WEBVTT

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Prof: Okay,
so what's coming for the next

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exam?

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Well we've been looking at
atoms, and at the idea of

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orbitals for many-electron
atoms, which we showed last time

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is wrong.

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So today we want to recover
from the orbital approximation.

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Okay?

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Then we're going to look at
molecules.

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And first we're going to look
at them in a very unconventional

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way,
from the point of view of

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molecules as plum-puddings,
what's called the "United

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Atom" limit,
to apply what we know from

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atoms to molecules.

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But then we're going to look at
a very, very different way of

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looking at it,
to try to understand bonds in

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terms of what are called linear
combinations --

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that means weighted sums -- of
atomic orbitals to make

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molecular orbitals;
but we're going to try

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understanding bonds.

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And then these terms that don't
mean much to you now but will

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mean a lot later on:
"energy-match"

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and "overlap."

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And then we're going to get to
reality --

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all this stuff is theory -- but
we're going to look at something

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real: at XH_3 molecules,
with different atoms for X,

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at their structure and at their
dynamics,

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and how that ties in with our
understanding of bonding.

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And then we'll go on to
reactivity, which is of course

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the main goal.

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We'll talk about HOMOs and
LUMOs, and you'll see what those

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are, and how to recognize
functional groups and their

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reactivity;
and that's the real prize,

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is to be able to --
you've been memorizing

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functional groups for this
previous exam,

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but what I want you to be able
to do is look at a molecule and

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recognize when it has a
functional group,

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even if you've never seen it
before,

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and how it might react,
what it would react with.

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And then we're going to see how
organic chemistry really

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developed, the real thing;
how it developed from the time

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of Lavoisier.

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Okay, so that's what we're
going to.

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And here's where we were last
time.

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We were wondering,
when it comes to a two-electron

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wave function,
might it be possible to write

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it as a product of one-electron
wave functions?

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Because we know how to write
1-electron wave functions;

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we've got that table,
we can write the real thing,

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at least for an atom.

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So if it were possible to write
the six-variable,

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two-electron wave function as a
product of one-electron wave

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functions,
that would be a fantastic

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simplification,
and all we'd have to do is

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square and we'd get the joint
probability;

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not really that you care very
much about the joint

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probability.

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I suspect you haven't stayed up
late nights worrying about joint

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probability.

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But anyhow, if you want to
handle things with more than one

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electron, you have to have a
two-electron wave function.

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And if we could do it from
one-electron wave functions,

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then we're really in a good
position.

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But we ended the first quarter
of the semester on a downer,

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the idea that there's no way
electrons can be independent.

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They repel one another,
so orbitals are fundamentally

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wrong.

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Okay?

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So forget that.

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And here's our paradise;
it's gone.

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Okay?

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But are there tricks that would
allow us to salvage orbitals,

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to use them,
even though we know they're

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wrong?

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Okay, well the first,
the simple-minded trick,

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is Z-effective.

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We talked about how you could
scale the one-electron wave

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functions, the atomic orbitals,
for nuclear charge;

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if you increase the nuclear
charge they contract;

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and we saw what different
properties are proportional to.

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So if you have other electrons
in the atom,

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it could be that they could be
sort of approximated,

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as if they were just reducing
the nuclear charge,

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right?

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There's a certain repulsion,
as well as an attraction taking

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place, if there are other
electrons there.

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But maybe we can just reduce
the attraction -- right?

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-- and things will be sort of
-- at least corrected in the

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right direction.

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Right?

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So pretend that the other
electrons, or some fraction of

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them, are concentrated at the
nucleus.

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So from the point of view of an
electron you're thinking about,

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they just reduce the nuclear
charge;

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and that's a problem we know
how to handle,

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what happens if you change the
nuclear charge.

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Okay?

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So might there be an effective
nuclear charge that we could

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use?

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So we pretend the other
electrons just reduce the

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nuclear charge for the electron
we're interested in,

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the orbital,
the one-electron wave function

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that we want to find.

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So we're going to just to try
to find a one-electron wave

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function in a many-electron
problem.

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Okay?

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And this is what we know,
that 1s looks like that

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and ρ scales with
Z.

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So if I just reduce Z a
little bit, maybe I get

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something at least that's
corrected in the right

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direction.

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Okay now, these guys,
Clemente and Raimondi,

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who worked at IBM in the early
days,

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when IBM was the only place
that had computers that were as

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powerful as your laptop,
or almost as powerful as your

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laptop,
they did things like this,

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where they did good quality
calculations,

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better than this,
and then tried to match the

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wave functions they got,
by adjusting Z,

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so they looked like the wave
functions that you get with

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better approximations.

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Right?

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So they got a best fit to
better calculation.

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So they found that helium --
which has two electrons;

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so it has a nuclear charge of
two;

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Z=2 -- you get electrons
distributed, more or less right,

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if you pretend that the nuclear
charge is 1.69,

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instead of 2.

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Okay?

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So each electron sort of what's
called 'screens' the electron

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from the nucleus.

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Right?

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Okay, now in the case of
carbon, which has Z=6,

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you can also use Z_eff,
but it's different depending on

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which electron you're talking
about.

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Why would that be so?

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Why would it be that you'd use
a different screening constant,

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or a different effective
Z, if you're talking

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about the 1s orbital,
than if you're talking about

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the 2s orbital say?

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Why would it be different,
how much you want to reduce the

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nuclear charge?

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Sherwin?

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Student:
>.

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Prof: Yes,
the electrons that are outside,

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you don't care about.

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It's the electrons that are
between you and the nucleus that

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are screening you from the
nucleus.

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Right?

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So a 1s is already
really close to the nucleus.

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There's very little other
electron density that's inside

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it.

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Right?

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That's why in helium it wasn't
screened very much.

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But in the case of carbon,
the 1s electrons,

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which are way down near the
nucleus, have an effective

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charge that's almost 6;
5.67.

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But the 2s have 3.22.

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Right?

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The 1s electrons are all
down inside and making the

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nuclear charge appear small.

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Right?

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And to a certain extent the
other, the 2p electrons,

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are also inside;
to a certain extent,

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but not as much inside as the
1s is.

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So 3.22.

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And the Z_eff for
2p is different than for

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2s;
it's 3.14.

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What do you notice about those
two numbers?

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You might think they'd be more
or less the same;

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and they are more or less the
same.

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But 2s is slightly less
screened.

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It sees more of the nucleus
than the 2p does.

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Okay?

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So it looks like 2s gets
more down inside than 2p

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does.

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Does that surprise you?

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Russell?

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Student:  No,
because the 2p orbital

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dumbbell doesn't have anything
at all.

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Prof: Yes,
the dumbbell doesn't have

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anything at the nucleus.

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Right?

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It's got a node at the nucleus.

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Whereas 2s has that
first little core,

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down in -- the stuff that's
inside its first spherical node

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is down quite close to the
nucleus.

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Now we could look at that.

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There's what 2s looks
like, and here's the probability

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distribution that we talked
about: r^2 times the

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radial function squared.

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So this stuff that's in here is
inside.

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If we compare it with the
2p orbital,

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you see that this part is way
down inside.

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It's not being screened by the
2p electrons.

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Right?

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In fact, we could also look at
the 1s.

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This is how the 1s is
distributed.

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Right?

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So the 1s will screen
this.

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And of course it screens this
part of the 2s,

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and it screens this of the
2p.

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But it's not really trivial to
look at this and figure out

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which one would be more
screened,

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the 2s or the 2p,
because this part is further in

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than this part,
but this part is way down

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inside.

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So it's a balancing act as to
exactly which one would be

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bigger, and they're not very
different.

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But as it turned out,
according to Clemente and

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Raimondi and their calculations,
the Z_eff that you

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should use for 2s is a
little bit bigger than the one

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you should use for 2p.

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Kevin?

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Student:  Why are there
two peaks for 2s instead

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of 1?

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Prof: Because there's a
radial node;

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remember 2-ρ,
the function is 2-ρ.

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So here's a_o;
2 a_o,

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when the distance is two;
two minus two is zero;

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there's a node there.

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Okay?

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And it's been squared of course.

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The wave function changes sign,
but here we square it.

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Any other questions about this?

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Okay, now let's look at sodium.

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So sodium has a nuclear charge
of eleven,

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and you won't be surprised to
see that Z_eff_

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for the 1s
electrons of sodium is 10.63;

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almost 11, they are very little
screened.

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2s is 6.57.

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2p is 6.8.

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But 3s is only 2 and a
half, because it's further out

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still.

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So there are more electrons
inside, hiding the nucleus.

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Everybody got the idea of
what's going on here?

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You'll notice one thing sort of
funny here.

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Do you notice what?

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<<Students speak over one
another>>

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Prof: John?

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Student:  The 2p
is higher than --

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Prof: Ah,
it turns around!

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This time it's vice-versa.

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Right?

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So it's a balancing act,
and who knows?

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And there's nothing fundamental
about this.

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This was just adjusted by
Clemente and Raimondi,

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in order to get shapes that
looked pretty much like better

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quality shapes.

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Okay, so it's a very subtle
thing.

10:47.330 --> 10:50.000
But this is very crude. Right?

