WEBVTT

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Prof: Okay,
let's get started.

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The stuff we're doing these few
days are really the focus of the

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first half of the semester.

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So pay attention and think
about it.

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Okay, so the topics today are
Overlap and Energy-Match,

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as determinants of how strong
bonds are.

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So we'll start with overlap.

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We saw last time that overlap
is what creates the difference

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density,
what focuses some small amount

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of the electron density between
the atoms and serves to hold

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them together.

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The overlap integral is the
total amount of that product of

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A and B.

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Remember, you square
(A+B) and you get A^2

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+ B^2 + 2AB.

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That 2AB term is the
overlap.

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It's called overlap because it
has values when both functions

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have appreciable magnitude,
at the same point in space;

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they overlap.

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And summing that over all
space, or integrating it,

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gives what's called the overlap
integral.

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It'll depend on the distance.

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Obviously if they're very far
apart,

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they don't give simultaneous
values to the same point in

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space,
because it's far from either

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one or the other,
or from both.

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So it depends on distance.

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It also depends on
hybridization,

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and that's what we're going to
talk about first.

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Okay, first let's just look for
scale.

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Here are the 2s orbitals
of carbon atom,

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and you can work out,
with your formulas,

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that the diameter of the
spherical node is 0.7 angstroms.

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Right?

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That means we have a distance
scale here.

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Now, the distance between two
carbons is roughly 1.4,1.5,

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when they're forming a bond
from one another -- between one

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another.

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Okay.

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So that means we can use that
0.7 angstrom scale to see how

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far we have to slide these
together, in order to get them

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to bonded distance.

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If we superimposed the X's --
remember those diameters are

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each 0.7 --
so if we slide them together

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until we superimpose the X's,
then they're 1.4 angstroms

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apart.

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So that's about the distance
that these two orbitals are,

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in the proper scale,
when they're in a carbon-carbon

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bond.

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Now, how big is the overlap
integral, the sum of the product

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of A times B,
over all space?

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Now I want you to guess,
guess how big that integral is.

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And we have a hint.

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What would the overlap integral
be if we moved them until they

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superimposed on one another
exactly?

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That's the maximum we could
get, if the distance were zero.

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So if pulled them all together,
then A and B would be measured

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from the same center.

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So it would be the same thing
as A^2.

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Okay?

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So what would the integral of
A^2^( )be, over all space?

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Student:  One.

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Prof: One;
that's normalization.

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Okay, so the maximum value you
can get when they're right on

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top of one another -- negatives
multiply negatives to get

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positive;
positive areas,

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multiply positive areas to get
positive.

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Everything is optimum,
is going to be one.

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Now I want somebody to give me
an idea, a guess,

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of how big it's going to be
here, or at least some

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considerations that would go
into that.

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The maximum value you can get
is one.

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How big do you think it would
be here?

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What's something that would
hurt it;

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that is, make it less than one?

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Russell?

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Student:  When there's
very little overlap on the

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opposite sides.

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Prof: Yeah,
out beyond the red areas,

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there's very little overlap
altogether.

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So you lose whatever overlap
you'd get from there,

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in doing the normalization.

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Now, in the middle,
right in the middle,

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you have blue on top of blue.

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That's good.

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But notice there's also a node
that's going to interfere with

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some of the wave functions'
overlapping,

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and there's going to be a
little bit of blue on top of

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red.

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Everybody got that?

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So I just want people to guess.

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Okay, I know what I'll do.

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We'll do a poll.

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This is a democracy after all;
we're about to get into

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November.

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So I'm going to start with 0.1,
as what you think it would be.

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Right?

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The maximum,
if you put them right on top of

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one another, is one.

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So I'm going to start with 0.1,
and move up and see hands.

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Okay, how many people think
it's 0.1?

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0.2?

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0.3?

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0.4?

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0.5?

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0.6?

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7,8, 9, one?

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Obviously not one.

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Now, I think,
my sense of this is it peaked

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between 0.3 and 0.4,
but there were some votes for

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0.2, and I would've voted for
0.2,

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because it looks like you've
lost an awful lot here.

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But in fact I personally am
surprised that it's pretty big.

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It's 0.41.

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It's almost half as big as it
would be if they were right on

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top of one another,
despite the fact that you're

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losing great parts of it,
that they don't overlap at all,

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and there's negative overlap
where red's on top of blue.

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Still the fact is it's 0.41;
so pretty big.

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Now that's going to depend on
distance obviously.

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Right?

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So here's how it depends on
distance.

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There's that point;
at 1.4 angstroms it's 0.41.

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And I've put on here for
reference the distance of a

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carbon-carbon single bond,
the distance of a carbon-carbon

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double bond,
and a carbon-carbon triple bond.

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And zero, notice,
would be way down,

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out in the hallway someplace.

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Right?

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We're only looking at the
region that's carbon-carbon

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bonds -- or maybe just about to
the far wall.

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Okay?

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Here's half an angstrom;
we got to go two more halves.

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Right?

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Okay, so here's how -- how do
you think -- what should the

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line look like?

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Should it be sine wave?

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Should it be increasing as we
go to the left?

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Should it be decreasing as we
go to the left?

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What do you think?

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Some guess.

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<<Students speak over one
another>>

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Prof: Should probably
decrease.

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Because we know it's going to
approach one.

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Right?

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And there it is.

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Yes, it decreases -- pardon me,
it increases as you go to the

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left, and ultimately it'll go up
to one, when you get out to the

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hallway.

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Okay?

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So that's s with
s.

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Now let's look at some other
orbitals that might overlap,

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and see how big their overlap
integrals would be.

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Okay, suppose you have s
on one, overlapping with

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p on the other.

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Now that p is called,
for this purpose,

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π.

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p is the Greek version of
p;

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in fact, in Greek you pronounce
it 'pee,' not 'pie'.

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Right?

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So that's the Greek letter
p.

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Am I right about that?

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Student:  It depends on
classical or modern Greek.

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Prof: Okay,
which way is it?

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Student:  It'd be 'pee'
in modern Greek;

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it'd be 'pie' in classical.

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Prof: Uh-huh, okay, good.

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So anyhow, ‘pie'.

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>

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Prof: Now,
so that symbol is used to talk

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about molecular orbitals,
or parts of molecular orbitals;

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where p is used to talk
about atomic orbitals.

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They correspond to one another.

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Now what is it that
characterizes a p atomic

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orbital?

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Has a nodal plane.

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And in the same way a
π orbital has a nodal

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plane that contains the nuclei
that are in question.

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Right?

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So you see, if those two came
together there'd be a nodal

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plane in the orbital on the
right that contains the two

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nuclei.

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Okay?

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So that's called a π
orbital.

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Okay?

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Now, but notice the
contributions to the overlap.

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Below that plane,
you're going to have blue

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overlapping with blue.

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So that's going to be a
positive contribution.

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But above, you're going to have
blue overlapping with red;

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that's going to be a negative
contribution,

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and it's symmetrical.

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So what will the total be?

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What will the total overlap
integral be;

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the sum of all the overlaps?

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Student:  One.

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Prof: One?

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Prof: It'll be zero,
because for every contribution

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on the bottom that's positive,
there'll be one on the top

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that's the same value and is
negative.

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So it's going to be zero.

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So the one on the left is
called σ.

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Now σ is the Greek
equivalence of 's.'

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Right?

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So it means,
for a molecular orbital,

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what s meant for an
atomic orbital;

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there are no nodes that contain
the nucleus.

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Okay?

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Okay, so if you have
σ overlapping with

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a π,
there's going to be this

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symmetry that causes a
cancellation.

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So you can never get any
overlap, by overlapping a

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σ orbital with a
π orbital.

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Does that make sense to
everyone?

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It does make sense,
doesn't make sense?

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If you have one that doesn't
have a horizontal node,

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that's the same top and bottom,
and one that does have a node

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that's plus on the top,
minus on the bottom,

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then you're going to get this
cancellation everywhere,

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and zero.

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And that's called
‘orthogonal'.

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Okay, so you won't get any net
interaction there.

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Okay, but if you turn the
p orbital so that it is

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also σ --
notice it has a plane,

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a nodal plane,
that contains that nucleus,

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but it doesn't contain the
other nucleus you're interested

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in.

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So to call σ and
π, we need a plane

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that contains both
nuclei.

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That'll create this symmetry
that cancels things out.

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Okay, so this one will give
overlap.

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Now what would you expect for
the trend of this one,

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as you go from great distance
to small distance?

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If they're very,
very far apart,

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how big will the overlap
integral be;

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like ten meters apart?

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Student:  Zero.

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Prof: Zero. Okay?

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Now, as you bring it together,
as they start coming together,

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will you get positive or
negative overlap?

