 01:06:10,880
the Davisson-Germer
discovery that electron beams

1271
01:06:10,880 --> 01:06:12,140
can be refracted.

1272
01:06:12,140 --> 01:06:15,050
So electrons, when
you shoot electrons,

1273
01:06:15,050 --> 01:06:17,840
they are not behaving
as classical particle.

1274
01:06:17,840 --> 01:06:20,480
And they either sort of hit
something and bounce back,

1275
01:06:20,480 --> 01:06:21,950
or they go through.

1276
01:06:21,950 --> 01:06:26,048
But they actually diffract
and so behave as waves.

1277
01:06:26,048 --> 01:06:27,590
And sort of these
were sort of if you

1278
01:06:27,590 --> 01:06:30,260
want one of the key issues.

1279
01:06:30,260 --> 01:06:32,750
And how do we summarize this?

1280
01:06:32,750 --> 01:06:37,260
Well, this is actually sort
of one of fundamental ideas,

1281
01:06:37,260 --> 01:06:40,350
so one of the ideas that
I want you to remember

1282
01:06:40,350 --> 01:06:42,220
from quantum mechanics.

1283
01:06:42,220 --> 01:06:44,640
This is what is called
de Broglie relation.

1284
01:06:44,640 --> 01:06:46,500
It's from 1923.

1285
01:06:46,500 --> 01:06:49,490
It's what I call
actually the shortest PhD

1286
01:06:49,490 --> 01:06:51,330
thesis in the world.

1287
01:06:51,330 --> 01:06:53,370
There are only three
letters in that.

1288
01:06:53,370 --> 01:06:56,850
But sort of what it's
telling us is this.

1289
01:06:56,850 --> 01:06:59,310
There is a fundamental constant.

1290
01:06:59,310 --> 01:07:00,600
It's like gravitation.

1291
01:07:00,600 --> 01:07:02,850
In gravitation, you have a
fundamental constant that

1292
01:07:02,850 --> 01:07:04,920
is the attraction between mass.

1293
01:07:04,920 --> 01:07:07,830
Well, there is another
fundamental constant

1294
01:07:07,830 --> 01:07:10,980
that we call the
Planck constant.

1295
01:07:10,980 --> 01:07:14,010
And here is sort of
what its value is

1296
01:07:14,010 --> 01:07:16,170
in the international system.

1297
01:07:16,170 --> 01:07:21,240
And this is a relation that
all particles, all objects,

1298
01:07:21,240 --> 01:07:22,770
satisfy.

1299
01:07:22,770 --> 01:07:28,170
And it says that, if a particle
has a certain momentum, that is

1300
01:07:28,170 --> 01:07:31,320
has a certain mass
time velocity--

1301
01:07:31,320 --> 01:07:34,920
OK, so if a particle
has a certain momentum,

1302
01:07:34,920 --> 01:07:39,210
it will also display
wave-like properties.

1303
01:07:39,210 --> 01:07:43,770
And the wavelength typical
of those wave-like properties

1304
01:07:43,770 --> 01:07:45,670
is given by lambda.

1305
01:07:45,670 --> 01:07:47,610
So basically, you
have an electron.

1306
01:07:47,610 --> 01:07:49,440
You can sort of
plug in what would

1307
01:07:49,440 --> 01:07:52,350
be an average momentum
for that electron,

1308
01:07:52,350 --> 01:07:55,740
and you find out what
is the wavelength.

1309
01:07:55,740 --> 01:07:59,340
What it turns out is
that, if you plug here

1310
01:07:59,340 --> 01:08:03,840
the average momentum of a
valence electron, what you find

1311
01:08:03,840 --> 01:08:07,740
out is that the
wavelength that it has

1312
01:08:07,740 --> 01:08:11,490
is sort of comparable with
the interatomic distances.

1313
01:08:11,490 --> 01:08:14,940
So electrons behave
significantly

1314
01:08:14,940 --> 01:08:18,960
as waves at the length scale
that are relevant for you.

