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PROFESSOR: We began
our introduction

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to molecules last
time and tried to get

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a picture of the scales
that are involved.

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In these objects we spoke of
a lattice of nuclei and clouds

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of electrons in which a
molecule had some scale,

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A. Then we had electronic
energies E, electronic.

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We had vibrational energies.

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This is from the nuclei.

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And we had rotational energies.

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And they were one
bigger than the other

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and bigger than the last.

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In fact, the electronic
energies were

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bigger than the vibrational
energies of the nuclei

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and bigger than the rotational
energies of the whole molecule

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when it rotates as a solid body.

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In fact, the ratio was like 1 to
square root of little m over M,

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where little m represents
electron mass, and capital M,

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the nuclear mass.

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And so this number could
be 10 to the minus 2,

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and then you have m over M.
So that's the proportions.

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So this is larger than
the second one, like 1

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is larger than that, and
the ratio between these two

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is this.

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So that's what we've discussed.

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And we said that,
in some sense, there

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was an adiabatic approximation
in the vibration of the nuclei.

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As the nuclei vibrate, they pull
the electronic clouds with it

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in an adiabatic way.

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That is, if you solved
for the electronic cloud

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as a function of
position, that would

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be a family of eigenstates.

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Say the ground state
of the electronic cloud

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is a function of position, this
would be a family of states.

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If the positions of the
nuclei change in time,

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you could use those as
instantaneous eigenstates.

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And there is a sense in which
this is a good approximation,

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given that the
timescale associated

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to the vibrations
of the nuclei is

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much bigger than the
timescale associated

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to any variation in the
electronic configuration.

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This is just because of
the scales of the energies.

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So let's implement this idea
in an approximation that

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is used to solve molecules.

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And we'll discuss
it in all detail.

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I'll skip one step
of the derivation.

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One calculation will
be in the notes,

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but I don't want to go
through the details in class.

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And we will appreciate the form
of the nuclear Hamiltonian, how

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it behaves.

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So this is going to be the
Born-Oppenheimer approximation.

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And we will consider
the situation

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where we have N, nuclei--

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capital N, nuclei-- and
little m, electrons.

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So it's a many-body wavefunction
and a many-body situation.

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In such cases, your
notation is important.

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You have to define
labels that help

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you distinguish this situation.

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So here are the
labels we're going

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to use for the nuclei, P
alpha and R alpha, where alpha

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denotes which nuclei
you're talking about.

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So it goes from 1
up to capital N,

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because there are
capital N nuclei

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for each nucleus, the first,
the second, the third.

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There is a momentum operator
and a position operator,

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each one of which is three
components, because molecules

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live in three dimensions.

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So this is one vector
and another vector

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for each value of alpha.

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In fact, these are operators.

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We're doing quantum
mechanics, so these

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are our canonical
pairs for the nuclei,

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canonical pairs for the nuclei.

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We need similar variables
for the electrons,

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and we'll use little
p and little r,

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both vectors, both operators.

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And this time this i runs
from 1 to lower case n.

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And these are the canonical
pairs for the electrons.

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So when we write
the Hamiltonian,

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it should be a Hamiltonian that
depends on all those variables.

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And we can write
the Hamiltonian,

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because we know the
physics of this situation.

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We think of this is
a lattice of nuclei,

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and there is the cloud
of the electrons,

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and we have a coordinate system.

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Here is maybe capital
R1 and capital R2.

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They're all there.

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And well, when we
write Hamiltonian,

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we think of the electrons at
some points and write things.

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So let's write the Hamiltonian.

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So this is going to be
the total Hamiltonian.

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What should I include?

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I should include kinetic
terms for each of the nuclei.

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So I should put sum over alpha.

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I don't have to repeat
here from 1 to capital N.

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You know already
what alpha runs over.

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P alpha vector
squared over 2M alpha.

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M alpha is the mass
of alpha nucleus.

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It could be a collection
of protons and neutrons.

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Then there's going to be--

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and we get a little more
schematic-- a potential that

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depends nucleus with nucleus.

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So the nucleus,
among each other,

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have a Coulomb potential.

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So there's going to be a
potential that represents here,

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and it will depend
on the various R's.

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I could write--
this looks funny.

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You say, which R?

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Well, it depends on
all the capital R's.

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So I could write
depends on this set

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that, but it's a
little too cumbersome.

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I'll just write V
of R, like this.

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And this is, if the nuclei
lived alone, that would be it.

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This would be the kinetic
energies and the potential

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between the nuclei.