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Students:  Positive.

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Prof: Positive;
right, the blue will overlap

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with the blue.

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And you keep coming,
it'll get bigger and bigger.

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And then what will happen?

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Prof: Then the red is
going to start coming in and --

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well the blue is going to
start, the blue on the right,

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from the p orbital,
will start --

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and going to overlap that red
part on the left.

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And then, as it slides on over,
till they get exactly on top of

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one another -- when they get
exactly on top of one another,

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what will it be?

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Students:  Zero.

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Prof: Zero,
because it'll be canceling

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right to left.

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Okay, so here's a plot of that,
for s-p

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σ_ overlap.

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Right?

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It's almost the same as
s with s.

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In fact, it's a little bigger
at great distance,

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a little smaller at small
distance;

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except, at very small distance
what will happen?

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<<Students speak over one
another>>

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Prof: When you approach
zero?

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It'll go to zero.

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Whereas the other one went to
one.

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Right?

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The s with s went
to one.

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So, but in the region we're
interested in,

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the carbon-carbon bond
distances, it's about the same.

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Okay, now how about
p_σ with

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p_σ?

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Well we won't go through an
elaborate guessing game.

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If they're very far apart it'll
be zero.

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How about if they were on top
of one another?

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It's obviously going to grow as
blue gets on top of blue,

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and then as they get really
close, the blue up from the

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right will be on top of the red,
and the red on top of the blue.

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Right?

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So what will it approach?

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Zack, what do you say?

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Student:  Zero.

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Prof: Approach zero?

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What will it be when they're
right on top of one another?

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So red on blue here;
blue on red here.

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Will the overlap be zero?

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Student:  Negative one.

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Prof: Pardon me?

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Student:  Negative one.

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Prof: It'll be minus
one,

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because it would be like the
orbital on top of itself --

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that would be one -- except
you've changed the sign,

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of one of them. Right?

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So every contribution will be
negative.

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So that one starts there,
and it's heading toward minus

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one.

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So it's going down.

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Okay?

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So that's p_σ with
p_σ.

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Can you think of any other way
we should talk about it?

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We've talked about s
with s.

12:48.690 --> 12:51.610
We saw that s with
p_π_ was

12:51.613 --> 12:53.533
orthogonal;
s with

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p_σ_
we've got there;

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p_σ with
p_σ.

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Any others we need to think
about?

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Student:
p_σ with

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p_π?

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Prof: How about
p_σ_ with

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p_π?

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Can anybody guess that one?

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P_σ_ is
the same sign,

13:16.515 --> 13:19.415
top and bottom;
p_π_ is

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opposite signs,
top and bottom.

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Student:  So it's zero.

13:22.350 --> 13:23.300
Prof: That's going to be
zero.

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There'll be orthogonal.

13:24.668 --> 13:26.598
So have we done it now,
or is there anything else to

13:26.600 --> 13:27.130
think about?

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Student:  π,
π.

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Prof: We could do
p_π with

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p_π.

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Okay, so there's p_π
with p_π.

13:33.350 --> 13:35.380
Okay?

13:35.379 --> 13:37.699
Now at great distance they'll
be nothing.

13:37.700 --> 13:39.620
We'll bring them together,
they'll begin to overlap.

13:39.620 --> 13:41.760
What will it approach as we go
to zero?

13:41.759 --> 13:42.589
Student:  Zero.

13:42.590 --> 13:43.860
Prof: Zero?

13:43.860 --> 13:45.950
If they're on top of one
another, the product will be

13:45.947 --> 13:46.227
zero?

13:46.230 --> 13:47.400
<<Students speak over one
another>>

13:47.399 --> 13:48.289
Prof: It'll be one,
right?

13:48.288 --> 13:51.568
It'll be the orbital with
itself, normalized.

13:51.570 --> 13:54.150
So that'll be one. Right?

13:54.149 --> 13:56.599
But now which do you think is
going to be bigger,

13:56.600 --> 13:58.070
p_σ_
with

13:58.070 --> 14:00.390
p_σ,_ or
p_π with

14:00.394 --> 14:01.254
p_π?

14:01.250 --> 14:02.860
Any guesses?

14:02.860 --> 14:05.400
Well, I'll give you the answer.

14:05.397 --> 14:05.967
Right?

14:05.970 --> 14:08.500
p_π with
p_π is approaching

14:08.500 --> 14:08.830
one.

14:08.830 --> 14:10.860
At big distance,
it's smaller than

14:10.861 --> 14:11.781
σ.

14:11.784 --> 14:12.344
Right?

14:12.340 --> 14:15.140
But at small distance,
it must get much bigger,

14:15.144 --> 14:17.284
and in this region it's
crossing;

14:17.279 --> 14:18.589
so it's bigger.

14:18.590 --> 14:25.490
Okay, so there are the kinds of
overlaps of simple 2p

14:25.493 --> 14:26.783
orbitals.

14:26.779 --> 14:28.889
Now, there's a curiosity here.

14:28.889 --> 14:32.319
Over most of this range,
more than half of the range,

14:32.320 --> 14:34.910
2s overlaps with
2p_σ_

14:34.914 --> 14:38.264
better than either
2s with 2s or

14:38.259 --> 14:41.489
2p_σ_
with 2p_σ.

14:41.490 --> 14:43.140
That just seems curious to me.

14:43.139 --> 14:46.369
You'd think that one of them
would have a shape that gives

14:46.370 --> 14:47.220
better overlap.

14:47.220 --> 14:47.730
Right?

14:47.730 --> 14:50.700
So two of those together would
do the best job.

14:50.700 --> 14:52.380
But it's not that way.

14:52.379 --> 14:56.079
It's that an s with a
p is the best,

14:56.083 --> 14:58.053
over most of this range.

14:58.048 --> 15:00.918
I just think that's curious,
and it has an important

15:00.918 --> 15:03.278
implication, which you'll see
right now.

15:03.278 --> 15:06.938
What if we use hybrid orbitals,
that are partly s and

15:06.942 --> 15:08.062
partly p?

15:08.058 --> 15:11.238
Now, since this one is partly
s, it's s plus a

15:11.236 --> 15:12.796
certain amount of p.

15:12.798 --> 15:14.828
This one's s with a
certain amount of p.

15:14.830 --> 15:17.780
The overlap will be s
here with s here;

15:17.778 --> 15:20.728
s here with p
here;

15:20.730 --> 15:22.890
s here with p
here;

15:22.889 --> 15:24.429
and p here with p
here.

15:24.427 --> 15:24.767
Right?

15:24.769 --> 15:28.439
There'll be four contributions:
s with s;

15:28.440 --> 15:30.620
p with p;
s with p;

15:30.621 --> 15:31.711
p with s.

15:31.712 --> 15:32.142
Right?

15:32.139 --> 15:36.399
So if we took sp^3,
for example -- and we can make,

15:36.395 --> 15:40.275
remember, four such orbitals --
then you get that.

15:40.279 --> 15:44.169
It's better than any of the
pure hybrids.

15:44.171 --> 15:44.951
Right?

15:44.950 --> 15:48.830
And the reason is that it has
s with s,

15:48.832 --> 15:52.982
and p with p;
and also s with

15:52.982 --> 15:56.862
p, that comes in as a
bonus, twice.

15:56.860 --> 16:00.630
Okay, now how about if we use,
instead of sp^3,

16:00.629 --> 16:03.049
how about if we use
sp^2?

16:03.048 --> 16:06.618
Remember what happens as we
approach sp?

16:06.620 --> 16:09.640
The orbitals extend more.

16:09.639 --> 16:10.919
So what do you expect?

16:10.919 --> 16:12.339
Student:  More overlap.

16:12.340 --> 16:15.530
Prof: More overlap.

16:15.528 --> 16:19.188
So there's sp^2 with
sp^2, a little better.

16:19.190 --> 16:21.050
How about sp with
sp?

16:21.049 --> 16:25.069
Better still.

16:25.070 --> 16:26.800
Could we get any better?

16:26.798 --> 16:30.578
How about s^2p with
s^2p, or

16:34.447 --> 16:35.367
Right?

16:35.370 --> 16:39.740
It's a little better at short
distance, and a little not so

16:39.740 --> 16:41.550
good at long distance.

16:41.548 --> 16:45.138
So it optimizes around
sp hybridization.

16:45.139 --> 16:46.379
That gives the best overlap.

16:46.379 --> 16:50.469
Why do we think now you want
good overlap?

16:50.470 --> 16:52.770
Why would it be good to have
good overlap?

16:52.769 --> 16:53.659
Student:  Keep the bonds
strong.

16:53.658 --> 16:55.828
Prof: Because that's
what's building up the

16:55.830 --> 16:57.560
difference density,
as we saw last time.

