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ANNA MORALES MELGARES: Hi.

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My name is Anna.

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I'm from Barcelona, although
I'm starting an EPFL, which

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is in Switzerland.

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It's the Polytechnical
Federal School of Lausanne.

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And today, I wanted to show
you potential energy surfaces.

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But I thought that
might be a bit boring,

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and since it's my
video, and I do

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what I want, I'm
going to show you

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theme parks and slides,
because why not?

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Going to make
myself smaller here.

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OK.

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What do we have here?

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Well, this is the Water World.

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The Water World
is this theme park

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which is in Lloret de Mar,
a city next to Barcelona,

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kind of.

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And as you can see, they
have these amazing slides

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that end up in a
funnel, like you

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can see this guy at the
bottom of the funnel just fall

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through this hole.

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But what if I want
to simulate this?

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I would need a software that
enables me to do this, right?

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There is some software that
can do something like this,

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but I chose to use
Wolfram Mathematica.

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And now I will show
you my little funnel.

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Here it is.

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My humans are these
black particles here.

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Now, what if I let
some time pass?

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What happens to my humans?

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They get closer to the
center of the funnel.

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That makes sense, because
gravity pulls them

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to the lowest height, right?

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This height is actually related
to gravitational energy,

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since gravitational energy
is proportional to height.

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Therefore, this is
just exactly the shape

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that the potential
energy surface

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for the gravitational
energy looks like.

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So we have constructed actually
a potential energy surface.

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But these kind of
surfaces can represent

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many other types of energy,
not only gravitational energy,

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and can be plotted with other
coordinates rather than space.

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Here, we have x and y, which
only defined the space where

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these humans and
funnels sit, but I

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could use other coordinates.

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Like for example, the
distance between two items,

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or the state of the system,
or things like this.

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Things in which energy,
the energy that employing

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depends on.

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In order to simulate these
things-- to simulate the funnel

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that I did or what I'm
going to show you after,

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I need statistics.

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But don't run away,
they're easy to understand.

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So what I'm going to do
in order to simulate this

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is the following.

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Imagine that I have
this blue line, which

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will be like a
slide, and imagine

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that the bottom of this blue
line will represent the floor--

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the ground.

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We'll call it ground state.

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And the upper part
is like something

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which is higher in energy.

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Now, I'm going to separate this
slide into some finer parts.

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This is going to be the
discretization of the system.

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I'm going to
discretize my system.

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Here, I discretized
it in five steps.

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You can see these red steps.

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What I'm going to
tell my particles,

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which will represent
the state of the system,

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is the following.

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So my particle will be
in the initial state.

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I will tell my
particle the following.

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It will attempt to go
either down or either up.

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It will have 50% probabilities
of trying to go up

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and trying to go down.

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Now, there is something
we have to differentiate.

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One thing is the attempt.

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One thing is to try
to go somewhere,

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and another different thing
is actually going somewhere.

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So my particle is going
to try to go up and down

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with equal probability.

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Now, if it chooses
to go down, it

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will have probability
equal to one to go down.

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This means that if
it tries to go down,

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it will actually go down.

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Now, if it chooses
to go up, it's

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not that clear that it's
going to go up because--

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and I'm going to make my screen
bigger so I can show you--

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probability of going up will
be equal to this term here.

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This term depends
on energy two, which

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is the energy of
the upper state,

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and energy one, which is the
energy of the initial state.

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The more separated
they are, this

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means the bigger
this barrier is.

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This is actually
called energy barrier.

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The bigger this step
is, the more difficult

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it will be for my particle
to actually go up.

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There is another
term I didn't talk

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about yet, which is this one.

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This is temperature.

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Temperature, here, will--
if temperature increases,

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the probability of my particle
to go up will also increase.

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This means that the
more energy we give--

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the more thermal energy we
give the system, the more

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temperature we
give the system, it

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will be more probable for my
particle to be able to go up.

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In a system where my
temperature is close to zero,

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in a system where my
temperature is super low,

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my particle will tend to go
to the ground state, which

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is the bottom of my curve.

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If I give energy
enough, then particle

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will be able to go anywhere.

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Now, what I'm talking
about here is not a system

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with only one coordinate.

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I'm talking about a system
with two coordinates,

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therefore I'm not
talking about lines.

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I'm not talking about curves.

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I'm talking about surfaces.

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Then, the example
I showed before

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was basically a particle that
can go up and can go down,

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as you can see here
with this blue row.

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But now, I'm going
to have a surface,

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so my particle will be able
to go many other places.

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Therefore, my particle
will attempt--

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the attempts that it
can do are not two.

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There are going
to be eight which

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will describe this surface.

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Now, what I wanted to
show you is actually

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a real case where
these surfaces can

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be really useful to
understand the system,

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and these are the
transition states.