1315
01:08:18,960 --> 01:08:22,649
Because if you have two atoms
that are two angstrom far apart

1316
01:08:22,649 --> 01:08:25,710
and the electron is a
wavelength of 2 angstrom,

1317
01:08:25,710 --> 01:08:28,410
well, then we really
need to take into account

1318
01:08:28,410 --> 01:08:32,220
the wave-like
properties of electrons.

1319
01:08:32,220 --> 01:08:34,410
And you see, this is
why actually nuclei

1320
01:08:34,410 --> 01:08:37,500
we can get by just with
classical mechanics

1321
01:08:37,500 --> 01:08:41,340
because the mass of a
proton is 2,000 times

1322
01:08:41,340 --> 01:08:43,060
the mass of an electron.

1323
01:08:43,060 --> 01:08:49,200
So the mass of a nucleus is
at least 2,000 times larger

1324
01:08:49,200 --> 01:08:50,970
than the mass of an electron.

1325
01:08:50,970 --> 01:08:55,660
And you actually have sort of
tens of protons and neutrons.

1326
01:08:55,660 --> 01:08:59,310
So the typical
momentum is going to be

1327
01:08:59,310 --> 01:09:03,960
thousands, tens of
thousands, larger

1328
01:09:03,960 --> 01:09:06,060
than the momentum
of the electron.

1329
01:09:06,060 --> 01:09:10,470
And so the typical wavelength
is going to be at least a 3

1330
01:09:10,470 --> 01:09:14,100
or 4 orders of
magnitude smaller.

1331
01:09:14,100 --> 01:09:18,840
And so, now, the wavelength
of a nucleus start

1332
01:09:18,840 --> 01:09:22,859
to be 10 to the minus 14,
10 to the minus 15 meters,

1333
01:09:22,859 --> 01:09:29,229
is not relevant anymore for the
distance between the nuclei.

1334
01:09:29,229 --> 01:09:33,689
So for all practical purposes,
the wavelength of nuclei

1335
01:09:33,689 --> 01:09:36,149
is so small that
we can treat nuclei

1336
01:09:36,149 --> 01:09:38,250
as a point-like particles.

1337
01:09:38,250 --> 01:09:41,560
But the wave of electrons is
comparable to the distance,

1338
01:09:41,560 --> 01:09:45,270
and so we need to describe
electrons as waves.

1339
01:09:45,270 --> 01:09:49,319
And this relation is actually
very simple to sort of look

1340
01:09:49,319 --> 01:09:51,319
at in practical terms.

1341
01:09:51,319 --> 01:09:55,080
Instead of using the
international system of meters

1342
01:09:55,080 --> 01:09:57,240
and seconds and
so on, we actually

1343
01:09:57,240 --> 01:10:00,120
use a different system
of unit of measure.

1344
01:10:00,120 --> 01:10:02,490
That is what we
call atomic units.

1345
01:10:02,490 --> 01:10:06,570
Of course, no one forces
us to use the meter

1346
01:10:06,570 --> 01:10:07,800
as a unit of measure.

1347
01:10:07,800 --> 01:10:11,580
That was actually a byproduct
of the French Revolution.

1348
01:10:11,580 --> 01:10:18,240
So sort of a lot of people in
the general arena of chemistry

1349
01:10:18,240 --> 01:10:21,690
and solid state use different
sets of units of measure.

1350
01:10:21,690 --> 01:10:24,600
One of the most
difficult is atomic units

1351
01:10:24,600 --> 01:10:30,300
where the unit of length is
actually the typical radius

1352
01:10:30,300 --> 01:10:32,290
of 1 hydrogen atom.

1353
01:10:32,290 --> 01:10:36,480
So the sort of standard
measure gives us

1354
01:10:36,480 --> 01:10:43,080
0.53 angstrom or what is
called 1 Bohr for the radius

1355
01:10:43,080 --> 01:10:46,240
of an hydrogen atom.

