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TROY VAN VOORHIS: All right,
well, good morning, everyone.

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I'm not Bob Field, but I'm Troy.

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00:00:27,100 --> 00:00:30,910
Nice to see everyone.

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So I'm here.

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00:00:31,610 --> 00:00:33,610
We're going to spend the
next couple of lectures

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talking about electronic
structure theory.

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And at this point, I'll
give you the big reveal,

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which is that at this point,
you have already encountered

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virtually every problem
that you can solve by hand

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00:00:47,160 --> 00:00:48,700
in electronic structure theory.

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So there's a handful
of other things

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you could do if you
really, like, knew

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00:00:51,903 --> 00:00:54,970
your hypergeometric functions
and some things like this,

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but basically, every
electronic structure problem

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00:00:58,090 --> 00:01:01,150
that you can solve by hand
you've already encountered.

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00:01:01,150 --> 00:01:03,580
And so if chemistry
relied on us being

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00:01:03,580 --> 00:01:06,756
able to solve things by hand,
we wouldn't get very far.

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We'd be pretty limited
in what we could do.

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So we end up using the fact
that we have computers.

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And computers are far more good
at doing tedious repetitive

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things that human beings are.

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They don't complain at all.

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And so we can use computers
to do fairly neat things.

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So I'll show you just a
couple of quick pictures here

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00:01:27,730 --> 00:01:29,120
of things that we can do.

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00:01:29,120 --> 00:01:32,560
So if you open up
any research paper

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these days, even if it's
just synthesis-- you know,

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00:01:34,900 --> 00:01:36,250
we made this catalyst.

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00:01:36,250 --> 00:01:37,880
We did this thing, whatever.

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00:01:37,880 --> 00:01:39,940
There's always some part
of it where they said,

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well, we did a
calculation, and this is

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00:01:42,050 --> 00:01:43,300
what the structure looks like.

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00:01:43,300 --> 00:01:44,920
Or this is what the
orbitals look like

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00:01:44,920 --> 00:01:46,040
or something like this.

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00:01:46,040 --> 00:01:47,620
So this was actually
a calculation.

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00:01:47,620 --> 00:01:50,604
We are computational
chemists, I guess.

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00:01:50,604 --> 00:01:52,780
I'll use this screen over here.

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00:01:54,245 --> 00:01:56,620
Is this the one that you're--
or you're filming this one.

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00:01:56,620 --> 00:01:57,190
This one?

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00:01:57,190 --> 00:02:00,220
All right. so I'll film
this one-- go to this one.

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00:02:00,220 --> 00:02:02,540
Breaking the fourth wall there.

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00:02:02,540 --> 00:02:04,680
So this is the
picture of the HOMO,

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00:02:04,680 --> 00:02:06,832
the reactive orbital
water-splitting catalyst

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00:02:06,832 --> 00:02:07,540
that we computed.

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00:02:07,540 --> 00:02:10,870
So you can see that
there's 50, 60 atoms here,

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probably several
hundred electrons.

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We did not do this by hand.

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We didn't draw the
surfaces by hand.

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The computer did
all of this for us.

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And you can also do things
like look at chemical reactions

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and run dynamics.

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So we can do--

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there we go.

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So you can run
dynamics of molecules.

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So this is a particular
molecule in solution.

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And we can use the
electronic energy

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in order to govern the
dynamics of the molecule.

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And the computer
can actually predict

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for us whether this
molecule is going to react,

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what it's going to
do, how it's going

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to behave in one
solution, say, versus

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another or at one particular
temperature versus another,

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those kinds of things.

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And all of these, again,
are based off of the idea

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that we can use computers
to solve problems

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in electronic structure, or
at least approximately solve

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problems we couldn't do by hand.

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And so what the goal over
the next couple lectures

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is for us to learn
enough of the basics

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so that you can do
some calculations

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like this yourselves and then
see what you can do with that.

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So now I will switch over.

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You'll find where the
other differences, which

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is that I don't like the
feeling of chalk on my fingers,

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so I use my iPad
as a chalkboard.

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So the rest of this
will all be on the iPad.

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There we go.

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So for those of
you who want to--

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you may not have
gotten in this time.

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But I've literally
posted what I'm

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going to be writing on here,
these blank notes, online.

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You can download them,
print them off, if you want.

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If you have an
iPad or a computer,

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and you like to
take notes on that,

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you can download them
to your iPad or computer

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and take notes that way.

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But I find it's good because
there are some things that

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take a long time for me
to write at the board

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and that it would take
a long time for you

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to write in your notes.

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00:04:03,590 --> 00:04:06,640
And we can just have them
written down in advance.

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So the starting point
for all of these things

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that you can do on a computer
for electronic structure

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00:04:10,600 --> 00:04:12,250
theory, we start with
the Born-Oppenheimer

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approximations.

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So the Born-Oppenheimer
approximation

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was the idea that because
the nuclei are very heavy,

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you can clamp their positions
down to some particular values.

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So in this particular case
here, in this equation,

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I have it such that big R here
are the nuclear positions.

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00:04:30,430 --> 00:04:34,480
So there's going to be more
than one nucleus generally.

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So there's going
to be R1, R2, R3.

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All those crammed together,
I'm just going to denote big R

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00:04:42,357 --> 00:04:43,690
And that's where the nuclei are.

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00:04:46,772 --> 00:04:48,480
And so in Born-Oppenheimer
approximation,

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we clamp the nuclei down.

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00:04:49,590 --> 00:04:51,880
And then what we're left
with is the electrons.

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00:04:51,880 --> 00:04:53,160
The electrons whiz around.

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00:04:53,160 --> 00:04:56,676
They move in the field
dictated by those nuclei.

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So we have a
Schrodinger equation

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that describes the
motion of the electrons

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in the presence of the nuclei.

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00:05:01,600 --> 00:05:05,320
And so here I'm using the lower
case letters, lowercase r.

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00:05:05,320 --> 00:05:09,690
So they'll be, again, many
electrons, so r1, r2, r3, so on

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00:05:09,690 --> 00:05:10,320
and so forth.

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00:05:14,670 --> 00:05:19,892
And it's the electrons that are
really doing all the dirty work

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00:05:19,892 --> 00:05:20,600
in this equation.

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They're the ones that
are moving around.

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I've clamped the nuclei down.

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00:05:23,180 --> 00:05:25,602
They're just parameters
in this equation.

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And so what I have is I
have an electronic wave

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00:05:27,560 --> 00:05:30,170
function that describes the
distribution of electrons.

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00:05:30,170 --> 00:05:33,204
I have an electronic
Hamiltonian that governs

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00:05:33,204 --> 00:05:34,370
the motion of the electrons.

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Then I have an electronic
energy after I've solved

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this Schrodinger equation.

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And the nice thing about
this is that for an arbitrary

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number of electrons and an
arbitrary number of nuclei,

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I can in one line,
in about 10 seconds,

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00:05:46,760 --> 00:05:48,961
write down the Hamiltonian.

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00:05:48,961 --> 00:05:50,960
So the Hamiltonian is
pretty easy to write down.

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00:05:50,960 --> 00:05:52,700
I've even written it right here.

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00:05:52,700 --> 00:05:56,135
So I did use the cheat
of using atomic units.

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So the reason there's no H
bars or mass of the electrons

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00:06:02,842 --> 00:06:04,300
or any of that
appearing in here is

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because I chose atomic units.

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But then the ingredients
of my Hamiltonian

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00:06:09,550 --> 00:06:10,840
are all pretty well-defined.

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So I'll have some
kinetic energy.

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00:06:13,690 --> 00:06:16,100
And that kinetic energy will
just be for the electrons

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because the nuclei are fixed.

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I'll have an electron-electron
repulsion term.

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00:06:23,330 --> 00:06:26,750
So of course, the
electrons repel each other.

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I'll have an electron
nuclear attraction term

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because the nuclei are
fixed in positions,

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and the electrons feel
a potential due to that.

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And then I have a
nuclear repulsion term.

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And in this case, because
the nuclei are fixed,

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that nuclear repulsion
is just a number

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because I know where
all my nuclei are.

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They have some particular
repulsion energy.

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That's just some number that I
add onto my Hamiltonian there.

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But if I put all
those things together,

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I get the Hamiltonian.

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00:07:03,130 --> 00:07:04,750
So I can write down
the Hamiltonian

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00:07:04,750 --> 00:07:07,270
without too much difficulty.

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00:07:07,270 --> 00:07:08,920
But of course, the
interesting thing

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here is this Hamiltonian depends
on these nuclear positions.

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So I have to know what--

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depending on where
the nuclei are,

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I'm parametrically
changing this Hamiltonian.

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That changes H, and it also
changes the eigenvalue,

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the electronic eigenvalue
E. And so there's

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an obvious question
of, well, what

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is this electronic eigenvalue?

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It turns out to be a
very important thing.

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We will typically call this
thing a potential energy

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00:07:40,400 --> 00:07:40,900
surface.

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And it turns out to
govern a whole host

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of chemical phenomena.

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00:07:50,790 --> 00:07:53,780
So the first thing
that it could generate

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that we can look at--
we've already actually seen

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00:07:57,510 --> 00:07:58,680
potential energy surfaces.

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00:07:58,680 --> 00:08:02,640
So on the left-hand side
here, I have the picture

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00:08:02,640 --> 00:08:05,580
from the case of H2-plus.

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00:08:05,580 --> 00:08:13,900
So here we had the energy
of the sigma orbital.

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And as a function of
R, it forms some bonds.

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00:08:16,020 --> 00:08:20,590
So as we made R shorter, the
sigma orbital formed a bond.

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00:08:20,590 --> 00:08:26,520
Then we had also this
sigma star orbital

187
00:08:26,520 --> 00:08:29,100
as a function of R. When we
brought the atoms together,

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00:08:29,100 --> 00:08:31,183
in that one there was no
bond, no swarming energy.

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00:08:31,183 --> 00:08:32,490
It just went straight up.

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So we've got two
of these things,

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and I should label my axes.

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00:08:35,429 --> 00:08:41,770
So what I'm plotting
here is the energy

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00:08:41,770 --> 00:08:45,080
as a function of
the bond distance R.

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00:08:45,080 --> 00:08:47,810
So this was our first
example of a potential energy

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00:08:47,810 --> 00:08:50,630
surface, the electronic
energy as a function of R.

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00:08:50,630 --> 00:08:53,150
And we see that it taught
us about the bond, the bond

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strength, so forth.

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00:08:54,860 --> 00:08:58,810
There's one other thing that
this picture reminds me of,

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00:08:58,810 --> 00:09:00,560
and that's that for
the same problem here,

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00:09:00,560 --> 00:09:02,851
I actually had two different
potential energy surfaces.

201
00:09:02,851 --> 00:09:06,170
I have the sigma potential
energy surface and sigma star.

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00:09:06,170 --> 00:09:08,870
And that's because I left
something out in my Schrodinger

203
00:09:08,870 --> 00:09:10,710
equation up here.