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Then there's going to be what
we can call a Hamiltonian that

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only involves the electrons.

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In some sense, that gives
dynamics to the electrons--

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not only involves
electrons, gives

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dynamics to the electrons.

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And this Hamiltonian, H
e, is going to depend on--

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well, this big Hamiltonian
for all the molecules

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depends on the two
canonical pairs

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for times N times little n.

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This Hamiltonian for
the electron will depend

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on the p's, will depend on the
r's, and it will also depend

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on the capital R's.

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And you can think of it,
and that's reasonable.

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Suppose you're an electron.

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Who affects you?

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Well, you get affected
by your electron friends,

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and you get affected
by the nuclei

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and therefore, by the positions
of the nuclei, as well.

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So this is the electron
part of the Hamiltonian.

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And it would be given by a sum
of kinetic energy, as usual.

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So i, sum over i this time,
little p, i squared over 2m.

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Let's assume, of course, all
the electrons are the same mass.

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And then we would have, just
in this shorthand, a potential

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that represents the
interaction of the electrons

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with the nuclei.

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And that potential would
depend on the R's, on the R's.

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And finally-- my picture maybe
should be moved the little

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to the right--

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there's a term, the electron
potential that just depends

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on the R's.

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So all these potentials
are Coulomb potentials,

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Coulomb from nucleus-nucleus,
nucleus-electron,

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electron-electron.

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So here it is.

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You've written the Hamiltonian.

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And if you have three
nuclei and five electrons,

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you could write
all the equations.

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And it's a nice thing that
you can write the Hamiltonian,

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and you could dream of
putting it into a computer,

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and it will tell you
what the molecule is,

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and that's roughly true.

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But even for a good computer
nowadays, this is difficult.

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So you have to try to think how
you can simplify this problem.

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So one way to think about
it is to think again

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of the physics of the situation.

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We'll have a
separation of scales.

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It's lucky we have that
separation of scales,

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very light electrons,
very heavy nuclei.

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So let's think of a
fixed nuclear skeleton,

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and consider electron
states associated

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to that fixed nuclear skeleton.

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The nuclear skeleton
is not fixed.

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In principle, the nuclei
are not classical particles

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with fixed positions.

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They're going to vibrate.

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But we're trying to
understand this problem,

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and to some approximation,
we can roughly

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think of them localized.

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So let's exploit that
and use the vibrations.

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So for large M alpha, this
[INAUDIBLE] to large M

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alpha, consider a
fixed nuclear skeleton.

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So that means fixed R alpha,
all the R alphas, and fix.

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And now calculate
the electron states

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as a function of our alpha.

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So you simplify the problem.

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Ignore all this
dynamics of the nuclei,

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all this nuclear-nuclear
interaction.

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Focus on the electrons
as if the nuclear

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are completely fixed, and try
to figure out the dynamics.

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So there are going to be
many electronic states.

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This is electrons in
some fixed potential.

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Any particle in quantum
mechanics in a fixed potential.

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There are many
energy eigenstates.

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So these electrons are going to
have many energy eigenstates.

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So let's try to decide on a
name for this wavefunction.

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So I will call them
phi for electrons.

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If there are wavefunctions
that are wavefunctions

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of the electrons,
naturally, they just

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depend on the electron
positions, nothing else.

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A wavefunction for a particle
depends on the position.

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Now, this r hides a little of--

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thing.

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This really means that
phi of r1, r2, r3, r4,

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because they're
little n electrons,

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so it's not just one variable.

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So I can say this is r1,
r2, all of them, r little n.

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That's a wavefunction.

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Now, we said there are
many of those states.

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So there will be the
ground state, the next one,

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the next one, the next one.

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So we should put an i in this.

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Maybe an i in this
is the wrong letter,

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given that they have i there--

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k in this.

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But there is more
dependence here.

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There is implicit dependence on
the positions of the lattice,

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because these wave
functions depend

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on what lattice square
did you place the nuclei.

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00:15:15,560 --> 00:15:17,950
At this moment, you're
placing them arbitrarily.

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So this means that this
wavefunction really

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depends, of course, of how
did you build the lattice?

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Did you build the
lattice this way,

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or did you build it this way?

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It makes a difference.

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So it depends on the
capital R's, as well,

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which is the position
of the lattice.

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So here is our wavefunction.

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I will simplify the
writing by writing

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phi of capital R, little
r, and k here, see.

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So what equation do
we demand from this?

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Well we have the
electron Hamiltonian.