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Nobody argues that it's the
last word.

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So isn't there a better way to
do it?

10:55.700 --> 11:00.370
And there is a better way to do
it, to get back to orbitals,

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even for many-electron
problems, and that's called

11:04.240 --> 11:06.930
Self-Consistent Field,
or SCF.

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And it's a recipe for
calculating orbitals;

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for calculating better orbitals
than you would get with an

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effective-Z.

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So first you go through all the
electrons in an atom,

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or a molecule,
and you find an approximate

11:24.349 --> 11:26.889
form of the orbitals;
for example,

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you could use Z_eff,
to get something that's sort of

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approximate.

11:30.610 --> 11:31.410
Okay?

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So you have approximate wave
functions, square them;

11:34.928 --> 11:37.678
you have approximate
distributions for all the

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electrons.

11:38.418 --> 11:40.868
Now with Z_eff_
what we were pretending

11:40.873 --> 11:43.553
was that a certain fraction of
the other electrons are on the

11:43.549 --> 11:46.859
nucleus,
and the rest of them we forget;

11:46.860 --> 11:48.810
that's pretty crude.

11:48.809 --> 11:49.389
Okay?

11:49.389 --> 11:52.159
Now what we're going to do is
pretend we know,

11:52.163 --> 11:55.253
at least approximately,
how the other electrons are

11:55.245 --> 11:58.105
distributed in a cloud;
the other electrons.

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We're interested in one
electron, an orbital.

12:01.325 --> 12:01.905
Right?

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Now how will knowing how the
other electrons are distributed

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help us find the orbital we're
interested in,

12:08.745 --> 12:11.065
the one-electron wave function?

12:11.070 --> 12:13.670
What do you need in order to
solve a quantum mechanical

12:13.668 --> 12:14.148
problem?

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You need the mass of the
electron -- that's easy.

12:17.379 --> 12:19.299
What else do you need?

12:19.299 --> 12:21.809
The potential law;
what its energy is,

12:21.808 --> 12:23.938
its potential energy at
different positions.

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But if you know where the
nucleus is,

12:25.820 --> 12:30.880
or nuclei, and you know the
cloud of the other electrons,

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and assume they're just static
clouds,

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then you can -- it's laborious,
you need a computer to do it --

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but you can calculate the
potential that the electron

12:39.787 --> 12:42.807
you're interested in would have,
at different positions,

12:42.807 --> 12:45.767
if there were this fixed cloud
of other electrons and the

12:45.769 --> 12:46.299
nuclei.

12:46.298 --> 12:49.298
So you have the potential law,
which means you can find your

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one-electron orbital.

12:50.470 --> 12:52.850
Okay?

12:52.850 --> 12:55.020
Does everybody see the strategy
here?

12:55.019 --> 12:57.749
So we're going to do it one
electron at a time.

12:57.750 --> 13:01.790
So we fix all the other
electrons, all but one,

13:01.788 --> 13:05.198
from some crummy estimate,
like Z_eff_ or

13:05.201 --> 13:08.551
something like that,
calculate the potential for the

13:08.552 --> 13:10.782
one electron we're interested
in,

13:10.778 --> 13:14.088
and then we get -- use that
potential to calculate an

13:14.091 --> 13:17.721
orbital for that one electron
that's better than if we had

13:17.721 --> 13:18.871
used Z_eff.

13:18.869 --> 13:19.569
Right?

13:19.570 --> 13:22.230
Because it's not just putting a
certain fraction of the other

13:22.229 --> 13:24.799
electrons at the nucleus,
it's treating them as a cloud.

13:24.799 --> 13:26.069
Okay?

13:26.070 --> 13:30.880
So now we have a much better
guess for that one electron than

13:30.883 --> 13:34.173
we would've had earlier with
Z_eff,

13:34.173 --> 13:36.583
or whatever type of guess.

13:36.580 --> 13:38.810
What do you do next?

13:38.808 --> 13:41.418
Student:  Do it for
every orbital.

13:41.419 --> 13:42.269
Prof: Dana?

13:42.269 --> 13:43.219
Student:  You do it for
every other orbital,

13:43.215 --> 13:43.475
in sequence.

13:43.480 --> 13:44.780
Prof: Ah, one at a time.

13:44.779 --> 13:48.129
You now know how that one is
distributed in the cloud,

13:48.130 --> 13:48.890
much better.

13:48.889 --> 13:49.459
Right?

13:49.460 --> 13:52.720
And you take all the others but
one, fix them in their

13:52.717 --> 13:54.007
approximate clouds.

13:54.009 --> 13:57.529
Now you calculate a potential
for the second electron,

13:57.530 --> 13:58.860
and do that trick.

13:58.860 --> 14:02.460
Okay, so we repeat steps two
and three to improve the orbital

14:02.460 --> 14:03.840
for another electron.

14:03.840 --> 14:07.540
Then what do we do?

14:07.538 --> 14:09.508
Will, what would you do,
at this point?

14:09.509 --> 14:11.269
Student:  See how it
changes shape?

14:11.269 --> 14:12.369
Prof: Well yes,
it'll change shape.

14:12.370 --> 14:15.160
That second electron will have
a better shape now.

14:15.159 --> 14:16.059
So you've got that.

14:16.058 --> 14:18.508
You've got a good shape for the
first electron.

14:18.509 --> 14:21.749
Now you've got an improved
shape for the second electron.

14:21.750 --> 14:22.910
What do you do?

14:22.908 --> 14:25.318
Student:  Start from
step one or the first electron.

14:25.320 --> 14:27.110
Prof: Or you go,
if there's -- what if there are

14:27.111 --> 14:27.711
three electrons?

14:27.710 --> 14:30.520
Student:  Do all three
again.

14:30.519 --> 14:32.319
Prof: No,
you do the third one.

14:32.320 --> 14:33.170
Student:  Oh yes,
definitely.

14:33.169 --> 14:33.719
Prof: Right?

14:33.720 --> 14:34.730
And then the fourth,
fifth, sixth,

14:34.726 --> 14:36.006
until you get through all the
electrons.

14:36.009 --> 14:36.909
And then?

14:36.909 --> 14:37.559
Student:  Start over.

14:37.559 --> 14:38.659
Prof: Start over.

14:38.658 --> 14:41.798
So you improve all the
orbitals, one by one,

14:41.798 --> 14:45.958
and then you cycle back to
improve the first one again;

14:45.960 --> 14:46.930
go through them all.

14:46.929 --> 14:48.109
Then what do you do?

14:48.110 --> 14:50.070
<<Students speak over one
another>>

14:50.070 --> 14:51.830
Prof: Start again.

14:51.830 --> 14:52.490
Then what do you do?

14:52.490 --> 14:54.420
This is why it's good to have a
computer, right?

14:54.419 --> 14:57.129
Then what do you do?

14:57.129 --> 14:58.809
When do you stop?

14:58.808 --> 15:00.188
<<Students speak over one
another>>

15:00.190 --> 15:00.780
Student:  When they stop
changing.

15:00.778 --> 15:01.838
Prof: They'll stop
changing;

15:01.840 --> 15:04.080
at least -- it'll be like
Erwin meets Goldilocks

15:04.076 --> 15:06.806
where it's out in eighth decimal
place that things are changing.

15:06.808 --> 15:09.778
So then you know you've got it
as close as is reasonable to go.

15:09.779 --> 15:10.769
So you stop.

15:10.769 --> 15:14.379
And what do you call it,
when you get to that situation

15:14.379 --> 15:15.449
when you stop?

15:15.450 --> 15:18.360
The system is self-consistent.

15:18.360 --> 15:19.040
Right?

15:19.038 --> 15:22.178
That's why it's a
self-consistent field.

15:22.179 --> 15:23.469
Okay?

15:23.470 --> 15:26.960
So you quit when the orbital
steps -- shapes stop changing.

15:26.960 --> 15:32.610
So now you have the right wave
functions, the right orbitals.

15:32.610 --> 15:34.860
So now we've got the real thing.

15:34.860 --> 15:38.120
Right, or wrong?

15:38.120 --> 15:38.910
What could be wrong?

15:38.909 --> 15:41.929
We've got it self-consistent.

15:41.928 --> 15:45.718
Where's the weakness in the
assumption on which we've been

15:45.716 --> 15:46.576
doing this?

15:46.580 --> 15:47.640
Kevin?

15:47.639 --> 15:50.989
Student:  Well you're
assuming fixed positions for the

15:50.985 --> 15:51.705
other ones.

15:51.710 --> 15:52.630
Prof: I'm assuming what?

15:52.629 --> 15:54.409
Student:  You're
assuming fixed positions for the

15:54.408 --> 15:54.828
other ones.

15:54.830 --> 15:57.540
Prof: You're assuming
that all those other electrons,

15:57.538 --> 16:00.108
except the one you're working
on, are fixed in clouds.

16:00.110 --> 16:03.230
But they're not fixed. Right?

16:03.230 --> 16:05.460
The electrons can move around.

16:05.460 --> 16:09.720
So depending on where your
electron happens to be,

16:09.720 --> 16:13.550
those other electrons may
change their shape,

16:13.546 --> 16:15.716
at any given instant.

