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PROFESSOR: The molecules and
Born-Oppenheimer approximation.

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OK, we all know
that molecules are

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a lot harder to
solve than atoms,

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and even atoms are
not that easy once you

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have more than one
electron because

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of the electrostatic
repulsion, but molecules

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are significantly different
in that one of the greatest

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simplicities that
we had with atoms

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is that the atom is such
that the potential created

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by the nucleus is
spherically symmetric.

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When you have a molecule
you have separate nuclei

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and therefore your
spherical symmetry is gone,

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and whether you
have one electron

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or more than one electron,
there is no spherical symmetry.

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All our tools of angular
momentum don't help us much.

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We have to start
the problem anew.

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So the difficulty with
molecules is basically

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that the potential for the
electrons, where they move

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is not spherically symmetric.

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There is another
thing that helps

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us, however, is that there's a
nice separation of mass scales.

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You have the mass,
little m, of the electron

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and the nuclear mass, this
could be the mass of one nucleus

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or a mass of a few
nuclei or doesn't matter,

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but this is small, and typically
it is 10 to the minus 4.

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You know, an electron is
about 2,000 times lighter

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than a single proton.

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If you already have a nice
nucleus with a couple of them,

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you're several
thousandths already there,

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so this is a nice number.

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So we want to understand the
scales of the physics of this,

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so let's assume we
have a molecule,

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maybe a has few centers and
then there's electronic cloud,

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and it has a size
of the order of a.

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So we can estimate
at first sight what

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are the energies involved here?

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What is the energy of the
electron, for example?

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And you can say the
momentum of the electron

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is of the order of h bar
times the size of the cloud.

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It's an estimate, and the
energies, the electron

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energies, are
basically determined

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by the size in which
there are confined,

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which is a, h bar, and
the mass of the electron.

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Those are the electronic
energy, so h squared over

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m a squared, those
are the right units,

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is p squared over m, the units,
so it's natural like this.

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So this is the
electronic configuration

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and typical electronic
energy, so what

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is the approximate picture
of the physics of a molecule?

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Basically you have nuclei, and
the nuclei repel each other,

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they want to be away.

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Now the electrons
come in and then they

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fly around the nuclei, and the
electrons attract the nuclei

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and attract the nuclei, so the
electrons, reasonably placed,

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compensate the
repulsion of the nuclei

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by creating some new attraction.

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At the end you get some
kind of equilibrium.

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It's not classical because
you cannot create stable

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equilibrium between
classical particles.

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This is one of the amazing
things about quantum mechanics.

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You know very well
an atom would not

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exist without quantum
mechanics, you

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would have stable classical
orbits, circular orbits,

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but they radiate
energy so they decay.

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If you have a molecule you
have two repelling positive

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charged particles.

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In [? E&M ?] you
saw the case when

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you put the negative
charge in the middle

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and you create equilibrium,
but that's absolutely unstable.

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You move that electric charge
a micron and then off it goes,

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the whole thing is unstable.

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But quantum mechanics
solves those problems

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and our physical picture is
that of repelling nuclei,

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an attracting cloud
of electrons creating

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an equilibrium in which
the nuclei are reasonably

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well-localized, they're
quantum particles

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but they're reasonably
well-localized,

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while the electrons are
completely delocalized.

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That is our picture, they
are moving in that way.

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So our picture will
be-- we have that

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and we have slow vibrations
of the nuclei that

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are mostly localized.

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And you can imagine calculating
the electronic configuration

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for any particular
separation of the nuclei

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and that's the cloud
for this separation.

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As the nuclei vibrate, the
electronic configuration

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probably adjusts
adiabatically to this thing

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in the instantaneous
eigenstates,

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and they work that way.

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We're going to do that in
trying to understand things.

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I also want to
emphasize this kind

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of remarkable property
of an electron

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to create the bound state.

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So we are not amazed on the
fact that if we have a proton

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and we can have an electron
here, we create a bound state,

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and that's a hydrogen atom.

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But suppose I bring in
another proton from infinity.

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Well, I-- if I
bring it, I'm going

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to start bringing it here,
this kind of is neutral

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but eventually this proton
is gonna polarize this thing.

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OK, I'm going to bring it
and then if I come too close

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it repels it so it will
be a force repelling it

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and we have to figure
out what it does.

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Suppose I try to bring it in.

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Do you think that an electron
is capable of stabilizing

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two protons?

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Can they create a cloud in which
an electron stabilizes the two

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protons?

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Is that possible
or not possible?

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There's a lot of protons
for one single electron.

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Is that possible or not?

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All right, you say yes,
yes, any other opinions?

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Can that happen?

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Can the electric produce a bound
state of two protons like that?

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AUDIENCE: Isn't
that just hydrogen?

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PROFESSOR: Sorry?

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AUDIENCE: Isn't
that just hydrogen?

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PROFESSOR: It is what?

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AUDIENCE: Isn't
that just hydrogen?

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Or [INAUDIBLE]
hydrogen [INAUDIBLE]??

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PROFESSOR: OK, yes,
it can produce it,

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and this would be called,
with an electron here,

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the hydrogen
molecule ion, so H2+.

