WEBVTT

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Prof: Okay,
let's get started.

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So last time we looked at
methane and ammonia,

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and saw something interesting
about molecular orbitals:

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we could give them the same
name as atomic orbitals;

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that is, if we,
especially if we look at a low

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electron density contour of the
molecular orbitals,

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we see that what it looks like
is an atom where the nucleus is

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split into pieces.

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Right?

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Now that splitting into pieces
changes the potential energy for

00:39.005 --> 00:40.105
the electron.

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So you expect that to distort
the orbital.

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The electrons will move in the
directions that pieces of the

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nucleus have gone.

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But then, in addition to
potential energy controlling the

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shape of orbitals,
kinetic energy also does.

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And that's the same thing as it
is in an atom.

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You have a thing with no nodes,
a thing with one node --

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and when you have one node you
can have either a spherical

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node,
or a planar node,

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and there can be three planar
nodes;

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so a 2s and three
2p's.

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Exactly the same considerations
apply in a molecular orbital as

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in an atomic orbital.

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You have the kinetic energy,
which comes with curvature,

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which comes with nodes.

01:20.739 --> 01:25.239
Now, as you split the nucleus
up and pull pieces in different

01:25.239 --> 01:27.219
directions,
it doesn't have the same

01:27.223 --> 01:29.483
symmetry it had when it was all
together in the nucleus,

01:29.480 --> 01:30.860
a spherical kind of symmetry.

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So the nodes get distorted.

01:32.780 --> 01:34.300
But still you can see them
there.

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And we saw them last time and
went through all the occupied

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and vacant valance orbitals of
ammonia and methane,

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and saw how they looked like
atomic orbitals.

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This, not surprisingly,
because it's so fundamental,

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the potential energy and the
kinetic energy,

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applies to every system.

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It applies to you,
viewed as a single atom --

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right?

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-- with a zillion electrons.

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Okay?

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But pieces have moved around,
so the orbitals change.

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We'll look at two more
complicated cases and then we'll

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get on to a different way of
looking at bonding.

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So we'll look at ethane and
methanol.

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And we use -- I didn't tell you
last time, explicitly,

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where I got the molecular
orbitals from.

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I got them from my laptop.

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There's a program -- the
particular one is called

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Spartan,
that I use -- and it calculates

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what molecular orbitals look
like,

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using approximate molecular
orbital (that is,

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Okay, so let's just look at the
ones of ethane and methanol.

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Now both of these have seven
pairs of valence electrons.

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There are also core electrons,
and if we were looking at the

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orbitals for all the electrons,
we'd include those.

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And exactly how we're going to
count those --

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you could do it one way or the
other --

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whether you consider the core
electrons,

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the 1s electrons to just
be part of the nucleus and then

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treat the rest as the electrons
you're interested in;

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or whether you want to count
the core electrons too.

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You can do it either way,
but what you analogize to what

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depends on whether you count
them as part of the nuclei.

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So anyhow, we're going to
compare these molecular orbitals

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to the atomic orbitals of argon,
which has also seven electron

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pairs.

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Okay, so there's the 2s
orbital.

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I'm going to start with that,
because I'm going to pretend

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that that red part is the core
-- that there's a 1s

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which is core electrons.

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But in this case -- here's my
pedantic note on this

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subject which I just added.

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So if you have -- before we had
just one heavy atom,

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carbon in methane,
nitrogen in ammonia.

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Right?

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So there was one 1s
orbital.

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Right?

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Now we have two heavy atoms,
carbon and carbon in methane,

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carbon and oxygen in methanol.

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So there are two heavy atoms
and therefore two boring core

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orbitals.

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So for purposes of making
analogies,

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we'll use the atomic 1s
orbital,

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of the atom that we're
analogizing things to,

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to stand for all the molecular
core orbitals.

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You can do it any way you want
to.

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We're not really interested in
that.

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We're interested in the valence
orbitals.

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So whether I start with the
1s or the 2s

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depends on how I'm handling the
core electrons;

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it's not a big deal.

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Anyhow, let's pretend it's the
2s of argon here,

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and we're going to compare it
with this lowest valence level

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molecular orbital of ethane.

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Now we can -- as I do this,
on the left side of the

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pictures I'm going to show one
view,

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and then, because it's more
complicated than methane and

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ammonia were,
I'm going to also show a

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picture rotated by ninety
degrees.

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So I'm going to rotate around
that axis, and on the right I'll

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show a different view of the
same orbital.

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Okay, so that's the lowest
valence level molecular orbital

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of ethane.

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And it's not very exciting;
it's just a distorted sphere.

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And you can see the way in
which it's distorted.

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It's distorted vertically,
because the two carbons are

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pulled apart;
so it's got sort of a narrow

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waist to it.

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And then it's pulled out,
where each of the protons left

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the middle atom to come out and
be hydrogens.

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Okay?

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Now if we look at methanol,
it'll be a little different.

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Can you anticipate how it might
be different if,

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on the bottom we'll have CH_3
again, but on the top,

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instead of having CH_3,
we're going to have OH.

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How do you think it might be
different, in the way it looks?

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Student:  It might be
skewed more towards the OH

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because oxygen is --
Prof: Okay,

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so oxygen has a bigger nuclear
charge.

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So that's going to pull the
lowest energy orbital toward the

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oxygen.

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That'll be one thing.

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We expect it to be bigger,
top-heavy.

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Okay?

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What else, about how the top
will look?

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How will it be different from
if it were a CH_3?

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Yes, Alex?

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I can't hear very well.

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Student:  Oxygen has
lone pairs.

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Prof: Oxygen has lone
pairs.

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Now, how is that going to
change things?

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Or another way of saying the
same thing is it has only one

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hydrogen up there.

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How's that going to change it,
do you think?

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You have to speak up.

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Student:  It's not going
to be symmetrical.

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Prof: It's not going to
be symmetrical.

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Which way is it going to be
distorted to be unsymmetrical?

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You have to speak up.

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Student:  Towards the
electron pair.

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Prof: Toward the
electron pair did you say?

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I couldn't hear.

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Student:  Away from
hydrogen.

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Prof: Away from hydrogen?

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Well how about -- the proton
goes out;

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does it bring electrons with
it, or does it repel electrons,

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the proton?

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Speak up.

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Student:  It brings them.

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Prof: It pulls,
so it should distort toward the

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hydrogen.

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So here's what it looks like.

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Right?

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It's top-heavy,
as Angela said,

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and it's distorted out toward
the hydrogen;

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there are no protons pulling it
to the top left.

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And you see the same thing
end-on there,

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on the right.

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Okay, this is the next orbital.

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What does that look like?

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Obviously you can peek in the
middle and see.

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It's obviously a 2p_z
orbital, with the node across

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the middle.

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Okay?

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In both cases.

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Now the ethane case is
symmetrical.

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The other case is unsymmetrical.

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Why is it unsymmetrical?

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Because the first orbital
pulled electrons to the top.

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Right?

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So the next- in the next
orbital, electrons aren't going

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to want to be at the top so
much,

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because they're going to be
repelled by the other electrons;

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so there'll be more toward the
bottom, because the first ones

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went to the top.

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Okay?

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Okay, then here's the next
orbital.

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You can see there are energies
marching up here.

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The lowest one was s,
then 2p_z,

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now 2p_x,
if we define the horizontal

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axis here on the left as the x
axis.

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So again, it's what you expect,
and it's pulled out,

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stretched vertically from being
a dumbbell by where the nuclei

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went.

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And here's 2p_y,
which we can see more clearly

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on the right,
perpendicular to the

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2p_x.

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Okay?

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And notice that it's obviously
so, that the methanol orbital

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will be less symmetric,
in all cases,

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than the one for ethane.

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But still we can recognize the
nodes, because they must go that

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way.

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It must be no nodes,
one node of three different

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kinds, and so on.

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Okay, now this is 3s.

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You notice it has -- the node
that we don't see,

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the one that's down near the
nuclei,

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in fact two nodes down near the
nuclei,

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one for each of the heavy atoms.

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But then this now has another
spherical node,

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or approximately spherical
node.

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So we have that extra red lump
in the middle on the top,

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or blue, in the middle.

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Just to review,
what's the difference between

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red and blue?

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On the top, the computer
decided to draw it with red in

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the middle.

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On the bottom it decided to
draw it with blue in the middle.

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What do those colors mean?

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Yes, Cathy?

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Student:  Positive and
negative signs.

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Prof: The mean positive
and negative.

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So which one is right?

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Should it be positive in the
middle or negative in the

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middle?

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Student:  Positive.

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Prof: Positive is a
guess.

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Wilson, what do you say?

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Student:  It doesn't
matter.

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Prof: Why not?

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Student:  Because it's
not really positive or negative;

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it's just kind of a phase of it.

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Prof: Yes,
it's the sign of the wave

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function.

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But you can multiply the wave
function by minus one,

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any constant,
and it's just as good as it was

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before.

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So it's arbitrary,
and the computer was arbitrary

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in choosing the colors.

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I think there's actually a
function that I could've changed

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it, if I'd thought to do so,
so they'd be the same.

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But actually it tells a story
if I leave them different.

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Okay, so here's the next one.

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Now this is -- if you look on
the left now,

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it's more clear where the nodes
are, that that's a d_xz

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orbital.

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The name 'xz' means that
the product of x and z appears

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in the wave function.