16:15.720 --> 16:18.480
Okay, it's still wrong,
because real electrons are not

16:18.480 --> 16:21.500
fixed in clouds;
they keep out of each other's

16:21.495 --> 16:23.785
way by correlating their motion.

16:23.787 --> 16:24.357
Right?

16:24.360 --> 16:26.980
They don't move independently
so that this one is a cloud and

16:26.976 --> 16:28.936
when this moves around it,
nothing happens.

16:28.940 --> 16:32.000
As this one moves,
when it gets near this place,

16:31.995 --> 16:33.355
this one gets away.

16:33.360 --> 16:34.270
Okay?

16:34.269 --> 16:37.799
So they keep out of other ways
by correlating their motion.

16:37.798 --> 16:43.398
So the true energy must be
lower, more favorable,

16:43.399 --> 16:45.849
than you calculate by
self-consistent field,

16:45.850 --> 16:48.960
because the orbitals are able
to get away from where you think

16:48.961 --> 16:49.831
they should be.

16:49.830 --> 16:52.740
Is everybody clear on why the
limit goes in that direction;

16:52.740 --> 16:54.720
why the true energy is lower?

16:54.720 --> 16:58.330
Because the electrons are
smarter than you are.

16:58.330 --> 17:00.360
They know to get out of each
other's way;

17:00.360 --> 17:03.710
or at least than you pretended.

17:03.710 --> 17:06.040
Okay?

17:06.039 --> 17:07.749
So what do you do?

17:07.750 --> 17:11.860
You hide the residual error so
that people won't embarrass you

17:11.863 --> 17:15.513
by saying, "What the heck
were you thinking?"

17:15.505 --> 17:16.175
Right?

17:16.180 --> 17:20.140
You say that when you go to the
Hartree-Fock limit --

17:20.140 --> 17:22.560
which is a fancy name,
named after two people that

17:22.560 --> 17:25.180
thought of doing this
self-consistent field thing,

17:25.180 --> 17:27.070
or thought of a method for
doing it --

17:27.068 --> 17:29.938
when you get to the
Hartree-Fock limit,

17:29.940 --> 17:34.270
that's a self-consistent field,
but there's going to be an

17:34.272 --> 17:36.632
error because of correlation.

17:36.630 --> 17:39.830
So you give it a fancy name so
people will think,

17:39.827 --> 17:42.757
"ah, what in the heck is
that?";

17:42.759 --> 17:43.929
they must really know a lot.

17:43.928 --> 17:44.218
Right?

17:44.220 --> 17:44.920
>

17:44.920 --> 17:46.960
Prof: So what do you
call the error?

17:46.960 --> 17:50.140
Do you call it error?

17:50.140 --> 17:50.820
No way.

17:50.818 --> 17:53.338
Sophie, do you know what you
call it?

17:53.338 --> 17:54.328
Student:  Correlation
energy.

17:54.328 --> 17:56.298
Prof: Correlation
energy.

17:56.300 --> 17:56.810
Right?

17:56.808 --> 17:59.338
There is no such thing as
correlation energy.

17:59.339 --> 18:01.309
That's not a fundamental energy.

18:01.308 --> 18:04.778
It's not like Coulomb's Law or
gravity or something like that.

18:04.775 --> 18:05.225
Right?

18:05.230 --> 18:08.380
It's just the error you make
when you do self-consistent

18:08.376 --> 18:10.606
field, that the true energy is
lower.

18:10.608 --> 18:12.978
Now if you want to measure
correlation energy,

18:12.983 --> 18:14.413
what do you have to know?

18:14.410 --> 18:16.070
Student:  True energy.

18:16.068 --> 18:18.598
Prof: You got to know
the true energy,

18:18.597 --> 18:20.947
so that you know how bad your
estimate is.

18:20.954 --> 18:21.534
Right?

18:21.528 --> 18:24.238
And where do you get the
correct energy,

18:24.242 --> 18:28.212
or the true electron density
that you have an error in?

18:28.210 --> 18:29.160
Student:  Experiment.

18:29.160 --> 18:31.110
Prof: You get it from
experiment.

18:31.108 --> 18:35.598
Or you get it from some
whopping calculation that's not

18:35.599 --> 18:37.679
as simple as SCF,
Okay?

18:37.680 --> 18:39.370
For example,
there's a thing called

18:39.366 --> 18:41.196
"configuration
interaction",

18:41.202 --> 18:43.832
which we won't know anything
about, and don't need to.

18:43.834 --> 18:44.434
Right?

18:44.430 --> 18:47.220
But it's a much more
complicated calculation.

18:47.220 --> 18:50.410
It takes into account the fact
that electrons keep out of each

18:50.407 --> 18:51.137
other's way.

18:51.140 --> 18:54.490
Or there's another one called
"density functional

18:54.491 --> 18:57.151
theory",
which is also approximate.

18:57.150 --> 18:58.460
They're all approximations.

18:58.460 --> 19:00.630
You can't solve the real
equation.

19:00.630 --> 19:03.100
But these are better
approximations than just

19:03.101 --> 19:04.451
self-consistent field.

19:04.450 --> 19:05.920
Okay?

19:05.920 --> 19:09.250
But they're hard to think
about, because they involve so

19:09.249 --> 19:12.639
much manipulation that it's hard
to reason about them.

19:12.640 --> 19:15.840
Self-consistent field is easier
to understand.

19:15.839 --> 19:17.079
Okay?

19:17.078 --> 19:21.748
So if we're really lucky
though, correlation energy might

19:21.746 --> 19:24.636
be negligible;
it might be so small we don't

19:24.644 --> 19:26.864
care about it,
in which case we're golden,

19:26.856 --> 19:28.256
for practical purposes.

19:28.259 --> 19:31.149
We can use orbitals,
self-consistent field orbitals,

19:31.147 --> 19:34.147
treat things as if they were
independent electrons.

19:34.150 --> 19:35.480
Okay?

19:35.480 --> 19:37.690
And then we're in business.

19:37.690 --> 19:38.470
Okay?

19:38.470 --> 19:42.220
So we should think about the
magnitude of energies and

19:42.215 --> 19:45.035
whether we care about
correlation energy,

19:45.042 --> 19:47.732
this error that is orbital
theory.

19:47.730 --> 19:52.510
What do I mean by saying
orbital theory?

19:52.509 --> 19:53.839
What's an orbital?

19:53.838 --> 19:55.278
<<Students speak over one
another>>

19:55.279 --> 19:56.679
Prof: A one-electron
wave function.

19:56.680 --> 19:59.600
But we're trying to understand
many-electron problems.

19:59.598 --> 20:03.508
Can we analyze many-electron
problems as sums of

20:03.506 --> 20:06.906
one-electrons,
as sums of things that come

20:06.913 --> 20:08.413
from orbitals?

20:08.410 --> 20:10.520
Is the whole equal to the sum
of the parts,

20:10.516 --> 20:11.866
the parts being orbitals?

20:11.868 --> 20:14.718
We know that orbitals have to
be fundamentally wrong,

20:14.721 --> 20:17.851
but if the error is -- if the
correlation energy is really

20:17.847 --> 20:19.217
small, we don't care.

20:19.220 --> 20:22.050
So should we care about the
error in orbital theory?

20:22.048 --> 20:25.368
So here is a scale,
a logarithmic scale,

20:25.365 --> 20:29.355
of energy changes that occur
when things happen,

20:29.363 --> 20:32.853
and how big are these energy
changes.

20:32.849 --> 20:34.279
Let's start at the beginning.

20:34.279 --> 20:37.569
So we take a bunch of neutrons
and protons and bring them

20:37.570 --> 20:40.390
together to make a nucleus,
and energy comes out.

20:40.392 --> 20:40.982
Right?

20:40.980 --> 20:46.270
And the amount of energy that
is given off when a C-12 nucleus

20:46.268 --> 20:49.648
is formed is 2*10^9
kilocalories/mol.

20:49.650 --> 20:50.840
How do I know?

20:50.838 --> 20:54.428
Because I look at the mass,
the rest mass of the proton and

20:54.430 --> 20:57.090
the neutron, and I look at the
mass of C-12,

20:57.092 --> 20:59.512
and mass was lost when it came
together.

20:59.507 --> 21:00.247
Right?

21:00.250 --> 21:05.150
And the amount of mass lost,
E=mc^2, is 0.1 atomic mass

21:05.147 --> 21:09.227
units, which is 2*10^9^(
)kilocalories/mol.

21:09.230 --> 21:11.570
So that's a lot of energy.

21:11.569 --> 21:12.319
Okay?

21:12.319 --> 21:16.239
Now so we got C^+6.

21:16.240 --> 21:18.470
Now we're going to put two
electrons on it,

21:18.468 --> 21:20.218
the 1s electrons of
carbon.

21:20.219 --> 21:20.749
Right?

21:20.750 --> 21:21.780
Bingo!

21:21.778 --> 21:25.698
And that gives us 2*10^4
kilocalories/mol,

21:25.700 --> 21:28.570
given off when that happens.

21:28.569 --> 21:29.179
Okay?

21:29.180 --> 21:32.060
Now that is our old,
familiar friend.