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00:09:10,710 --> 00:09:13,250
There's, of course, an index.

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00:09:13,250 --> 00:09:16,190
I have different eigenfunctions
of the Schrodinger equation.

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00:09:16,190 --> 00:09:17,970
So I have an
electronic Hamiltonian.

207
00:09:17,970 --> 00:09:19,950
It'll have many
different eigenfunctions.

208
00:09:19,950 --> 00:09:23,050
So I have an index n for
those eigenfunctions.

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00:09:23,050 --> 00:09:26,730
And each of those eigenfunctions
will have a different energy.

210
00:09:26,730 --> 00:09:29,281
And each of those energies is
a different potential energy

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00:09:29,281 --> 00:09:29,780
surface.

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00:09:29,780 --> 00:09:33,025
So the sigma state was
the lowest solution.

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00:09:33,025 --> 00:09:35,150
The sigma start state was
the next lowest solution.

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00:09:35,150 --> 00:09:36,733
They had different
potential surfaces.

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00:09:39,060 --> 00:09:42,000
But I will often not refer
to potential energy surfaces

216
00:09:42,000 --> 00:09:46,410
plural but instead to potential
energy surface singular

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00:09:46,410 --> 00:09:51,570
because most chemistry occurs
on the lowest potential energy

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00:09:51,570 --> 00:09:53,220
surface.

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00:09:53,220 --> 00:09:56,970
So it involves just the lowest
potential energy surface.

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00:09:56,970 --> 00:09:59,550
And the reason for that is
because electronic energies

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00:09:59,550 --> 00:10:01,534
tend to be very big
in chemical terms.

222
00:10:01,534 --> 00:10:03,450
So the difference between
sigma and sigma star

223
00:10:03,450 --> 00:10:06,990
is several EV, typically.

224
00:10:06,990 --> 00:10:10,500
And chemical reactions
on that same scale

225
00:10:10,500 --> 00:10:13,830
are usually tenths
or hundreds of an EV.

226
00:10:13,830 --> 00:10:16,500
So the chemical reaction energy
might be somewhere way down

227
00:10:16,500 --> 00:10:19,260
here, way way, way below
the amount of energy

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00:10:19,260 --> 00:10:22,542
you would need to get all
the way up to sigma star.

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00:10:22,542 --> 00:10:24,500
And so the sigma star,
potential energy surface

230
00:10:24,500 --> 00:10:25,399
is just irrelevant.

231
00:10:25,399 --> 00:10:27,690
It's there, but it doesn't
participate in the reactions

232
00:10:27,690 --> 00:10:30,310
at all.

233
00:10:30,310 --> 00:10:33,350
So in most cases, knowing about
the lowest potential energy

234
00:10:33,350 --> 00:10:37,640
surface gets you
most of chemistry.

235
00:10:37,640 --> 00:10:40,850
But then we can also,
sort of, schematically

236
00:10:40,850 --> 00:10:43,490
talk about how these potential
energy surfaces will look

237
00:10:43,490 --> 00:10:45,600
for a more complicated example.

238
00:10:45,600 --> 00:10:51,230
So let's say I had the
water molecule, HOH.

239
00:10:53,780 --> 00:10:55,520
So this is a more general case.

240
00:10:55,520 --> 00:11:01,310
We have at least two
coordinates here, R1 and R2.

241
00:11:01,310 --> 00:11:03,500
There's also going
to be an angle,

242
00:11:03,500 --> 00:11:06,275
but I can't really
plot three-dimensional

243
00:11:06,275 --> 00:11:09,620
hypersurfaces, even
with my computer.

244
00:11:09,620 --> 00:11:10,880
So I'll stick with two.

245
00:11:10,880 --> 00:11:12,327
There's a third
coordinate there,

246
00:11:12,327 --> 00:11:14,660
which is theta, which I'm not
going to play around with.

247
00:11:14,660 --> 00:11:16,800
But we'll play around
with just R1 and R2.

248
00:11:16,800 --> 00:11:19,580
So I can change
individually the OH bond

249
00:11:19,580 --> 00:11:21,580
lengths in this molecule.

250
00:11:21,580 --> 00:11:24,680
So then this is going
to give me the energy

251
00:11:24,680 --> 00:11:27,794
as a function of R1 and R2.

252
00:11:27,794 --> 00:11:28,710
So it'll be a surface.

253
00:11:28,710 --> 00:11:30,710
So it's a function
of two variables.

254
00:11:30,710 --> 00:11:32,030
That's a surface.

255
00:11:32,030 --> 00:11:35,960
And that surface might look
like what I have plotted there,

256
00:11:35,960 --> 00:11:39,830
where it would have various
peaks and valleys on it.

257
00:11:39,830 --> 00:11:44,720
The very, very lowest valley,
the very lowest minimum,

258
00:11:44,720 --> 00:11:47,090
just like we found for
H2, that lowest minimum

259
00:11:47,090 --> 00:11:49,160
was the equilibrium bond--

260
00:11:49,160 --> 00:11:50,664
bonding configuration.

261
00:11:50,664 --> 00:11:52,580
So for this two-dimensional
potential surface,

262
00:11:52,580 --> 00:11:54,621
that lowest minimum's
going to be the equilibrium

263
00:11:54,621 --> 00:11:56,090
configuration.

264
00:11:56,090 --> 00:11:59,246
But there's also the
possibility of other minima

265
00:11:59,246 --> 00:12:00,620
on this potential
energy surface.

266
00:12:00,620 --> 00:12:02,690
Those would be metastable
intermediates, things

267
00:12:02,690 --> 00:12:04,231
that you could get
trapped in, things

268
00:12:04,231 --> 00:12:06,890
that would live for a while,
but eventually go down

269
00:12:06,890 --> 00:12:08,765
towards the lowest minimum.

270
00:12:08,765 --> 00:12:10,390
Then there'd also be
reaction barriers.

271
00:12:10,390 --> 00:12:13,310
So these are the things that
we learned about governing

272
00:12:13,310 --> 00:12:16,340
how fast things convert
from these metastable states

273
00:12:16,340 --> 00:12:18,830
into the most stable states.

274
00:12:18,830 --> 00:12:21,602
So that's what you would get if
you had just two coordinates.

275
00:12:21,602 --> 00:12:24,060
And so then you can sort of
try to generalize in your head,

276
00:12:24,060 --> 00:12:29,570
well, what if we had 3,
4, 5, 6, 127, 368, lots

277
00:12:29,570 --> 00:12:30,774
and lots of coordinates?

278
00:12:30,774 --> 00:12:32,690
In some many dimensional
space, it would still

279
00:12:32,690 --> 00:12:33,690
have the same qualities.

280
00:12:33,690 --> 00:12:36,330
There'd be minima and
barriers and so forth.

281
00:12:36,330 --> 00:12:38,780
And we would get those out
of the Schrodinger equation,

282
00:12:38,780 --> 00:12:40,740
the electronic
Schrodinger equation.

283
00:12:40,740 --> 00:12:43,770
And so that is the
electronic structure problem.

284
00:12:43,770 --> 00:12:48,050
How do we accurately solve
for the electronic eigenvalue

285
00:12:48,050 --> 00:12:51,860
and the electronic wave function
for an arbitrary molecule?

286
00:12:51,860 --> 00:12:56,054
And the key word
here is accurately.

287
00:12:56,054 --> 00:12:57,470
I didn't just say
how can we solve

288
00:12:57,470 --> 00:13:00,410
for electronic energy and
the electronic wave function

289
00:13:00,410 --> 00:13:04,780
because, in general here,
exact solutions are impossible.

290
00:13:04,780 --> 00:13:07,010
And I don't mean
impossible for 5.61.

291
00:13:07,010 --> 00:13:11,280
I mean impossible for
humans or computers.

292
00:13:11,280 --> 00:13:13,310
So in general, electronic
structure theorists

293
00:13:13,310 --> 00:13:14,960
love approximations.

294
00:13:14,960 --> 00:13:17,750
We love to make approximations
because that's the only way

295
00:13:17,750 --> 00:13:19,230
that you can make progress.

296
00:13:19,230 --> 00:13:22,160
And so what we're going to learn
about today and next Monday

297
00:13:22,160 --> 00:13:25,280
are the, sort of, different
categories of approximations

298
00:13:25,280 --> 00:13:29,180
that we make and what the
pluses and minuses of those are.

299
00:13:29,180 --> 00:13:31,720
So I'll pause there and see
if anybody has any questions.

300
00:13:37,160 --> 00:13:40,220
All right, so
we'll move on then.

301
00:13:40,220 --> 00:13:43,820
So we have an out--

302
00:13:43,820 --> 00:13:48,110
so there is actually an
outline of how different

303
00:13:48,110 --> 00:13:51,770
modern electronic structure
methods make approximations.

304
00:13:51,770 --> 00:13:54,410
And these follow the
same kinds of steps

305
00:13:54,410 --> 00:13:57,900
that you would use in a
molecular orbital calculation.

306
00:13:57,900 --> 00:13:59,930
So the first thing
that you typically

307
00:13:59,930 --> 00:14:02,390
had to do in these
calculations was

308
00:14:02,390 --> 00:14:06,440
to choose a set of
basis functions.

309
00:14:06,440 --> 00:14:09,160
So you had to choose like
1s function over here

310
00:14:09,160 --> 00:14:12,310
or 1s function over there
or a 2s function or 2p.

311
00:14:12,310 --> 00:14:14,690
You had to choose some
set of atomic orbitals

312
00:14:14,690 --> 00:14:16,747
that you're going to use
as the starting point

313
00:14:16,747 --> 00:14:17,580
of your calculation.

314
00:14:17,580 --> 00:14:20,270
Then you made linear
combinations of those things

315
00:14:20,270 --> 00:14:22,820
to get better functions.

316
00:14:22,820 --> 00:14:26,250
And so then after you'd
chosen that basis,

317
00:14:26,250 --> 00:14:28,580
you had to build some matrices.

318
00:14:28,580 --> 00:14:30,530
Then you had to solve
the eigenvalues problem

319
00:14:30,530 --> 00:14:31,370
for that matrix.

320
00:14:31,370 --> 00:14:34,070
Then you had to pick out which
orbitals you were actually

321
00:14:34,070 --> 00:14:35,930
going to occupy.

322
00:14:35,930 --> 00:14:37,820
Those were the next few steps.

323
00:14:37,820 --> 00:14:41,500
And then finally, you had
to compute the energy.

324
00:14:41,500 --> 00:14:44,327
Now, that's already kind
of a detailed outline.

325
00:14:44,327 --> 00:14:45,910
If it seemed like
we had to understand

326
00:14:45,910 --> 00:14:48,640
the nuances of every
one of those five steps,

327
00:14:48,640 --> 00:14:51,010
we'd be kind of--
well, you can do that.

328
00:14:51,010 --> 00:14:52,730
It would be an entire
course unto itself.

329
00:14:52,730 --> 00:14:54,730
And that's not what
we're going to be doing.