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But in order to
do this, I'm going

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to talk to you about the
Lennard-Jones potential first.

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The Lennard-Jones potential
is a very well known potential

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energy curve that
describes the interaction--

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describes the behavior of energy
of the system as two atoms

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get closer, and closer,
and closer together.

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Imagine, here, two atoms, which
could be two hydrogen atoms

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and are teared apart.

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They're super far
from each other,

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and then the energy is zero.

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When they are getting closer,
and closer, and closer,

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there is a point where energy
will be at its minimum.

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This minimum energy is the most
stable state of the system.

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This distance will be
the optimal distance--

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will be the equilibrium state.

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Now, if we put these atoms even
closer, and closer, and closer,

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the energy of the
system will increase.

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An increase in energy means
that the state is less happy.

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The state is less stable.

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Therefore, the most
stable part of the system

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will be the minimum.

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The state will always try
to be at it's minimum.

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Here, this minimum would be in
a distance between these two

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atoms.

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What I wanted to
talk to you about,

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actually, is triatomic
hypothetic molecules.

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This molecule, especially HHF.

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HHF is a molecule which is--

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it cannot be isolated.

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There are some experiments
that show its existence

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for nano or femtoseconds.

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This exists for a super,
super short amount of time.

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We could even say
they are virtual,

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but it's a state where the
system has to pass through

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in order to change.

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I'll explain after.

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The thing that we have here is
one hydrogen atom which has--

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we will plot.

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One of the coordinates will be--

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the distance between
this hydrogen atom,

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and the other hydrogen atom,
and the other coordinate

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that we will be plotting is the
distance between this hydrogen

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atom and fluorine.

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This means that we will have
two Lennard-Jones potentials.

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One between HH and
one between HF.

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This will describe a surface.

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First of all, I'm going
to explain you what

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this transition state means.

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We will have different
states of our system.

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First of all, we
will have a molecule

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which is H2, and some
fluorine somewhere.

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This particles-- the
molecule and fluorine--

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will collide, and we
will end up with an HF

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molecule and a hydrogen
atom somewhere else.

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In order for this to
happen, the system

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has to undergo through
another state, which

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is called a transition state,
which is actually our HHF

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virtual or hypothetic molecule.

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If a system wants
to go from H2 plus F

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to HF plus H or visa versa,
it will preferentially

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go through this
transition state.

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How can we actually see this
in a potential energy surface?

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Here, I show you how
the potential energy

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surface of HHF system is.

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[INAUDIBLE] described
by the HF distance,

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where this minimum represents
the state of the system where

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the molecule HF exists,
and the other hydrogen atom

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is somewhere else.

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Here, we have another
Lennard-Jones potential,

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described by the HH distance.

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So in this minimum,
the state of the system

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is an H2 molecule and some
fluorine somewhere else.

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Now, the preferential
path for this change

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to happen from HH to
HF, or vise versa,

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will be through the transition
state, which is a subtle point.

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What is a subtle point?

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A subtle point is
somewhere in the surface

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where the derivative
of energy is zero,

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but it's not a minimum,
and it's not a maximum.

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It's actually a minimum
in one direction,

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and a maximum in
another direction,

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and this will be the
HHF state of the system,

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as you can see in this
contour plot on the left.

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To illustrate this,
I'm going to show you.

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Now, my particles--
I'm going to show

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some particles in the surface.

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Now, this particle means it
represents the system, not

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atoms.

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It's the system.

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Here, I have an HF
molecule because the system

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is in this minimum.

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Now, if the particle
for some reason,

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when given enough temperature,
it goes through this state--

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this is the transition state.

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The system is HHF, but
if this particle falls

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into this minimum,
then the system is HH--

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this hydrogen molecule and
the fluorine somewhere else.

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Knowing this, I'm
going to show you

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what happens when I have
lots of particles which

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are lots of representations
of the same system

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because I want to see the
probability of the system to be

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in one state or the other.

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I'm going to give [INAUDIBLE].

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This is an histogram.

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An histogram is
this plot with bars

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that shows me the population.

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As you can see
here, the particles

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are just running
all over the place

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because I gave energy enough
for them to overcome any energy

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barriers.

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But if I decrease
temperature, [INAUDIBLE]..

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OK, now, here you can
see that we actually

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have some preferential
places for these particles--

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these states of
the system to be.

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They are in the minimums because
they cannot jump in the minimum

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because they cannot jump
over the energy barriers.

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Now, as you can see,
they are also pretty--

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like, in the
histogram, you can see

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that if I increase
temperature a little bit,

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the preferential path
they take, more or less,

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is the transition state,
which is done at this point.

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This is one of the things that
we can use Mathematica for,

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but it's much more powerful.

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It can actually do
many other things,

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and it can solve many other
science related problems.

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And not even
science related, you

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can even do art
with Mathematica.