10:16.097 --> 10:16.977
Right?

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So when both x and z are
positive, then the product is

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positive;
on the top right, red. Right?

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When x is negative,
and z is positive,

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the top left,
it's negative,

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the product of them.

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So that's why it has the name
xz.

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Okay, and there's d_yz,
which you see on the right,

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turned ninety degrees.

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And here's the d_z^2,
which is that thing that has a

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doughnut that goes around the
middle.

10:46.798 --> 10:47.418
Right?

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It's hard to see the doughnut.

10:49.490 --> 10:50.050
Can you see?

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It's blue on top.

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You can easily see the red on
the top left,

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which is what's blue in the
middle;

10:57.039 --> 10:58.909
the sign has changed. Right?

10:58.908 --> 11:01.508
But the doughnut is highly
distorted,

11:01.509 --> 11:04.619
because as you go around,
first a proton on the top pulls

11:04.620 --> 11:06.340
it up,
then a proton on the bottom

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pulls it down,
then up, down.

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So it's like a crown,
the doughnut has been made into

11:11.306 --> 11:15.046
a crown around the end,
by the protons pulling in that

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way.

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Okay.

11:16.360 --> 11:19.900
Then here's the 3p_z.

11:19.899 --> 11:24.069
So it has the horizontal nodal
plane, but also it has a

11:24.068 --> 11:27.308
spherical node,
which you can see in either

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picture really.

11:28.779 --> 11:29.969
Okay?

11:29.970 --> 11:32.170
Then here's the 3p _y
orbital,

11:32.168 --> 11:34.078
which again has that spherical
node,

11:34.080 --> 11:37.020
but now the planar node,
or approximately planar node,

11:37.019 --> 11:39.509
is vertical instead of
horizontal.

11:39.509 --> 11:43.969
So say on the top right there
you have red on the right and

11:43.969 --> 11:45.429
blue on the left.

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There's a vertical node over
which it changes sign,

11:49.346 --> 11:51.066
going right to left.

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Or here's the 3_px.

11:54.559 --> 11:59.979
Or here's the 3d_xy;
which you don't see so well

11:59.976 --> 12:02.456
here;
well, but if you turned it down

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you would.

12:03.230 --> 12:06.880
And here's the
3d_x^2-y^2;

12:06.879 --> 12:08.999
which again,
to see it well,

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you'd have to turn it.

12:10.879 --> 12:14.819
And here's the 4f
orbital.

12:14.820 --> 12:18.500
So you can see,
especially say in the top ones,

12:18.500 --> 12:21.570
compared with the atomic
orbital, that it's exactly the

12:21.566 --> 12:26.156
same general pattern of nodes,
slightly distorted.

12:26.158 --> 12:29.778
And incidentally,
remember all the n

12:29.775 --> 12:32.355
equals whatever,
n=3;

12:32.360 --> 12:35.630
all the orbitals had the same
energy before.

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Now they don't have all the
same energy.

12:37.759 --> 12:39.709
Notice that they all have
different energies.

12:39.710 --> 12:41.190
None of them are degenerate.

12:41.190 --> 12:42.460
Why?

12:42.460 --> 12:46.710
Because if you broke the
nucleus apart,

12:46.710 --> 12:49.510
how stable a thing that has
that general shape,

12:49.509 --> 12:52.389
like a dumbbell or a cloverleaf
or something,

12:52.389 --> 12:57.779
how stable the electrons are in
those lumps depends on whether a

12:57.779 --> 13:00.689
proton got pulled into the lump.

13:00.690 --> 13:03.810
If there happens to be a proton
in the lump, where kinetic

13:03.812 --> 13:06.002
energy, the node pattern,
wants it to be,

13:06.003 --> 13:08.033
then that'll be unusually
stable.

13:08.028 --> 13:12.018
If the protons have been pulled
someplace where there is a node,

13:12.024 --> 13:15.644
because of kinetic energy,
then it won't be stabilized.

13:15.639 --> 13:19.979
So that breaks the degeneracy
of these different patterns that

13:19.975 --> 13:22.245
have the same number of nodes.

13:22.250 --> 13:25.650
It depends on where the -- how
the potential energy changed.

13:25.649 --> 13:27.919
Okay, now just finally and very
quickly,

13:27.918 --> 13:30.208
I want to look at
1-flouoroethanol,

13:30.210 --> 13:31.460
which is a molecule that would
have,

13:31.460 --> 13:34.380
I think, no stability at all,
but you can put it into the --

13:34.379 --> 13:36.919
as a practical matter,
you're never going to put it in

13:36.923 --> 13:38.843
a bottle,
but you can easily put it into

13:38.841 --> 13:41.481
the computer and calculate what
its molecular orbitals would

13:41.477 --> 13:42.057
look like.

13:42.058 --> 13:45.278
And I did it to show you
something that's very,

13:45.279 --> 13:48.779
very unsymmetrical,
and has atoms of very different

13:48.779 --> 13:50.109
nuclear charge.

13:50.110 --> 13:50.940
Okay?

13:50.940 --> 13:55.200
So what will the very,
very, very lowest orbital look

13:55.198 --> 13:58.228
like, do you think,
for this thing?

13:58.230 --> 14:02.250
It has a fluorine,
an oxygen, two carbons and five

14:02.254 --> 14:03.244
hydrogens.

14:03.240 --> 14:04.800
So what do you think?

14:04.798 --> 14:08.258
If you were an electron,
and you had that set of nuclei,

14:08.264 --> 14:11.294
arranged this way,
where would you want to go?

14:11.289 --> 14:11.949
Elizabeth?

14:11.950 --> 14:14.850
Student:  Very biased to
the left, especially around the

14:14.846 --> 14:15.356
fluorine.

14:15.360 --> 14:15.960
Prof: Yes.

14:15.956 --> 14:17.636
Now what would the very,
very, very lowest one?

14:17.639 --> 14:21.439
It should be more toward the
left, and really close to the

14:21.437 --> 14:22.167
fluorine.

14:22.168 --> 14:25.248
It would be a 1s
orbital, and mostly on fluorine,

14:25.254 --> 14:27.724
because that's where most of
the protons are,

14:27.721 --> 14:29.631
the most concentrated protons.

14:29.629 --> 14:30.479
So let's look.

14:30.480 --> 14:34.320
There's a smaller scale ball
model,

14:34.320 --> 14:36.900
so that we can see really small
orbitals,

14:36.899 --> 14:39.099
and there that's,
you're absolutely right,

14:39.100 --> 14:42.490
it's essentially the 1s
orbital of fluorine,

14:42.490 --> 14:44.660
is the very lowest orbital.

14:44.659 --> 14:47.769
What would be next?

14:47.769 --> 14:50.239
Suppose this one,
you came up and this seat was

14:50.240 --> 14:52.230
already taken,
now where would you go,

14:52.227 --> 14:53.407
the next electron?

14:53.409 --> 14:53.519
Zack?

14:53.519 --> 14:54.179
Student:  Oxygen.

14:54.178 --> 14:57.558
Prof: To the oxygen,
because that's the next highest

14:57.557 --> 14:59.127
concentration of protons.

14:59.129 --> 15:02.699
And next?

15:02.700 --> 15:03.890
Where will you go next?

15:03.889 --> 15:05.669
Steve, what do you say?

15:05.668 --> 15:08.088
The 1s of fluorine is
taken.

15:08.090 --> 15:10.000
The 1s of oxygen is
taken.

15:10.000 --> 15:12.680
Where do we want to go next?

15:12.679 --> 15:14.309
Pardon me?

15:14.308 --> 15:16.218
Student:  One of the
carbons.

15:16.220 --> 15:18.450
Prof: Ah, which carbon?

15:18.450 --> 15:19.830
Student:  The left
carbon.

15:19.830 --> 15:20.940
Prof: Why?

15:20.940 --> 15:23.700
Student:  Because it's
closer to the fluorine than the

15:23.696 --> 15:24.106
oxygen.

15:24.110 --> 15:28.090
Prof: And what does the
proximity have to do with it?

15:28.090 --> 15:33.660
The proximity means that the
electrons that are on that atom

15:33.658 --> 15:38.008
will have been drawn toward --
and we'll talk about this more

15:38.009 --> 15:40.269
later --
will have been drawn toward the

15:40.269 --> 15:41.769
fluorine and the oxygen.

15:41.769 --> 15:44.979
Therefore that atom will have
fewer electrons,

15:44.976 --> 15:46.826
or lower electron density.

15:46.827 --> 15:47.467
Right?

15:47.470 --> 15:51.440
Therefore, it's a better place
for other electrons to go.

15:51.440 --> 15:51.950
Okay?

15:51.950 --> 15:55.240
So you'd expect the next one to
be the middle carbon.

15:55.243 --> 15:55.753
Right!

15:55.750 --> 15:58.160
And finally,
of course, it's going to be the

15:58.155 --> 16:00.615
second carbon,
the one that's remote from the

16:00.619 --> 16:03.249
electronegative,
high nuclear charge atoms.

16:03.250 --> 16:06.480
Okay, now we've done all the
1s's.

16:06.480 --> 16:08.570
So we can look at what's
interesting, the valence

16:08.573 --> 16:11.283
orbitals, the ones that are
going to be involved in bonding;

16:11.278 --> 16:13.858
these don't have anything to do
with bonding.