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So that's what we should solve.

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We should solve H
electron, on phi R, k of r.

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We'll have some energies,
and those energies

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will be electronic energies.

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That's for electronic energy.

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It will depend on k-- those are
the various energies, as well--

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and what else?

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Certainly the energies
don't depend on r.

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That's your eigenstate.

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But they can depend
and will depend

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on capital R. Capital R
is the parameters that

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define your lattice.

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Clearly they should
depend on that.

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And then you have phi k, R, r.

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So this is the equation
you should solve in order

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to find electronic states
associated with a skeleton.

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And now suppose you wanted
to find a complete solution

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of the Schrodinger equation.

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You say, ah, approximations.

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00:17:40,450 --> 00:17:41,960
Why should they
do approximations?

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I can solve things exactly,
which is almost never possible,

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00:17:47,900 --> 00:17:50,530
but we can imagine that.

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00:17:50,530 --> 00:17:57,190
So what would be a possible way
to write an [? n-set ?] would

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be the following.

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00:17:58,100 --> 00:18:02,680
You could write a psi for the
whole thing now that depends

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on the R's.

252
00:18:05,050 --> 00:18:09,820
And the R's wavefunction for
the whole degrees of freedom

253
00:18:09,820 --> 00:18:17,460
of the molecule could be
written as a sum over k

254
00:18:17,460 --> 00:18:33,300
of phi K, R of r times
solutions, eta k, that

255
00:18:33,300 --> 00:18:41,500
depend on capital R.

256
00:18:41,500 --> 00:18:45,310
That is, I'm saying
we can try to write

257
00:18:45,310 --> 00:18:51,550
the solution in which the full
wavefunction for the molecule

258
00:18:51,550 --> 00:18:56,650
is the sum of states of
this form, a solution here

259
00:18:56,650 --> 00:18:58,020
and a solution there.

260
00:19:07,710 --> 00:19:15,450
This is correct, but then how
do you determine the etas?

261
00:19:15,450 --> 00:19:17,790
The only way to
determine the etas

262
00:19:17,790 --> 00:19:22,200
is to plug into the full
Schrodinger equation--

263
00:19:22,200 --> 00:19:24,630
this is the full Hamiltonian.

264
00:19:24,630 --> 00:19:29,640
So you would have to plug
this into the full Schrodinger

265
00:19:29,640 --> 00:19:32,820
equation and see what you get.

266
00:19:32,820 --> 00:19:34,710
So what are you going to get?

267
00:19:34,710 --> 00:19:39,660
Presumably, you did
solve this first part.

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00:19:39,660 --> 00:19:42,720
So the phis are known.

269
00:19:42,720 --> 00:19:45,390
So if the phis are
known, you're going

270
00:19:45,390 --> 00:19:48,505
to find differential
equations for the etas.

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00:19:51,040 --> 00:19:55,060
So this problem has become now
a problem of finding solutions

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00:19:55,060 --> 00:19:57,010
for the etas.

273
00:19:57,010 --> 00:20:01,360
And there are many etas,
and they are all coupled

274
00:20:01,360 --> 00:20:03,590
by the Schrodinger equation.

275
00:20:03,590 --> 00:20:13,060
So by the time you
plug this into H,

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00:20:13,060 --> 00:20:18,880
the total H, capital
psi Rr equals

277
00:20:18,880 --> 00:20:25,500
sum Eq, sum energy eigenstate--

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00:20:25,500 --> 00:20:27,450
I don't have to put the thing--

279
00:20:27,450 --> 00:20:33,940
psi Rr, to find the energy
eigenstates of the molecules.

280
00:20:33,940 --> 00:20:39,840
This equation is going to
imply a set of differential

281
00:20:39,840 --> 00:20:44,010
equations, a couple differential
equations for the etas.

282
00:20:44,010 --> 00:20:47,730
And that's not so easy to do.

283
00:20:47,730 --> 00:20:51,530
That's pretty hard in general.

284
00:20:51,530 --> 00:20:57,050
So this is very difficult.

285
00:20:57,050 --> 00:21:01,300
On the other hand,
there is a way

286
00:21:01,300 --> 00:21:06,870
to think of this in
a simpler context.

287
00:21:06,870 --> 00:21:11,790
We can try-- and now
we are approximating--

288
00:21:11,790 --> 00:21:19,220
so try to believe that you can
get an approximate solution,

289
00:21:19,220 --> 00:21:21,620
approximate.