330
00:14:54,730 --> 00:14:58,570
The reason that we can avoid
going over every single nuance

331
00:14:58,570 --> 00:14:59,390
is twofold.

332
00:14:59,390 --> 00:15:04,390
So first, we can avoid, more
or less, steps two through four

333
00:15:04,390 --> 00:15:06,850
because they're
done automatically.

334
00:15:06,850 --> 00:15:10,870
So once you have chosen your
basis that you want to use,

335
00:15:10,870 --> 00:15:12,730
the computer knows
how to build matrices.

336
00:15:12,730 --> 00:15:15,490
It knows how to diagonalize
matrices and find eigenvalues.

337
00:15:15,490 --> 00:15:17,867
It knows how to pick out
which orbitals to occupy.

338
00:15:17,867 --> 00:15:19,450
It knows how to do
all of those things

339
00:15:19,450 --> 00:15:21,160
without you telling
it to do anything.

340
00:15:21,160 --> 00:15:24,720
You don't really have
to change anything.

341
00:15:24,720 --> 00:15:26,470
So the only things you
have to worry about

342
00:15:26,470 --> 00:15:29,440
are choosing a
good basis and then

343
00:15:29,440 --> 00:15:32,920
also telling the computer a
good way to compute the energy.

344
00:15:32,920 --> 00:15:35,380
The other reason that we can
actually do something useful

345
00:15:35,380 --> 00:15:38,134
here is because using
a computer to solve

346
00:15:38,134 --> 00:15:39,550
an electronic
structure problem is

347
00:15:39,550 --> 00:15:43,200
much like using an NMR to
get a spectrum of a compound.

348
00:15:43,200 --> 00:15:45,030
I doubt that
anybody in this room

349
00:15:45,030 --> 00:15:49,200
could build their
own functioning NMR.

350
00:15:49,200 --> 00:15:52,170
I know generally the
principles of how an NMR works.

351
00:15:52,170 --> 00:15:54,290
But building from
the ground up-- oh,

352
00:15:54,290 --> 00:15:55,620
so we do have any takers?

353
00:15:55,620 --> 00:15:57,524
Anybody able to
build and NMR, huh?

354
00:15:57,524 --> 00:15:58,432
AUDIENCE: [INAUDIBLE]

355
00:15:58,432 --> 00:15:58,932
[LAUGHTER]

356
00:15:58,932 --> 00:16:00,473
TROY VAN VOORHIS:
From the ground up.

357
00:16:00,473 --> 00:16:01,320
I'm betting-- no.

358
00:16:01,320 --> 00:16:03,026
See?

359
00:16:03,026 --> 00:16:05,400
You have to actually, like,
make your own superconducting

360
00:16:05,400 --> 00:16:06,150
magnet.

361
00:16:06,150 --> 00:16:08,580
You've got-- from the
ground up, actually

362
00:16:08,580 --> 00:16:11,160
building your own NMR would
be very, very difficult.

363
00:16:11,160 --> 00:16:12,960
But that doesn't
preclude you from knowing

364
00:16:12,960 --> 00:16:15,870
how to use an NMR
because you know,

365
00:16:15,870 --> 00:16:18,556
OK, well, this piece is-- this
dial is roughly doing this,

366
00:16:18,556 --> 00:16:19,680
so I change this over here.

367
00:16:19,680 --> 00:16:20,520
I shim this.

368
00:16:20,520 --> 00:16:21,915
That's how I make the NMR work.

369
00:16:21,915 --> 00:16:24,040
It's the same thing with
electronic structure code.

370
00:16:24,040 --> 00:16:26,631
You need to understand a little
bit of how they work in order

371
00:16:26,631 --> 00:16:27,630
to use them effectively.

372
00:16:27,630 --> 00:16:30,150
But you don't have to understand
every single line of code that

373
00:16:30,150 --> 00:16:31,740
went into building
it, which is good

374
00:16:31,740 --> 00:16:34,320
because there's several
hundred thousand lines of codes

375
00:16:34,320 --> 00:16:36,640
in many of these electronic
structure packages.

376
00:16:36,640 --> 00:16:39,067
So we're going to try to
understand the principles that

377
00:16:39,067 --> 00:16:40,650
go into this so that
we know how to be

378
00:16:40,650 --> 00:16:44,820
good users of
computational tools.

379
00:16:44,820 --> 00:16:46,680
And we can actually,
then, organize.

380
00:16:46,680 --> 00:16:49,560
So roughly speaking, we'll
be focusing on step one today

381
00:16:49,560 --> 00:16:51,810
and step five next Monday.

382
00:16:51,810 --> 00:16:53,670
But we can organize
these approximations

383
00:16:53,670 --> 00:16:56,100
on a sort of
two-dimensional plot, which

384
00:16:56,100 --> 00:16:59,160
just sort of summarizes
the whole idea of what

385
00:16:59,160 --> 00:17:00,480
we're going to be getting at.

386
00:17:00,480 --> 00:17:03,270
So the idea here
is that on one axis

387
00:17:03,270 --> 00:17:04,920
we have the step
one thing, which

388
00:17:04,920 --> 00:17:07,410
is the basis, the
atomic orbital basis.

389
00:17:07,410 --> 00:17:11,609
Then we can organize those
basis, those choices of basis,

390
00:17:11,609 --> 00:17:18,089
from, roughly speaking, on the
left hand, we have bad choices.

391
00:17:18,089 --> 00:17:22,631
And on the right hand,
we have good choices.

392
00:17:22,631 --> 00:17:25,089
Now, you might say, well, why
would we even include choices

393
00:17:25,089 --> 00:17:26,500
if we know they're bad?

394
00:17:26,500 --> 00:17:32,010
Because the bad ones also
happen to be the fast basis.

395
00:17:32,010 --> 00:17:36,250
And the good ones
tend to be slow.

396
00:17:36,250 --> 00:17:37,680
And so it's a time trade-off.

397
00:17:37,680 --> 00:17:39,510
So you have a lot
of time, you might

398
00:17:39,510 --> 00:17:41,154
choose a really good basis.

399
00:17:41,154 --> 00:17:42,570
If you have not a
lot of time, you

400
00:17:42,570 --> 00:17:44,430
might choose not
a very good one.

401
00:17:44,430 --> 00:17:47,940
And then we can do the
same thing with the energy.

402
00:17:47,940 --> 00:17:50,370
We have different choices of
how we compute the energy.

403
00:17:50,370 --> 00:17:57,210
And we can roughly arrange those
from bad choices to good ones.

404
00:17:57,210 --> 00:18:01,295
But again, these bad
choices are fast,

405
00:18:01,295 --> 00:18:02,420
and the good ones are slow.

406
00:18:05,880 --> 00:18:08,580
And so the general
thing that we want

407
00:18:08,580 --> 00:18:13,200
is we want to get up here into
the upper right-hand corner,

408
00:18:13,200 --> 00:18:14,340
where the exact answer is.

409
00:18:14,340 --> 00:18:16,798
Somewhere up there, with a very
good energy and a very good

410
00:18:16,798 --> 00:18:19,290
basis, we're going to
get the exact or nearly

411
00:18:19,290 --> 00:18:20,790
the exact answer.

412
00:18:20,790 --> 00:18:22,770
So that's where we want to get.

413
00:18:22,770 --> 00:18:27,000
But moving from the lower
left to the upper right.

414
00:18:27,000 --> 00:18:28,680
Every time we move
in that direction

415
00:18:28,680 --> 00:18:31,170
we're making the calculation
slower and slower and slower

416
00:18:31,170 --> 00:18:32,086
and slower and slower.

417
00:18:32,086 --> 00:18:34,230
Until eventually we
just lose patience.

418
00:18:34,230 --> 00:18:35,902
And so then we cut
it off and say,

419
00:18:35,902 --> 00:18:37,860
all right, this is as
far as I'm willing to go.

420
00:18:37,860 --> 00:18:39,234
What's the best
answer I can get?

421
00:18:39,234 --> 00:18:43,020
What's the most best-cost
benefit analysis I can do?

422
00:18:43,020 --> 00:18:45,540
And so in terms of
being good users

423
00:18:45,540 --> 00:18:47,250
of electronic
computational chemistry,

424
00:18:47,250 --> 00:18:49,410
it's about knowing,
OK, well, if I've only

425
00:18:49,410 --> 00:18:52,652
got three hours for
this calculation to run

426
00:18:52,652 --> 00:18:55,110
or two days or however long
I'm willing to let the computer

427
00:18:55,110 --> 00:18:58,200
run on this, what's the
best combination of a basis

428
00:18:58,200 --> 00:19:01,840
and an energy to get me a decent
result in that kind of time?

429
00:19:04,702 --> 00:19:06,410
So questions about
that before I move on?

430
00:19:12,520 --> 00:19:17,480
OK, so we'll focus on choosing
an atomic orbital basis.

431
00:19:17,480 --> 00:19:21,230
So we already have chosen an
atomic orbital basis before.

432
00:19:21,230 --> 00:19:28,040
So when we did H2-plus, which
is, sort of, a standard problem

433
00:19:28,040 --> 00:19:31,580
that you can actually
solve by hand if you choose

434
00:19:31,580 --> 00:19:36,310
the right basis, when
we did that calculation,

435
00:19:36,310 --> 00:19:39,870
we chose to have an
atomic orbital on A,

436
00:19:39,870 --> 00:19:45,040
the 1s function on A, and the
atomic orbital on B, 1s on B.

437
00:19:45,040 --> 00:19:47,770
And those two things
together formed our basis.

438
00:19:47,770 --> 00:19:49,454
So we wrote our
molecular orbitals

439
00:19:49,454 --> 00:19:51,370
as linear combinations
of those two functions.

440
00:19:51,370 --> 00:19:53,650
We got sigma and sigma star out.

441
00:19:53,650 --> 00:19:57,500
And we were able to
work out the energies.

442
00:19:57,500 --> 00:20:02,860
Now, this type of basis
is a useful basis.

443
00:20:02,860 --> 00:20:05,045
And it has a name, and it's
called a minimal basis.

444
00:20:07,960 --> 00:20:10,930
And it's minimal because you
could not possibly do less.

445
00:20:10,930 --> 00:20:13,690
So if you want to form a
bond between two atoms,

446
00:20:13,690 --> 00:20:17,160
you need at least one
function on each atom.

447
00:20:17,160 --> 00:20:19,990
If you had only one
function, period,

448
00:20:19,990 --> 00:20:21,670
you couldn't really form a bond.

449
00:20:21,670 --> 00:20:25,000
So it's as low as you can
go, can't go lower than this.

450
00:20:25,000 --> 00:20:28,630
Now, you could think
about the possibility

451
00:20:28,630 --> 00:20:30,580
of adding additional functions.

452
00:20:30,580 --> 00:20:34,582
So I could add in, say,
the 2s function on A,

453
00:20:34,582 --> 00:20:38,780
the 2s function on B,
dot, dot, dot, dot, dot.