1356
01:10:46,240 --> 01:10:49,350
And so this will
be an atomic unit.

1357
01:10:49,350 --> 01:10:55,200
So 1 angstrom is
roughly 2 atomic units.

1358
01:10:55,200 --> 01:10:57,330
In this scheme,
atomic unit system,

1359
01:10:57,330 --> 01:11:03,950
the Planck constant is
actually going to be 2 pi.

1360
01:11:03,950 --> 01:11:06,800
6.28, that's what
it turns out to be

1361
01:11:06,800 --> 01:11:09,140
when we sort of choose
appropriately all

1362
01:11:09,140 --> 01:11:11,240
the atomic units of measure.

1363
01:11:11,240 --> 01:11:17,090
And again, the atomic
unit of momentum, one

1364
01:11:17,090 --> 01:11:20,900
is going to be the
mass of an electron

1365
01:11:20,900 --> 01:11:26,900
times the velocity of
a 1s electron, again,

1366
01:11:26,900 --> 01:11:28,500
in a hydrogen atom.

1367
01:11:28,500 --> 01:11:31,910
So you see the electron
in the hydrogen atom

1368
01:11:31,910 --> 01:11:35,420
basically satisfy
perfectly Planck relation.

1369
01:11:35,420 --> 01:11:40,910
Because in atomic units, it's
momentum p is equal to 1.

1370
01:11:40,910 --> 01:11:45,440
And h, the Planck
constant, is equal to 2 pi.

1371
01:11:45,440 --> 01:11:50,220
So its wavelength is
going to be equal to 2 pi.

1372
01:11:50,220 --> 01:11:54,440
And if you think at the atom
as having a radius of 1,

1373
01:11:54,440 --> 01:11:57,140
you understand that
the wavelength of 2 pi

1374
01:11:57,140 --> 01:11:59,660
is the right
wavelength to basically

1375
01:11:59,660 --> 01:12:05,780
have along the diameter
1 single oscillation.

1376
01:12:05,780 --> 01:12:09,170
And so we sort of start
thinking at this ground

1377
01:12:09,170 --> 01:12:11,420
state of the
electron that doesn't

1378
01:12:11,420 --> 01:12:17,390
fall on the nucleus as
the fundamental harmonics

1379
01:12:17,390 --> 01:12:21,050
for this wave in the
presence of a potential.

1380
01:12:24,800 --> 01:12:25,370
OK.

1381
01:12:25,370 --> 01:12:30,170
There are cases in which the
quantum mechanical nature

1382
01:12:30,170 --> 01:12:34,280
of the nuclei actually
becomes relevant.

1383
01:12:34,280 --> 01:12:38,750
There is much less
modeling activity on that

1384
01:12:38,750 --> 01:12:43,100
because it's just fairly
computationally expensive.

1385
01:12:43,100 --> 01:12:46,010
One of the cases in
which it's important

1386
01:12:46,010 --> 01:12:49,550
is when you have a
very light nuclei.

1387
01:12:49,550 --> 01:12:52,670
So if you have a system
that has a lot of hydrogen,

1388
01:12:52,670 --> 01:12:56,030
well, it actually
might become relevant.

1389
01:12:56,030 --> 01:12:59,720
And the fact that your proton
has its own wavelength,

1390
01:12:59,720 --> 01:13:02,300
has its own delocalization,
could actually

1391
01:13:02,300 --> 01:13:03,980
become important.

1392
01:13:03,980 --> 01:13:08,330
And it's never going
to change dramatically

1393
01:13:08,330 --> 01:13:10,520
the physics of the
chemistry of your system,

1394
01:13:10,520 --> 01:13:13,160
but it's going to
introduce relevant effects.

1395
01:13:13,160 --> 01:13:15,980
And you know, this
is actually coming

1396
01:13:15,980 --> 01:13:20,420
from a Nature paper from 1999, a
very, very expensive simulation

1397
01:13:20,420 --> 01:13:23,130
in which both the electronics
degrees of freedom

1398
01:13:23,130 --> 01:13:23,990
were studied.