21:32.058 --> 21:36.148
What do you notice about the
ratio of these energies?

21:36.150 --> 21:38.080
10^5, right?

21:38.078 --> 21:41.068
So it's 10^5 smaller,
like the ratio of my hair to

21:41.067 --> 21:42.407
the width of the room.

21:42.411 --> 21:42.961
Right?

21:42.960 --> 21:46.720
So much, much smaller energy
involved in putting electrons

21:46.723 --> 21:50.293
into the 1s shell of
carbon, than in putting the

21:50.289 --> 21:52.139
carbon nucleus together.

21:52.140 --> 21:53.390
Okay?

21:53.390 --> 21:56.550
Then we're going to put the
four valence electrons onto

21:56.547 --> 21:57.597
carbon -- right?

21:57.599 --> 21:59.409
-- the 2s and 2p.

21:59.410 --> 22:03.630
And with that we get another
3*10^3.

22:03.630 --> 22:05.990
So it's an order of magnitude
less.

22:05.990 --> 22:08.970
The 1s electrons are
bound much more strongly than

22:08.971 --> 22:12.971
the 2s and 2p;
and you know that,

22:12.973 --> 22:17.653
from this scaling,
Z^2/n^2.

22:17.650 --> 22:18.200
Okay?

22:18.200 --> 22:21.170
Because you could use
Z_eff to guess how the

22:21.165 --> 22:24.665
1s's are affecting the
energies -- the nuclear charge

22:24.665 --> 22:26.085
for the 2s's.

22:26.088 --> 22:29.278
So you could lower the nuclear
charge from six to four,

22:29.277 --> 22:32.817
because there are already two
electrons way down in there;

22:32.818 --> 22:34.638
and then you have that
n^2^( )as well.

22:34.640 --> 22:37.980
So you could scale the energy
and find that it's an order of

22:37.978 --> 22:38.938
magnitude less.

22:38.940 --> 22:40.010
Okay.

22:40.009 --> 22:46.299
So then what are we going to do
next, now that we have atoms?

22:46.298 --> 22:48.258
Going to put them together to
make bonds.

22:48.259 --> 22:49.139
Okay.

22:49.140 --> 22:53.010
So we make four single bonds
from carbon, but they're to

22:53.005 --> 22:54.125
other carbons.

22:54.130 --> 22:56.930
So for this one carbon we
should count only half of each

22:56.932 --> 22:57.852
energy -- right?

22:57.848 --> 23:00.228
-- because of half of it we'll
assign to the other carbon.

23:00.230 --> 23:04.120
So half of four single bonds --
a single bond is order of

23:04.124 --> 23:06.424
magnitude 100 kilocalories/mol.

23:06.420 --> 23:08.650
So that's about 200
kilocalories/mol.

23:08.650 --> 23:12.260
So another order of magnitude
down, the energy in making

23:12.255 --> 23:12.775
bonds.

23:12.779 --> 23:13.809
Okay?

23:13.808 --> 23:16.458
And then we can have non-bonded
contacts.

23:16.460 --> 23:19.480
Now there are different kinds
of non-bonded interactions

23:19.480 --> 23:20.580
between molecules.

23:20.578 --> 23:23.518
But typically,
to be worth talking about,

23:23.519 --> 23:27.709
they're in the range of one to
twenty kilocalories/mol;

23:27.710 --> 23:32.530
so another order of magnitude
down, or more.

23:32.528 --> 23:36.108
Okay, and the weakest of all,
the weakest bond known,

23:36.105 --> 23:39.125
or interaction known,
attractive interaction,

23:39.130 --> 23:41.330
is between two helium atoms.

23:41.328 --> 23:44.598
It turns out that two helium
atoms, although they don't form

23:44.596 --> 23:46.996
a bond, are attractive;
you know, things at long

23:47.001 --> 23:49.391
distance are attractive and then
they become repulsive.

23:49.390 --> 23:54.390
So the minimum energy distance
is fifty-two angstroms -- right?

23:54.390 --> 23:58.390
-- thirty-some times as long as
a normal bond.

23:58.390 --> 24:03.920
And the depth of that well,
how favorable is that energy,

24:03.917 --> 24:08.167
is 2*10^-6 kilocalories/mol;
so that's nothing.

24:08.170 --> 24:12.210
But all these things we're
looking at down below here are

24:12.214 --> 24:14.314
all based on Coulomb's Law.

24:14.309 --> 24:16.219
The first one was not. Right?

24:16.220 --> 24:19.490
The nuclear binding energy is
not Coulomb's Law.

24:19.490 --> 24:22.160
But all these others are
Coulomb's Law.

24:22.160 --> 24:28.810
But they're all 10^5^(,) up to
10^15,^( )times weaker than what

24:28.807 --> 24:31.487
goes on in the nucleus.

24:31.490 --> 24:35.130
And that means if you made any
error,

24:35.130 --> 24:39.240
at all, in the energy of the
nucleus,

24:39.240 --> 24:43.800
it would completely wipe out
anything that had to do with

24:43.795 --> 24:45.175
Coulomb's energy.

24:45.176 --> 24:45.906
Right?

24:45.910 --> 24:47.650
Because it's so much bigger.

24:47.650 --> 24:51.250
So an infinitesimal error in
nuclear energy,

24:51.250 --> 24:54.440
or change in nuclear energy
during a reaction,

24:54.440 --> 24:57.650
would completely wipe out
everything that we're talking

24:57.647 --> 24:58.007
about.

24:58.005 --> 24:58.535
Right?

24:58.538 --> 25:01.888
So this sounds like we would
really worry about it,

25:01.885 --> 25:05.965
except -- so a 0.001% change in
nuclear energy would overwhelm

25:05.968 --> 25:07.368
everything Coulombic.

25:07.373 --> 25:08.113
Right?

25:08.108 --> 25:13.238
But fortunately nuclear energy
doesn't care about chemistry.

25:13.240 --> 25:17.100
If you change from one
arrangement of atoms to another,

25:17.096 --> 25:19.946
the nuclear energy doesn't
change at all.

25:19.954 --> 25:20.674
Right?

25:20.670 --> 25:24.120
So why is that good?

25:24.119 --> 25:27.389
You can just cancel it out.

25:27.390 --> 25:30.390
The starting material and
products of any reaction have

25:30.390 --> 25:32.280
exactly the same nuclear energy.

25:32.278 --> 25:36.198
So if you're not a physicist,
you can just forget nuclear

25:36.200 --> 25:38.930
energy, even though it's so
enormous.

25:38.930 --> 25:39.460
Okay?

25:39.460 --> 25:40.940
Is that clear?

25:40.940 --> 25:43.020
So forget nuclear energy,
we don't have to worry about

25:43.022 --> 25:43.222
it.

25:43.220 --> 25:45.760
All we have to worry about are
these Coulomb things.

25:45.759 --> 25:49.959
Now, so black that out.

25:49.960 --> 25:50.510
Okay.

25:50.509 --> 25:53.369
Now here's how big correlation
energy is.

25:53.368 --> 25:55.558
It depends on what problem
you're dealing with,

25:55.556 --> 25:58.166
how big the molecule is,
what kind of atoms there are in

25:58.171 --> 25:58.981
the molecule.

25:58.980 --> 26:02.710
The error you make in SCF is
different, for different cases.

26:02.710 --> 26:05.820
But for the kind of cases we're
interested in,

26:05.819 --> 26:09.209
normal organic molecules,
it's of the order of 100

26:09.208 --> 26:10.658
kilocalories/mol.

26:10.660 --> 26:13.070
Now that's an error.

26:13.069 --> 26:17.189
Do you care about that error?

26:17.190 --> 26:20.690
The error in nuclear energy,
an error, would have been

26:20.693 --> 26:23.673
enormous and completely wiped
anything out.

26:23.670 --> 26:26.320
Is this enormous?

26:26.318 --> 26:32.168
Do we care about an error of
100 kilocalories/mol?

26:32.170 --> 26:33.860
Do we?

26:33.858 --> 26:41.528
Would you care about an error
of -- Devin, what do you say?

26:41.529 --> 26:44.049
Would you care if you made an
error of 100 kilocalories/mol?

26:44.048 --> 26:49.308
How big is that in the scale of
things we're talking about?

26:49.308 --> 26:50.758
Student:  Doesn't seem
too big.

26:50.759 --> 26:52.309
Prof: Compared to what?

26:52.308 --> 26:53.168
Student:  That's the
question.

26:53.170 --> 26:53.930
>

26:53.930 --> 26:55.460
Prof: Yes.

26:55.460 --> 26:57.310
What are we interested in?

26:57.309 --> 26:57.969
Student:  Bonds.

26:57.970 --> 26:59.700
Prof: Bonds.

26:59.700 --> 27:00.460
How big is it compared to bonds?

27:00.460 --> 27:01.560
Student:  Fifty times as
big.

27:01.558 --> 27:02.368
Prof: It's as big as a
bond.

27:02.369 --> 27:04.359
A bond is 100 kilocalories/mol.

27:04.358 --> 27:09.058
That means that we're -- that's
a disaster -- right?

27:09.059 --> 27:10.919
-- unless what?