16:13.860 --> 16:15.880
Okay, so there's the first one.

16:15.879 --> 16:19.209
Russell, tell me something
about the shape of this?

16:19.210 --> 16:20.510
Does it surprise you?

16:20.509 --> 16:21.439
Student:  No.

16:21.437 --> 16:23.477
It's highly distorted towards
the fluorine.

16:23.480 --> 16:26.350
Prof: Yes,
it has no nodes;

16:26.350 --> 16:29.340
except it actually has little
nodes down near the nucleus,

16:29.340 --> 16:32.120
because it's actually more like
a 2s orbital in that

16:32.121 --> 16:34.311
respect,
because of the core electrons.

16:34.308 --> 16:37.538
So it has tiny nodes that we
don't see around the nuclei.

16:37.538 --> 16:41.218
But it has no nodes within the
valence;

16:41.220 --> 16:42.760
big orbitals that we're looking
at.

16:42.759 --> 16:44.799
So it's like an s
orbital.

16:44.798 --> 16:45.308
Right?

16:45.308 --> 16:48.948
And it's big where the nuclear
charge is big,

16:48.952 --> 16:50.032
as you say.

16:50.029 --> 16:51.329
Okay?

16:51.330 --> 16:56.710
What's the next one going to
look like?

16:56.710 --> 17:01.890
Any ideas?

17:01.889 --> 17:02.889
Pardon me?

17:02.889 --> 17:03.369
Student:  Towards the
oxygen.

17:03.370 --> 17:06.930
Prof: Toward the oxygen.

17:06.926 --> 17:10.136
Will it also have no nodes?

17:10.140 --> 17:10.330
No.

17:10.333 --> 17:13.243
The next highest orbital has to
have a node.

17:13.240 --> 17:14.540
Where do you think the node
will be?

17:14.538 --> 17:17.578
What orbital will it look like,
sort of?

17:17.579 --> 17:19.329
It'll be highly distorted.

17:19.328 --> 17:21.958
But what -- if this is the s
orbital, right?

17:21.960 --> 17:26.180
What's going to be next?

17:26.180 --> 17:27.450
Elizabeth?

17:27.450 --> 17:30.250
Student:  It's as if
it's switched but with a planar

17:30.250 --> 17:31.120
node in between.

17:31.119 --> 17:32.839
Prof: Oh let's see.

17:32.839 --> 17:34.339
Right!

17:34.338 --> 17:37.318
So the first one was -- you
were right Russell that this one

17:37.317 --> 17:38.627
should be big on oxygen.

17:38.630 --> 17:42.490
But notice it's like a p
orbital, it has that horizontal

17:42.490 --> 17:45.170
node, because it's higher
kinetic energy.

17:45.170 --> 17:45.690
Okay?

17:45.690 --> 17:49.120
And then this one,
that's on the carbons,

17:49.123 --> 17:49.813
right?

17:49.809 --> 17:50.379
Mostly.

17:50.380 --> 17:54.040
But it's a p_y,
it has a vertical node.

17:54.038 --> 17:57.698
So it's negative say on the
fluorine and oxygen and positive

17:57.695 --> 17:58.745
on the carbons.

17:58.750 --> 18:02.340
Or if we rotate this one around
a horizontal axis to look at it

18:02.344 --> 18:04.344
from the side,
it looks like that,

18:04.337 --> 18:07.307
and you can see that it's a
p_y kind of orbital.

18:07.309 --> 18:08.009
Elizabeth?

18:08.009 --> 18:09.689
Student:  I'm just
asking a point of clarification.

18:09.690 --> 18:11.540
These aren't technically
p orbitals,

18:11.538 --> 18:11.888
right?

18:11.890 --> 18:13.980
We're just saying they're
analogous to them.

18:13.980 --> 18:15.120
Prof: They're analogous
to p orbitals;

18:15.119 --> 18:18.419
because it has to be the same.

18:18.420 --> 18:22.360
The orbitals must go in order
of the number of nodes.

18:22.363 --> 18:22.973
Right?

18:22.970 --> 18:25.410
So the same thing is in an atom
or in a molecule.

18:25.410 --> 18:27.170
You go from no nodes to one
node.

18:27.170 --> 18:29.920
And there are three ways of
getting one node:

18:29.924 --> 18:33.684
spherical or distorted sphere,
planes, or distorted planes if

18:33.683 --> 18:36.443
the nuclei are pulled around;
and so on.

18:36.440 --> 18:39.680
So they're really very much
like the atomic orbitals.

18:39.680 --> 18:42.170
Okay, or this one.

18:42.170 --> 18:46.070
Now that's an interesting one
because that's like a hybrid

18:46.074 --> 18:46.764
orbital.

18:46.759 --> 18:48.369
It looks like the orbital up
here.

18:48.368 --> 18:50.238
It's a mixture of s with
p.

18:50.240 --> 18:52.030
Did everybody see how that is?

18:52.029 --> 18:55.289
It's a big sort of blue lobe on
the top and a small red one on

18:55.286 --> 18:55.976
the bottom.

18:55.980 --> 18:58.630
And also the next one is the
other one,

18:58.630 --> 19:00.720
the hybrid that points the
other direction,

19:00.720 --> 19:02.810
another combination of the
s and the p,

19:02.809 --> 19:04.859
that points down.

19:04.859 --> 19:06.979
Okay?

19:06.980 --> 19:09.620
And then this one looks like
d_xy again.

19:09.619 --> 19:10.079
Right?

19:10.078 --> 19:14.468
It's sort of a cloverleaf with
two nodes.

19:14.470 --> 19:16.700
Okay, so that's all I want to
do with that,

19:16.701 --> 19:18.721
and we won't go any further
with it.

19:18.720 --> 19:21.810
This is just an interesting way
of looking at how --

19:21.808 --> 19:24.868
that molecular orbitals are
really just like atomic

19:24.865 --> 19:27.155
orbitals,
and have energies for the same

19:27.161 --> 19:28.991
reason,
except the potential energy

19:28.993 --> 19:31.873
gets screwed up by breaking the
nucleus and pulling pieces

19:31.874 --> 19:34.734
around,
but in an understandable way,

19:34.733 --> 19:38.313
and the nodes get distorted
because of this.

19:38.308 --> 19:43.128
Okay now, now we're getting
into really -- we just looked at

19:43.126 --> 19:46.716
this view, at the single united
atom view.

19:46.720 --> 19:49.740
But the other view is the one
that's going to be more

19:49.737 --> 19:52.117
generalizable,
and that's the one where we

19:52.116 --> 19:53.156
looked at bonding.

19:53.161 --> 19:53.801
Right?

19:53.798 --> 19:57.368
So you have to probe a little
harder to get a qualitative

19:57.374 --> 20:00.124
understanding of what chemical
bonds are.

20:00.118 --> 20:03.648
And that's what we're going to
do now by choosing a higher

20:03.647 --> 20:06.307
contour with which to look at a
molecule.

20:06.308 --> 20:10.648
Now, true molecular orbitals,
to the extent that orbitals are

20:10.647 --> 20:14.767
true all together -- why aren't
they true all together;

20:14.769 --> 20:18.879
why aren't orbitals true all
together?

20:18.880 --> 20:20.360
Yes, Alex?

20:20.358 --> 20:21.638
Student:  Because you
have multiple electrons.

20:21.640 --> 20:22.740
Prof: Because you have
many electrons;

20:22.740 --> 20:25.680
you can't have independent
electrons, you can't have

20:25.675 --> 20:26.305
orbitals.

20:26.308 --> 20:29.048
But we're approximating things
by orbitals,

20:29.048 --> 20:32.128
trying to take electron
interaction into account in a

20:32.130 --> 20:35.330
sort of a left-handed way by
Self-Consistent Field,

20:35.329 --> 20:36.419
or something like that.

20:36.420 --> 20:39.610
Because it's much easier if we
can divide the whole into a

20:39.605 --> 20:41.555
bunch of parts,
each of which we can

20:41.560 --> 20:42.400
understand.

20:42.400 --> 20:44.900
So, to the extent that
molecular orbitals are true --

20:44.900 --> 20:46.360
the kinds of things I've just
been showing you,

20:46.358 --> 20:50.738
calculated with my laptop --
they extend over the whole

20:50.738 --> 20:54.358
molecule;
they're not local. Right?

20:54.358 --> 20:57.878
Except like the 1s of
fluorine was local,

20:57.878 --> 21:01.248
but mostly they go over the
whole molecule.

21:01.250 --> 21:05.230
But bonds are thought of,
and have always been thought

21:05.230 --> 21:08.610
of, as interactions between
pairs of atoms.

21:08.608 --> 21:14.378
So we want to divide things
completely differently and look

21:14.378 --> 21:18.658
at the bonds now,
at pairwise LCAO molecular

21:18.655 --> 21:19.945
orbitals.

21:19.950 --> 21:22.240
Now what's an LCAO?

21:22.240 --> 21:26.030
It's a sum, or a linear
combination.

21:26.032 --> 21:26.902
Right?

21:26.900 --> 21:30.220
A weighted sum of atomic
orbitals.

21:30.220 --> 21:31.460
So here's an example.

21:31.460 --> 21:34.400
Ψ, which is an orbital --
what's it a function of?

21:34.400 --> 21:35.690
Student:  Position.