16:57.557 --> 16:57.997
Right?

16:58.000 --> 17:01.540
That's putting the glue in the
orbital.

17:01.538 --> 17:05.708
Okay, so there's an interesting
observation here,

17:05.712 --> 17:10.672
that hybrids overlap about
twice as well as pure orbitals.

17:10.670 --> 17:11.540
Right?

17:11.538 --> 17:17.108
All the hybrids are roughly
twice as good as the pure ones.

17:17.109 --> 17:18.059
Okay?

17:18.058 --> 17:20.718
And they're not very,
very different from one

17:20.715 --> 17:21.315
another.

17:21.318 --> 17:25.278
But sp is about the
best, over much of this range.

17:25.278 --> 17:28.628
The trouble with sp is
you can only make two of them,

17:28.630 --> 17:31.980
because each involves 50% of
s, and you only have one

17:31.980 --> 17:34.800
s to go in it;
you have one s and three

17:34.798 --> 17:35.338
p's.

17:35.338 --> 17:38.608
So if you want to make just one
orbital, fine,

17:38.606 --> 17:41.866
use sp,
because you'll get good -- only

17:41.874 --> 17:45.924
one bond from the carbon;
fine use sp and get the

17:45.915 --> 17:46.815
best overlap.

17:46.818 --> 17:50.588
Or maybe even s^2p,
depending on the distance.

17:50.588 --> 17:51.168
Right?

17:51.170 --> 17:54.340
But if you want to make more
bonds, more than two bonds,

17:54.336 --> 17:57.786
then you're going to have to
cut back on the s in each

17:57.790 --> 17:58.310
bond.

17:58.308 --> 18:02.908
Because you only have 100% to
use, you can't use more than 50%

18:02.914 --> 18:05.864
in more than two bonds you're
making.

18:05.859 --> 18:06.359
Okay.

18:06.358 --> 18:09.428
And what you see here is that
sp^3^( )isn't much worse

18:09.428 --> 18:10.398
than sp^2.

18:10.400 --> 18:12.910
So why not make four -- right?

18:12.906 --> 18:14.576
-- because you can.

18:14.578 --> 18:18.508
Okay, anyhow,
that's the hybridization of

18:18.506 --> 18:19.386
carbon.

18:19.390 --> 18:22.300
So, and the reason they do
this, as I already said,

18:22.298 --> 18:25.618
is because they allow nearly
full measure of s with

18:25.623 --> 18:28.383
p overlap,
plus s with s,

18:28.375 --> 18:31.875
and p with p,
when you have mixtures.

18:31.880 --> 18:35.850
So that all depends on the fact
that s-p_σ

18:35.853 --> 18:38.223
is so good in its overlap.

18:38.220 --> 18:41.410
Now, so sp gives the
best overlap,

18:41.414 --> 18:45.334
but only allows two orbitals,
with 50% in each;

18:45.328 --> 18:48.618
sp^3 gives four,
and nearly as much overlap.

18:48.619 --> 18:49.869
Okay?

18:49.868 --> 18:52.988
Now we're going to look at the
influence of overlap on

18:52.990 --> 18:54.580
molecular orbital energy.

18:54.578 --> 18:56.888
But we're going to use Erwin
Meets Goldilocks,

18:56.894 --> 18:57.924
just for familiarity.

18:57.920 --> 19:00.350
So we're working here in just
one-dimension,

19:00.352 --> 19:01.712
to get the idea of it.

19:01.710 --> 19:07.310
And we'll also then think in
more general terms.

19:07.308 --> 19:11.338
And we've already done this,
so it's partly review.

19:11.338 --> 19:13.928
So we'll assume that there's
perfect energy match;

19:13.930 --> 19:16.240
that is, the two atoms we're
talking about.

19:16.240 --> 19:18.990
Of course, atoms would be
Coulombic potentials,

19:18.986 --> 19:21.786
not harmonic oscillator,
Hooke's Law potentials.

19:21.791 --> 19:22.391
Right?

19:22.390 --> 19:24.880
But the idea of a
double-minimum is the same.

19:24.880 --> 19:27.760
So we'll suppose that the atoms
are equivalent.

19:27.759 --> 19:31.579
So the bottom of the well is
the same on both sides.

19:31.579 --> 19:33.009
And guess an energy.

19:33.009 --> 19:36.679
And lo and behold we were right;
we got the solution,

19:36.681 --> 19:38.491
and you've seen that one.

19:38.490 --> 19:38.950
Okay?

19:38.950 --> 19:42.090
But you can also get that one,
one with one node.

19:42.086 --> 19:42.606
Right?

19:42.608 --> 19:45.358
And on the far left you can see
that they have the same

19:45.363 --> 19:46.743
curvature-over-amplitude.

19:46.740 --> 19:48.930
So they have essentially the
same energy.

19:48.930 --> 19:52.640
If the green energy is right,
the red energy is the same

19:52.635 --> 19:52.965
deal.

19:52.972 --> 19:53.582
Right?

19:53.578 --> 19:57.838
They're ‘degenerate'
orbitals, because they're far

19:57.839 --> 19:58.459
apart.

19:58.460 --> 19:58.970
Okay?

19:58.970 --> 20:01.230
No significant energy
difference.

20:01.230 --> 20:04.350
Now, suppose we increase the
overlap.

20:04.348 --> 20:06.968
What will happen if we shorten
the distance?

20:06.970 --> 20:09.370
You did this on a problem set
with Erwin.

20:09.371 --> 20:09.781
Right?

20:09.778 --> 20:13.078
So you bring them closer
together, and you have the one

20:13.080 --> 20:15.710
with no nodes,
and the one with one node.

20:15.710 --> 20:19.280
And now there's a little --
right halfway between there's

20:19.278 --> 20:22.778
more electron density in one
than there is in the other.

20:22.781 --> 20:23.421
Right?

20:23.420 --> 20:27.360
The antibonding one has zero
and the bonding one has some

20:27.363 --> 20:30.253
buildup of electron density in
between.

20:30.250 --> 20:35.000
But the energy hasn't changed
visibly on this scale that we're

20:34.997 --> 20:35.617
using.

20:35.619 --> 20:37.679
It looks just the same.

20:37.680 --> 20:38.790
Okay?

20:38.788 --> 20:41.688
Now we increase it more and now
there's a big difference in the

20:41.691 --> 20:42.021
middle.

20:42.018 --> 20:42.438
Right?

20:42.440 --> 20:45.430
And now the energies are
beginning to change.

20:45.430 --> 20:49.080
The unfavorable orbital is
going up and the favorable

20:49.075 --> 20:52.015
bonding orbital is going down in
energy.

20:52.019 --> 20:56.199
And if we get it still closer
together, we can see that

20:56.202 --> 20:59.612
there's now a big difference
between them.

20:59.609 --> 21:01.349
Okay?

21:01.348 --> 21:06.068
So that increasing overlap
creates the splitting that we

21:06.065 --> 21:07.175
talk about.

21:07.180 --> 21:09.260
This is just review,
as you know,

21:09.258 --> 21:12.628
but it's worth -- it's so
important that it's good to

21:12.634 --> 21:14.004
mention it again.

21:14.000 --> 21:16.810
Now that overlap holds the
atoms together.

21:16.808 --> 21:20.498
And this again is a picture we
saw before, except we're going

21:20.499 --> 21:23.019
to do it with writing-in the
equations.

21:23.019 --> 21:28.059
Remember, a good guess of the
form of the molecular orbital is

21:28.058 --> 21:32.518
1/√2 (A+B),
or 1/√2 (A-B).

21:32.519 --> 21:36.509
And that will give two new
orbitals, one better and one

21:36.509 --> 21:37.099
worse.

21:37.098 --> 21:39.748
And, in fact,
if you think about

21:39.753 --> 21:42.823
normalization,
it should be a little less than

21:42.819 --> 21:45.739
1/√2,
in the one that's going to have

21:45.738 --> 21:48.028
+2AB in it,
when you square it,

21:48.029 --> 21:50.819
and a little less than
1/√2 when it's

21:50.817 --> 21:53.887
going to have a minus sign for
the 2AB term.

21:53.890 --> 21:55.000
Okay?

21:55.000 --> 21:58.280
Now, because of that ‘a
little bit greater' and ‘a

21:58.275 --> 22:00.995
little bit less',
the energies aren't quite,

22:00.998 --> 22:05.038
as we saw them before,
equally split, up and down.

22:05.038 --> 22:07.358
The one that goes down doesn't
go down quite as much,

22:07.358 --> 22:10.358
because this is less than,
and the one that goes up,

22:10.358 --> 22:12.868
goes up a little more because
it's greater than.