27:10.920 --> 27:13.470
The nuclear would've been a
disaster too.

27:13.470 --> 27:15.550
Why isn't it?

27:15.548 --> 27:17.818
Errors in nuclear energy are
not a disaster.

27:17.819 --> 27:18.519
Why?

27:18.519 --> 27:20.979
<<Students speak over one
another>>

27:20.980 --> 27:23.640
Prof: Because they don't
change.

27:23.640 --> 27:26.010
The starting material and the
product have the same.

27:26.009 --> 27:27.859
So the same thing would be true
here.

27:27.858 --> 27:30.738
If correlation energy didn't
change,

27:30.740 --> 27:33.740
from one arrangement of atoms
to another arrangement of the

27:33.743 --> 27:35.963
same atoms,
then it would just cancel out

27:35.959 --> 27:37.139
and you wouldn't care.

27:37.140 --> 27:40.940
Everybody got that idea?

27:40.940 --> 27:45.000
Okay, so the correlation energy
is about equal to the magnitude

27:45.002 --> 27:45.792
of a bond.

27:45.788 --> 27:47.808
But how big are
changes in

27:47.805 --> 27:50.385
correlation energy,
when you change the arrangement

27:50.387 --> 27:51.057
of atoms?

27:51.058 --> 27:56.608
Well changes tend to be about
10 to 15% as big as bond energy,

27:56.607 --> 28:00.607
or 10 to 15% as big as
correlation energy.

28:00.608 --> 28:02.648
So correlation energy does
change.

28:02.650 --> 28:06.290
You make different errors for
different arrangements of the

28:06.292 --> 28:07.112
same atoms.

28:07.108 --> 28:11.208
But those errors are about 10
to 15% as big as a bond.

28:11.210 --> 28:15.430
Now do you care about
correlation?

28:15.430 --> 28:16.650
You don't care as much.

28:16.650 --> 28:19.030
So you'll get approximate ideas.

28:19.028 --> 28:24.098
If you don't care within -- if
you're satisfied with getting

28:24.102 --> 28:28.662
about 80% of the right answer,
then you don't care.

28:28.660 --> 28:29.880
Okay?

28:29.880 --> 28:32.930
So that implies that orbital
theory is fine,

28:32.926 --> 28:36.326
as long as what you're
interested in is answering

28:36.328 --> 28:40.368
qualitative, rather than fine
quantitative questions.

28:40.369 --> 28:40.989
Okay?

28:40.990 --> 28:43.240
So if you want to get numbers
really right,

28:43.240 --> 28:45.950
and the magnitude of the number
really makes a difference to

28:45.948 --> 28:47.698
you,
a few percent change,

28:47.699 --> 28:51.509
then you have to do something
better than use orbitals.

28:51.509 --> 28:52.299
Okay?

28:52.298 --> 28:55.588
But if you just want to
understand why bonds work,

28:55.589 --> 28:57.269
then orbitals are fine.

28:57.269 --> 28:58.299
Okay?

28:58.298 --> 29:02.628
Because the changes in
correlation energy aren't that

29:02.630 --> 29:03.130
big.

29:03.130 --> 29:07.620
So, but for these properties,
for non-bonded contacts,

29:07.617 --> 29:12.357
especially for helium-helium,
the correlation is the only

29:12.361 --> 29:14.311
game in town often.

29:14.308 --> 29:16.788
That's what holds helium atoms
together.

29:16.788 --> 29:19.508
Helium atoms are nuclei,
positively charged;

29:19.509 --> 29:21.589
electrons, negatively charged.

29:21.589 --> 29:26.069
So nucleus repels nucleus;
electrons repel electrons;

29:26.069 --> 29:30.449
nucleus attracts electrons;
nucleus attracts electrons.

29:30.446 --> 29:31.006
Right?

29:31.009 --> 29:33.889
At first, at fifty-three
angstroms, or fifty-two

29:33.893 --> 29:36.723
angstroms, far apart,
those essentially cancel.

29:36.715 --> 29:37.325
Right?

29:37.328 --> 29:42.548
But the motion of the electron
around this nucleus correlates

29:42.545 --> 29:47.235
with the motion of the electron
around this nucleus.

29:47.240 --> 29:49.350
They tend to be in-phase with
one another.

29:49.348 --> 29:53.488
So at any given time you have
plus, minus, plus,

29:53.490 --> 29:56.840
minus, and they attract one
another.

29:56.838 --> 30:02.558
So precisely what holds helium
to helium is correlation.

30:02.558 --> 30:07.308
So if what you're interested in
is bonding, then use orbitals,

30:07.305 --> 30:07.845
fine.

30:07.848 --> 30:11.228
But if you're interested in
non-bonded interactions,

30:11.234 --> 30:13.894
then correlation can be a big
problem.

30:13.890 --> 30:14.430
Okay?

30:14.430 --> 30:18.820
But we're talking about bonds
now, not correlation.

30:18.819 --> 30:20.809
Okay, so orbitals can't be true.

30:20.808 --> 30:23.348
That's clear,
because electrons influence one

30:23.347 --> 30:25.307
another, they repel one another.

30:25.308 --> 30:27.388
If you have just one electron,
fine;

30:27.390 --> 30:29.830
more electrons,
orbitals can't be true.

30:29.828 --> 30:33.878
But still we'll use them,
to understand bonding and

30:33.881 --> 30:36.961
structure and energy and
reactivity.

30:36.960 --> 30:40.490
And we know we won't get
precise values,

30:40.493 --> 30:44.393
we'll be off by one to ten
kilocalories/mol,

30:44.391 --> 30:48.471
but we'll get insight that's
very useful.

30:48.470 --> 30:49.330
Okay.

30:49.328 --> 30:54.348
Now what gives atomic orbitals
their shape?

30:54.348 --> 30:59.428
Why does this particular
orbital have a node,

30:59.432 --> 31:01.052
for example?

31:01.048 --> 31:04.368
We know Coulomb's Law tends to
attract the electrons to the

31:04.366 --> 31:04.936
nucleus.

31:04.940 --> 31:11.480
Why does it have a node and
spread out?

31:11.480 --> 31:13.630
Because of kinetic energy --
right?

31:13.630 --> 31:16.020
-- this curvature of wave
functions that's required to

31:17.140 --> 31:20.050
So it's kinetic energy that
gives them their shape.

31:20.048 --> 31:23.658
Or if you double the nuclear
charge, the thing gets half as

31:23.664 --> 31:24.044
big.

31:24.038 --> 31:27.158
That's Coulomb's Law,
sucking it in.

31:27.156 --> 31:27.866
Right?

31:27.868 --> 31:32.488
So the potential energy scales
the radius through the formula

31:32.487 --> 31:34.767
for ρ;
we've seen that.

31:34.769 --> 31:38.139
But the kinetic energy is what
creates nodes.

31:38.140 --> 31:41.240
So the 2s has that
spherical node.

31:41.240 --> 31:45.380
Or this orbital here has a
conical, two cones,

31:45.375 --> 31:49.325
and a spherical node;
it's the 4d orbital.

31:49.328 --> 31:52.138
So kinetic energy is what
creates the shapes.

31:52.140 --> 31:56.720
The charge just scales things
in and out.

31:56.720 --> 32:01.950
Now if we use orbitals,
how should we calculate the

32:01.952 --> 32:04.572
total electron density?

32:04.568 --> 32:07.358
Well we have two one-electron
wave functions.

32:07.358 --> 32:10.658
We know the density of electron
one, at this position,

32:10.663 --> 32:12.663
by squaring its wave function.

32:12.660 --> 32:15.810
We know the density of electron
two at that position,

32:15.809 --> 32:17.749
by squaring its wave function.

32:17.750 --> 32:21.860
How do we get the total
electron density at that same

32:21.859 --> 32:22.729
position?

32:22.730 --> 32:23.880
How would you get it?

32:23.880 --> 32:25.880
You know how much of electron
one is there,

32:25.881 --> 32:27.121
its probability density.

32:27.119 --> 32:28.589
You know two.

32:28.589 --> 32:30.109
How do you get the total?

32:30.109 --> 32:34.239
ilana?

32:34.240 --> 32:37.070
If you know how much of
electron one is there,

32:37.068 --> 32:39.958
and you know how much of
electron two is there,

32:39.960 --> 32:42.790
how much total electron density
is there?

32:42.789 --> 32:45.069
How would you get it?

32:45.069 --> 32:46.389
Can't hear very well.

32:46.390 --> 32:47.170
Student:  You'd just add
them together.

32:47.170 --> 32:48.890
Prof: Yes,
add them together.

32:48.890 --> 32:50.580
A whole is the sum of its parts.

32:50.579 --> 32:50.979
Okay?

32:50.980 --> 32:54.400
So the total density is the sum
of those two squared wave

32:54.396 --> 32:56.206
functions;
just right.

32:56.210 --> 32:59.570
But notice it's a sum;
it's not a product.

32:59.568 --> 33:01.878
This is not a question of joint
probability.

33:01.880 --> 33:05.750
It's not what's the probability
that electron one and electron

33:05.747 --> 33:07.837
two are there at the same time?

33:07.839 --> 33:09.369
That's not the question.