21:35.690 --> 21:36.650
Prof: Position of what?

21:36.650 --> 21:37.270
Student:  One electron.

21:37.269 --> 21:37.939
Prof: One electron.

21:37.940 --> 21:40.220
So it's a function of
x_1,y_1,z_1;

21:40.220 --> 21:42.630
we're talking just about
electron one.

21:42.630 --> 21:46.640
So the wave function for
electron one we say is

21:46.644 --> 21:51.364
1/√2 times one
atomic orbital plus another

21:51.359 --> 21:52.669
atomic orbital.

21:52.669 --> 21:53.629
Right?

21:53.630 --> 21:58.170
Now, have you ever seen adding
orbitals like that before?

21:58.170 --> 22:01.960
That's what hybridization is;
we added s and p.

22:01.960 --> 22:04.880
But this is different,
because when we added s

22:04.877 --> 22:08.367
and p before,
they were on the same nucleus,

22:08.366 --> 22:12.246
and we did it to get a new
orbital for that particular

22:12.248 --> 22:15.368
nucleus for that atom;
to distort it one way or the

22:15.367 --> 22:17.297
other, for example,
or to rotate a p

22:17.300 --> 22:17.670
orbital.

22:17.669 --> 22:18.129
Right?

22:18.130 --> 22:22.500
But this is very different,
because we're adding orbitals

22:22.500 --> 22:26.330
that are on different
nuclei: A, nucleus A,

22:26.325 --> 22:27.725
and nucleus B.

22:27.730 --> 22:30.600
See the difference?

22:30.598 --> 22:34.048
Adding is just -- wave
functions are numbers,

22:34.049 --> 22:36.089
we just add the numbers.

22:36.088 --> 22:38.318
But in the first case,
hybridization,

22:38.323 --> 22:41.243
those two functions were on the
same nucleus.

22:41.240 --> 22:44.460
Now they're on different
nuclei, what we're adding

22:44.464 --> 22:45.194
together.

22:45.190 --> 22:50.270
Okay, now why is it sensible to
think that you might get

22:50.266 --> 22:55.986
pairwise molecular orbitals that
can be expressed like this?

22:55.990 --> 22:58.750
How do you interpret an orbital?

22:58.750 --> 23:00.990
Corey?

23:00.990 --> 23:01.860
What good is an orbital?

23:01.859 --> 23:03.709
What do you use it for?

23:03.710 --> 23:06.320
Student:  It's a
one-electron wave function.

23:06.318 --> 23:08.538
Prof: Well what do you
use it for?

23:08.538 --> 23:10.738
Student:  For
probability.

23:10.740 --> 23:14.110
Prof: And how do you get
probability?

23:14.108 --> 23:16.518
From the wave function -- if
you have the wave function,

23:16.520 --> 23:18.320
how do you get the probability
density?

23:18.319 --> 23:19.019
Student:  You square it.

23:19.019 --> 23:20.109
Prof: You square it.

23:20.108 --> 23:25.278
So do you see why we have a
1/√2 in this?

23:25.278 --> 23:28.838
Because when we square it,
that's going to be half,

23:28.843 --> 23:32.763
and we're going to get half of
atomic orbital A squared,

23:32.763 --> 23:35.333
and of atomic orbital B
squared;

23:35.329 --> 23:36.979
so it's a half of each of them.

23:36.980 --> 23:39.050
That's why we have
1/√2.

23:39.049 --> 23:40.649
So let's go on with this.

23:40.650 --> 23:44.570
Suppose we have a hydrogen
molecule, and suppose that the

23:44.568 --> 23:48.068
nuclei are at a great distance
from one another.

23:48.068 --> 23:52.558
So far they don't interact,
or negligible interaction;

23:52.559 --> 23:53.889
they're very far apart.

23:53.890 --> 23:57.180
What would you expect the
lowest energy,

23:57.178 --> 24:00.718
one-electron wave function to
look like?

24:00.720 --> 24:05.750
One possibility is that the
electron could sit exactly

24:05.746 --> 24:08.686
halfway between the two nuclei?

24:08.685 --> 24:09.535
Right?

24:09.538 --> 24:12.568
Is that a reasonable place,
is that the low energy place

24:12.565 --> 24:13.385
for it to be?

24:13.390 --> 24:14.830
Lucas, what do you say?

24:14.828 --> 24:17.848
Student:  No,
because the added probability

24:17.849 --> 24:20.069
density there is not the
greatest.

24:20.069 --> 24:20.719
Prof: Why not?

24:20.720 --> 24:22.430
Student:  It needs to be
one or the other.

24:22.430 --> 24:24.410
Prof: Why shouldn't the
electrons sit -- if the two

24:24.407 --> 24:27.167
nuclei are this far apart;
and I don't mean two angstroms

24:27.171 --> 24:30.751
apart, I mean two meters apart
-- there's a proton here and a

24:30.753 --> 24:31.653
proton here.

24:31.650 --> 24:35.140
Is the electron most likely to
be here, halfway between?

24:35.140 --> 24:35.840
Student:  No.

24:35.838 --> 24:37.268
Prof: Where would it be
most likely?

24:37.269 --> 24:39.329
Student:  Probably
nearer to one of the two atoms.

24:39.328 --> 24:43.268
Prof: And which of the
two?

24:43.269 --> 24:44.539
<<Students speak over one
another>>

24:44.538 --> 24:46.958
Prof: Suppose you took
the long view.

24:46.960 --> 24:52.480
Suppose you averaged it over
eighteen zillion millennia --

24:52.480 --> 24:54.030
time averaged.

24:54.029 --> 24:57.679
Sometimes it would be near this
one, sometimes it would be near

24:57.682 --> 24:58.332
this one.

24:58.329 --> 24:59.089
Okay?

24:59.088 --> 25:01.208
What would it be if you took a
really long view?

25:01.210 --> 25:02.210
Student:  Both.

25:02.210 --> 25:05.520
Prof: And half here,
and half here.

25:05.519 --> 25:08.409
So the wave function,
when you square it,

25:08.414 --> 25:11.674
you want it to be half looking
like this atom,

25:11.673 --> 25:14.283
and half looking like this
atom.

25:14.278 --> 25:16.438
Then you see the time average.

25:16.435 --> 25:16.935
Right?

25:16.940 --> 25:19.640
It might take a long time to
achieve that average,

25:19.637 --> 25:22.887
because it'd take a long time
for the electron to tunnel two

25:22.887 --> 25:23.877
meters, right?

25:23.880 --> 25:26.110
But in the very,
very long time it would look

25:26.111 --> 25:26.721
like that.

25:26.720 --> 25:28.710
So we know what we want it to
look like.

25:28.710 --> 25:31.620
What we want is that the
probability density,

25:31.618 --> 25:34.308
the square of this one-electron
wave function,

25:34.308 --> 25:38.028
should look half of the time
like the atomic orbital A

25:38.032 --> 25:42.322
squared and half of the time
like atomic orbital B squared.

25:42.318 --> 25:45.348
So on time average it's half of
one and half of the other.

25:45.349 --> 25:47.509
Everybody with me?

25:47.509 --> 25:49.529
So that's what the -- yes, Nate?

25:49.529 --> 25:54.649
Student:  Why isn't
there a 2A -- or a 2AB?

25:54.650 --> 25:56.890
Prof: Oh,
because I'm telling you what it

25:56.890 --> 25:57.510
looks like.

25:57.509 --> 26:00.329
It's got to look half like this
and half like this.

26:00.327 --> 26:00.777
Right?

26:00.778 --> 26:04.538
So the square of it has to be
this.

26:04.538 --> 26:08.238
So all we have to do to find
the wave function is what,

26:08.238 --> 26:10.428
if we know what its square is?

26:10.430 --> 26:13.130
All we got to do is take the
square root and we've got the

26:13.131 --> 26:13.891
wave function.

26:13.890 --> 26:14.810
Bingo!

26:14.809 --> 26:20.169
There's the square root. Right?

26:20.170 --> 26:25.150
So that is a reasonable way to
write the wave function,

26:25.150 --> 26:28.750
1/√2 (AO_A +
AO_B). Claire,

26:28.746 --> 26:30.956
you have a question.

26:30.960 --> 26:33.950
Student:  From my small
understanding,

26:33.951 --> 26:37.491
with the math that I've got --
and this may be wrong,

26:37.486 --> 26:39.386
correct me if I'm wrong.

26:39.390 --> 26:42.820
But don't you,
if you have a bracket and two

26:42.819 --> 26:46.169
things inside it,
you square outside of it,

26:46.171 --> 26:50.241
don't you have four things that
come out of it.

26:50.240 --> 26:51.530
Prof: You want that?

26:51.529 --> 26:52.559
Student:  Yes.

26:52.559 --> 26:53.649
>

26:53.650 --> 26:57.350
Prof: Okay,
now Claire you're going to help

26:57.349 --> 26:58.029
me out.

26:58.029 --> 27:00.479
How big is -- that's a number.

27:00.480 --> 27:04.570
It's the product of -- atomic
orbital A assigns numbers

27:04.566 --> 27:07.816
everywhere in space,
everywhere in space.

27:07.818 --> 27:11.268
Atomic orbital B assigns
numbers everywhere in space.