22:12.868 --> 22:18.478
The mathematical requirement
for that isn't immediately

22:18.482 --> 22:21.812
obvious to you,
but it's true.

22:21.808 --> 22:24.778
Okay, so there are the new
orbitals you get when you bring

22:24.776 --> 22:25.866
the atoms together.

22:25.869 --> 22:27.149
Okay?

22:27.150 --> 22:30.480
And if you had more overlap,
there'd be a bigger difference.

22:30.480 --> 22:32.410
So that's why overlap is
important.

22:32.410 --> 22:34.090
The more overlap,
the more splitting.

22:34.088 --> 22:36.978
We just saw that in
one-dimension with Erwin

22:36.976 --> 22:38.416
Meets Goldilocks.

22:38.420 --> 22:42.350
Now, how many electrons do we
have to put in these orbitals?

22:42.348 --> 22:44.248
That's going to make a big
difference.

22:44.250 --> 22:47.190
So suppose we have just one
electron for these two orbitals.

22:47.190 --> 22:50.910
When we bring the two wells
together, when we bring the two

22:50.909 --> 22:53.729
atoms together,
that electron will go down in

22:53.730 --> 22:56.360
energy, to the new molecular
orbital.

22:56.359 --> 22:57.849
So that's going to be good.

22:57.848 --> 22:59.218
And if we tried to pull them
apart,

22:59.220 --> 23:01.200
the electron would have to go
back up in energy,

23:01.200 --> 23:05.360
and that would require energy,
and that would oppose the

23:05.357 --> 23:06.187
breaking.

23:06.190 --> 23:07.760
So that's the bond strength.

23:07.758 --> 23:08.148
Right?

23:08.150 --> 23:10.170
You have to put an electron up
that much.

23:10.170 --> 23:12.860
Now suppose there were two
electrons.

23:12.858 --> 23:14.898
Now a second one goes down the
same way.

23:14.900 --> 23:17.210
So it's even stronger.

23:17.210 --> 23:21.340
It won't, in truth,
be quite twice as strong.

23:21.339 --> 23:24.429
Do you see why?

23:24.430 --> 23:25.070
Yes?

23:25.068 --> 23:25.948
Student:  Electron
repulsion.

23:25.950 --> 23:27.420
Prof: Because the
electrons will repel one

23:27.423 --> 23:27.733
another.

23:27.730 --> 23:30.950
We haven't taken that into
account when we think about the

23:30.945 --> 23:31.845
energies here.

23:31.848 --> 23:35.378
But it'll be certainly
stronger, considerably stronger.

23:35.380 --> 23:36.380
Okay.

23:36.380 --> 23:39.530
Now suppose we had three
electrons.

23:39.529 --> 23:43.559
What's going to happen?

23:43.558 --> 23:46.538
Prof: One of them is
going to have to go up instead

23:46.537 --> 23:47.057
of down.

23:47.058 --> 23:50.218
And, in fact,
it will go up further than

23:50.222 --> 23:54.682
either of the others went down,
even neglecting electron

23:54.683 --> 23:56.553
repulsion -- right?

23:56.548 --> 23:58.978
-- because of that
‘greater than/less than'

23:58.984 --> 24:01.324
thing that shifted the pair up a
little bit.

24:01.319 --> 24:01.979
Okay?

24:01.980 --> 24:05.480
So that's going to be even
weaker than having one electron

24:05.478 --> 24:08.238
to hold them together,
when you have three.

24:08.240 --> 24:10.230
How about if you have four?

24:10.230 --> 24:13.110
Student:  No more bonds.

24:13.109 --> 24:13.769
Prof: Wilson?

24:13.769 --> 24:15.079
Student:  They'll be
antibonding, so --

24:15.078 --> 24:17.718
Prof: Yeah,
you're now definitely losing,

24:17.723 --> 24:20.993
because the two that go up,
go up more than the two that go

24:20.987 --> 24:21.997
down, go down.

24:22.000 --> 24:22.950
So that's bad.

24:22.950 --> 24:25.480
So we can summarize that here,
the effect on the bond.

24:25.480 --> 24:27.590
If you have one electron,
it's bonding;

24:27.588 --> 24:29.438
two electrons is strongly
bonding;

24:29.440 --> 24:32.830
three electrons is weakly
bonding, probably;

24:32.828 --> 24:35.748
and four electrons will
definitely be antibonding.

24:35.750 --> 24:38.810
So that's why helium bounces
off helium,

24:38.808 --> 24:41.618
until it gets fifty-two
angstroms apart,

24:41.618 --> 24:45.408
and then there's this very,
very, very weak bond,

24:45.410 --> 24:48.380
that we talked about,
due to the correlation of the

24:48.381 --> 24:48.971
electrons.

24:48.974 --> 24:49.514
Right?

24:49.509 --> 24:53.449
But as soon as you get overlap
with helium, it's repulsive,

24:53.450 --> 24:56.100
when you start to split these
levels.

24:56.098 --> 25:00.098
Okay, now let's look at the
other end.

25:00.098 --> 25:03.918
Why doesn't increasing overlap,
as you bring things closer

25:03.919 --> 25:08.139
together, why doesn't that make
these plum-puddings collapse?

25:08.140 --> 25:11.910
The electrons are getting more
and more stable as you increase

25:11.910 --> 25:14.940
the overlap by bringing things
closer together.

25:14.940 --> 25:22.670
So why doesn't it just collapse?

25:22.670 --> 25:28.060
In fact, the electron -- if a
hydrogen molecule collapsed to

25:28.055 --> 25:30.315
helium --
it would be a funny helium

25:30.317 --> 25:32.107
because it would have no
neutrons --

25:32.108 --> 25:34.038
but if it collapsed,
the electrons,

25:34.038 --> 25:37.878
indeed, would become 55% more
stable.

25:37.880 --> 25:42.120
The electrons in helium are
lower in energy than the

25:42.117 --> 25:43.527
electrons in H_2.

25:43.530 --> 25:44.280
Right?

25:44.279 --> 25:47.239
So that would tend to pull them
together, all the way,

25:47.239 --> 25:48.299
to distance zero.

25:48.299 --> 25:50.729
Why don't they?

25:50.730 --> 25:51.400
Elizabeth?

25:51.400 --> 25:52.300
Student:  The protons
repel.

25:52.298 --> 25:54.258
Prof: Because the
protons will repel one another.

25:54.259 --> 25:56.339
That depends on 1/r.

25:56.339 --> 25:56.879
Right?

25:56.880 --> 25:59.580
And that increases much more
dramatically.

25:59.578 --> 26:04.338
In fact, the amount by which
the electrons become more stable

26:04.343 --> 26:07.923
is 650 kcal/mole,
or would become more stable.

26:07.915 --> 26:08.705
Right?

26:08.710 --> 26:13.200
But the nuclear repulsion
increases by that much by the

26:13.195 --> 26:18.265
time they get to 0.3 angstroms,
let alone the last little bit,

26:18.265 --> 26:21.085
when 1/r gets enormous.

26:21.089 --> 26:21.619
Okay?

26:21.619 --> 26:24.999
So it's not worth it;
unless you have glue to hold

26:25.003 --> 26:26.443
the neutrons together.

26:26.444 --> 26:26.974
Right?

26:26.970 --> 26:31.120
So on the sun you can take
deuterium and fuse it into --

26:31.118 --> 26:33.668
to make a helium -- you have
neutrons there to hold the

26:33.667 --> 26:36.527
protons together --
and then you get 200 million

26:36.532 --> 26:38.332
kcal/mole;
which is, of course,

26:38.327 --> 26:40.387
nuclear energy,
not electronic energy.

26:40.390 --> 26:43.420
It is possible,
but not under the chemical

26:43.423 --> 26:46.463
circumstances that we're talking
about.

26:46.460 --> 26:49.730
Okay, so here's the Morse
potential.

26:49.730 --> 26:51.120
Yes, Lucas?

26:51.118 --> 26:54.328
Student:  Have we been
taking proton-proton repulsion

26:54.325 --> 26:56.765
into account before,
with all our energies?

26:56.769 --> 27:00.969
Prof: No,
or only sort of subliminally.

27:00.971 --> 27:01.721
Right?

27:01.720 --> 27:05.120
Because if you take -- at first
glance there's going to be

27:05.118 --> 27:07.918
nuclear-nuclear repulsion,
and electron-electron

27:07.922 --> 27:12.122
repulsion, between the atoms;
nuclear-electron attraction;

27:12.119 --> 27:13.739
nuclear-electron attraction.

27:13.740 --> 27:17.000
So to first approximation,
those will all cancel out.

27:16.998 --> 27:17.498
Right?