33:09.368 --> 33:11.848
The question is,
what's the total probability of

33:11.848 --> 33:13.378
finding any electron there?

33:13.380 --> 33:13.800
Okay?

33:13.798 --> 33:16.848
So it's a sum,
it's not a product.

33:16.849 --> 33:17.729
Okay?

33:17.730 --> 33:20.190
Now we had this question you
looked at before:

33:20.186 --> 33:21.986
How lumpy is the nitrogen atom?

33:21.990 --> 33:25.360
This picture is taken from a
recent organic text and they had

33:25.364 --> 33:28.634
a fancy graphic program or an
artist or something that drew

33:28.625 --> 33:30.595
something that looks very
realistic.

33:30.595 --> 33:31.265
Right?

33:31.269 --> 33:33.109
But we can check it because we
know the formulas.

33:33.108 --> 33:35.988
And if we want to get the total
electron density,

33:35.990 --> 33:38.730
we add the electron density of
the electron that's in the

33:38.730 --> 33:41.570
p_x and the one that's in
the p_y and the one

33:41.568 --> 33:43.378
that's in the p_z
orbital.

33:43.380 --> 33:45.000
So we square them,
and we get this,

33:44.997 --> 33:47.517
and we sum it up to get the
total electron density;

33:47.519 --> 33:51.599
and it's some constant times
x^2+y^2+z^2 times

33:51.602 --> 33:56.372
e^-ρ^( );
after we've squared. Right?

33:56.369 --> 33:58.589
And how can you simplify that?

33:58.589 --> 34:02.269
What's x^2+y^2+z^2?

34:02.269 --> 34:03.489
It's r^2.

34:03.490 --> 34:05.150
Okay?

34:05.150 --> 34:08.650
So how does it depend on
θ?

34:08.650 --> 34:10.270
How does it depend on
φ?

34:10.269 --> 34:11.279
Elizabeth?

34:11.280 --> 34:11.750
Student:  It doesn't.

34:11.750 --> 34:12.710
Prof: It doesn't depend.

34:12.708 --> 34:13.448
What does it look like?

34:13.449 --> 34:13.909
Student:  A sphere.

34:13.909 --> 34:15.069
Prof: It's a sphere.

34:15.070 --> 34:15.720
It's a ball.

34:15.719 --> 34:18.059
It doesn't look like this thing.

34:18.059 --> 34:18.849
So it's spherical.

34:18.849 --> 34:21.609
So forget, for all the elegance
of that picture,

34:21.606 --> 34:25.146
forget it;
they're not showing you the

34:25.152 --> 34:25.892
truth.

34:25.889 --> 34:28.589
Okay, or this problem we had
before of looking at the

34:28.592 --> 34:30.622
cross-section of the CN Triple
Bond,

34:30.619 --> 34:33.479
where we sliced it and turned
it and thought it might look

34:33.483 --> 34:36.473
like a cloverleaf,
or maybe some kind of diamond

34:36.474 --> 34:37.814
shape or something.

34:37.809 --> 34:39.029
But in fact it's round.

34:39.030 --> 34:40.070
Why?

34:40.070 --> 34:43.290
Because
(2p_x)^2+(2p_y)^2 depends

34:43.289 --> 34:48.239
on x^2+y^2,^( )which is
r^2, in two dimensions.

34:48.244 --> 34:49.074
Right?

34:49.070 --> 34:53.960
So it's cylindrical,
doesn't depend on the angle

34:53.963 --> 34:56.363
around the bond axis.

34:56.360 --> 34:57.540
Okay, so this is what we've
seen.

34:57.539 --> 35:01.229
We've seen three-dimensional
reality, hydrogen-like atoms.

35:01.230 --> 35:03.040
We talked about hybridization.

35:03.039 --> 35:06.889
We saw that orbitals are
fundamentally wrong but that we

35:06.885 --> 35:10.865
can recover from the orbital
approximation and use it,

35:10.869 --> 35:14.429
if we -- for example,
with self-consistent field;

35:14.429 --> 35:17.969
as long as we're not interested
in getting the very finest

35:17.972 --> 35:21.022
energy but can be satisfied with
approximation.

35:21.018 --> 35:23.588
Now we're going to move on to
something more interesting,

35:23.586 --> 35:24.546
which is molecules.

35:24.550 --> 35:28.600
We're going to look first at
Plum-Pudding molecular orbitals,

35:28.597 --> 35:31.357
and then understanding bonds,
and overlap,

35:31.362 --> 35:32.782
and energy-match.

35:32.780 --> 35:35.860
Now there are lots of different
ways of looking at the electron

35:35.862 --> 35:36.612
distribution.

35:36.610 --> 35:39.670
You know this story,
presumably, about different

35:39.670 --> 35:43.120
blind men doing experiments on
an elephant and getting

35:43.121 --> 35:45.141
completely different ideas.

35:45.139 --> 35:48.199
The same is true of the
electrons in a molecule.

35:48.199 --> 35:51.259
So here's the electron density
in a hydrogen molecule.

35:51.260 --> 35:53.770
It's calculated,
but there are ways to get that

35:53.766 --> 35:56.216
kind of information
experimentally as well.

35:56.219 --> 35:57.799
So, and you know how contours
work;

35:57.800 --> 36:03.900
it's a lot more dense near the
origin, near the nuclei.

36:03.900 --> 36:09.360
So which contour do we want to
look at in order to understand

36:09.364 --> 36:09.824
it?

36:09.820 --> 36:13.260
Well we can choose our contour,
and we get different pictures,

36:13.264 --> 36:16.264
different understanding,
depending on which contour we

36:16.257 --> 36:16.877
choose.

36:16.880 --> 36:19.320
For example,
we could choose,

36:19.318 --> 36:22.628
way down, a very high electron
density.

36:22.628 --> 36:23.498
Right?

36:23.500 --> 36:26.990
And then we see just a set of
two atoms;

36:26.989 --> 36:28.739
it just looks like two atoms.

36:28.737 --> 36:29.157
Right?

36:29.159 --> 36:31.219
We don't see the fact that it's
a molecule.

36:31.219 --> 36:34.309
Where have we done this before?

36:34.309 --> 36:37.049
Student:
>

36:37.050 --> 36:40.440
Prof: Andrew?

36:40.440 --> 36:42.100
Shai?

36:42.099 --> 36:42.729
Pardon me?

36:42.730 --> 36:43.430
Student:  Difference
density.

36:43.429 --> 36:44.689
Prof: Not difference
density.

36:44.690 --> 36:45.820
Student:  Well,
we were subtracting to get the

36:45.822 --> 36:46.282
difference density.

36:46.280 --> 36:47.140
Prof: Yes.

36:47.141 --> 36:49.021
We looked at total electron
density;

36:49.019 --> 36:50.429
it just looked like atoms.

36:50.429 --> 36:53.129
We had to do the difference in
order to see that it was not

36:53.125 --> 36:53.725
just atoms.

36:53.730 --> 36:55.480
It was essentially just atoms.

36:55.480 --> 36:58.170
So if you look at high density,
you see just a set of atoms;

36:58.170 --> 36:59.880
no excitement there.

36:59.880 --> 37:04.950
So that's the molecule as a set
of atoms, just a set of atoms.

37:04.949 --> 37:05.879
Okay?

37:05.880 --> 37:09.380
Or we could take a somewhat
lower electron density contour

37:09.380 --> 37:12.880
and we'd see that the atoms are
a little bit distorted.

37:12.880 --> 37:14.800
Or we could do a difference
map, as Shai says,

37:14.800 --> 37:17.060
in order to see that they're a
little bit distorted,

37:17.059 --> 37:20.649
because bonding distorts the
shape of the atoms a little bit.

37:20.650 --> 37:23.610
But first -- and we'll,
this is what we're going to

37:23.614 --> 37:27.164
look at shortly --
but first I want to get some

37:27.155 --> 37:31.865
insight by looking at the very
lowest electron density.

37:31.869 --> 37:35.549
So that would be molecules
formed from a set of atoms,

37:35.550 --> 37:37.630
rather than as a set of atoms.

37:37.634 --> 37:38.264
Right?

37:38.260 --> 37:39.630
There are little changes
because of the bonds.

37:39.630 --> 37:41.790
But how about if we look way
out there?

37:41.789 --> 37:45.699
Now, if you don't look really
close, it looks spherical,

37:45.695 --> 37:49.455
it looks just like an atom,
or almost like an atom.

37:49.460 --> 37:52.360
And if you went even further
out, it would get spherical,

37:52.356 --> 37:53.646
for all you could tell.

37:53.650 --> 37:54.540
Okay?

37:54.539 --> 37:57.999
So that's the molecule looking
like an atom,

37:57.996 --> 38:02.256
with whatever number of
electrons the molecule has.

38:02.260 --> 38:05.130
So that's the molecule as one
atom.

38:05.130 --> 38:06.890
The only difference is that the
nucleus,

38:06.889 --> 38:08.869
which for an atom would be in
the middle,

38:08.869 --> 38:12.559
has been split,
to give two nuclei,

38:12.559 --> 38:16.239
which of course distorts the
shape of the electrons a little

38:16.242 --> 38:18.512
bit,
if you move -- cut the nucleus

38:18.507 --> 38:20.027
in two and split it out.