454
00:20:38,780 --> 00:20:41,900
I could come up
with various things.

455
00:20:41,900 --> 00:20:44,514
Now, you might ask, why
would I want to do this?

456
00:20:44,514 --> 00:20:46,180
And there's a very
good reason for doing

457
00:20:46,180 --> 00:20:47,680
this, which is one
of the principles

458
00:20:47,680 --> 00:20:51,640
that we need to learn in
constructing basis functions,

459
00:20:51,640 --> 00:20:57,220
choosing our basis, which is
that adding basis functions

460
00:20:57,220 --> 00:20:59,980
always improves the calculation.

461
00:20:59,980 --> 00:21:04,300
So even though I think
that adding 2s A and 2s B,

462
00:21:04,300 --> 00:21:05,730
why would those be important?

463
00:21:05,730 --> 00:21:08,320
I can't really say.

464
00:21:08,320 --> 00:21:12,001
Adding basis functions
always make things better.

465
00:21:12,001 --> 00:21:13,750
And the reason for
that is because we were

466
00:21:13,750 --> 00:21:15,460
doing a variation calculation.

467
00:21:15,460 --> 00:21:19,060
We're trying to approximate the
lowest energy of the system.

468
00:21:19,060 --> 00:21:23,110
And just choosing the 1s A and
1s B functions doesn't give you

469
00:21:23,110 --> 00:21:24,040
the lowest energy.

470
00:21:24,040 --> 00:21:26,490
It may be close,
but it's not exact.

471
00:21:26,490 --> 00:21:29,700
So then by adding
the 2s functions,

472
00:21:29,700 --> 00:21:32,910
the result could get
better by adding a little--

473
00:21:32,910 --> 00:21:38,790
by making C3 and C4
not quite 0 here,

474
00:21:38,790 --> 00:21:40,814
maybe the answer gets
a little bit better.

475
00:21:40,814 --> 00:21:42,480
Maybe it doesn't, but
it can't get worse

476
00:21:42,480 --> 00:21:46,290
because the calculation could
always choose C3 and C4 0.

477
00:21:46,290 --> 00:21:47,970
So adding basis
functions always makes

478
00:21:47,970 --> 00:21:52,550
things either better, or at
least not worse than they were.

479
00:21:52,550 --> 00:21:54,530
And so what you'll
find is that when

480
00:21:54,530 --> 00:21:57,560
we're talking about AO
basis, choosing an AO basis,

481
00:21:57,560 --> 00:21:59,150
we're going to be
choosing bases that

482
00:21:59,150 --> 00:22:01,580
are much bigger than your
chemical intuition would

483
00:22:01,580 --> 00:22:02,720
suggest.

484
00:22:02,720 --> 00:22:04,522
So for H2, I think,
oh, 1s A, 1s B,

485
00:22:04,522 --> 00:22:06,230
that should more or
less describe things.

486
00:22:06,230 --> 00:22:08,646
And if I was doing things by
hand, that's what I would do.

487
00:22:08,646 --> 00:22:10,670
But if the computer's
doing the work,

488
00:22:10,670 --> 00:22:13,070
well, no skin off my
nose if this computer

489
00:22:13,070 --> 00:22:16,497
wastes some time doing some
2s or 3s or 4s intervals.

490
00:22:16,497 --> 00:22:18,080
I'll let the computer
do that, as long

491
00:22:18,080 --> 00:22:20,910
as it gives me a better answer.

492
00:22:20,910 --> 00:22:25,004
So the bases we have will be
much bigger because of this.

493
00:22:25,004 --> 00:22:26,420
But the other thing
I want to note

494
00:22:26,420 --> 00:22:33,800
is that when I talk
about these 1s functions,

495
00:22:33,800 --> 00:22:36,810
you probably all
have in your head--

496
00:22:36,810 --> 00:22:38,450
you know what 1s
functions look like.

497
00:22:38,450 --> 00:22:43,630
They look like e to
the minus a times r.

498
00:22:43,630 --> 00:22:46,840
So they just look like
exponential decay functions.

499
00:22:46,840 --> 00:22:49,770
Turns out that for
practical reasons,

500
00:22:49,770 --> 00:22:52,200
these are inconvenient
things to use on a computer.

501
00:22:52,200 --> 00:22:55,890
And that is because integrals
involving exponentials

502
00:22:55,890 --> 00:22:58,590
are not analytic in three
dimensions primarily because

503
00:22:58,590 --> 00:23:00,919
of the cusp that
occurs at r equal 0.

504
00:23:00,919 --> 00:23:03,210
So if you multiply two of
these things times each other

505
00:23:03,210 --> 00:23:05,376
and try to do an integral,
you have a cusp over here

506
00:23:05,376 --> 00:23:06,650
and a cusp over here.

507
00:23:06,650 --> 00:23:08,490
And the integral is
just not something

508
00:23:08,490 --> 00:23:10,510
that can be worked out.

509
00:23:10,510 --> 00:23:15,660
And so when the going gets
tough, the tough get empirical.

510
00:23:15,660 --> 00:23:19,200
And so instead of using these
exponential functions, what we

511
00:23:19,200 --> 00:23:21,150
use in practice are Gaussians.

512
00:23:21,150 --> 00:23:23,990
So Gaussians are things
instead of looking

513
00:23:23,990 --> 00:23:27,150
at like e to the minus ar,
they look like e to the minus

514
00:23:27,150 --> 00:23:29,050
alpha r squared.

515
00:23:29,050 --> 00:23:33,450
So if I was to plot a
Gaussian here on the same axes

516
00:23:33,450 --> 00:23:37,380
and choose the alpha
value appropriately,

517
00:23:37,380 --> 00:23:41,532
I could get a Gaussian
that might look like that.

518
00:23:41,532 --> 00:23:46,320
So it would be similar to
that actual 1s function.

519
00:23:46,320 --> 00:23:48,720
But particularly
near the origin,

520
00:23:48,720 --> 00:23:52,050
where it doesn't have a cusp,
and particularly in the tails,

521
00:23:52,050 --> 00:23:55,570
where e to the minus alpha r
squared decays very quickly,

522
00:23:55,570 --> 00:23:58,490
they don't actually
look that similar.

523
00:23:58,490 --> 00:24:00,355
So whole reason
to do this is not

524
00:24:00,355 --> 00:24:01,730
that we have some
physical reason

525
00:24:01,730 --> 00:24:03,730
to think that Gaussians
describe atoms is better

526
00:24:03,730 --> 00:24:04,830
than hydrogenic functions.

527
00:24:04,830 --> 00:24:05,560
They don't.

528
00:24:05,560 --> 00:24:09,050
It's just that we
can do the integrals.

529
00:24:09,050 --> 00:24:11,580
There's easy integrals here.

530
00:24:11,580 --> 00:24:14,990
And if we really do want
a hydrogen-like function,

531
00:24:14,990 --> 00:24:17,660
we can get that
by just including

532
00:24:17,660 --> 00:24:18,892
more than one Gaussian.

533
00:24:18,892 --> 00:24:21,350
So one Gaussian doesn't look
very much like an exponential.

534
00:24:21,350 --> 00:24:24,440
But if I take two Gaussians
and choose their exponents

535
00:24:24,440 --> 00:24:27,020
and their coefficients
appropriately,

536
00:24:27,020 --> 00:24:29,090
I can make a linear
combination of two Gaussians.

537
00:24:29,090 --> 00:24:36,332
So this was the one
Gaussian result.

538
00:24:36,332 --> 00:24:38,040
With two Gaussians I
could make something

539
00:24:38,040 --> 00:24:42,790
that might look like that.

540
00:24:42,790 --> 00:24:44,914
And with three
Gaussians I might be

541
00:24:44,914 --> 00:24:46,580
able do something
that would look like--

542
00:24:50,244 --> 00:24:54,280
and then four Gaussians and five
Gaussians and six Gaussians.

543
00:24:54,280 --> 00:24:56,050
But it's clear,
then, that by using

544
00:24:56,050 --> 00:24:58,180
a large number of Gaussians,
I can get whatever

545
00:24:58,180 --> 00:25:00,550
I want just by brute force.

546
00:25:00,550 --> 00:25:04,090
And so those are the two
things about building basis,

547
00:25:04,090 --> 00:25:06,472
choosing basis, AO bases.

548
00:25:06,472 --> 00:25:07,930
And at this point,
you should start

549
00:25:07,930 --> 00:25:10,810
to feel a little bit
intimidated because I've

550
00:25:10,810 --> 00:25:14,980
said that for atoms you're
going to need more basis

551
00:25:14,980 --> 00:25:17,290
functions than you
thought just to try to get

552
00:25:17,290 --> 00:25:18,760
a variation a lower energy.

553
00:25:18,760 --> 00:25:22,789
And those basis functions are
likely to be constructed out

554
00:25:22,789 --> 00:25:25,080
of Gaussians, of which you'll
need a bunch of Gaussians

555
00:25:25,080 --> 00:25:28,990
to even really approximate
one hydrogen-like orbital.

556
00:25:28,990 --> 00:25:32,050
So you're going to have lots
and lots and lots of Gaussians

557
00:25:32,050 --> 00:25:33,000
on every atom.

558
00:25:33,000 --> 00:25:35,500
And you're going to choose the
exponents of every single one

559
00:25:35,500 --> 00:25:37,480
of those Gaussians yourself.

560
00:25:37,480 --> 00:25:41,057
And that would be just
horrible to have to do.

561
00:25:41,057 --> 00:25:42,640
And the thing that
comes to our rescue

562
00:25:42,640 --> 00:25:45,520
is a thing that's
known as a basis set.

563
00:25:45,520 --> 00:25:49,490
So a basis set is constructed
in the following way.

564
00:25:49,490 --> 00:25:52,150
A graduate student, probably
long before you were bored,

565
00:25:52,150 --> 00:25:56,330
spent years of their life
going through and figuring out,

566
00:25:56,330 --> 00:26:00,100
OK, for carbon, what
is a good combination

567
00:26:00,100 --> 00:26:01,900
of Gaussian exponents?

568
00:26:01,900 --> 00:26:09,720
OK, they're 113.6
74.2, 11.3, and 1.6.

569
00:26:09,720 --> 00:26:10,900
They wrote that down.

570
00:26:10,900 --> 00:26:12,850
They said, OK,
now, for nitrogen,

571
00:26:12,850 --> 00:26:15,100
what are a good set
of Gaussian exponents?

572
00:26:15,100 --> 00:26:16,600
And then they went
through, and they

573
00:26:16,600 --> 00:26:19,320
did this for every
element, or at least many,

574
00:26:19,320 --> 00:26:21,070
many, many elements
in the periodic table.

575
00:26:21,070 --> 00:26:23,390
Then they wrote a paper
that said here is--

576
00:26:23,390 --> 00:26:24,210
I did it.

577
00:26:24,210 --> 00:26:27,340
So I'm going to say here
is the Troy basis set.