27:11.269 --> 27:14.969
So at some point in space
atomic orbital B assigns a

27:14.974 --> 27:18.394
number and atomic orbital A
assigns a number;

27:18.390 --> 27:22.840
and A times B is the product of
those two numbers.

27:22.839 --> 27:25.959
How big is that product?

27:25.960 --> 27:30.130
How big is atomic orbital A
here?

27:30.130 --> 27:32.290
This, a meter away from the
proton?

27:32.289 --> 27:33.159
Student:  Not very big.

27:33.160 --> 27:36.090
Prof: And how big is
atomic orbital B there?

27:36.089 --> 27:36.599
Student:  Not very big.

27:36.598 --> 27:39.658
Prof: Now how big is
atomic orbital A here?

27:39.660 --> 27:40.780
Student:  Very big.

27:40.779 --> 27:42.779
Prof: Okay.

27:42.781 --> 27:49.571
So now, how big an error do we
make if we neglect A times B?

27:49.569 --> 27:50.999
Where do we make an error?

27:51.000 --> 27:53.120
Do we make an error here?

27:53.119 --> 27:54.719
Student:  No.

27:54.720 --> 27:56.590
Prof: Do we make an
error here?

27:56.589 --> 27:57.059
Student:  No.

27:57.058 --> 27:58.078
Prof: Do we make an
error here?

27:58.079 --> 27:58.809
Student:  Yes.

27:58.809 --> 28:00.009
Prof: No!

28:00.009 --> 28:01.919
We make an error;
yes, sure enough,

28:01.919 --> 28:03.019
we make an error.

28:03.019 --> 28:04.759
How big is the error?

28:04.759 --> 28:05.749
Student:  Not very big.

28:05.750 --> 28:07.940
Prof: Ah,
it's negligible,

28:07.940 --> 28:10.200
because it's at great distance.

28:10.200 --> 28:13.520
Okay, so at great distance we
can forget that.

28:13.519 --> 28:18.619
So now it's easier to take the
square root, right?

28:18.619 --> 28:19.969
Okay?

28:19.970 --> 28:21.670
The old fox up here, huh?

28:21.670 --> 28:26.030
>

28:26.028 --> 28:29.948
Okay, now your problem
is what happens if H_2 is at the

28:29.949 --> 28:31.169
bonding distance?

28:31.170 --> 28:34.220
What if they're only an
Angstrom apart?

28:34.220 --> 28:39.690
Now there should be a problem,
because A times B is not going

28:39.690 --> 28:42.790
to be negligible everywhere now.

28:42.788 --> 28:44.768
Okay, so now that's going to
come back.

28:44.769 --> 28:47.979
So now we got an error, right?

28:47.980 --> 28:49.830
Or do we?

28:49.829 --> 28:52.589
Let's think what that does.

28:52.588 --> 28:57.358
Okay, so we if approximate the
molecular orbital as the sum of

28:57.362 --> 29:00.802
atomic orbitals,
this way, then it looks very

29:00.804 --> 29:02.764
good near the nuclei.

29:02.759 --> 29:09.479
Because near A it looks like A;
near B it looks like B.

29:09.480 --> 29:14.080
And if we want to square it to
find the electron density,

29:14.080 --> 29:15.150
we do this.

29:15.150 --> 29:19.670
But if we then subtract,
what the atoms would give for

29:19.669 --> 29:21.289
electron density.

29:21.288 --> 29:25.368
Now what does this remind you
of, where we look at the total

29:25.373 --> 29:28.283
electron density and subtract
the atoms?

29:28.278 --> 29:29.178
Student:  Difference
density.

29:29.180 --> 29:31.120
Prof: So we're actually
looking for the difference

29:31.123 --> 29:31.473
density.

29:31.470 --> 29:33.040
Everybody with me on this?

29:37.962 --> 29:38.572
Right?

29:38.569 --> 29:42.089
If we subtract, what do we get?

29:42.088 --> 29:45.468
What do we get for a difference
density?

29:45.470 --> 29:49.150
We get A times B.

29:49.150 --> 29:52.410
So we get the difference
electron density,

29:52.411 --> 29:54.481
which is due to overlap.

29:54.480 --> 29:56.140
And what do I mean by overlap?

29:56.140 --> 30:00.930
I mean that is only important
in regions where both of them

30:00.925 --> 30:02.575
have a finite value.

30:02.576 --> 30:03.316
Right?

30:03.318 --> 30:06.458
The product of A and B is
negligible if A is zero;

30:06.460 --> 30:09.210
it's negligible if B is very,
very small.

30:09.210 --> 30:09.760
Right?

30:09.759 --> 30:13.189
So it's only where they
overlap, the two functions have

30:13.194 --> 30:15.554
simultaneous values,
that you care.

30:15.548 --> 30:21.358
Okay, now that thing,
the thing that's the bonding,

30:21.358 --> 30:25.388
the difference density,
is really a byproduct --

30:25.390 --> 30:28.870
that's a little bit of a pun
because it's a product --

30:28.868 --> 30:32.378
but it's a byproduct of
squaring the sum --

30:32.380 --> 30:34.280
of what Claire didn't like
about it.

30:34.279 --> 30:38.149
So the very thing you didn't
like is what's going to give

30:38.152 --> 30:39.952
rise to bonding density.

30:39.950 --> 30:44.020
Isn't that neat?

30:44.019 --> 30:47.839
Okay, but notice that here
we're multiplying A times B.

30:47.838 --> 30:50.498
But this is a completely
different instance of

30:50.500 --> 30:52.570
multiplying from what we had
before.

30:52.567 --> 30:53.157
Right?

30:53.160 --> 30:55.600
Before we multiplied two
orbitals to try to get a

30:55.596 --> 30:57.066
two-electron wave function.

30:57.068 --> 31:00.028
This has nothing to do with
this, because both of these are

31:00.032 --> 31:03.742
functions of the same electron;
it's like one electron that

31:03.743 --> 31:05.323
we're squaring here.

31:05.318 --> 31:08.158
So this A times B,
this product,

31:08.162 --> 31:11.832
this overlap,
comes from the squaring.

31:11.828 --> 31:15.118
It was when we squared it that
we got that.

31:15.119 --> 31:18.609
Okay?

31:18.608 --> 31:21.178
Now, because we have this extra
term,

31:24.338 --> 31:28.258
which would -- what's the
probability of A^2^( )summed

31:28.259 --> 31:30.399
over all space,
or integrated,

31:30.401 --> 31:31.561
if it's normalized?

31:31.559 --> 31:32.339
Students:  One.

31:32.339 --> 31:33.089
Prof: And this one?

31:33.089 --> 31:33.989
Students:  One.

31:33.990 --> 31:35.560
Prof: And what's this
whole quantity?

31:35.559 --> 31:36.129
Students:  One.

31:36.130 --> 31:37.620
Prof: One,
because of the half.

31:37.618 --> 31:40.708
But actually we've got it
bigger than that,

31:40.711 --> 31:43.581
because we added the overlap
term to it.

31:43.583 --> 31:44.323
Right?

31:44.318 --> 31:46.038
So it's actually not going to
be half;

31:46.038 --> 31:48.898
it'll have to be something a
little less than half,

31:48.895 --> 31:52.205
so that it'll sum up to one,
if we want to normalize it.

31:52.210 --> 31:52.750
Okay?

31:52.750 --> 31:55.640
So there's going to be those
less than halves there.

31:55.640 --> 31:56.780
Okay.

31:56.779 --> 31:59.079
Now what does that do?

31:59.078 --> 32:01.388
That shifts electron density,
right?

32:01.390 --> 32:05.480
We're taking electron density
away from where the nuclei are,

32:05.479 --> 32:08.269
from A^2 and B^2,
and where's the electron

32:08.272 --> 32:09.502
density going?

32:09.500 --> 32:12.900
Because we're using less than
half to begin with,

32:12.898 --> 32:15.518
we're taking electron density
away.

32:15.519 --> 32:18.819
There's going to be -- we're
subtracting more than was there

32:18.817 --> 32:19.767
at the beginning.

32:19.767 --> 32:20.267
Right?

32:20.269 --> 32:23.079
Which means that we're going to
have negative electron density

32:23.076 --> 32:24.176
in the difference map.

32:24.180 --> 32:26.090
Electrons are going away from
the atoms.

32:26.089 --> 32:28.369
Where are they going to?

32:28.369 --> 32:29.719
Student:  Into it.

32:29.720 --> 32:31.950
Prof: Into the region
where there's overlap.

32:31.952 --> 32:32.312
Right?

32:32.308 --> 32:36.448
So they go away from there,
into the overlap region.

32:36.450 --> 32:39.870
So this is just like what we
were seeing with X-ray.

32:39.868 --> 32:43.028
Okay, so that overlap,
the A times B term,

32:43.032 --> 32:45.042
is what creates bonding.

32:45.039 --> 32:46.769
And we've seen this before.

32:46.769 --> 32:50.489
Remember when you have wells
far apart, the wave function is

32:50.493 --> 32:53.793
the sum of the two;
we saw this in one-dimension.

32:53.785 --> 32:54.255
Right?

32:54.259 --> 32:56.169
But if they come close
together, you get a wave

32:56.174 --> 32:58.744
function that looks like that,
which we looked before,

32:58.743 --> 33:01.033
just from the point of view of
the energy,

33:01.028 --> 33:03.528
and saw that that would
stabilize the particle,

33:03.528 --> 33:05.918
because it's got less
curvature, less kinetic energy.