38:20.030 --> 38:21.570
So we want to look at a few
molecules,

38:21.570 --> 38:24.690
from this point of view,
as single atoms --

38:24.690 --> 38:28.550
because we know about atoms now
-- but distorted by the fact

38:28.548 --> 38:31.098
that the nuclei has been split
apart.

38:31.099 --> 38:36.179
So this is nuclei embedded in a
cloud of electrons.

38:36.179 --> 38:40.429
Now what gave those electrons
their shape, the shape of the

38:40.434 --> 38:43.374
cloud in which this thing is
embedded;

38:43.369 --> 38:46.039
what gave them their shape?

38:46.039 --> 38:48.239
We just talked about this a
short time ago.

38:48.239 --> 38:49.749
What gives orbitals their shape?

38:49.750 --> 38:51.300
Dana?

38:51.300 --> 38:52.230
Can't hear very well.

38:52.230 --> 38:54.080
Student:  A desire to be
as far apart as possible?

38:54.079 --> 38:56.409
Prof: That's potential
energy, right.

38:56.409 --> 39:00.059
But that's not what gives --
that's not what creates nodes.

39:00.059 --> 39:01.849
That just causes things to
spread out.

39:01.849 --> 39:04.439
What gives them their
characteristic shape of nodes,

39:04.438 --> 39:06.568
planar nodes,
spherical nodes and so on?

39:06.570 --> 39:07.770
Elizabeth?

39:07.769 --> 39:08.359
Student:  Kinetic energy.

39:08.360 --> 39:09.330
Prof: It's the kinetic
energy.

39:09.329 --> 39:12.639
And the same kinetic energy
considerations will apply in a

39:12.637 --> 39:14.087
molecule as in an atom.

39:14.090 --> 39:16.470
It's always curvature of the
wave function divided by the

39:16.469 --> 39:17.149
wave function.

39:17.150 --> 39:21.270
So the electrons are dispersed
by electron repulsion,

39:21.273 --> 39:25.083
and noded, given nodes,
by the kinetic energy;

39:25.079 --> 39:28.109
and the kinetic energy also
causes them to spread out,

39:28.112 --> 39:29.432
as we've seen before.

39:29.429 --> 39:32.299
Okay, but you see then that.

39:32.300 --> 39:35.360
although Thomson was wrong
about the atom,

39:35.355 --> 39:39.525
it wasn't a plum pudding of
nuclei embedded in a cloud of

39:39.529 --> 39:41.019
positive charge.

39:41.018 --> 39:43.998
But a molecule is
a plum pudding!

39:44.000 --> 39:48.530
It's nuclei embedded in cloud
of negative charge,

39:48.532 --> 39:53.542
but that cloud of negative
charge is given its form by

39:53.539 --> 39:57.739
kinetic energy largely;
also potential energy,

39:57.739 --> 39:58.639
as you say.

39:58.639 --> 40:01.139
So it's backwards from what
Thomson thought.

40:01.139 --> 40:05.109
But his idea wasn't a silly
one, it just happened to be

40:05.112 --> 40:06.512
wrong, for atoms.

40:06.510 --> 40:09.850
So here's a piece of plum
pudding;

40:09.849 --> 40:12.419
which you know is like
fruitcake.

40:12.420 --> 40:16.660
So how do the plums distort the
pudding?

40:16.659 --> 40:17.529
Right?

40:17.530 --> 40:21.990
We know what atoms would look
like now.

40:21.989 --> 40:22.929
Right?

40:22.929 --> 40:25.429
But if you split the nucleus
into several nuclei,

40:25.429 --> 40:27.979
and moved them around as the
plums in the pudding,

40:27.983 --> 40:30.383
they'll change the shape of the
electrons.

40:30.380 --> 40:30.730
How?

40:30.730 --> 40:34.060
That's what we want to look at
today.

40:34.059 --> 40:37.819
Okay, so first we're going to
look at methane and ammonia,

40:37.824 --> 40:40.604
and we want to understand them
visually.

40:40.599 --> 40:45.809
Why do the orbitals in these
molecules have the shapes they

40:45.809 --> 40:46.259
do?

40:46.260 --> 40:50.520
Okay, so there are four pairs
of valence electrons -- there

40:50.516 --> 40:54.696
are also two core electrons,
1s electrons on carbon

40:54.701 --> 40:56.171
and on nitrogen.

40:56.170 --> 40:59.790
So we want to compare the
molecular orbitals to the atomic

40:59.786 --> 41:02.636
orbitals of neon,
which has the same number of

41:02.641 --> 41:05.191
electrons;
four electron pairs with

41:05.190 --> 41:07.200
n=2 in the neon atom.

41:07.199 --> 41:08.959
So there are the same number of
electrons as neon.

41:08.960 --> 41:10.970
So they should,
at a big distance,

41:10.974 --> 41:14.274
if you look at very low
contours, it should look like a

41:14.273 --> 41:15.133
Neon atom.

41:15.130 --> 41:17.640
But what does it look like as
you get closer?

41:17.639 --> 41:21.229
Okay, so here's a 1s
orbital of the Neon atom.

41:21.228 --> 41:21.778
Right?

41:21.780 --> 41:24.860
And here's the 1s
orbital of methane;

41:24.860 --> 41:27.300
it's that net that's shown,
the red net.

41:27.300 --> 41:29.050
Everybody see it?

41:29.050 --> 41:32.440
And here's the -- the contour
level that we're drawing is

41:32.438 --> 41:35.218
where the orbital,
you know, which level are we

41:35.222 --> 41:36.012
choosing?

41:36.010 --> 41:39.950
We're choosing 0.001
electrons/cubic angstrom is the

41:39.947 --> 41:43.037
level we've chosen to draw,
of the onion.

41:43.036 --> 41:43.806
Right?

41:43.809 --> 41:45.129
And here it is for nitrogen.

41:45.130 --> 41:50.080
The blue one there is the
lowest molecular orbital.

41:50.079 --> 41:51.779
But they're just little
spheres, right?

41:51.780 --> 41:59.270
They're like the 1s of
neon.

41:59.268 --> 42:03.868
Okay, so the core orbitals are
like the 1s of the carbon

42:03.869 --> 42:05.279
or nitrogen atom.

42:05.280 --> 42:09.030
They're tightly held,
they aren't distorted very much

42:09.027 --> 42:12.267
because the nuclei have split
apart, right?

42:12.269 --> 42:14.489
And they're boring.

42:14.489 --> 42:16.839
So forget them,
we won't talk about them

42:16.836 --> 42:17.436
anymore.

42:17.440 --> 42:22.030
We'll focus on the valence
orbitals where the bonding

42:22.032 --> 42:23.802
action will occur.

42:23.800 --> 42:26.110
Now, there are eight valence
electrons.

42:26.110 --> 42:28.590
That means two to an orbital.

42:28.590 --> 42:32.350
So there'll be four molecular
orbitals that have electrons in

42:32.349 --> 42:33.539
them, in methane.

42:33.539 --> 42:36.379
And they are arranged in energy
this way.

42:36.380 --> 42:39.140
There's a lowest energy,
and then there are three that

42:39.137 --> 42:40.747
have exactly the same energy.

42:40.750 --> 42:43.820
Do you remember what we call
those when several orbitals have

42:43.818 --> 42:44.738
the same energy?

42:44.739 --> 42:45.359
Students:  Degenerate.

42:45.360 --> 42:47.090
Prof: Degenerate. Okay?

42:47.090 --> 42:50.160
So there are three degenerate
molecular orbitals for CH_4 that

42:50.159 --> 42:51.719
have exactly the same energy.

42:51.719 --> 42:54.539
In the case of ammonia it's a
little different.

42:54.539 --> 42:57.849
There's one lowest one,
and then there are three,

42:57.849 --> 43:01.089
but one of them is higher than
the other two;

43:01.090 --> 43:02.830
there are only two degenerate
ones there.

43:02.829 --> 43:05.149
Incidentally,
what does that remind you of,

43:05.150 --> 43:08.470
to have one that's a little bit
lower and then three that are

43:08.465 --> 43:09.565
the same energy?

43:09.570 --> 43:11.000
Have you ever seen that before?

43:11.000 --> 43:15.630
Student:  2p and
--

43:15.630 --> 43:16.300
Prof: Ah!

43:16.300 --> 43:17.640
2s and 2p,
right?

43:17.639 --> 43:21.129
There's one 2s and three
2p's, in an atom.

43:21.132 --> 43:21.642
Right?

43:21.639 --> 43:24.739
Here's a molecule and there's
one that's lower and then three

43:24.735 --> 43:25.865
that are equivalent.

43:25.869 --> 43:29.989
Okay, now there's the lowest
one, and it's a 2s

43:29.992 --> 43:30.772
orbital.

43:30.768 --> 43:35.858
It's the neon's 2s
orbital that's been distorted by

43:35.856 --> 43:41.296
taking four protons out of the
nucleus and pulling them out to

43:41.302 --> 43:44.072
be where the hydrogens are.

43:44.070 --> 43:44.910
Okay?