578
00:26:27,340 --> 00:26:28,840
And the Troy basis
means when you

579
00:26:28,840 --> 00:26:30,548
say you're doing the
Troy basis set means

580
00:26:30,548 --> 00:26:32,890
you're using those
exponents that I wrote down

581
00:26:32,890 --> 00:26:36,130
for carbon or for nitrogen
or for oxygen or for fluorine

582
00:26:36,130 --> 00:26:39,130
or for hydrogen. And
it's all predefined.

583
00:26:39,130 --> 00:26:41,044
It's all laid out.

584
00:26:41,044 --> 00:26:42,460
And if I was really
diligent, it's

585
00:26:42,460 --> 00:26:44,260
laid out for every element
in the periodic table.

586
00:26:44,260 --> 00:26:45,676
If I was less
diligent, maybe it's

587
00:26:45,676 --> 00:26:47,920
only the first two rows
or something like that.

588
00:26:47,920 --> 00:26:50,380
But the result is that
all you have to say

589
00:26:50,380 --> 00:26:52,720
is I want the Troy
basis set because Troy

590
00:26:52,720 --> 00:26:54,550
makes really good basic sets.

591
00:26:54,550 --> 00:26:56,000
And so I'm going to use that.

592
00:26:56,000 --> 00:26:57,760
And then the computer can
go and just say, OK, well,

593
00:26:57,760 --> 00:26:59,480
there's some file that
has all those exponents.

594
00:26:59,480 --> 00:27:01,938
And the computer looks up the
numbers and says, for carbon,

595
00:27:01,938 --> 00:27:05,340
you need this; for oxygen,
this; for nitrogen, this.

596
00:27:05,340 --> 00:27:10,090
And so the key idea here is
that these are predefined

597
00:27:10,090 --> 00:27:12,170
sets or AO basis functions.

598
00:27:12,170 --> 00:27:13,737
So somebody already
defined these.

599
00:27:13,737 --> 00:27:15,070
You don't have to make a choice.

600
00:27:15,070 --> 00:27:17,890
Other than choosing the
set, there's not other knobs

601
00:27:17,890 --> 00:27:19,180
that you have to turn.

602
00:27:19,180 --> 00:27:21,220
So you can obviously
see the benefit

603
00:27:21,220 --> 00:27:23,054
of this, which is you
don't have to put down

604
00:27:23,054 --> 00:27:24,761
hundreds and hundreds
of numbers in order

605
00:27:24,761 --> 00:27:26,160
to get the calculation to run.

606
00:27:26,160 --> 00:27:28,090
So that's a big win,
which hopefully you're

607
00:27:28,090 --> 00:27:30,260
remember for the
next 25 minutes.

608
00:27:30,260 --> 00:27:32,360
Because the downside,
then, is that you just

609
00:27:32,360 --> 00:27:35,560
noticed I called my basis
set the Troy basis set, which

610
00:27:35,560 --> 00:27:38,680
would give you absolutely no
idea of what was in that basis

611
00:27:38,680 --> 00:27:41,320
set, just that I
made the basis set.

612
00:27:41,320 --> 00:27:42,850
And then somebody
else might call it

613
00:27:42,850 --> 00:27:46,720
the Cambridge basis set because
they did it in Cambridge.

614
00:27:46,720 --> 00:27:49,480
And somebody else might
number their basis sets.

615
00:27:49,480 --> 00:27:52,960
You know, this is
basis set number 17.

616
00:27:52,960 --> 00:27:54,820
And none of those
things actually

617
00:27:54,820 --> 00:27:56,830
tell you what's actually
in the basis set

618
00:27:56,830 --> 00:27:58,060
and how they designed it.

619
00:27:58,060 --> 00:27:59,710
They just give it a name.

620
00:27:59,710 --> 00:28:03,700
But then that name is-- you have
to know the name in order to be

621
00:28:03,700 --> 00:28:04,990
able to specify the basis set.

622
00:28:04,990 --> 00:28:06,845
And you have to know what--

623
00:28:06,845 --> 00:28:08,470
you have to sort of
memorize, oh, well,

624
00:28:08,470 --> 00:28:09,590
this basis that does this.

625
00:28:09,590 --> 00:28:10,631
This basis set does this.

626
00:28:10,631 --> 00:28:12,520
Or at least remember
where to look up

627
00:28:12,520 --> 00:28:14,200
what those basis sets do.

628
00:28:14,200 --> 00:28:15,752
So for the next
20 minutes or so,

629
00:28:15,752 --> 00:28:17,710
we're going to talk about
some of those things.

630
00:28:17,710 --> 00:28:20,470
And you'll be
annoyed at the fact

631
00:28:20,470 --> 00:28:23,650
that these basis sets have these
funky, weird names and design

632
00:28:23,650 --> 00:28:24,490
principles, I think.

633
00:28:24,490 --> 00:28:27,196
But just remember, I don't
have to put in 100 numbers.

634
00:28:27,196 --> 00:28:28,570
This is the price
you pay for not

635
00:28:28,570 --> 00:28:30,850
having to put in 100 numbers.

636
00:28:30,850 --> 00:28:36,730
So the first thing that
we'll talk about here--

637
00:28:36,730 --> 00:28:40,934
so I'll say that basis sets
are typically grouped by row.

638
00:28:40,934 --> 00:28:43,100
Of course, the basis sets
are going to be different.

639
00:28:43,100 --> 00:28:47,500
So hydrogen's going to need
a different number of basis

640
00:28:47,500 --> 00:28:49,594
functions than carbon.

641
00:28:49,594 --> 00:28:51,760
Argon's going to need a
different number from carbon

642
00:28:51,760 --> 00:28:53,380
or hydrogen just
because they have

643
00:28:53,380 --> 00:28:55,570
different numbers
of valence functions

644
00:28:55,570 --> 00:28:58,330
and different numbers
of core functions.

645
00:28:58,330 --> 00:29:01,420
And I'm also going to
introduce a shorthand.

646
00:29:04,030 --> 00:29:10,080
So first of all, I'll note
that we already have--

647
00:29:10,080 --> 00:29:12,230
so actually, I'll just
introduce the shorthand.

648
00:29:12,230 --> 00:29:18,540
So, say, for nitrogen
and for the basis set

649
00:29:18,540 --> 00:29:21,080
we've talked about so far,
the minimal basis set,

650
00:29:21,080 --> 00:29:23,360
the smallest basis
set I can come up

651
00:29:23,360 --> 00:29:25,250
with for nitrogen
would need to have

652
00:29:25,250 --> 00:29:30,670
the 1s function, the 2s
function, and the 2p function.

653
00:29:30,670 --> 00:29:32,770
I need at least those
functions to just

654
00:29:32,770 --> 00:29:35,800
have places to put all
of my nitrogen electrons.

655
00:29:35,800 --> 00:29:38,177
Now, I'll get
writer's cramp if I

656
00:29:38,177 --> 00:29:39,760
have to write all
of these things out.

657
00:29:39,760 --> 00:29:41,510
So I'm going to develop
a shorthand, which

658
00:29:41,510 --> 00:29:44,680
is when I have 1s
and 2s, I'm just

659
00:29:44,680 --> 00:29:48,490
going to denote that as 2s.

660
00:29:48,490 --> 00:29:50,659
So I'm not indicating that
2s is the last function.

661
00:29:50,659 --> 00:29:52,450
It's Indicating that
there are two of them.

662
00:29:52,450 --> 00:29:54,489
So I have two s-type
functions, one

663
00:29:54,489 --> 00:29:57,030
that's sort of core like and
one that's sort of valence like.

664
00:29:57,030 --> 00:29:58,290
But there's two of them.

665
00:29:58,290 --> 00:30:00,640
And then I have one
p-like function.

666
00:30:00,640 --> 00:30:02,730
So I'll put 1p So
that's just telling me

667
00:30:02,730 --> 00:30:04,500
how many of each of
these things I have.

668
00:30:07,050 --> 00:30:14,250
So again, this is the
number of s-type functions.

669
00:30:14,250 --> 00:30:17,520
This is the number
of p-type function.

670
00:30:20,565 --> 00:30:21,940
And so then we
can go, and we can

671
00:30:21,940 --> 00:30:26,590
talk about the one basis
set we've dealt with so far,

672
00:30:26,590 --> 00:30:27,930
which is the minimal basis set.

673
00:30:27,930 --> 00:30:30,510
So for hydrogen and
helium, it's just

674
00:30:30,510 --> 00:30:33,824
going to be that 1s
function on each of those.

675
00:30:33,824 --> 00:30:35,490
And then when I go
down to the next row,

676
00:30:35,490 --> 00:30:37,031
it looks just like
nitrogen. They all

677
00:30:37,031 --> 00:30:40,950
are going to need in an s
core function and an s valence

678
00:30:40,950 --> 00:30:42,750
function and a p
valence function.

679
00:30:42,750 --> 00:30:46,200
So there'll be 2s, 1p-like.

680
00:30:46,200 --> 00:30:50,340
And then I go down one more
row for sodium through argon,

681
00:30:50,340 --> 00:30:51,910
and I'll need
another s function.

682
00:30:51,910 --> 00:30:54,010
So it'll be total of
three s-type functions.

683
00:30:54,010 --> 00:30:57,940
And then I need two sets of
p, a core p and a valence p.

684
00:30:57,940 --> 00:30:58,580
So there's 2p.

685
00:31:01,590 --> 00:31:03,240
So that's the minimal basis.

686
00:31:03,240 --> 00:31:07,082
That's the smallest basis that I
could conceive of for any atom.

687
00:31:07,082 --> 00:31:08,790
And then we're going
to have various ways

688
00:31:08,790 --> 00:31:11,940
of trying to make these things
more elaborate, more broke.

689
00:31:11,940 --> 00:31:14,400
And the most common
way to do this

690
00:31:14,400 --> 00:31:17,760
is to note that, well, when
I bring atoms together,

691
00:31:17,760 --> 00:31:21,330
it's the valence functions
that actually either contract

692
00:31:21,330 --> 00:31:24,990
or expand in order to describe
the electrons moving around.

693
00:31:24,990 --> 00:31:27,730
The core functions don't
really change very much.

694
00:31:27,730 --> 00:31:31,530
And so in order to give the
valence functions flexibility

695
00:31:31,530 --> 00:31:36,100
to change, it makes sense to
add more valence-like functions.

696
00:31:36,100 --> 00:31:38,040
So if the valence
was s, it makes sense

697
00:31:38,040 --> 00:31:39,596
to add another s function.

698
00:31:39,596 --> 00:31:41,220
Or if the valence
was p, it makes sense

699
00:31:41,220 --> 00:31:43,570
to add another p function.

700
00:31:43,570 --> 00:31:51,000
And so we get a name here, this
concept, which is minimal basis

701
00:31:51,000 --> 00:31:55,830
is what's known as a single zeta
basis set because it has just

702
00:31:55,830 --> 00:31:58,152
one set of valence functions.