33:05.922 --> 33:06.292
Right?

33:06.288 --> 33:10.738
But also the electron density
grows in the middle.

33:10.741 --> 33:11.471
Right?

33:11.470 --> 33:15.820
So from the point of view of
the electron distribution,

33:15.816 --> 33:19.596
that was the glue holding the
atoms together.

33:19.598 --> 33:22.298
So it's held together,
both because the energy goes

33:22.298 --> 33:24.888
down and because you put this
glue in the middle,

33:24.892 --> 33:27.432
which is what causes the energy
to go down.

33:27.430 --> 33:28.150
Okay?

33:28.150 --> 33:29.460
So that's bonding.

33:29.460 --> 33:31.350
And remember we also had this.

33:31.348 --> 33:35.218
So as the energy went up in the
middle one, the energy is lower

33:35.223 --> 33:37.603
here than it was in the atoms
apart.

33:37.598 --> 33:41.868
So the nuclei push one another
apart now, without the glue in

33:41.865 --> 33:44.705
the middle, and that was
anti-bonding.

33:44.710 --> 33:46.570
So we've seen it before in
one-dimension,

33:46.567 --> 33:48.607
but it's true in
three-dimensions as well.

33:48.608 --> 33:50.178
Now let's think about this
again.

33:50.180 --> 33:53.780
So here's atom A.

33:53.779 --> 33:58.679
Now where is the square of that
function significant?

33:58.680 --> 34:00.950
Is it significant there?

34:00.950 --> 34:01.650
No.

34:01.650 --> 34:04.990
Is it significant there?

34:04.990 --> 34:06.850
Yes, it's a little bit
significant at least.

34:06.849 --> 34:09.889
How about there?

34:09.889 --> 34:12.389
A little bit. Right?

34:12.389 --> 34:13.479
Okay.

34:13.480 --> 34:17.430
Now suppose we have another
atomic orbital there.

34:17.429 --> 34:23.609
Now, where is the product
significant, of A times B?

34:23.610 --> 34:24.880
Okay?

34:24.880 --> 34:28.150
So is the product A times B
significant there?

34:28.150 --> 34:30.320
No.

34:30.320 --> 34:39.460
Is it significant there?

34:39.460 --> 34:40.990
Nick, what do you say?

34:40.989 --> 34:41.829
Student:  No.

34:41.829 --> 34:44.029
Prof: Why not?

34:44.030 --> 34:44.820
Speak up.

34:44.820 --> 34:46.100
Student:  It's very
small near A.

34:46.099 --> 34:47.129
Prof: What's very small?

34:47.130 --> 34:49.110
Student:  The value of A.

34:49.110 --> 34:50.020
Prof: No,
no.

34:50.018 --> 34:52.318
The value of A there is
something -- we said that

34:52.315 --> 34:55.085
before, when we were looking
only at A -- it's not very big

34:55.090 --> 34:56.910
but there's a significant value.

34:56.909 --> 34:58.629
But how about the product?

34:58.630 --> 34:59.120
Josh?

34:59.119 --> 35:00.229
Student:  The value of B
is very small.

35:00.230 --> 35:02.170
Prof: The value of B is
very small there.

35:02.170 --> 35:04.170
So the product is going to be
very small.

35:04.170 --> 35:06.720
How about there?

35:06.719 --> 35:08.129
Student: There's going
to be a change.

35:08.130 --> 35:09.970
Prof: Ah,
now they're equal;

35:09.969 --> 35:11.629
halfway between they're equal.

35:11.630 --> 35:14.980
So both of them are a little
bit small, but their product is

35:14.983 --> 35:16.013
still significant.

35:16.007 --> 35:16.517
Right?

35:16.518 --> 35:19.918
Only in this region,
where they overlap,

35:19.922 --> 35:22.542
is that product significant.

35:22.539 --> 35:23.189
Okay.

35:23.190 --> 35:25.780
So at the center,
notice that the number

35:25.784 --> 35:29.714
Ψ_A assigns and the
number Ψ_B assigns

35:29.711 --> 35:31.311
are the same number.

35:31.309 --> 35:35.529
So 2(Ψ_A Ψ_B) is
as large as

35:35.525 --> 35:40.135
(Ψ_A)^2+(Ψ_B)^2;
because Ψ_A times

35:40.143 --> 35:42.473
Ψ_B is the same as
(Ψ_A)^2.

35:42.469 --> 35:45.849
So the electron density is
nearly doubled in the middle

35:45.847 --> 35:49.537
from what it would've been if it
had just been two atoms.

35:49.539 --> 35:51.679
So that's the region of
significant overlap,

35:51.684 --> 35:53.284
and that's what we care about.

35:53.280 --> 35:55.920
So the overlap integral,
summing this produc -- or

35:55.922 --> 35:59.322
integrating it -- over all that
space, that's a certain density.

35:59.320 --> 35:59.860
Right?

35:59.860 --> 36:01.490
We squared in order to get
that.

36:01.489 --> 36:01.909
Right?

36:01.909 --> 36:03.919
That's part of the density.

36:03.920 --> 36:07.130
So we sum that,
the density that comes from

36:07.130 --> 36:09.270
that product,
over all space,

36:09.271 --> 36:12.561
and that's called the overlap
integral.

36:12.559 --> 36:15.989
If the atoms are very far
apart, the overlap integral is

36:15.992 --> 36:17.182
essentially zero.

36:17.179 --> 36:19.569
If the atoms are close
together, the overlap orbital

36:19.574 --> 36:23.454
will be finite,
and the better the -- the more

36:23.447 --> 36:27.757
the orbitals overlap,
the bigger the overlap

36:27.755 --> 36:29.645
integral, obviously.

36:29.650 --> 36:33.890
And that measures the net
change that arises on bonding,

36:33.889 --> 36:37.359
the difference density,
as we've just seen.

36:37.360 --> 36:42.320
Now let's look at some
theoretical examples here.

36:42.320 --> 36:47.240
So let's look at the total
electron density as calculated

36:47.240 --> 36:52.340
for two -- adding two 1s
orbitals of hydrogen at the

36:52.338 --> 36:55.238
appropriate distance for H_2.

36:55.239 --> 36:57.889
This was this was done forty
years ago and published in the

36:57.894 --> 36:59.594
Israel Journal of
Chemistry.

36:59.590 --> 37:02.240
Okay now, and here's -- so on
the left we have the total

37:02.235 --> 37:04.635
electron density that you'd
calculate from that.

37:04.639 --> 37:10.969
A very simple thing:
1/√2(A+B).

37:10.969 --> 37:13.939
And you square it,
and you get the density,

37:13.936 --> 37:16.826
and that's the density,
contoured at 0.025

37:16.833 --> 37:20.583
electrons/cubic a_o,
the unit of distance.

37:20.579 --> 37:24.259
Okay, now, on the right is the
difference density.

37:24.260 --> 37:28.130
So from that thing on the left
we've subtracted the atomic

37:28.132 --> 37:30.922
orbital, the atomic electron
densities.

37:30.920 --> 37:33.380
And you see exactly what you
expect.

37:33.380 --> 37:37.400
It builds up in the middle,
where there's overlap,

37:37.400 --> 37:40.190
and at the expense of the
atoms.

37:40.190 --> 37:40.850
Yes Russell?

37:40.849 --> 37:42.829
Student:  Shouldn't it
be H_2^+ ion?

37:42.829 --> 37:47.409
Prof: Oh,
I think it's the H_2_

37:47.411 --> 37:52.441
molecule,
I'm sure it's the H_2_.

37:52.440 --> 37:55.260
They aren't so
fantastically different,

37:55.262 --> 37:58.602
because the two electrons are
in the same orbital;

37:58.599 --> 38:00.219
two electrons can be in the
same orbital.

38:00.219 --> 38:02.059
So one'll be twice as big as
the other.

38:02.059 --> 38:04.259
Qualitatively they'll look very
similar.

38:04.260 --> 38:06.270
And I think this is the H_2
molecule;

38:06.268 --> 38:08.488
but it might be the ion,
I'm not sure.

38:08.489 --> 38:13.409
Okay, at any rate the contours
on the right are much smaller.

38:13.409 --> 38:16.649
Remember, difference density is
much smaller than total density.

38:16.650 --> 38:20.320
So what you see is that it's
contoured at 0.004.

38:20.320 --> 38:22.880
So you have one,
two, three, four,

38:22.878 --> 38:24.118
five contours.

38:24.119 --> 38:26.659
So you get up,
in the middle,

38:26.655 --> 38:30.275
to 0.02 electrons per cubic
angstrom .

38:30.280 --> 38:33.960
That's how much bonding has
changed things,

38:33.960 --> 38:35.450
at the maximum.

38:35.449 --> 38:38.979
Okay, now the energy that's
calculated,

38:38.980 --> 38:42.190
with this very,
very, very crude wave function,

38:42.190 --> 38:45.620
just one half --
1/√2 times the sum

38:45.621 --> 38:47.741
of the two atomic orbitals.

38:47.739 --> 38:54.019
The energy, you calculate that,
is 92.9% of the true energy.

38:54.019 --> 38:57.079
That's pretty darn good, right?

38:57.079 --> 39:01.009
But almost all of that energy
that you calculate was already

39:01.014 --> 39:03.154
present in the separate atoms.