27:17.500 --> 27:21.350
But then you get -- on top of
that you have a little change

27:21.348 --> 27:24.338
from the fact that the
nuclear-nuclear goes as

27:24.336 --> 27:27.926
1/r and gets quite big,
when they get really close,

27:27.933 --> 27:30.723
and that the electrons are
coming down because of overlap.

27:30.720 --> 27:33.400
And we've been focusing on this
overlap thing that causes the

27:33.396 --> 27:36.116
bonding, but obviously things
aren't going to collapse because

27:36.118 --> 27:37.188
of the nuclear part.

27:37.190 --> 27:40.490
But we haven't been talking
about it explicitly;

27:40.487 --> 27:41.537
you're right.

27:41.538 --> 27:44.358
Okay, so anyhow this is the
form of the bonding potential,

27:44.358 --> 27:47.308
the Morse potential,
which was just cobbled together

27:47.313 --> 27:50.383
artificially to make it
convenient to calculate things

27:50.382 --> 27:53.222
and to have reasonable
properties for a bond.

27:53.220 --> 27:57.390
But now we understand why it
should look like that.

27:57.390 --> 27:59.490
This first part,
the attractive part,

27:59.492 --> 28:02.532
is because the electron pair
becomes more stable with

28:02.529 --> 28:05.509
increasing overlap,
as the atoms come together.

28:05.509 --> 28:09.099
And then the nuclear repulsion
becomes dominant,

28:09.103 --> 28:11.093
when it becomes too close.

28:11.090 --> 28:11.780
Right?

28:11.779 --> 28:15.179
So all this is Coulomb's Law.

28:15.180 --> 28:16.550
There's nothing special.

28:16.548 --> 28:18.378
There's not
‘correlation

28:18.383 --> 28:21.193
energy', which is some
completely new physical

28:21.189 --> 28:21.819
phenomenon.

28:21.818 --> 28:22.448
Right?

28:22.450 --> 28:24.720
It's all Coulombic.

28:24.720 --> 28:28.480
Correlation energy is just,
remember, an excuse for the

28:28.482 --> 28:31.482
error you're making,
or a way to hide it.

28:31.480 --> 28:35.220
Okay, so it all comes from
Coulomb's Law for potential

28:35.223 --> 28:35.863
energy.

28:35.858 --> 28:39.408
But the funny thing was the
kinetic energy of the electrons,

28:41.218 --> 28:41.758
Right?

28:41.759 --> 28:44.569
Because as you make things --
as you concentrate things,

28:44.570 --> 28:46.360
you change the curvature as
well.

28:46.358 --> 28:48.628
So you have to take that into
account.

28:48.630 --> 28:54.180
But this curve provides the
potential for studying molecular

28:54.180 --> 28:57.200
vibration;
that is, once you know the form

28:57.195 --> 28:59.615
of this curve,
then you can use quantum

28:59.616 --> 29:02.086
mechanics for the motion of the
atoms --

29:02.088 --> 29:03.968
not of the electrons but of the
atoms --

29:03.970 --> 29:06.120
and figure out how atoms
vibrate;

29:06.118 --> 29:09.168
which we've already done,
when we were doing Erwin

29:09.170 --> 29:10.520
Meets Goldilocks.

29:10.519 --> 29:14.079
Okay, but here,
finally we understand the

29:14.078 --> 29:15.858
atom-atom force law.

29:15.858 --> 29:16.658
Right?

29:16.660 --> 29:18.210
That's what we've been doing
the whole time,

29:18.210 --> 29:21.010
for four weeks or five weeks or
whatever it's been,

29:21.009 --> 29:24.679
is trying to understand where
-- what the force law is.

29:24.680 --> 29:26.040
Wouldn't Newton be happy?

29:26.038 --> 29:27.948
Remember, he said,
"There are therefore

29:27.952 --> 29:30.362
agents in Nature able to make
particles of bodies stick

29:30.355 --> 29:32.175
together by very strong
attractions,

29:32.180 --> 29:34.760
and it's the business of
Experimental Philosophy to find

29:34.760 --> 29:35.560
them out."

29:35.559 --> 29:37.599
And now we've found it out.

29:37.598 --> 29:42.678
Now, can we use this
understanding then to gain

29:42.682 --> 29:46.222
greater purchase on chemistry?

29:46.220 --> 29:47.460
So we've looked at overlap.

29:47.460 --> 29:50.620
But there's one other thing we
have to look at,

29:50.618 --> 29:53.638
in order to apply it,
and that's what we call

29:53.642 --> 29:54.812
energy-match.

29:54.808 --> 29:57.888
I don't think anybody else
talks about energy-match,

29:57.885 --> 29:58.605
but we do.

29:58.608 --> 30:01.748
Okay, so here's what we talked
about already.

30:01.749 --> 30:02.319
Right?

30:02.318 --> 30:05.828
And we have this splitting,
which is about proportional to

30:05.833 --> 30:06.453
overlap.

30:06.450 --> 30:09.420
It's not numerically strictly
proportional to overlap,

30:09.423 --> 30:12.513
but it certainly is close to
proportional to overlap.

30:12.509 --> 30:16.649
Okay, now what would happen if
the two atoms you started with

30:16.650 --> 30:19.890
were not the same,
if one was a better place for

30:19.894 --> 30:25.924
electrons to be than the other;
like there for example?

30:25.920 --> 30:29.250
So why would you -- remember
what happened is you mix these.

30:29.250 --> 30:32.000
You get A+B and A-B.

30:32.000 --> 30:35.460
But why would you put
any A together

30:35.455 --> 30:38.625
with B, if B is already much
better than A?

30:38.630 --> 30:41.540
If they're equal,
you don't care where it is,

30:41.544 --> 30:43.734
then fine, use a 50/50 mixture.

30:43.730 --> 30:49.000
But if one's better than the
other, why use any of the

30:48.998 --> 30:50.488
inferior one?

30:50.490 --> 30:53.580
Why not just use the one that's
good?

30:53.578 --> 30:58.618
So we already actually saw this
in the case of the 1s

30:58.616 --> 31:01.796
atomic orbitals,
which did remain pure and

31:01.797 --> 31:05.347
unmixed, when we made the
molecular orbitals of that funny

31:05.346 --> 31:09.606
molecule last time,
the fluoroethanol.

31:09.609 --> 31:10.549
Remember, it looked like that.

31:10.548 --> 31:12.888
One of them was just an
s orbital on F;

31:12.890 --> 31:15.410
another one was an s
orbital on O;

31:15.410 --> 31:17.890
s orbital on one carbon,
s orbital on the other

31:17.887 --> 31:18.257
carbon.

31:18.259 --> 31:23.459
The didn't mix. Right?

31:23.460 --> 31:27.500
So the compact 1s core
orbitals did remain pure and

31:27.500 --> 31:31.610
unmixed, but the valence level
atomic orbitals were heavily

31:31.611 --> 31:32.321
mixed.

31:32.318 --> 31:35.008
For example,
the next one was biggest on

31:35.009 --> 31:39.149
fluorine, but also substantial
on oxygen, and carbon even got

31:39.147 --> 31:40.937
into the act a little bit.

31:40.941 --> 31:41.701
Right?

31:41.700 --> 31:47.970
Now why is it that the core
orbitals didn't mix and the

31:47.972 --> 31:51.112
valence orbitals did mix?

31:51.109 --> 31:54.389
What's different?

31:54.390 --> 31:58.830
The core orbitals are very
small, very compact.

31:58.829 --> 32:00.409
These orbitals are bigger.

32:00.410 --> 32:02.590
So what happens?

32:02.589 --> 32:05.309
They overlap. Right?

32:05.308 --> 32:09.858
So it's overlap that causes
things to mix.

32:09.858 --> 32:11.718
If there's no overlap they
don't mix.

32:11.720 --> 32:15.150
If they overlap they mix.

32:15.150 --> 32:17.930
So why use any of an inferior
orbital?

32:17.930 --> 32:21.050
Well suppose the energy of A is
much higher, less favorable,

32:21.046 --> 32:22.416
than that of B. Right?

32:22.420 --> 32:25.540
So now we make a sum,
a weighted sum,

32:25.540 --> 32:29.010
a parts of A,
b parts of B,

32:29.009 --> 32:30.829
and we square it.

32:30.828 --> 32:34.418
So we get a^2^( )parts
of the A density,

32:34.416 --> 32:38.936
b^2^( )parts of the B
density, and also this overlap

32:38.942 --> 32:40.582
term, 2ab.

32:40.578 --> 32:44.988
Now, can you profit from
shifting electron density toward

32:44.990 --> 32:49.560
the inter-nuclear AB region,
without paying too much of the

32:49.560 --> 32:51.610
high energy cost of A?