703
00:31:58,152 --> 00:32:00,360
It might seem like would
make better sense to call it

704
00:32:00,360 --> 00:32:03,060
a single valence basis set.

705
00:32:03,060 --> 00:32:04,920
But history made a
different choice.

706
00:32:04,920 --> 00:32:06,090
It said that it's zeta.

707
00:32:06,090 --> 00:32:08,520
I don't know why they chose
the name zeta, but they did.

708
00:32:08,520 --> 00:32:09,978
So the zeta just
means that there's

709
00:32:09,978 --> 00:32:11,580
one set of valence functions.

710
00:32:11,580 --> 00:32:14,119
And then you can think
about making a double zeta.

711
00:32:14,119 --> 00:32:15,660
So you'd take every
valence function,

712
00:32:15,660 --> 00:32:18,120
just add another
valence function

713
00:32:18,120 --> 00:32:19,710
that has a different
exponent so it

714
00:32:19,710 --> 00:32:21,690
would allow more flexibility.

715
00:32:21,690 --> 00:32:26,750
So that'll be a Double
Zeta basis set, DZ.

716
00:32:26,750 --> 00:32:30,480
So for hydrogen and
helium, it's all valence.

717
00:32:30,480 --> 00:32:32,510
So we'd just get
two s functions.

718
00:32:32,510 --> 00:32:37,130
For things in the same
row as lithium and neon,

719
00:32:37,130 --> 00:32:40,574
the valence functions
are in s and a p.

720
00:32:40,574 --> 00:32:41,990
And so when we
double those, we're

721
00:32:41,990 --> 00:32:44,160
going to add another
s-type function

722
00:32:44,160 --> 00:32:45,380
and another p-type function.

723
00:32:45,380 --> 00:32:50,580
So we would go 3s
2p here because we'd

724
00:32:50,580 --> 00:32:52,860
add one s function
and one p function.

725
00:32:52,860 --> 00:32:58,779
And then for sodium through
argon, we would have 4s 3p.

726
00:32:58,779 --> 00:33:00,820
And then we could go on
and talk about, oh, well,

727
00:33:00,820 --> 00:33:04,150
what about a triple zeta?

728
00:33:04,150 --> 00:33:10,510
So triple zeta would be
3s for hydrogen helium.

729
00:33:10,510 --> 00:33:12,860
And then we add an s and
a p for the next row.

730
00:33:12,860 --> 00:33:15,970
So it's 4s 3p.

731
00:33:15,970 --> 00:33:19,370
And then we add s and p for
this, and we end up with 5s 4p.

732
00:33:24,960 --> 00:33:28,660
And so again, the idea
here is that for carbon, we

733
00:33:28,660 --> 00:33:32,610
might have 1s that's the core.

734
00:33:32,610 --> 00:33:41,060
And then we'd have 2s and 2p as
the first valence and then 3p.

735
00:33:46,110 --> 00:33:47,550
And these would
respectively be--

736
00:33:50,690 --> 00:33:54,610
so the single zeta or--

737
00:33:54,610 --> 00:33:56,520
and then you include
the next cell.

738
00:33:56,520 --> 00:33:58,900
And it becomes-- or
sorry, single zeta

739
00:33:58,900 --> 00:34:00,464
would be all of this, sorry.

740
00:34:04,160 --> 00:34:05,784
Oh, it's the core.

741
00:34:05,784 --> 00:34:07,450
And then you could
say, all right, well,

742
00:34:07,450 --> 00:34:11,280
but then I include another set.

743
00:34:11,280 --> 00:34:14,690
And this gives me a
double zeta basis set.

744
00:34:14,690 --> 00:34:18,460
And then I could
include another set,

745
00:34:18,460 --> 00:34:21,570
which would give me a triple
zeta kind of basis set.

746
00:34:21,570 --> 00:34:23,630
Then you can go
on and on and on.

747
00:34:23,630 --> 00:34:26,340
So you could just make a
quadruple zeta and quintuple

748
00:34:26,340 --> 00:34:27,270
zeta basis set.

749
00:34:27,270 --> 00:34:29,969
And then this is where
the names get annoying.

750
00:34:29,969 --> 00:34:34,530
So the name of the most common
minimal basis set is STO-3G.

751
00:34:41,179 --> 00:34:45,639
And that stands for Slater-Type
Orbitals 3 Gaussians,

752
00:34:45,639 --> 00:34:49,239
just in case you're
wondering why it's STO-3G.

753
00:34:49,239 --> 00:34:52,719
And then for the
double zeta basis set,

754
00:34:52,719 --> 00:34:57,790
there are ones that
are called 3-21G,

755
00:34:57,790 --> 00:35:02,200
6-31G, another one that
is somewhat confusingly

756
00:35:02,200 --> 00:35:04,780
just called DZ, Double
Zeta, as if there

757
00:35:04,780 --> 00:35:07,810
was only one such basis set.

758
00:35:07,810 --> 00:35:16,600
And then for a triple zeta,
you can do 6-311G triple zeta

759
00:35:16,600 --> 00:35:17,500
valence.

760
00:35:17,500 --> 00:35:20,140
And there are others.

761
00:35:20,140 --> 00:35:23,740
So there are other
abbreviations out there as well.

762
00:35:23,740 --> 00:35:26,980
But that's how you would
do these kinds of-- how you

763
00:35:26,980 --> 00:35:30,829
change the zeta, the valence.

764
00:35:30,829 --> 00:35:31,870
So questions about those?

765
00:35:36,740 --> 00:35:37,490
Yeah?

766
00:35:37,490 --> 00:35:40,236
AUDIENCE: So is
there [INAUDIBLE]

767
00:35:40,236 --> 00:35:42,610
in the example you're including
as your additional zetas,

768
00:35:42,610 --> 00:35:44,520
like 3s 3p 4s.

769
00:35:44,520 --> 00:35:46,850
But would there be an
advantage to including

770
00:35:46,850 --> 00:35:51,524
things that weren't atomic
orbital-like things?

771
00:35:51,524 --> 00:35:52,440
TROY VAN VOORHIS: Yes.

772
00:35:52,440 --> 00:35:54,270
So there are
occasionally times where

773
00:35:54,270 --> 00:35:57,630
people include basic
functions that don't

774
00:35:57,630 --> 00:35:59,140
look like atomic orbitals.

775
00:35:59,140 --> 00:36:01,830
So the most common one is to
make bond-centered functions.

776
00:36:01,830 --> 00:36:03,290
Like say, I want a bond
here, and you could

777
00:36:03,290 --> 00:36:04,373
put basis functions there.

778
00:36:05,620 --> 00:36:08,390
There's a couple of reasons
that becomes undesirable.

779
00:36:08,390 --> 00:36:11,210
One is that, then, you can't
use a standard basis set

780
00:36:11,210 --> 00:36:13,986
because basis sets have to be
sort of anchored to the atoms.

781
00:36:13,986 --> 00:36:15,110
That's how we catalog them.

782
00:36:15,110 --> 00:36:16,610
If you say, oh, I'm going to put
one in the middle of the bond,

783
00:36:16,610 --> 00:36:18,860
you say, well, it would
depend on both atoms involved

784
00:36:18,860 --> 00:36:21,840
in the bond and how long the
bond was and things like that.

785
00:36:21,840 --> 00:36:23,780
So that's a little
bit undesirable.

786
00:36:23,780 --> 00:36:27,200
The other one is that unless
you pin that bond function down

787
00:36:27,200 --> 00:36:30,410
to a particular position, if
you let the center of that bond

788
00:36:30,410 --> 00:36:33,440
function move, there's actually
numerical instability that

789
00:36:33,440 --> 00:36:37,370
happens when two Gaussians
come on top of each other.

790
00:36:37,370 --> 00:36:39,530
And that can sometimes
cause calculations--

791
00:36:39,530 --> 00:36:42,280
the computer to choke.

792
00:36:42,280 --> 00:36:45,590
So that in practice, we
usually stick to increasing

793
00:36:45,590 --> 00:36:48,770
the atomic basis sets.

794
00:36:48,770 --> 00:36:51,230
Oh, but there is
also one other case.

795
00:36:51,230 --> 00:36:57,840
In physics, they are much
more fond of using solids.

796
00:36:57,840 --> 00:37:00,260
That's sort of one of
the things that chemists

797
00:37:00,260 --> 00:37:02,600
tend to think about
molecules as the example.

798
00:37:02,600 --> 00:37:04,310
Physicists tend to
think about solids.

799
00:37:04,310 --> 00:37:05,990
And in solids,
they're much more fond

800
00:37:05,990 --> 00:37:08,420
of using plane waves
as their basis set.

801
00:37:08,420 --> 00:37:10,240
So they start off
with the free--

802
00:37:10,240 --> 00:37:11,710
instead of starting off
with atoms as the reference,

803
00:37:11,710 --> 00:37:14,180
they start off with free
electrons as their reference.

804
00:37:14,180 --> 00:37:15,440
And then say, oh,
and then we introduce

805
00:37:15,440 --> 00:37:17,140
these potentials which
are perturbations

806
00:37:17,140 --> 00:37:18,056
to the free electrons.

807
00:37:18,056 --> 00:37:19,919
So you get plane waves
as your basis set.

808
00:37:19,919 --> 00:37:21,710
You make linear
combinations of plane waves

809
00:37:21,710 --> 00:37:23,481
instead of atomic functions.

810
00:37:26,348 --> 00:37:26,848
Yeah?

811
00:37:26,848 --> 00:37:28,772
AUDIENCE: [INAUDIBLE]
atomic orbitals

812
00:37:28,772 --> 00:37:31,449
do we need [INAUDIBLE]?

813
00:37:31,449 --> 00:37:33,240
TROY VAN VOORHIS: That's
exactly the next--

814
00:37:33,240 --> 00:37:35,330
you've hit on exactly
the next thing,

815
00:37:35,330 --> 00:37:39,550
which is that none of those
include any d functions.

816
00:37:39,550 --> 00:37:41,440
I mean, I can go
up to like 18 zeta,

817
00:37:41,440 --> 00:37:43,120
and I would never
have any d functions.

818
00:37:43,120 --> 00:37:45,880
And on hydrogen and helium,
I can go [INAUDIBLE]..

819
00:37:45,880 --> 00:37:47,800
I wouldn't even have
any p functions.

820
00:37:47,800 --> 00:37:49,360
And certainly there
are situations

821
00:37:49,360 --> 00:37:53,140
where you would want these
higher angular momenta.

822
00:37:53,140 --> 00:37:56,570
One of them is if you're
in an electric field.

823
00:37:56,570 --> 00:38:00,477
So that polarizes electrons
and tends to actually require

824
00:38:00,477 --> 00:38:02,560
slightly higher angular
momenta functions in order

825
00:38:02,560 --> 00:38:04,860
to describe that polarization.

826
00:38:04,860 --> 00:38:07,360
And the other one is for just
the directionality of bonding.