39:03.150 --> 39:05.530
We're not interested in the
energy of the separate atoms,

39:05.530 --> 39:08.900
we're interested in how much it
changes when you make a bond,

39:08.900 --> 39:12.050
which is a small difference
between large numbers.

39:12.050 --> 39:16.020
So it turns out that although
we're within 7% of the true

39:16.023 --> 39:18.663
total energy,
this simple model only

39:18.664 --> 39:22.634
calculates about 50% of the
change in energy that came from

39:22.632 --> 39:24.962
putting them together -- right?

39:24.960 --> 39:28.330
-- which is much smaller.

39:28.329 --> 39:32.199
Okay, so high accuracy is
required to calculate a correct

39:32.202 --> 39:34.072
value of the bond energy.

39:34.070 --> 39:35.510
This simple thing won't do it.

39:35.510 --> 39:38.840
Well it's in the right
direction, you're halfway there,

39:38.838 --> 39:40.568
so it's a pretty good start.

39:40.565 --> 39:41.115
Right?

39:41.119 --> 39:44.719
But to do the difference,
as in the same way you needed

39:44.721 --> 39:47.791
high precision to do X-ray
difference maps,

39:47.789 --> 39:49.609
you need better orbitals than
this,

39:49.610 --> 39:51.730
if you want to calculate good
bond energies.

39:51.730 --> 39:54.830
So you need to make the
orbitals better.

39:54.829 --> 39:55.329
Okay?

39:55.329 --> 39:59.719
Now -- so but already we can
take heart that the very crudest

39:59.724 --> 40:03.894
model shows most,
52%, of the energy of the bond,

40:03.894 --> 40:08.434
and it shows the electron
density building up by 0.02

40:08.434 --> 40:11.494
electrons per cubic bohr radius.

40:11.489 --> 40:15.999
And what we saw qualitatively
was there was a shift from the

40:16.001 --> 40:19.061
atom to the bond,
of electron density.

40:19.059 --> 40:22.909
Okay, now we can adjust the
molecular orbital to get a

40:22.909 --> 40:25.889
better approximation of the true
thing.

40:25.889 --> 40:28.669
How will we know when we've
adjusted it and it's gotten

40:28.666 --> 40:29.126
better?

40:29.130 --> 40:32.870
If we adjust it and get a
lower, calculate a lower average

40:32.867 --> 40:35.107
energy --
I should've said a lower

40:35.106 --> 40:37.966
average energy,
because if we don't have a true

40:37.965 --> 40:41.415
wave function we get different
values for the total energy;

40:41.420 --> 40:44.570
the kinetic won't exactly
offset the potential as you move

40:44.565 --> 40:45.775
from place to place.

40:45.780 --> 40:48.380
But if you get the lowest
average energy,

40:48.380 --> 40:51.750
then that is,
by definition almost,

40:51.750 --> 40:55.860
more realistic,
because you can easily prove

40:55.856 --> 41:00.916
that the true energy is the
lowest possible energy;

41:00.920 --> 41:02.860
that makes a certain amount of
sense.

41:02.860 --> 41:06.640
The lowest possible calculated
energy is the true energy.

41:06.639 --> 41:10.699
So if you change your wave
function and get a lower average

41:10.699 --> 41:13.289
energy, you're closer to the
truth.

41:13.289 --> 41:14.019
Okay?

41:14.018 --> 41:17.428
That's called the Variational
Principle.

41:17.429 --> 41:22.409
Okay, so here we've changed the
form of the molecular orbital.

41:22.409 --> 41:23.729
And how did we change it?

41:23.730 --> 41:26.290
What does a 1s orbital
look like?

41:26.289 --> 41:30.959
Kate, do you remember the form
for a 1s atomic orbital?

41:30.960 --> 41:31.540
I can't hear.

41:31.539 --> 41:32.609
Student:  A sphere.

41:32.610 --> 41:34.670
Prof: Angularly it's a
sphere.

41:34.670 --> 41:37.740
How does it change as you go
out, do you remember?

41:37.739 --> 41:41.259
How does it depend on
ρ?

41:41.260 --> 41:44.590
The same way they all depend on
ρ.

41:44.590 --> 41:47.120
Anybody remember?

41:47.119 --> 41:49.019
Prof: e^-ρ.

41:49.018 --> 41:52.158
So it falls off exponentially
at a certain rate.

41:52.159 --> 41:54.199
And that rate,
how fast it falls off,

41:54.197 --> 41:56.347
is determined by the nuclear
charge.

41:56.349 --> 41:56.869
Okay?

41:56.869 --> 42:01.509
Now one way to change that
shape would be change how fast

42:01.507 --> 42:02.577
it falls off.

42:02.583 --> 42:03.333
Right?

42:03.329 --> 42:05.449
It wouldn't be correct for the
atom anymore.

42:05.449 --> 42:07.939
We've already got the correct
solution for the atom.

42:07.940 --> 42:11.420
But we could change the shape
of the thing by changing how

42:11.423 --> 42:13.323
fast that exponent falls off.

42:13.320 --> 42:17.390
And we could vary that,
in the molecule,

42:17.391 --> 42:21.881
using one half --
1/√2(A+B).

42:21.880 --> 42:24.710
But those A+B are no
longer true atomic functions,

42:24.706 --> 42:26.976
they're a little fatter,
a little skinnier.

42:26.980 --> 42:27.530
Okay?

42:27.530 --> 42:30.260
And we can change how fat or
skinny it is,

42:30.257 --> 42:33.117
until we get the lowest
molecular energy.

42:33.119 --> 42:36.989
See, that's a way you can vary
it and find the best value.

42:36.989 --> 42:40.209
And that's what was done here
to optimize the exponent.

42:40.210 --> 42:42.680
And now you get a total
electron density that looks

42:42.681 --> 42:44.661
essentially the same as it did
before.

42:44.659 --> 42:48.009
And if you look at the
difference density,

42:48.007 --> 42:51.027
how has it changed,
if you do this?

42:51.030 --> 42:54.140
First, notice that the energy
got lower.

42:54.139 --> 42:58.519
We're now to 73% of the
lowering of the bond energy.

42:58.518 --> 43:00.818
So the total energy's gotten
lower, it's better.

43:00.820 --> 43:04.320
And how has the electron
density changed?

43:04.320 --> 43:06.040
It got higher in the middle.

43:06.039 --> 43:09.589
Because what we did was spread
the exponent out a little bit,

43:09.592 --> 43:11.962
so you had more overlap in the
middle.

43:11.960 --> 43:13.370
Okay?

43:13.369 --> 43:16.209
So this wouldn't have been good
for the single atom,

43:16.206 --> 43:18.816
to spread it out,
but it gives a better function

43:18.820 --> 43:19.990
for the molecule.

43:19.989 --> 43:21.679
And it's still very,
very simple.

43:21.679 --> 43:25.679
And what you see it did is it
increases the bonding density

43:25.677 --> 43:27.537
and the bonding strength.

43:27.539 --> 43:30.319
You get a larger shift from the
atoms to the bond.

43:30.320 --> 43:35.850
Now, how else could you change
the shape of the atomic orbital

43:35.847 --> 43:39.017
in order to increase the
overlap;

43:39.018 --> 43:42.128
some way other than making a
single exponential and having it

43:42.126 --> 43:43.366
get fatter or thinner?

43:43.369 --> 43:44.699
Can you think of some other way?

43:44.699 --> 43:47.429
Here you have an atomic
orbital, a sphere,

43:47.429 --> 43:51.289
and you want to change its
shape so that it overlaps better

43:51.291 --> 43:51.961
over here.

43:51.956 --> 43:52.686
Right?

43:52.690 --> 43:56.830
How could you change the shape
of an atomic orbital,

43:56.833 --> 44:00.003
without doing really gross
damage to it,

44:00.001 --> 44:04.471
making it a cube or something
like that, or a line?

44:04.469 --> 44:08.049
How could you change it so it
looked pretty much still like an

44:08.052 --> 44:10.762
atom did, has a lot of the
virtues of the atom,

44:10.755 --> 44:12.455
but is shifted over here?

44:12.460 --> 44:17.950
Sam?

44:17.949 --> 44:18.609
I can't hear.

44:18.610 --> 44:20.730
Student:  Can't you just
allow the electrons to shift?

44:20.730 --> 44:22.850
Prof: Yes,
how am I going to write a

44:22.851 --> 44:25.321
function that allows the
electrons to shift in the

44:25.324 --> 44:26.794
direction I want them to?

44:26.789 --> 44:27.749
Lexy?

44:27.750 --> 44:29.070
Student:  You could
hybridize it.

44:29.070 --> 44:30.540
Prof: Hybridize it!

44:30.539 --> 44:32.419
We could hybridize to shift the
electrons.

44:32.420 --> 44:33.780
So that's the next one.

44:33.780 --> 44:37.500
So here we're going to --
instead of this we're going to

44:37.503 --> 44:38.523
hybridize it.

44:38.518 --> 44:41.038
Now, this particular
calculation did the

44:41.041 --> 44:44.661
hybridization and also did a
little self-consistent field

44:44.664 --> 44:45.704
calculation.

44:45.699 --> 44:50.719
And the hybridization left it
96.7% 1s.

44:50.719 --> 44:53.939
So essentially it's still a
normal 1s orbital.