32:51.608 --> 32:55.168
That is, A^2^( )is higher
energy than B^2.

32:55.173 --> 32:55.873
Right?

32:55.868 --> 32:58.928
The electron density around A
is higher energy than the

32:58.930 --> 33:00.460
electron density about B.

33:00.460 --> 33:04.540
And if you mix A with B,
you're going to get a certain

33:04.540 --> 33:07.010
amount of that bad,
or less good,

33:07.005 --> 33:08.695
distribution,
A^2.

33:08.700 --> 33:12.840
But at the same time you're
going to get some of AB,

33:12.838 --> 33:17.708
which puts stuff in the middle,
at the expense of outside.

33:17.710 --> 33:19.200
That's good. Right?

33:19.200 --> 33:24.080
At a certain distance from the
nuclei, you'd rather be between

33:24.078 --> 33:27.678
the two nuclei than out beyond
one of them.

33:27.680 --> 33:31.190
So as you increase a,
you increase -- oops sorry

33:31.194 --> 33:33.284
>

33:33.278 --> 33:37.248
-- you increase this bad part,
but you also increase that good

33:37.249 --> 33:37.899
part.

33:37.900 --> 33:41.710
Can it be worthwhile to make
a non-0?

33:41.710 --> 33:43.150
Which one is going to help?

33:43.150 --> 33:44.630
But notice this.

33:44.630 --> 33:49.430
If you put only a small amount
of A in, then the amount of

33:49.431 --> 33:52.881
A^2^( )probability density,
this bit here,

33:52.884 --> 33:57.084
is a^2 parts of that;
really, really tiny,

33:57.082 --> 33:58.662
if a is tiny.

33:58.660 --> 34:02.250
But the amount of ab you
put in is a times

34:02.250 --> 34:03.880
b, not a^2.

34:03.882 --> 34:04.472
Right?

34:04.470 --> 34:05.430
So it's much bigger.

34:05.430 --> 34:12.240
So you get this good stuff,
without getting very much of

34:12.244 --> 34:14.104
this bad stuff.

34:14.103 --> 34:15.223
Right?

34:15.219 --> 34:17.869
For example,
suppose you used,

34:17.873 --> 34:21.723
a was 0.03 and b
was 0.98.

34:21.719 --> 34:23.909
So you square them,
to find out how much --

34:23.909 --> 34:26.269
<<technical
adjustment>>

34:26.266 --> 34:29.296
-- you square them,
and you find out that the

34:29.304 --> 34:32.044
amount of a^2 you get is
only 0.001.

34:32.039 --> 34:34.669
The amount of b^2 is
0.96.

34:34.670 --> 34:37.270
And the amount of 2ab is
0.06.

34:37.268 --> 34:41.448
So you're getting sixty times
as much of ab as you get

34:41.454 --> 34:42.574
of a^2.

34:42.570 --> 34:45.680
So that's why it can be useful
to put a little bit in.

34:45.679 --> 34:49.559
And if you're pedantic,
you can look at the footnote

34:49.561 --> 34:50.171
there.

34:50.170 --> 34:53.580
So let's look at Erwin Meets
Goldilocks in the case where

34:53.577 --> 34:56.877
the two wells are not matched,
where the energies are a little

34:56.876 --> 34:58.966
bit different;
because that's what we're

34:58.974 --> 34:59.654
talking about.

34:59.650 --> 35:04.380
So here are non-degenerate
atomic orbitals.

35:04.380 --> 35:07.850
So the one on the left is a
little bit lower than the one on

35:07.849 --> 35:09.849
the right, but only a teeny bit.

35:09.849 --> 35:11.049
Okay?

35:11.050 --> 35:13.260
Now we're going to look at the
wave functions here.

35:13.260 --> 35:16.010
So there's the lowest energy
wave function.

35:16.005 --> 35:16.525
Right?

35:16.530 --> 35:20.690
And notice it's the sum,
it's a weighted sum,

35:20.688 --> 35:22.198
with no nodes.

35:22.199 --> 35:25.719
But there's almost no mixing at
all, just a really,

35:25.715 --> 35:28.525
really, really tiny amount of A
in it.

35:28.530 --> 35:32.690
So essentially it's just the
solution you'd get in the left

35:32.693 --> 35:33.053
well.

35:33.052 --> 35:33.702
Right?

35:33.699 --> 35:36.769
There'd be one with one node as
well.

35:36.768 --> 35:41.048
What will it look like,
can you guess?

35:41.050 --> 35:43.970
This is the one with zero nodes.

35:43.969 --> 35:46.489
What's the wave function with
one node look like?

35:46.489 --> 35:49.999
Any guesses?

35:50.000 --> 35:51.370
Yes, Josh?

35:51.369 --> 35:53.159
Student:  It'll just
have a node in the middle of

35:53.164 --> 35:55.194
that first potential and then be
straight on the second one.

35:55.190 --> 35:56.000
Prof: And be what?

35:56.000 --> 36:00.270
Student:  And be sort of
straight on the second one.

36:00.268 --> 36:01.328
Prof: Right.

36:01.326 --> 36:03.096
So the first one,
the lowest one,

36:03.103 --> 36:04.553
is the one on the left.

36:04.550 --> 36:06.580
The second one,
the next highest energy

36:06.576 --> 36:09.666
orbital, is the one on the
right, with just a little bit of

36:09.670 --> 36:12.790
the one on the left in it;
negligible amount of mixing.

36:12.789 --> 36:13.169
Right?

36:13.170 --> 36:16.160
And their energies,
you might think their energies

36:16.159 --> 36:19.089
would be the same,
because they're not mixing.

36:19.090 --> 36:24.980
But the green one is in this
well, and the red one is in that

36:24.983 --> 36:28.033
well, and that well's higher.

36:28.030 --> 36:28.730
Okay?

36:28.730 --> 36:30.220
So it's got to be higher in
energy.

36:30.219 --> 36:33.909
That spitting is due only to
the original offset between the

36:33.909 --> 36:34.409
wells.

36:34.409 --> 36:37.029
There's no shift of it. Right?

36:37.030 --> 36:39.890
It's just you have A or you
have B, and you have what you

36:39.885 --> 36:41.105
would expect for them.

36:41.110 --> 36:46.730
Now suppose we increase the
overlap between these.

36:46.730 --> 36:52.030
What do you think is going to
happen?

36:52.030 --> 36:55.790
You should get a little more
mixing because AB is going to

36:55.786 --> 36:58.286
count for more,
if you have overlap.

36:58.289 --> 36:59.779
Okay, so here they are close
together.

36:59.780 --> 37:03.440
You still don't have very much
mixing, just a little bit,

37:03.443 --> 37:06.653
and you haven't seen any
appreciable change in the

37:06.648 --> 37:07.498
energies.

37:07.500 --> 37:10.620
You still have that same
splitting that's due to the

37:10.619 --> 37:11.719
original offset.

37:11.719 --> 37:12.279
Okay?

37:12.280 --> 37:15.220
Now we'll bring them still
closer together.

37:15.219 --> 37:18.999
And now it's getting worthwhile
to put it in the wrong well,

37:19.003 --> 37:21.253
because the AB term is so
important.

37:21.248 --> 37:21.888
Right?

37:21.889 --> 37:24.779
So the antibonding energy is
rising,

37:24.780 --> 37:27.560
the bonding energy is falling
-- that's why they're

37:27.561 --> 37:31.901
antibonding and bonding --
and if we bring them still

37:31.902 --> 37:35.672
closer together,
we get that. Right?

37:35.670 --> 37:40.840
Which, if you looked at it
casually, they look like a

37:40.838 --> 37:42.228
single well.

37:42.230 --> 37:46.270
But notice that they're still
quite unsymmetrical,

37:46.271 --> 37:48.171
they're still biased.

37:48.170 --> 37:52.210
The one without any nodes is
still mostly in the left well,

37:52.213 --> 37:55.983
and the one with a node is
mostly in the right well.

37:55.980 --> 37:57.610
But they're heavily mixed.

37:57.610 --> 37:58.050
Right?

37:58.050 --> 38:00.890
So what's happening is this
increasing overlap,

38:00.887 --> 38:04.337
as you bring them together,
is fighting the effect of the

38:04.344 --> 38:06.324
difference in the two wells.

38:06.320 --> 38:10.300
So when we had the core
electrons in fluoroethanol,

38:10.295 --> 38:13.715
if we had tiny orbitals that
didn't overlap,

38:13.715 --> 38:15.465
then they stayed pure.

38:15.465 --> 38:16.335
Right?