1399
01:13:23,990 --> 01:13:27,650
And also, the quantum
delocalization effects

1400
01:13:27,650 --> 01:13:30,470
on the proton were
studied, basically,

1401
01:13:30,470 --> 01:13:34,050
for what is called a
hydrated proton in water.

1402
01:13:34,050 --> 01:13:37,550
So if you have water and
you add a proton, well,

1403
01:13:37,550 --> 01:13:43,160
the first thing that will
form is a hydronium ion, H3O+.

1404
01:13:43,160 --> 01:13:46,130
And for 20, 30 years,
people had actually

1405
01:13:46,130 --> 01:13:51,650
discussed what was the general
coordination of this proton.

1406
01:13:51,650 --> 01:13:55,490
There was sort of a hypothesis
in which people were discussing

1407
01:13:55,490 --> 01:13:58,340
about an Eigen
compound, from Eigen,

1408
01:13:58,340 --> 01:14:00,200
the person that
first suggested it,

1409
01:14:00,200 --> 01:14:05,210
a proton bound to two water
molecules basically creating

1410
01:14:05,210 --> 01:14:09,140
an H5O 2+ system.

1411
01:14:09,140 --> 01:14:11,990
And then there was the Zundel
hypothesis in which this

1412
01:14:11,990 --> 01:14:16,790
hydronium ion, H3O+, was
surrounded by three water

1413
01:14:16,790 --> 01:14:17,930
molecules.

1414
01:14:17,930 --> 01:14:20,450
And you know-- nothing
better than doing an accurate

1415
01:14:20,450 --> 01:14:22,670
simulation to find it out.

1416
01:14:22,670 --> 01:14:26,060
And being finally a quantum
mechanical simulation,

1417
01:14:26,060 --> 01:14:28,190
the result was
that it's actually

1418
01:14:28,190 --> 01:14:31,280
the proton is in what is
called a functional state.

1419
01:14:31,280 --> 01:14:35,780
Sort of it stays half of the
time in an Eigen state, that

1420
01:14:35,780 --> 01:14:39,110
is a state predicted
by Eigen, and half

1421
01:14:39,110 --> 01:14:41,000
of the time in a Zundel state.

1422
01:14:41,000 --> 01:14:42,710
And you see this is
actually interesting,

1423
01:14:42,710 --> 01:14:46,160
sort of comparing the
quantum mechanical prediction

1424
01:14:46,160 --> 01:14:48,110
and the classical prediction.

1425
01:14:48,110 --> 01:14:51,980
That is for a classical proton
in a quantum mechanical field

1426
01:14:51,980 --> 01:14:52,880
of electrons.

1427
01:14:52,880 --> 01:14:55,760
And you see the sort
of average position

1428
01:14:55,760 --> 01:14:57,890
in the proton
between two molecules

1429
01:14:57,890 --> 01:15:01,760
are sort of more
delocalized, more spread out,

1430
01:15:01,760 --> 01:15:03,940
than in the classical case.

1431
01:15:03,940 --> 01:15:07,250
So that is, I think,
the structure of water

1432
01:15:07,250 --> 01:15:09,150
is interesting.

1433
01:15:09,150 --> 01:15:11,810
There are sort of
other cases in which

1434
01:15:11,810 --> 01:15:17,990
you have a very shallow energy
barrier between two states

1435
01:15:17,990 --> 01:15:21,770
in which the quantum
delocalization and the quantum

1436
01:15:21,770 --> 01:15:24,170
tunneling could be important.

1437
01:15:24,170 --> 01:15:27,750
And something like that happened
actually in strontium titanite.

1438
01:15:27,750 --> 01:15:33,110
That is a perovskite very
similar to the lead titanite

1439
01:15:33,110 --> 01:15:35,480
that we have seen
in the first slide.