44:53.940 --> 44:57.480
But they added .6% of
2s, which expanded it a

44:57.480 --> 45:01.370
little bit, because 2s is
-- goes further out than

45:01.367 --> 45:02.337
1s.

45:02.340 --> 45:06.350
And they added 2.7% of
2p.

45:06.349 --> 45:10.549
Now why was 2p much more
helpful than 2s?

45:10.550 --> 45:12.730
Lucas?

45:12.730 --> 45:14.620
Student:  It has those
lobes.

45:14.619 --> 45:16.999
Prof: Yes,
so it takes density from one

45:17.001 --> 45:18.801
side and shifts it to the other.

45:18.800 --> 45:21.570
In fact, this is precisely what
we saw before.

45:21.570 --> 45:29.190
Notice what it did was -- how
much it increased the density in

45:29.186 --> 45:30.806
the middle.

45:30.809 --> 45:33.429
And notice now,
it's not taking electron

45:33.429 --> 45:37.129
density away from the nuclei,
which was a good place for

45:37.126 --> 45:38.466
electrons to be.

45:38.469 --> 45:40.349
Where does it take it away from?

45:40.349 --> 45:41.759
Student: The left one.

45:41.760 --> 45:44.560
Prof: Out beyond the
nuclei, and shift it to the

45:44.561 --> 45:45.031
middle.

45:45.030 --> 45:47.420
So even before,
when it was an atom --

45:47.420 --> 45:50.680
here's the nucleus,
a certain distance out here,

45:50.679 --> 45:54.029
and a certain distance out
here, were the same in energy,

45:54.030 --> 45:55.360
for that atom.

45:55.360 --> 45:59.630
Now we've taken it from a place
which is -- out here,

45:59.634 --> 46:00.954
and put it here.

46:00.949 --> 46:01.689
Right?

46:01.690 --> 46:03.640
Great idea.

46:03.639 --> 46:05.839
Well done Lexy.

46:05.840 --> 46:06.780
So there's what it looked like.

46:06.780 --> 46:09.170
Remember, if it's 100 percent
at 1s,

46:09.170 --> 46:11.680
it looks like that,
and if you change it to be

46:11.684 --> 46:14.834
hybridized that way,
with Atom-in-a-Box,

46:14.827 --> 46:16.687
it looks like this.

46:16.686 --> 46:17.466
Right?

46:17.469 --> 46:20.759
It's not much shift,
but it shifts from the left to

46:20.759 --> 46:23.259
the right, and gives better
overlap.

46:23.260 --> 46:29.230
Okay, but it requires the atom
to be a little bit less happy as

46:29.233 --> 46:34.923
an atom, because it's partly
2s and 2p now.

46:34.920 --> 46:38.190
The electrons are further from
the nucleus, but you make up for

46:38.186 --> 46:39.816
that by having a better bond.

46:39.820 --> 46:43.430
Okay, now notice that didn't
change the energy very much.

46:43.429 --> 46:46.819
It went from 73% to 76%. Right?

46:46.820 --> 46:49.880
We were already about the right
energy, but it changed the

46:49.882 --> 46:52.302
density a lot,
to be what we want it to be.

46:52.300 --> 46:57.240
Okay, so there's a much bigger
shift, and it's now from beyond

46:57.242 --> 46:59.352
the nucleus into the bond.

46:59.349 --> 47:00.079
Right?

47:00.079 --> 47:04.269
And now we're going to do the
last thing.

47:04.268 --> 47:07.008
We've done SCF already,
but now you have to do a higher

47:07.005 --> 47:09.485
level calculation that'll do
correlation, take the

47:09.489 --> 47:11.819
correlation of the electrons
into account.

47:11.820 --> 47:16.640
And if you add some correlation
calculation to this,

47:16.641 --> 47:19.481
you now get 90% of the bond.

47:19.480 --> 47:21.980
If you did complete
correlation, you'd get 100% of

47:21.978 --> 47:22.538
the bond.

47:22.539 --> 47:27.449
What does it mean to do a
calculation with complete

47:27.445 --> 47:28.815
correlation?

47:28.820 --> 47:30.900
It means you don't have an
error anymore,

47:30.900 --> 47:32.980
you've used a really good
calculation.

47:32.980 --> 47:36.580
So already at this level,
that was done fifty years ago

47:36.581 --> 47:38.851
almost, you get 90% of the bond.

47:38.849 --> 47:42.139
And the bond density notice,
what about the electron

47:42.143 --> 47:42.793
density?

47:42.789 --> 47:44.129
Student:  No change.

47:44.130 --> 47:45.570
Prof: It hasn't really
changed.

47:45.570 --> 47:48.760
All that happened was that the
electrons kept apart from one

47:48.757 --> 47:51.347
another, but the average density
was the same.

47:51.349 --> 47:53.849
So already, with just that
hybridization,

47:53.849 --> 47:58.149
we got very close to the truth
in electron distribution,

47:58.150 --> 48:01.660
and three-quarters of the way
to the truth in how strong a

48:01.663 --> 48:02.283
bond is.

48:02.280 --> 48:04.290
So not a bad approximation.

48:04.289 --> 48:06.179
So hybridization,
to give better overlap,

48:06.181 --> 48:07.081
is a great thing.

48:07.079 --> 48:10.639
So the density wasn't changed,
but you got a much better

48:10.637 --> 48:11.217
energy.

48:11.219 --> 48:13.619
And how so?

48:13.619 --> 48:16.809
Because the electrons kept
apart from one another,

48:16.811 --> 48:18.701
when you allow correlation.

48:18.699 --> 48:23.209
Okay, so here's a pairwise
atomic orbital:

48:23.210 --> 48:27.790
1/√2(A+B);
and you can use hybridized

48:27.788 --> 48:28.278
orbitals.

48:28.280 --> 48:30.860
And the virtues are it's very
easy to formulate and to

48:30.856 --> 48:31.486
understand.

48:31.489 --> 48:35.159
And it looks like atoms,
especially when you get down

48:35.159 --> 48:36.429
near the nuclei.

48:36.429 --> 48:39.519
And you don't want that to
change because that's the main

48:39.516 --> 48:40.726
event for electrons.

48:40.730 --> 48:43.190
You get much more energy
forming atoms,

48:43.193 --> 48:46.183
as we saw before,
than you do making new bonds,

48:46.175 --> 48:48.505
once you have the atoms
already.

48:48.510 --> 48:49.860
Okay.

48:49.860 --> 48:52.520
It builds up electron density
between the nuclei,

48:52.518 --> 48:55.288
through overlap,
which is the source of bonding.

48:55.289 --> 49:00.629
It smoothes Ψ,
to lower the kinetic energy.

49:00.630 --> 49:04.190
And then there's a pedantic
footnote here that actually

49:04.190 --> 49:07.160
there's a thing called the
Virial Theorem,

49:07.159 --> 49:09.229
which I'm not going to stress
you with,

49:09.230 --> 49:12.470
but that little bit there,
it happens to be true.

49:12.469 --> 49:16.299
But still we have the proper
understanding of what's going

49:16.302 --> 49:16.642
on.

49:16.639 --> 49:20.509
Hybridizing AOs provides
flexibility that gives you

49:20.505 --> 49:21.815
better overlap.

49:21.820 --> 49:25.040
And if you use all the H-like
Atomic Orbitals,

49:25.043 --> 49:28.993
you have perfect flexibility,
you can make any shape you

49:28.985 --> 49:29.625
want.

49:29.630 --> 49:32.150
Okay, but we're going to keep
it simple,

49:32.150 --> 49:35.930
use only 2s and
2p orbitals to hybridize,

49:35.929 --> 49:38.559
because that'll get you most of
the way there and it's much

49:38.561 --> 49:40.971
simpler,
rather than to try to mix

49:40.965 --> 49:43.475
5f orbitals into it also.

49:43.480 --> 49:45.110
Okay, so that's great.

49:45.110 --> 49:46.840
And when we square it,
we get this,

49:46.835 --> 49:49.215
which has the overlap part that
helps us out.

49:49.219 --> 49:51.639
We have the atoms,
plus the bond,

49:51.641 --> 49:54.821
which is the overlap,
that product part.

49:54.820 --> 49:59.040
But we could've done the same
thing to get the same product,

49:59.039 --> 50:02.229
as far as the atoms go,
if we'd used a minus sign

50:02.233 --> 50:05.233
instead of a plus sign in
combining things;

50:05.230 --> 50:08.550
although then we would've
changed the sign of the overlap

50:08.545 --> 50:09.015
thing.

50:09.019 --> 50:10.429
It would become minus.

50:10.429 --> 50:13.979
So we would change it from
being less than to being greater

50:13.981 --> 50:17.231
than, in order to have it be
normalized at the bottom.

50:17.226 --> 50:17.836
Right?

50:17.840 --> 50:19.320
And that's the anti-bond.

50:19.320 --> 50:24.030
So we get both the bond and the
anti-bond by doing this.

50:24.030 --> 50:26.920
And now we're going to go on,
next time, to overlap,

50:26.922 --> 50:29.992
which we've already introduced,
and also the concept of

50:29.985 --> 50:30.945
energy-match.

50:30.949 --> 50:33.429
And when you put these two
together, you'll really

50:33.434 --> 50:34.504
understand bonding.

50:34.500 --> 50:40.000