43:44.909 --> 43:47.239
So you can see how this has
been distorted,

43:47.237 --> 43:50.337
by distorting the electron
cloud into the direction where

43:50.338 --> 43:52.388
the protons have been pulled
out.

43:52.389 --> 43:54.509
Do you see that?

43:54.510 --> 43:57.840
Okay, now you don't see the --
you have to remove the ball

43:57.836 --> 44:01.276
there, that's the carbon or the
nitrogen, to see that little

44:01.280 --> 44:02.390
spherical node.

44:02.389 --> 44:05.209
It's way down near the origin
that made it a 2s

44:05.206 --> 44:05.626
orbital.

44:05.630 --> 44:06.110
Right?

44:06.110 --> 44:07.710
That's the spherical node.

44:07.710 --> 44:10.820
And it doesn't look like a
sphere because of the algorithm

44:10.818 --> 44:14.088
the computer uses to draw this,
connecting dots with straight

44:14.092 --> 44:14.422
lines.

44:14.418 --> 44:15.018
Right?

44:15.018 --> 44:16.658
But it is more or less
spherical;

44:16.659 --> 44:19.089
a little bit distorted by the
fact that you've pulled these

44:19.094 --> 44:19.644
things out.

44:19.639 --> 44:23.049
Okay, now what do these
molecular -- these are molecular

44:23.047 --> 44:23.727
orbitals.

44:23.730 --> 44:26.720
But that, you see,
is a 2p_x orbital,

44:26.719 --> 44:29.079
that's been a little bit
distorted.

44:29.083 --> 44:29.783
Right?

44:29.780 --> 44:35.070
Notice that the difference
between the CH_4 and the NH_3,

44:35.067 --> 44:41.107
there was a proton that went up
in CH_4, which pulled them up.

44:41.110 --> 44:43.960
But if you didn't have that,
it didn't go up as high.

44:43.960 --> 44:44.400
Right?

44:44.400 --> 44:48.050
But then essentially this is
the 2p_x orbital of the

44:48.045 --> 44:51.685
molecule, distorted from the
atom by pulling protons out of

44:51.693 --> 44:52.703
the nucleus.

44:52.699 --> 44:53.259
Okay?

44:53.260 --> 44:55.860
Or that one, what's that one?

44:55.860 --> 44:58.180
It's 2p_y,
but you have to rotate it to

44:58.184 --> 44:58.654
see it.

44:58.650 --> 45:01.770
If we rotate it 90 degrees
here, you can see that it's a

45:01.771 --> 45:03.021
2p_y orbital.

45:03.018 --> 45:06.798
And again, the CH_4 is
distorted at the top because two

45:06.804 --> 45:09.614
protons came out and stretched
it out.

45:09.610 --> 45:11.620
Same thing in nitrogen.

45:11.619 --> 45:15.929
Now we go to the third of these
2p orbitals,

45:15.932 --> 45:19.782
which is different in nitrogen;
for the nitrogen case,

45:19.775 --> 45:20.585
higher in energy.

45:20.590 --> 45:21.450
There they are.

45:21.449 --> 45:25.859
Why is that orbital higher in
energy than the others in

45:25.860 --> 45:26.760
Nitrogen?

45:26.760 --> 45:27.760
Why isn't it so good?

45:27.760 --> 45:32.180
Because you have electrons up
in that red region that don't

45:32.175 --> 45:35.315
have a proton there;
it's not being stabilized.

45:35.320 --> 45:41.050
Here, the electron density that
went up is around a proton.

45:41.050 --> 45:44.800
There the top lobe of the
p orbital doesn't have a

45:44.802 --> 45:46.412
proton stabilizing it.

45:46.409 --> 45:49.529
So it's higher in energy.

45:49.530 --> 45:51.430
And it turns out to be more
reactive.

45:51.429 --> 45:52.509
What do you call it?

45:52.510 --> 45:55.750
<<Students speak over one
another>>

45:55.750 --> 45:58.040
Prof: That's the
unshared pair.

45:58.039 --> 46:01.329
The high-energy reactive
electrons are the ones that

46:01.333 --> 46:04.243
don't have a proton stabilizing
them there.

46:04.239 --> 46:07.519
Now there are also vacant
orbitals.

46:07.518 --> 46:08.288
Right?

46:08.289 --> 46:11.379
Remember, you could have any
number of orbitals.

46:11.380 --> 46:15.170
We're looking at the ones that
you can make from the 2s

46:15.173 --> 46:16.173
and 2p.

46:16.170 --> 46:24.550
But we have a number of atomic
orbitals that we can use.

46:24.550 --> 46:27.430
We're looking at the lowest
set, the ones that come just

46:27.434 --> 46:29.484
from 1s orbitals of
Hydrogen,

46:29.480 --> 46:33.720
and 2s and 2p
orbitals of carbon or nitrogen.

46:33.719 --> 46:36.609
But there are eight such
orbitals: four,

46:36.610 --> 46:40.020
2s and three
2p's, on the central

46:40.021 --> 46:44.611
atom;
four on the hydrogen for CH_4.

46:44.610 --> 46:48.330
So there'll be a number of
other combinations you can make

46:48.326 --> 46:51.386
from mixing those,
which won't have electrons to

46:51.391 --> 46:52.371
go in them.

46:52.369 --> 46:53.519
But let's look at them.

46:53.518 --> 46:57.488
So here are the next four
orbitals made from the valence

46:57.492 --> 47:00.312
level orbitals of the atoms
involved.

47:00.309 --> 47:04.059
So what you see on CH_4 that
this is the 3s orbital.

47:04.059 --> 47:06.029
It has a radial node. Right?

47:06.030 --> 47:08.620
Or pardon me,
yes, a radial or a spherical

47:08.617 --> 47:11.077
node here, that surrounds the
sphere.

47:11.079 --> 47:14.839
So it's red sign inside,
blue sign outside.

47:14.840 --> 47:20.060
The same thing is true of the
NH_3, but you have to rotate in

47:20.063 --> 47:21.633
order to see it.

47:21.630 --> 47:25.170
That's the lowest vacant,
the lowest unoccupied

47:25.172 --> 47:26.512
molecular orbital.

47:26.510 --> 47:32.410
And if we rotate it here,
you can see the inner part.

47:32.409 --> 47:35.859
And bear in mind that there are
two spherical nodes.

47:35.860 --> 47:40.350
There's one that's really tiny,
that's inside anything we can

47:40.347 --> 47:41.167
see here.

47:41.170 --> 47:44.230
Okay, then there's this
orbital, which is the

47:44.233 --> 47:47.433
3d_x^2-y^2,
looked at from a funny angle

47:47.434 --> 47:50.294
where it's not very clear what
it is.

47:56.027 --> 47:59.917
like that cross,
or whatever you call it,

47:59.918 --> 48:02.738
of the 3d_x^2-y^2.

48:02.739 --> 48:05.129
But why doesn't it look just
like that?

48:05.130 --> 48:06.980
Why are they distorted?

48:06.980 --> 48:09.760
Why does it have this funny
U-shape here.

48:09.760 --> 48:13.230
Instead of this lobe,
this lobe, here and here,

48:13.231 --> 48:15.271
the lobes moved up there.

48:15.269 --> 48:17.309
Why?

48:17.309 --> 48:18.179
Angela?

48:18.179 --> 48:18.849
Pardon me?

48:18.849 --> 48:19.569
Student:  There are
protons.

48:19.570 --> 48:21.690
Prof: That's where the
protons pulled them,

48:21.693 --> 48:22.693
the potential energy.

48:22.690 --> 48:26.010
But the fundamental shape,
before it got distorted by the

48:26.010 --> 48:28.680
potential energy of where the
protons are,

48:28.679 --> 48:31.959
the fundamental shape was from
the kinetic energy.

48:31.960 --> 48:34.520
It was like the
3d_x^2-y^2.

48:34.519 --> 48:36.079
Or how about this one?

48:36.079 --> 48:40.689
There you can see -- but it's
clearer still if we rotate it --

48:40.688 --> 48:43.558
that again it's that X-shaped
thing.

48:43.559 --> 48:48.039
But the ones on top have been
pulled out by the protons.

48:48.039 --> 48:49.009
Or this one.

48:49.010 --> 48:52.030
That's the one that has a
doughnut around the middle.

48:52.030 --> 48:56.500
But the doughnut has been
pulled down by the three

48:56.501 --> 48:59.971
hydrogens that aren't on the z
axis.

48:59.969 --> 49:02.649
Or there it is, rotated.

49:02.650 --> 49:03.480
Okay?

49:03.480 --> 49:07.360
Now I hoped to get -- let's
just start ethane and methanol,

49:07.364 --> 49:11.124
and then we can get to more
interesting things in it next

49:11.117 --> 49:11.717
time.

49:11.719 --> 49:15.229
It has a lot more pairs of
electrons, ethane and methanol.

49:15.230 --> 49:18.330
So we can compare those MOs to
the AOs of Argon,

49:18.327 --> 49:21.357
which has the same number of
electron pairs.

49:21.360 --> 49:25.310
Actually I should quit now and
let you get to your next

49:25.313 --> 49:26.343
appointment.

49:26.340 --> 49:32.000