1440
01:15:35,480 --> 01:15:39,590
And we remember there were
two phases, a cubic phase

1441
01:15:39,590 --> 01:15:42,380
and a distorted
ferroelectric phase.

1442
01:15:42,380 --> 01:15:45,710
Well, it turns out that in the
case of strontium titanite,

1443
01:15:45,710 --> 01:15:48,350
the energy barrier
between those two phases

1444
01:15:48,350 --> 01:15:52,130
is so shallow that
all the ions actually

1445
01:15:52,130 --> 01:15:56,840
tunnel back and forth between
all the possible structure.

1446
01:15:56,840 --> 01:16:00,080
And so a system that
would be ferroelectric

1447
01:16:00,080 --> 01:16:04,100
in a classical simulation of
the ions, when we actually

1448
01:16:04,100 --> 01:16:08,690
introduce a quantum mechanical
effect for the nucleus as well,

1449
01:16:08,690 --> 01:16:11,240
it turns out to be the
state that keeps tunneling

1450
01:16:11,240 --> 01:16:13,230
between all the possibilities.

1451
01:16:13,230 --> 01:16:17,030
And so it doesn't
display ferroelectricity.

1452
01:16:17,030 --> 01:16:19,670
The last case that is
probably the most relevant--

1453
01:16:19,670 --> 01:16:21,240
and we'll see that
in more detail,

1454
01:16:21,240 --> 01:16:22,760
so I haven't presented here--

1455
01:16:22,760 --> 01:16:25,370
in which we really need
to take into account

1456
01:16:25,370 --> 01:16:28,460
the quantum mechanical
nature of the ions

1457
01:16:28,460 --> 01:16:31,790
has to do with
structural excitation.

1458
01:16:31,790 --> 01:16:35,240
When you put temperature, say,
in something like a solid,

1459
01:16:35,240 --> 01:16:40,910
you don't excite vibrational
states in a continuum way.

1460
01:16:40,910 --> 01:16:45,380
But you actually excite
discrete vibrational states

1461
01:16:45,380 --> 01:16:46,610
of your system.

1462
01:16:46,610 --> 01:16:49,520
And this is sort of
fundamental to understand

1463
01:16:49,520 --> 01:16:54,260
many thermodynamical properties
of solid, of molecules,

1464
01:16:54,260 --> 01:16:57,320
something like the
specific heat of a solid.

1465
01:16:57,320 --> 01:17:01,010
But we'll see examples
of that later on.

1466
01:17:01,010 --> 01:17:04,100
And those are cases
in which we actually

1467
01:17:04,100 --> 01:17:07,640
know very well on how to deal
with this specific problem

1468
01:17:07,640 --> 01:17:11,810
of the quantization of
the vibrational excitation

1469
01:17:11,810 --> 01:17:14,090
of your system.

1470
01:17:14,090 --> 01:17:14,690
OK.

1471
01:17:14,690 --> 01:17:17,840
So with all of this, the
first thing to remember

1472
01:17:17,840 --> 01:17:23,400
is that electrons
actually behave as waves.

1473
01:17:23,400 --> 01:17:26,390
So in reality, what
is quantum mechanics?

1474
01:17:26,390 --> 01:17:29,180
It's nothing else
than the mechanics

1475
01:17:29,180 --> 01:17:30,980
of electrons as waves.

1476
01:17:30,980 --> 01:17:33,020
So we don't consider
it a mechanics

1477
01:17:33,020 --> 01:17:35,900
of classical particle,
but it's the mechanics

1478
01:17:35,900 --> 01:17:38,330
of wave-like objects.

1479
01:17:38,330 --> 01:17:41,960
And there are going to be very
specific laws if you want them.

1480
01:17:41,960 --> 01:17:44,330
There's going to be the
Schrodinger equation that

1481
01:17:44,330 --> 01:17:48,330
is the Newton equivalent
of the equation of motion.