38:16.340 --> 38:18.810
But once we went to the valence
level orbitals,

38:18.809 --> 38:21.279
that were much bigger,
so that they overlapped,

38:21.280 --> 38:23.430
then they started mixing,
like this.

38:23.429 --> 38:29.859
Okay, now what if one partner
is lower in energy than A?

38:29.860 --> 38:31.510
That's the case we're dealing
with.

38:31.510 --> 38:34.020
But what will these ultimate
energies be,

38:34.018 --> 38:37.578
when you have this competition
between overlap,

38:37.579 --> 38:41.759
which is trying to mix them,
and energy mismatch,

38:41.760 --> 38:44.760
which is trying to keep them
separate from one another?

38:44.760 --> 38:47.820
So here's what you get.

38:47.820 --> 38:51.770
You get again splitting,
but not as much shift as you

38:51.771 --> 38:52.761
got before.

38:52.760 --> 38:55.960
The lower level will look
mostly like C,

38:55.960 --> 38:58.340
because C is better than A.

38:58.340 --> 39:00.860
The antibonding will look
mostly like A,

39:00.862 --> 39:03.192
with just a little bit of C in
it.

39:03.190 --> 39:07.880
And if they had had the same
energy, you'd get a big energy

39:07.876 --> 39:10.056
shift because of overlap.

39:10.059 --> 39:13.229
But now you get a smaller
energy shift,

39:13.228 --> 39:16.318
because the mismatch is
fighting that.

39:16.315 --> 39:17.145
Right?

39:17.150 --> 39:20.660
You don't want to -- you lose
as you try to mix,

39:20.663 --> 39:23.733
because of the difference
between A and C,

39:23.728 --> 39:24.998
in this case.

39:25.000 --> 39:30.660
Now, so that one looks mostly
like C, both in shape and in

39:30.655 --> 39:31.545
energy.

39:31.550 --> 39:35.070
That one looks mostly like A,
both in shape and energy.

39:35.070 --> 39:38.150
But it's a little worse than A,
and the first one is a little

39:38.148 --> 39:38.968
better than C.

39:38.969 --> 39:46.069
Now how much smaller is that
bonding shift?

39:46.070 --> 39:48.620
When they were mismatched,
they didn't shift as much with

39:48.623 --> 39:49.493
the same overlap.

39:49.489 --> 39:51.259
How much less?

39:51.260 --> 39:52.710
Exactly where does it end up?

39:52.710 --> 39:55.400
Well that's not so crucially
important to you,

39:55.400 --> 39:58.390
but there's a neat trick you
can do to see that.

39:58.389 --> 40:02.359
Suppose we have that much
energy mismatch.

40:02.360 --> 40:02.940
Okay?

40:02.940 --> 40:06.680
And the red dot is halfway
between B and C,

40:06.681 --> 40:10.691
or between A and C,
since A and B are the same

40:10.690 --> 40:11.670
energy.

40:11.670 --> 40:14.260
Now, so when they were
perfectly matched,

40:14.264 --> 40:17.574
A and B perfectly matched in
energy, that's how much

40:17.574 --> 40:20.824
splitting you got from that
amount of overlap.

40:20.820 --> 40:24.010
So these are the two factors
that go into it:

40:24.012 --> 40:28.152
how much splitting you get when
they have the same energy,

40:28.146 --> 40:31.626
and how different the original
energies are.

40:31.630 --> 40:36.220
How can you put those together,
to find out how much shift you

40:36.221 --> 40:37.351
actually get?

40:37.349 --> 40:40.579
Well what we do is slide that
over,

40:40.579 --> 40:48.109
and then bend down the blue
one, so it's perpendicular to

40:48.105 --> 40:51.925
that,
and make a rectangle around it,

40:51.929 --> 40:53.969
and draw the diagonal.

40:53.969 --> 40:59.679
And that diagonal will be the
new energy difference between

40:59.677 --> 41:01.347
the new levels.

41:01.349 --> 41:02.119
Okay?

41:02.119 --> 41:04.569
You say, why do I make this
construction?

41:04.570 --> 41:07.160
It's because there's a
quadratic formula that comes in.

41:07.159 --> 41:11.569
So the diagonal is the square
root of the squares of the two

41:11.574 --> 41:15.244
sides, the sum of the squares of
the two sides.

41:15.239 --> 41:18.979
Okay, so we can rotate that;
put a tack in there and rotate

41:18.981 --> 41:19.291
it.

41:19.289 --> 41:23.019
And that's going to be the new
energy difference.

41:23.019 --> 41:23.859
Okay?

41:23.860 --> 41:27.000
And it's bigger than the
original energy difference,

41:26.996 --> 41:30.806
because the diagonal has to be
bigger than one of the sides.

41:30.809 --> 41:32.719
But because of the
normalization,

41:32.721 --> 41:35.471
because one of them is a little
less than --

41:35.469 --> 41:38.559
one of them is a little smaller
and one's a little greater --

41:38.559 --> 41:42.379
that was the less than/greater
than √2,

41:42.380 --> 41:45.470
1/√2 thing -- it's
going to shift up a little bit,

41:45.469 --> 41:48.029
both levels will shift up a
little bit, like that.

41:48.030 --> 41:48.450
Right?

41:48.449 --> 41:49.689
I'll do it again so you see it.

41:49.690 --> 41:50.970
Both will shift up.

41:50.969 --> 41:53.679
So there are the new levels.

41:53.679 --> 41:56.009
That's how you do it.

41:56.010 --> 41:57.100
Okay?

41:57.099 --> 42:03.289
Now, so for a given overlap,
the bonding shift,

42:03.289 --> 42:06.499
how much the electrons changed
when the atoms came together,

42:06.500 --> 42:10.680
is reduced if the energies
aren't well matched,

42:10.679 --> 42:13.059
if you have the same overlap.

42:13.059 --> 42:16.969
You don't get as strong a bond,
from that point of view.

42:16.969 --> 42:20.769
The energies didn't shift down
as much, or up as much.

42:20.768 --> 42:29.258
But still A+C will be lower in
energy than the original one

42:29.259 --> 42:30.869
was, A+B.

42:30.869 --> 42:31.679
Okay?

42:31.679 --> 42:35.569
And we had that construction,
which shows us an interesting

42:35.574 --> 42:38.034
thing,
which is the splitting is not

42:38.034 --> 42:41.494
very sensitive to the one of
these contributions that's

42:41.485 --> 42:42.185
smaller.

42:42.190 --> 42:45.350
There are two contributions:
the overlap part,

42:45.349 --> 42:47.719
that's the black arrow -- or
pardon me,

42:47.719 --> 42:51.859
that's the blue arrow -- and
the energy difference,

42:51.860 --> 42:55.110
the energy mismatch,
which is the black arrow.

42:55.110 --> 42:59.310
But if one of those is very
small, like here,

42:59.311 --> 43:01.991
the mismatch,
which is black,

43:01.987 --> 43:04.657
is very big,
and the overlap,

43:04.661 --> 43:07.911
the blue one,
is very small.

43:07.909 --> 43:11.649
But you can see that the length
of the diagonal is not going to

43:11.652 --> 43:14.252
be very sensitive to what the
overlap is.

43:14.250 --> 43:18.660
You could increase that blue
overlap, the width of the

43:18.661 --> 43:23.491
rectangle, which wouldn't change
the diagonal very much.

43:23.489 --> 43:26.739
So it's sensitive to the one
that's bigger.

43:26.739 --> 43:29.149
Okay, don't worry too much
about that part.

43:29.150 --> 43:31.910
I just think it's fun.

43:31.909 --> 43:34.519
Okay, so we can generalize from
this.

43:34.518 --> 43:38.168
Mixing two overlapping orbitals
gives one composite orbital

43:38.172 --> 43:41.512
that's lower in energy than
either of the parents,

43:41.510 --> 43:45.100
and one that's higher in energy
than either of the parents.

43:45.099 --> 43:45.649
Okay?

43:45.650 --> 43:48.810
The lower energy combination
looks -- that is,

43:48.813 --> 43:52.473
its shape -- is most like the
lower energy parent;

43:52.469 --> 43:54.579
and the same is true for
energy.

43:54.576 --> 43:55.116
Right?

43:55.119 --> 43:58.979
And the higher one looks like
the higher parent.

43:58.980 --> 44:02.400
For a given overlap,
increasing energy mismatch

44:02.396 --> 44:06.926
decreases the amount of mixing,
and decreases the magnitude of

44:06.925 --> 44:08.555
the energy shifts.

44:08.559 --> 44:11.549
Now what does this have to do
with bonding?

44:11.550 --> 44:16.660
Okay, the AC electrons are
clearly lower in energy.

44:16.659 --> 44:19.189
But that's not really what
we're interested in.