827
00:38:07,360 --> 00:38:09,700
That if you had, like, an
SN2 reaction or something

828
00:38:09,700 --> 00:38:12,295
like this, where temporarily
something was either actually

829
00:38:12,295 --> 00:38:16,000
hypervalent or sort
of hypervalent,

830
00:38:16,000 --> 00:38:18,370
you would need these higher
angular momentum functions

831
00:38:18,370 --> 00:38:21,490
to describe the hypervalency.

832
00:38:21,490 --> 00:38:23,870
And so those are called
polarization functions.

833
00:38:23,870 --> 00:38:25,720
So here polarization
functions are

834
00:38:25,720 --> 00:38:30,380
things that add higher
angular momentum and valence.

835
00:38:30,380 --> 00:38:33,220
So you need this for
things like polarizability,

836
00:38:33,220 --> 00:38:35,740
which is why they're called
polarization functions.

837
00:38:35,740 --> 00:38:38,770
But you also need it for
things like hypervalency.

838
00:38:38,770 --> 00:38:44,280
And so the nomenclature
for polarization functions

839
00:38:44,280 --> 00:38:47,670
is also a bit weird.

840
00:38:47,670 --> 00:38:50,740
But the simplest one is that
you just add the letter P,

841
00:38:50,740 --> 00:38:54,039
so P standing for
Polarization to the basis set.

842
00:38:54,039 --> 00:38:55,580
And then the idea
is that you're just

843
00:38:55,580 --> 00:38:58,844
going to add one
function with that P.

844
00:38:58,844 --> 00:39:00,260
You when you add
the P, that means

845
00:39:00,260 --> 00:39:03,470
you're adding one function that
has one unit higher angular

846
00:39:03,470 --> 00:39:04,030
momentum.

847
00:39:04,030 --> 00:39:05,780
So if P was your highest
angular momentum,

848
00:39:05,780 --> 00:39:07,156
you're going to
add a d function.

849
00:39:07,156 --> 00:39:08,946
And if s was your
highest angular momentum,

850
00:39:08,946 --> 00:39:10,334
you're going to
add a p function.

851
00:39:10,334 --> 00:39:11,750
So for hydrogen
and helium, that's

852
00:39:11,750 --> 00:39:15,680
going to add a P to lithium.

853
00:39:15,680 --> 00:39:17,480
To argon, it's going to add a D.

854
00:39:17,480 --> 00:39:19,490
If you did this also the
transition metal atoms,

855
00:39:19,490 --> 00:39:22,010
you would note the transition
metal atoms have d functions

856
00:39:22,010 --> 00:39:22,820
already.

857
00:39:22,820 --> 00:39:25,640
So the polarization
would add an f function,

858
00:39:25,640 --> 00:39:28,540
just adding one higher
angular momentum.

859
00:39:28,540 --> 00:39:36,970
So then going back to our table
up here, let me use my magic,

860
00:39:36,970 --> 00:39:39,715
copy the double zeta
basis results here-- copy.

861
00:39:43,590 --> 00:39:45,950
There's my double zeta.

862
00:39:45,950 --> 00:39:49,190
Now I'm just going to add
the polarization to this, so

863
00:39:49,190 --> 00:39:52,300
double zeta plus polarization.

864
00:39:52,300 --> 00:39:54,200
Now I add my
polarization functions.

865
00:39:54,200 --> 00:39:55,720
And so for hydrogen
and helium, I'm

866
00:39:55,720 --> 00:40:01,090
going to add one p function
to get a DZP basis set.

867
00:40:01,090 --> 00:40:04,500
Lithium to neon, I'm
going to add 1 d function.

868
00:40:04,500 --> 00:40:07,570
For sodium to argon, I'm
going to add 1 d function.

869
00:40:07,570 --> 00:40:15,510
And then I can do the same
thing with triple zeta.

870
00:40:19,990 --> 00:40:25,900
So if I make a TZP basis set,
then I'll add 1p, 1d, and 1d.

871
00:40:30,710 --> 00:40:35,810
And I'll note that you could
also do a TZ 2p basis set.

872
00:40:35,810 --> 00:40:37,730
And then you're
adding-- the 2p just

873
00:40:37,730 --> 00:40:40,190
indicates that I'm adding even
more polarization functions.

874
00:40:40,190 --> 00:40:43,760
So I'm adding two sets of d
functions instead of just one

875
00:40:43,760 --> 00:40:47,840
or two sets of p functions
instead of just one.

876
00:40:47,840 --> 00:40:51,790
And then these
things have names.

877
00:40:51,790 --> 00:41:06,030
So there's a 6-31G star, which
is a very common DZP basis set.

878
00:41:06,030 --> 00:41:10,430
Or there's also that DZ
basis that I mentioned above

879
00:41:10,430 --> 00:41:14,510
has a DZP generalization of it.

880
00:41:14,510 --> 00:41:18,560
Or you could have for the
triple zeta, the TZVP basis,

881
00:41:18,560 --> 00:41:28,765
or the longest basis
set name, 6-311G (d, p).

882
00:41:34,602 --> 00:41:36,560
This is where the annoyance
started to come in.

883
00:41:36,560 --> 00:41:37,940
You're like this is annoying.

884
00:41:37,940 --> 00:41:39,810
But it's less annoying
than the alternative.

885
00:41:39,810 --> 00:41:40,601
So we deal with it.

886
00:41:46,230 --> 00:41:50,770
So questions about that?

887
00:41:50,770 --> 00:41:51,630
Yep?

888
00:41:51,630 --> 00:41:53,191
AUDIENCE: [INAUDIBLE] 1p--

889
00:41:53,191 --> 00:41:53,690
I'm sorry.

890
00:41:53,690 --> 00:41:57,060
That 1p, they're just like,
we're going add DZ squared.

891
00:41:57,060 --> 00:41:58,394
Or [INAUDIBLE]?

892
00:41:58,394 --> 00:42:00,310
TROY VAN VOORHIS: It's
one set of d functions.

893
00:42:00,310 --> 00:42:00,600
AUDIENCE: Oh, OK.

894
00:42:00,600 --> 00:42:01,516
TROY VAN VOORHIS: Yes.

895
00:42:01,516 --> 00:42:03,290
Yeah.

896
00:42:03,290 --> 00:42:04,480
Yeah?

897
00:42:04,480 --> 00:42:05,190
Yep?

898
00:42:05,190 --> 00:42:08,749
AUDIENCE: Why can't we
have a single [INAUDIBLE]??

899
00:42:08,749 --> 00:42:10,040
TROY VAN VOORHIS: Oh, we could.

900
00:42:10,040 --> 00:42:11,980
But, yeah, we could do that.

901
00:42:11,980 --> 00:42:13,570
So part of this has to do with--

902
00:42:13,570 --> 00:42:15,870
and there are a couple of
basis sets that do that.

903
00:42:15,870 --> 00:42:17,700
This mainly represents
the hierarchy

904
00:42:17,700 --> 00:42:21,670
of empirically what people
have found is most important.

905
00:42:21,670 --> 00:42:26,220
So if you just start with a
minimal single zeta basis set,

906
00:42:26,220 --> 00:42:28,770
you say, well, what's the
next most important thing?

907
00:42:28,770 --> 00:42:31,750
Usually almost always
going to double zeta

908
00:42:31,750 --> 00:42:34,050
is more important than the
polarization functions.

909
00:42:34,050 --> 00:42:39,517
So very few people bother making
single zeta plus polarization.

910
00:42:39,517 --> 00:42:41,100
They'll say, OK,
we'll do double zeta.

911
00:42:41,100 --> 00:42:42,724
And then after you
do double zeta, then

912
00:42:42,724 --> 00:42:44,310
maybe you want to
go to polarization.

913
00:42:44,310 --> 00:42:45,934
Or maybe you want to
go to triple zeta,

914
00:42:45,934 --> 00:42:47,235
depending on what you're doing.

915
00:42:47,235 --> 00:42:48,151
Does that makes sense?

916
00:42:48,151 --> 00:42:49,840
AUDIENCE: [INAUDIBLE]

917
00:42:49,840 --> 00:42:52,210
TROY VAN VOORHIS:
It would be 1s 1p

918
00:42:52,210 --> 00:42:56,540
for hydrogen, 2s 1d for carbon.

919
00:42:56,540 --> 00:42:59,410
It would just be to add that
extra polarization function

920
00:42:59,410 --> 00:43:01,850
on top of the existing set.

921
00:43:01,850 --> 00:43:05,740
AUDIENCE: And would the
polarization be the--

922
00:43:05,740 --> 00:43:09,602
which AO did that come from?

923
00:43:09,602 --> 00:43:11,060
TROY VAN VOORHIS:
What do you mean?

924
00:43:11,060 --> 00:43:16,340
AUDIENCE: So for hydrogen, would
it be adding the 2p basis or--

925
00:43:16,340 --> 00:43:18,260
TROY VAN VOORHIS: It
would be like adding--

926
00:43:18,260 --> 00:43:21,070
now, so for hydrogen, it
would be like adding--

927
00:43:21,070 --> 00:43:21,570
yeah.

928
00:43:21,570 --> 00:43:23,430
For hydrogen, it would
be like the 2p function.

929
00:43:23,430 --> 00:43:25,055
But again, when we
go back, we remember

930
00:43:25,055 --> 00:43:28,370
that we're not actually using
the hydrogenic functions.

931
00:43:28,370 --> 00:43:30,920
We're using Gaussians
to approximate them.

932
00:43:30,920 --> 00:43:33,920
And so all it really means is
that it has the same angular

933
00:43:33,920 --> 00:43:35,510
distribution as a p function.

934
00:43:35,510 --> 00:43:38,940
But the radial part
is whatever the person

935
00:43:38,940 --> 00:43:40,710
who made the basis set
decided to make it.

936
00:43:45,460 --> 00:43:46,460
All right, then I have--

937
00:43:46,460 --> 00:43:50,145
so then when I teach this class,
I often have clicker questions.

938
00:43:50,145 --> 00:43:52,020
So this would have been
the clicker question.

939
00:43:52,020 --> 00:43:54,270
So we'll do it by show of
hands instead of by clicker,

940
00:43:54,270 --> 00:43:56,160
since we're small enough.

941
00:43:56,160 --> 00:43:59,270
So here's a test of
whether we understood--

942
00:43:59,270 --> 00:44:02,360
I can't show-- well, anyway.

943
00:44:02,360 --> 00:44:04,922
So I'll go back
and put things up

944
00:44:04,922 --> 00:44:06,380
in just a second,
if you need them.

945
00:44:06,380 --> 00:44:09,170
But my question is, how many
basis functions will there

946
00:44:09,170 --> 00:44:14,270
be for C60 in a DZP basis set?

947
00:44:14,270 --> 00:44:17,780
I guess should say [INAUDIBLE].

948
00:44:17,780 --> 00:44:19,890
So option A is there'll
be 60 basis functions.

949
00:44:19,890 --> 00:44:22,870
So that would be how many basis
functions per carbon atom?