290
00:21:21,620 --> 00:21:28,290
And I will justify
this solution using

291
00:21:28,290 --> 00:21:39,000
just one term in this
equation, and psi phi R of r.

292
00:21:39,000 --> 00:21:46,890
And this may be the ground
state of the electronic system.

293
00:21:46,890 --> 00:21:48,890
That's why I don't
put an index here.

294
00:21:48,890 --> 00:21:52,190
I could put a 0
there, but let's think

295
00:21:52,190 --> 00:21:54,990
of this as the ground state.

296
00:21:54,990 --> 00:21:58,060
And then, well, this
would be accompanied,

297
00:21:58,060 --> 00:22:04,526
if there is some
solution, by sum eta of R.

298
00:22:04,526 --> 00:22:20,050
And I can try to say that
your wavefunction is this,

299
00:22:20,050 --> 00:22:24,700
one term in this equation, the
one in which I pick the ground

300
00:22:24,700 --> 00:22:28,930
state and leave it there.

301
00:22:28,930 --> 00:22:32,320
Now, this is
definitely not going

302
00:22:32,320 --> 00:22:37,310
to be an exact solution ever
of the Schrodinger equation.

303
00:22:37,310 --> 00:22:41,440
So you really are--

304
00:22:41,440 --> 00:22:45,340
if you just take
one of these terms--

305
00:22:45,340 --> 00:22:49,330
out of luck in terms of
solving this exactly,

306
00:22:49,330 --> 00:22:53,230
because this
differential equation,

307
00:22:53,230 --> 00:22:57,580
this Hamiltonian,
has terms mixing

308
00:22:57,580 --> 00:22:59,500
the two degrees of freedom.

309
00:22:59,500 --> 00:23:01,000
It doesn't separate.

310
00:23:01,000 --> 00:23:04,750
You cannot show that the
Schrodinger equation has

311
00:23:04,750 --> 00:23:08,680
a solution which is one thing
that solves an equation and one

312
00:23:08,680 --> 00:23:12,530
thing that solves another
equation that are products like

313
00:23:12,530 --> 00:23:13,030
that.

314
00:23:13,030 --> 00:23:15,710
It will not happen.

315
00:23:15,710 --> 00:23:19,420
That's why in this
equation when you plug in,

316
00:23:19,420 --> 00:23:22,300
the various etas get coupled.

317
00:23:22,300 --> 00:23:29,080
But this is the spirit of
the adiabatic approximation,

318
00:23:29,080 --> 00:23:34,540
in which we sort of
have an electronic cloud

319
00:23:34,540 --> 00:23:37,570
and a nuclear state, and when
the nuclear state changes

320
00:23:37,570 --> 00:23:40,630
slowly, the electronic
cloud adjusts,

321
00:23:40,630 --> 00:23:42,940
and you don't have
to jump to a state

322
00:23:42,940 --> 00:23:45,990
with another electronic cloud.

323
00:23:45,990 --> 00:23:47,500
The electronic cloud adjusts.

324
00:23:47,500 --> 00:23:52,960
So this is in the spirit of
the adiabatic approximation,

325
00:23:52,960 --> 00:23:57,280
to try to find the solution
of this kind, in which

326
00:23:57,280 --> 00:23:59,620
an electronic
cloud is not forced

327
00:23:59,620 --> 00:24:02,830
to jump, because the coupling
between those states,

328
00:24:02,830 --> 00:24:08,920
saying that if you start
with one eta of one phi,

329
00:24:08,920 --> 00:24:12,280
you need all the rest, as the
Schrodinger equation tells you.

330
00:24:12,280 --> 00:24:15,520
It is a statement that the
electronic cloud just cannot

331
00:24:15,520 --> 00:24:19,150
stay by itself where it is.

332
00:24:19,150 --> 00:24:21,700
So here we go.

333
00:24:21,700 --> 00:24:24,840
This is what we're going to try.

334
00:24:24,840 --> 00:24:26,430
And you can say,
well, all right.

335
00:24:26,430 --> 00:24:32,580
So you're giving up the
Schrodinger equation,

336
00:24:32,580 --> 00:24:36,370
because exact Schrodinger
equation is not solved by this.

337
00:24:36,370 --> 00:24:40,500
How are you going to try
to understand now the eta?

338
00:24:40,500 --> 00:24:44,130
Because we found these
guys, so how about the etas?

339
00:24:44,130 --> 00:24:47,500
How are we going to find them?

340
00:24:47,500 --> 00:24:49,630
That is our question.