1482
01:17:48,330 --> 01:17:49,820
But there are also
going to be very

1483
01:17:49,820 --> 01:17:52,850
specific computational
constraints

1484
01:17:52,850 --> 01:17:54,920
on how we describe a wave.

1485
01:17:54,920 --> 01:17:55,610
OK.

1486
01:17:55,610 --> 01:17:57,050
What you are very
familiar is sort

1487
01:17:57,050 --> 01:17:58,570
of the mechanics of a particle.

1488
01:17:58,570 --> 01:18:00,110
So in the mechanics
of a particle,

1489
01:18:00,110 --> 01:18:02,150
you have Newton
equation of motion.

1490
01:18:02,150 --> 01:18:05,570
You have mass times acceleration
is equal to the force.

1491
01:18:05,570 --> 01:18:08,720
That's an ordinary second-order
differential equation.

1492
01:18:08,720 --> 01:18:11,900
That basically
means that what you

1493
01:18:11,900 --> 01:18:16,490
are going to obtain
by integrating

1494
01:18:16,490 --> 01:18:19,010
this equation of
motion is actually

1495
01:18:19,010 --> 01:18:21,860
the trajectory as
a function of time

1496
01:18:21,860 --> 01:18:24,290
and the first derivative
that is the velocity.

1497
01:18:24,290 --> 01:18:28,430
And because its second-order
differential equation,

1498
01:18:28,430 --> 01:18:30,800
what you really need
to define that is, say,

1499
01:18:30,800 --> 01:18:32,480
something like the
initial condition.

1500
01:18:32,480 --> 01:18:35,030
You need to define
where your particle is

1501
01:18:35,030 --> 01:18:38,360
at a certain instant and
what is its velocity.

1502
01:18:38,360 --> 01:18:40,820
And once you know that
and you know, say,

1503
01:18:40,820 --> 01:18:45,380
what is the force, something
like the gravitational force

1504
01:18:45,380 --> 01:18:50,040
in this case, well, you are able
to integrate your trajectory.

1505
01:18:50,040 --> 01:18:50,630
OK.

1506
01:18:50,630 --> 01:18:53,760
So this is sort of integration
of the equation of motion.

1507
01:18:53,760 --> 01:18:58,550
But what is really important
is that your dynamical system

1508
01:18:58,550 --> 01:19:03,600
that is your particle moving
around is perfectly described.

1509
01:19:03,600 --> 01:19:06,530
So we know everything
about that particle.

1510
01:19:06,530 --> 01:19:11,180
If I have every instant in
time, I give you two vectors,

1511
01:19:11,180 --> 01:19:11,720
basically.

1512
01:19:11,720 --> 01:19:15,530
I give the position,
and I give the velocity.

1513
01:19:15,530 --> 01:19:20,160
And in a sort of three
dimension, that's six numbers.

1514
01:19:20,160 --> 01:19:20,750
OK.

1515
01:19:20,750 --> 01:19:22,310
So it's very easy to remember.

1516
01:19:22,310 --> 01:19:25,340
And it's very easy to
store in a computer.

1517
01:19:25,340 --> 01:19:28,490
And this is my last
slide for today.

1518
01:19:28,490 --> 01:19:32,750
The amazing
computational complexity

1519
01:19:32,750 --> 01:19:37,190
that we have when we actually
need to describe a wave

1520
01:19:37,190 --> 01:19:40,920
is that to describe a
wave that is basically

1521
01:19:40,920 --> 01:19:47,240
a delocalized excitation a
wave is an amplitude everywhere

1522
01:19:47,240 --> 01:19:48,390
in space.

1523
01:19:48,390 --> 01:19:53,630
So even to describe a single
electron, what we need to say

1524
01:19:53,630 --> 01:19:59,240
is that at every instant what
is the amplitude of that wave

1525
01:19:59,240 --> 01:20:01,890
everywhere in space.