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It's a hydrogen molecule
that had these two things

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and it lost one electron,
and yes, this can be done

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and we'll analyze it.

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And now you can
get greedy and say,

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can you get an electron to
stabilize three protons?

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Yes or no?

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AUDIENCE: [INAUDIBLE]

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PROFESSOR: No, cannot get
it, it's just too much work

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for an electron.

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It cannot do it.

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There are theorems
like that, Elliott

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Lieb, a mathematical
physicist, has proven

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all kinds of these
things, and we

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might discuss some of
that in the homework,

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it's kind of nice stuff.

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So we will be looking at
the physics of those things

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with a little detail.

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So now of let's go
back to our molecule

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here and estimate the
vibrational nuclear energy.

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So we have this picture,
the electrons are there,

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the nuclei are kind
of semi-classical,

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they are roughly localized
and they're varying,

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and how would we estimate the
nuclear vibration frequency?

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So estimate nuclear vibration.

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So we think of a
harmonic oscillator,

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b squared over 2
mass of the nucleons,

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plus a restoring force,
H subnuclear oscillation.

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So this is an estimate so
it's not very rigorous,

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but I think it will
be clear enough.

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Now this restoring
force has to do

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with the cloud of the electrons
that adjusts and does things.

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As you separate
the nuclei you have

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to restore the
restoring energies

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due to the electrons
and the Coulomb forces,

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but it does not depend on the
mass, M, the restoring force.

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So this k depends on the
electrons and Coulomb forces

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but definitely not on the
mass of this quantity.

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So k has units of
E over L squared,

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and for energy we
have this energy

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of the electron and
length, the length,

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the scale of the molecule,
so k is proportional to h

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squared over ma to the fourth.

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It just cannot depend on
the mass of the nuclei,

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that's definitely not the
origin of the restoring force,

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so if the only mass you can use
is the mass of the electron it

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can only be that.

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But then the frequency,
omega, is square root

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of k over M, that is
the mass of the nuclei,

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so the nuclear
frequency is this,

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and it's therefore
equal to h squared

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over ma fourth one
over M, that was k,

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and this is square root of
little m over capital M times

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square root of h squared over
m squared a to the fourth,

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I borrowed a little m
up and a little m down.

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So from here we get
that h bar omega

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nuclear is square root of
little m over capital M,

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h squared over ma squared.

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If you take the square root here
you get h bar over ma squared,

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but there was another
h bar I put in.

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And now we get a
sensible equation

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in which we see that the
nuclear energies for oscillation

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are much smaller,
significantly smaller,

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so E nuclear oscillations go
like square root of m over M

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is 10 to the minus 4,
we say square root is 10

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to the minus 2,
times E electronic.

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So it's considerably less which
is this, so if you think--

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I want to make another
observation here on timescales,

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so omega n, we have
it here, omega n,

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or we have it here as well,
is equal to this times

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this quantity, so let
me say it this way.

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The omega electronic
for vibrations

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00:15:23,330 --> 00:15:30,680
is the electronic energy divided
by h bar, I need h bar over ma

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squared, but here we see
that that's precisely

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that square root, so
the omega nuclear is

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equal to little m over
M omega electronic.

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Now that is nice because it says
that the frequency associated

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00:16:01,300 --> 00:16:05,260
to the nuclear thing
is much smaller

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than the timescales associated
with electronic changes.

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00:16:11,030 --> 00:16:15,490
So this is a good reason
for our idiomatic thinking

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00:16:15,490 --> 00:16:21,190
that as the nuclei
slowly oscillate,

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the electronic cloud remains
in that particular eigenstate

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and thus and jump.

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00:16:27,310 --> 00:16:31,250
It's our use of the
adiabatic approximation.

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00:16:31,250 --> 00:16:37,090
A last comment is that
the rotational energy

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00:16:37,090 --> 00:16:42,670
of the molecule goes like
the angular momentum squared

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00:16:42,670 --> 00:16:45,670
over twice the
moment of inertia,

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00:16:45,670 --> 00:16:49,900
L squared is h squared
times some numbers,

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00:16:49,900 --> 00:16:53,170
as usual, L times
L plus 1, and here

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00:16:53,170 --> 00:16:57,250
you get the mass of the
molecule times the size squared

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00:16:57,250 --> 00:16:59,200
of the molecule.

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00:16:59,200 --> 00:17:09,099
These energies are like
little m over capital M times,

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00:17:09,099 --> 00:17:16,780
this is roughly h squared
over little ma squared,

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00:17:16,780 --> 00:17:19,060
and therefore the
rotational energies

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are even smaller than
the electronic energies.

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00:17:22,930 --> 00:17:28,270
The rotational energies
go like little m

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00:17:28,270 --> 00:17:31,375
over capital M times
the electronic energies.

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00:17:36,580 --> 00:17:40,860
So you have the electronic
energies, the nuclear energies

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00:17:40,860 --> 00:17:44,890
of oscillation, which
go like square root,

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00:17:44,890 --> 00:17:47,590
and then finally
the smallest of them

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are the rotation, in
which the molecule rotates

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00:17:50,470 --> 00:17:52,540
like a solid object.

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00:17:52,540 --> 00:17:55,900
We'll continue this next
time and calculate a few more

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things and molecules.