44:19.190 --> 44:22.330
We're interested in the
change of energy

44:22.329 --> 44:25.589
that comes when you break the
bond, or make the bond.

44:25.590 --> 44:29.950
So which bond is stronger,
AB or AC?

44:29.949 --> 44:34.159
How many think AB is stronger,
given this scheme?

44:34.159 --> 44:36.619
How many think AC is stronger?

44:36.619 --> 44:40.069
It's clearly got lower energy
electrons, AC.

44:40.070 --> 44:44.210
But this is a classic example
of "compared to what?"

44:44.210 --> 44:45.130
Okay?

44:45.130 --> 44:47.870
How are we going to break the
bond?

44:47.869 --> 44:51.489
Suppose what we're going to do
is break it, to put two

44:51.485 --> 44:54.625
electrons in C,
because that's the lower energy

44:54.625 --> 44:56.805
atom;
or in the other case,

44:56.809 --> 44:58.829
to put two electrons in B.

44:58.829 --> 44:59.219
Okay?

44:59.219 --> 45:01.499
So both electrons go the same
way.

45:01.500 --> 45:04.710
Can you see anything bad about
that?

45:04.710 --> 45:08.330
Why might it be better to put
one electron each way?

45:08.329 --> 45:10.369
Kevin?

45:10.369 --> 45:13.559
Student:  Well in terms
of spin, they have to be --

45:13.559 --> 45:14.419
Prof: Not spin.

45:14.420 --> 45:16.960
You can put two electrons in an
orbital with opposite spin.

45:16.960 --> 45:20.790
So spin's not a problem,
if you only have two.

45:20.789 --> 45:23.619
If you had three it would be a
problem, but with two,

45:23.621 --> 45:24.931
spin is not a problem.

45:24.929 --> 45:27.349
But there's an obvious problem
with having both the electrons

45:27.346 --> 45:27.826
go to one.

45:27.829 --> 45:28.599
Dana?

45:28.599 --> 45:29.149
Student:  They'll repel
each other.

45:29.150 --> 45:30.360
Prof: They'll repel each
other.

45:30.360 --> 45:31.620
If you could put them on
opposite ones,

45:31.619 --> 45:32.249
so much the better.

45:32.251 --> 45:32.551
Right?

45:32.550 --> 45:36.580
But it would be possible to do
this, and it might plausible in

45:36.581 --> 45:39.621
the case of AC,
where C is lower in energy than

45:39.623 --> 45:41.853
A;
although the repulsion might

45:41.847 --> 45:44.717
make it better to go one on C
and one on A.

45:44.719 --> 45:46.849
Okay, but anyhow,
suppose you break it this way.

45:46.849 --> 45:48.639
Which one would be easier to
break?

45:48.639 --> 45:52.119
Which requires less energy,
red or blue?

45:52.119 --> 45:52.959
Students:  Red.

45:52.960 --> 45:54.890
Prof: Red requires less
energy.

45:54.889 --> 45:59.359
Okay, so AB is stronger,
if you're forming A^+ B^-,

45:59.364 --> 46:03.304
if both the electrons are going
the same way.

46:03.304 --> 46:04.204
Right?

46:04.199 --> 46:08.649
So mismatch aids heterolysis.

46:08.650 --> 46:11.110
Heterolysis,
(uneven breaking) is to put

46:11.108 --> 46:14.448
them both on the same side,
rather than one on each side

46:14.445 --> 46:17.545
(that's called homolysis,
as you'll see in a second).

46:17.550 --> 46:21.300
So if you put them both on one
side, then it's easier to break

46:21.302 --> 46:22.782
if they're mismatched.

46:22.780 --> 46:25.710
The bond is stronger if they're
well matched.

46:25.710 --> 46:26.910
Okay?

46:26.909 --> 46:28.719
That's because that.

46:28.719 --> 46:34.619
But now suppose you do
homolysis and put one on each.

46:34.619 --> 46:37.579
Now which one's easier?

46:37.579 --> 46:38.189
Student:  The blue one.

46:38.190 --> 46:42.690
Prof: Now the blue is
less energy than the red.

46:42.690 --> 46:43.410
Okay?

46:43.409 --> 46:47.319
So now the AC is going to be
stronger.

46:47.320 --> 46:51.470
So mismatch hinders homolysis.

46:51.471 --> 46:52.441
Right?

46:52.440 --> 46:55.290
So you can't really say which
bond is stronger,

46:55.293 --> 46:58.033
unless you say,
"Compared to what -- how

46:58.025 --> 47:00.255
are you breaking it apart?"

47:00.260 --> 47:00.670
Okay?

47:00.670 --> 47:03.270
Is all this true?

47:03.268 --> 47:06.238
So I've been weaving this
fairytale for you,

47:06.237 --> 47:09.477
for the last couple of weeks,
but is it true?

47:09.480 --> 47:11.870
Well, we can check it with
experiment.

47:11.869 --> 47:14.629
For example, compare HH with HF.

47:14.630 --> 47:19.130
HH has perfect matching between
the two atoms.

47:19.130 --> 47:22.480
In HF, there's a much bigger
nuclear charge on F;

47:22.480 --> 47:25.970
electrons in the valence
orbital are lower in energy on F

47:25.969 --> 47:27.279
than they are on H.

47:27.280 --> 47:29.800
Okay, so if we look at the
σ orbital,

47:29.800 --> 47:31.540
the valence σ
orbital,

47:31.539 --> 47:35.229
it's symmetrical in the case of
H_2 and quite unsymmetrical in

47:35.231 --> 47:36.261
the case of HF.

47:36.260 --> 47:40.650
It's big on F and small on H,
just as we were speaking of.

47:40.650 --> 47:43.880
That's the σ,
big on F. Right?

47:43.880 --> 47:48.270
But there's also an antibonding
orbital, σ,

47:48.271 --> 47:49.791
which has a node.

47:49.789 --> 47:54.279
So that one is the contrary;
it's big on H, small on F.

47:54.280 --> 47:57.310
This is just what we were
talking about a few minutes ago.

47:57.309 --> 48:00.019
And the electrons,
of course, are only in the

48:00.023 --> 48:02.863
σ*,
not in the σ*;

48:02.860 --> 48:05.310
there are only two electrons
and that one's empty.

48:05.309 --> 48:09.779
And the * means that it's
antibonding.

48:09.782 --> 48:10.752
Right?

48:10.750 --> 48:14.600
So it has the node between the
nuclei, planar in the

48:14.597 --> 48:18.067
symmetrical case and bent in the
case of HF;

48:18.070 --> 48:20.390
but it's still the node.

48:20.389 --> 48:23.529
Now, how about the experiment?

48:23.530 --> 48:25.880
Heterolysis,
where you break,

48:25.882 --> 48:28.912
in order to get H^+ H^-,
in one case,

48:28.907 --> 48:31.677
or H^+ F^- in the other case.

48:31.679 --> 48:34.719
In the case of HH,
it costs 400 kilocalories to

48:34.717 --> 48:35.837
break the bond.

48:35.840 --> 48:39.760
In the case of HF,
it costs only 373.

48:39.760 --> 48:41.960
Incidentally,
this is a lot of energy,

48:41.960 --> 48:44.930
to break this bond,
because you're putting both the

48:44.934 --> 48:46.664
electrons the same place.

48:46.659 --> 48:49.569
It can happen in solution,
but it's not so easy in the gas

48:49.572 --> 48:51.772
phase, which is what we're
talking about.

48:51.768 --> 48:57.178
Okay, so indeed we were right,
that mismatch weakens the bond;

48:57.179 --> 49:01.679
for heterolysis,
the HF bond is weaker.

49:01.679 --> 49:04.699
That's why you call it
hydrofluoric acid,

49:04.704 --> 49:06.894
because it can easily break off
an H.

49:06.889 --> 49:12.549
You don't call H_2 hydrogenic
acid or something like that.

49:12.545 --> 49:13.335
Right?

49:13.340 --> 49:17.340
But if you break it into two
atoms,

49:17.344 --> 49:22.444
two H atoms or an H atom and an
F atom, then it costs only 104

49:22.436 --> 49:24.686
for HH, but 136 for HF.

49:24.690 --> 49:28.320
So the HF bond is stronger,
if you're breaking to atoms.

49:28.320 --> 49:30.230
So that's the first
verification,

49:30.226 --> 49:33.446
experimental verification,
of what I've been talking to

49:33.445 --> 49:34.275
you about.

49:34.280 --> 49:37.520
And we're going to go on next
time to talk about XH_3,

49:37.516 --> 49:40.876
which will have lots of
experimental data that will show

49:40.875 --> 49:42.825
that this stuff is sensible.

49:42.829 --> 49:48.999