1526
01:20:01,890 --> 01:20:05,570
And so you see we have gone
from a classical particle that

1527
01:20:05,570 --> 01:20:09,050
was basically six
numbers at every instant

1528
01:20:09,050 --> 01:20:13,490
to a quantum mechanical
particle that at every instant,

1529
01:20:13,490 --> 01:20:18,150
in principle, requires a field
that covers all the space.

1530
01:20:18,150 --> 01:20:22,970
So we need to specify the
amplitude everywhere in space.

1531
01:20:22,970 --> 01:20:26,960
And this becomes worse
and worse as the number

1532
01:20:26,960 --> 01:20:29,810
of particle increases.

1533
01:20:29,810 --> 01:20:32,270
Suppose that we
had two electrons.

1534
01:20:32,270 --> 01:20:34,580
Well, what we need
to give is now

1535
01:20:34,580 --> 01:20:38,030
an amplitude at a
certain instant of time

1536
01:20:38,030 --> 01:20:44,120
that would be a function of one
vector r and another vector r2.

1537
01:20:44,120 --> 01:20:48,110
So if, by any chance,
you were satisfied

1538
01:20:48,110 --> 01:20:52,700
by sort of describing this
in a discretized form--

1539
01:20:52,700 --> 01:20:56,180
you see sort of discretizing
space and giving the amplitude

1540
01:20:56,180 --> 01:20:57,620
at every point in space.

1541
01:20:57,620 --> 01:20:59,990
And say to describe
an electron, you

1542
01:20:59,990 --> 01:21:04,310
could be happy by just
giving, say, a million points.

1543
01:21:04,310 --> 01:21:07,110
Well, if you want to
describe two electrons,

1544
01:21:07,110 --> 01:21:09,860
you have to give 1 million
times 1 million points.

1545
01:21:09,860 --> 01:21:12,380
And three electrons--
1 million to the cube

1546
01:21:12,380 --> 01:21:13,980
and so on and so forth.

1547
01:21:13,980 --> 01:21:17,630
So just the computational
complexity of this object

1548
01:21:17,630 --> 01:21:18,860
explodes.

1549
01:21:18,860 --> 01:21:22,340
That is ultimately
why it is so difficult

1550
01:21:22,340 --> 01:21:26,960
to solve the equivalent of the
Newton's equation of motion

1551
01:21:26,960 --> 01:21:30,420
for a quantum particle that
is the Schrodinger equation.

1552
01:21:30,420 --> 01:21:33,680
And so with this, I'll conclude
that, if you need any reading

1553
01:21:33,680 --> 01:21:35,690
on sort of fundamental
quantum mechanics,

1554
01:21:35,690 --> 01:21:37,280
I can give you a suggestion.

1555
01:21:37,280 --> 01:21:40,220
In general, sort of
physical chemistry textbooks

1556
01:21:40,220 --> 01:21:41,600
do a very good job.

1557
01:21:41,600 --> 01:21:43,280
There is a quantum
mechanical book

1558
01:21:43,280 --> 01:21:45,950
that is called the Bransden
and Joachain Quantum

1559
01:21:45,950 --> 01:21:48,110
Mechanics that is very good.

1560
01:21:48,110 --> 01:21:50,870
But what you need
for this class you'll

1561
01:21:50,870 --> 01:21:52,880
sort of see in the
next few slides

1562
01:21:52,880 --> 01:21:54,950
and in the beginning
of next lecture.

1563
01:21:54,950 --> 01:21:57,710
And then really we'll
sort of start describing

1564
01:21:57,710 --> 01:21:59,140
electronic structure methods.

1565
01:21:59,140 --> 01:22:01,880
That is, how do we deal
with this complexity?

1566
01:22:01,880 --> 01:22:04,830
And how do we apply it
to practical system?

1567
01:22:04,830 --> 01:22:07,830
And so with this
appointment on Thursday--

1568
01:22:07,830 --> 01:22:11,120
same time, but in room 1115.

1569
01:22:11,120 --> 01:22:15,190
And then we meet again for
class here next Tuesday.