950
00:44:22,870 --> 00:44:23,720
One.

951
00:44:23,720 --> 00:44:27,120
All right, or 180,
that's three per carbon.

952
00:44:27,120 --> 00:44:29,730
360, that's six per carbon.

953
00:44:29,730 --> 00:44:36,102
840 is-- I can't--

954
00:44:36,102 --> 00:44:37,560
anybody do the math
better than me?

955
00:44:37,560 --> 00:44:39,120
Is that 14?

956
00:44:39,120 --> 00:44:39,690
No.

957
00:44:39,690 --> 00:44:40,760
No way.

958
00:44:40,760 --> 00:44:41,360
Yeah, it's 14.

959
00:44:41,360 --> 00:44:44,400
Yeah, 14-- 14 per carbon.

960
00:44:44,400 --> 00:44:51,519
This is-- huh?

961
00:44:51,519 --> 00:44:52,060
AUDIENCE: 18.

962
00:44:52,060 --> 00:44:52,935
TROY VAN VOORHIS: 18.

963
00:44:52,935 --> 00:44:55,200
And then this is
21, then, right?

964
00:44:55,200 --> 00:45:02,859
Yeah, so we've got,
1, 3, 6, 14, 18, 21

965
00:45:02,859 --> 00:45:04,400
as the number of
basis functions per.

966
00:45:04,400 --> 00:45:08,020
And I'll go back
and throw this up

967
00:45:08,020 --> 00:45:10,090
for those who want the table up.

968
00:45:10,090 --> 00:45:11,600
We didn't write all that down.

969
00:45:11,600 --> 00:45:14,140
So here's our EZP basis set.

970
00:45:14,140 --> 00:45:16,020
Carbon's in this group here.

971
00:45:25,910 --> 00:45:27,820
All right, so then
now I'll go back.

972
00:45:27,820 --> 00:45:28,920
Everybody feel like
they got their answer?

973
00:45:28,920 --> 00:45:29,110
OK.

974
00:45:29,110 --> 00:45:30,693
So by a show of
hands, how many people

975
00:45:30,693 --> 00:45:34,010
think the answer is A, 60?

976
00:45:34,010 --> 00:45:34,770
Nobody, all right.

977
00:45:34,770 --> 00:45:35,625
How about B?

978
00:45:38,662 --> 00:45:40,050
C?

979
00:45:40,050 --> 00:45:41,180
We go most on C, OK.

980
00:45:41,180 --> 00:45:43,555
D?

981
00:45:43,555 --> 00:45:45,274
I don't know, E?

982
00:45:45,274 --> 00:45:46,190
F?

983
00:45:46,190 --> 00:45:49,720
Everybody says C. C is
not the correct answer.

984
00:45:49,720 --> 00:45:56,260
And every year people
miss this question.

985
00:45:56,260 --> 00:45:58,824
So the key thing to get
this right-- and we're

986
00:45:58,824 --> 00:46:00,240
going to re-poll
in a minute here.

987
00:46:00,240 --> 00:46:01,040
But the key thing
to get this right

988
00:46:01,040 --> 00:46:02,456
was actually the
question that you

989
00:46:02,456 --> 00:46:07,460
asked, which is that you
said, when I add a d function,

990
00:46:07,460 --> 00:46:10,970
do I add DZ squared, or
do I-- and I said, no,

991
00:46:10,970 --> 00:46:14,240
you add a set of d
functions, which is how many

992
00:46:14,240 --> 00:46:15,690
functions in d?

993
00:46:15,690 --> 00:46:16,350
5.

994
00:46:16,350 --> 00:46:19,790
All right, so now
we'll go back and look.

995
00:46:19,790 --> 00:46:23,450
All right, but don't feel bad.

996
00:46:23,450 --> 00:46:28,530
Every year, the majority of
people choose that same answer.

997
00:46:28,530 --> 00:46:31,450
So now let's redo our math.

998
00:46:31,450 --> 00:46:33,110
AUDIENCE: I've never
gotten it right.

999
00:46:33,110 --> 00:46:34,170
TROY VAN VOORHIS:
Never gotten it right?

1000
00:46:34,170 --> 00:46:34,670
OK.

1001
00:46:40,486 --> 00:46:50,990
[INTERPOSING VOICES]

1002
00:46:50,990 --> 00:46:53,260
All right, so now
we'll go through.

1003
00:46:53,260 --> 00:46:54,530
OK, how many people think A?

1004
00:46:54,530 --> 00:46:55,497
How many B?

1005
00:46:55,497 --> 00:46:57,580
Nobody's going to say C
because I already told you

1006
00:46:57,580 --> 00:46:58,205
that was wrong.

1007
00:46:58,205 --> 00:46:59,530
How many people say D?

1008
00:46:59,530 --> 00:47:00,946
All right, good,
you're all right.

1009
00:47:00,946 --> 00:47:03,040
Yes, the answer is
D. In a DZP basis

1010
00:47:03,040 --> 00:47:06,760
there are 14 basis
functions per carbon atom.

1011
00:47:06,760 --> 00:47:08,950
Now, you can already
start to see even

1012
00:47:08,950 --> 00:47:10,990
for this DZP basis,
which isn't the biggest

1013
00:47:10,990 --> 00:47:13,030
basis we've talked
about, for C60,

1014
00:47:13,030 --> 00:47:15,880
you've got 840 basis functions.

1015
00:47:15,880 --> 00:47:17,240
That's a lot of basis functions.

1016
00:47:17,240 --> 00:47:19,615
So that's why we're really
glad that the computer's doing

1017
00:47:19,615 --> 00:47:24,380
the work and not us
because 840 integrals--

1018
00:47:24,380 --> 00:47:29,020
well, 840 by 840 matrices are
hard to diagonalize by hand.

1019
00:47:29,020 --> 00:47:32,170
Doing all those integrals
is really, really a pain.

1020
00:47:32,170 --> 00:47:35,610
So then going
beyond those basis--

1021
00:47:35,610 --> 00:47:37,780
so those two things are
the key basis set ideas.

1022
00:47:37,780 --> 00:47:41,110
I'll just touch on a
couple of other things

1023
00:47:41,110 --> 00:47:42,800
before we finish up.

1024
00:47:42,800 --> 00:47:44,710
So the first thing
is diffuse functions.

1025
00:47:44,710 --> 00:47:47,920
So occasionally,
if you have anion,

1026
00:47:47,920 --> 00:47:50,050
so you have an extra
electron, that extra electron

1027
00:47:50,050 --> 00:47:53,110
is not very well
described by the valence

1028
00:47:53,110 --> 00:47:57,580
because negative electrons tend
to spread out a great deal.

1029
00:47:57,580 --> 00:48:01,030
Bob is one of the world
experts in Rydberg states,

1030
00:48:01,030 --> 00:48:03,580
which is where these electrons
spread out a really, really

1031
00:48:03,580 --> 00:48:05,350
significant amount.

1032
00:48:05,350 --> 00:48:07,780
And so these Gaussian
functions that decay quickly

1033
00:48:07,780 --> 00:48:09,310
don't describe those well.

1034
00:48:09,310 --> 00:48:14,650
And so you have
to add a Gaussian

1035
00:48:14,650 --> 00:48:17,914
with a small value of alpha.

1036
00:48:17,914 --> 00:48:19,330
And it's mostly
useful for anions.

1037
00:48:22,060 --> 00:48:25,480
So you add these in order
to describe anions better.

1038
00:48:25,480 --> 00:48:31,350
And the notation here is
either you use the phrase aug

1039
00:48:31,350 --> 00:48:34,374
or plus to the basis set.

1040
00:48:34,374 --> 00:48:36,540
So if the basis set has the
word aug in front of it,

1041
00:48:36,540 --> 00:48:39,090
that means it has some diffuse
functions on it in order

1042
00:48:39,090 --> 00:48:42,210
to describe those
weakly bound electrons.

1043
00:48:42,210 --> 00:48:46,570
Or the plus indicates it has
some of those functions in it.

1044
00:48:46,570 --> 00:48:49,191
And again, those are added
with the same angular

1045
00:48:49,191 --> 00:48:50,190
momentum as the valence.

1046
00:48:50,190 --> 00:48:54,430
They're not usually polarization
and diffuse at the same time.

1047
00:48:54,430 --> 00:48:57,540
And then the final thing is
you can generalize these ideas

1048
00:48:57,540 --> 00:48:59,730
to transition metals.

1049
00:48:59,730 --> 00:49:01,950
It's a little bit hazy
because a lot of this

1050
00:49:01,950 --> 00:49:06,650
is predicated on us knowing what
the valence of the element is.

1051
00:49:06,650 --> 00:49:08,730
And for transition metals,
it's like, well, is s

1052
00:49:08,730 --> 00:49:09,630
in the valence?

1053
00:49:09,630 --> 00:49:10,560
Yeah, probably.

1054
00:49:10,560 --> 00:49:12,240
Is p in the valence?

1055
00:49:12,240 --> 00:49:13,349
Well, maybe.

1056
00:49:13,349 --> 00:49:13,890
I don't know.

1057
00:49:13,890 --> 00:49:15,990
D is definitely in the valence.

1058
00:49:15,990 --> 00:49:18,420
But depending on what
you think the valence is,

1059
00:49:18,420 --> 00:49:21,210
your definition of some of these
things is a little bit fuzzier.

1060
00:49:21,210 --> 00:49:22,660
Sometimes you add a p function.

1061
00:49:22,660 --> 00:49:25,220
Sometimes you don't when
you go from double zeta

1062
00:49:25,220 --> 00:49:26,640
to triples for a
transition metal.

1063
00:49:29,480 --> 00:49:31,160
But the overall idea
here is that we're

1064
00:49:31,160 --> 00:49:33,582
trying to approach this
particular limit, which

1065
00:49:33,582 --> 00:49:35,540
is known as the complete
basis set limit, which

1066
00:49:35,540 --> 00:49:37,760
is the result that you
would hypothetically

1067
00:49:37,760 --> 00:49:40,790
get with an infinite
number of atomic orbitals.

1068
00:49:40,790 --> 00:49:43,374
So you just crank up, include
all the atomic orbital

1069
00:49:43,374 --> 00:49:44,040
in the universe.

1070
00:49:44,040 --> 00:49:46,970
You would get to an answer
that would be the right answer.

1071
00:49:46,970 --> 00:49:49,640
And we're just trying to
asymptotically approach that

1072
00:49:49,640 --> 00:49:51,830
by making our basis
sets bigger and bigger

1073
00:49:51,830 --> 00:49:54,400
until we run out of steam.

1074
00:49:54,400 --> 00:49:57,100
All right, so that's
everything about basis sets.

1075
00:49:57,100 --> 00:49:59,510
And tomorrow-- or not
tomorrow, next Monday we

1076
00:49:59,510 --> 00:50:02,210
will talk about how
you compute the energy.

1077
00:50:02,210 --> 00:50:04,997
All right, Happy Thanksgiving.