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NICOLA MARZARI: To calculate
ensemble averages--

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that is to calculate
thermodynamical properties.

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And we have seen
also a few movies

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on how we can calculate
reaction barriers

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and how we can follow in
real time chemical kinetics.

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The last application
of molecular dynamics

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is likely more different.

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In particular, one can
use a final temperature,

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classic in molecular dynamics,
as an optimization scheme.

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In general, suppose that you
have a complex potential energy

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surface.

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This is obviously
in one dimension,

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but you could
think this as being

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in a multi-dimensional space.

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And actually, your
potential energy function

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could represent a
complex cost function.

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Again, it could be,
say, the problem

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of optimizing the
usage of your planes

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on several routes for
one airline company.

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And obviously, depending on
where you put your planes,

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you have a certain cost.

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And so this cost function
tends to be very complex

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depending on your coordinates,
how you arrange your planes.

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And finding either the
global minimum or sort of one

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of the lowest minima can
be a very complex affair.

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And one of the
powerful techniques

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that has been introduced
in the early '80s

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is something called a
simulated annealing.

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That is basically a
technique that basis itself

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on a thermodynamical analogy.

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That is suppose that you
want to find this minimum.

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And if you were to use
a deterministic recipe,

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it would be very
difficult. You would

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start from a certain
point and then

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go down according
to the gradient

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until you end up in what
would be a local minima.

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And so what, in particular,
Kilpatrick, Gelatt, and Vecchi

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introduced was a thermodynamical
technique in which you really

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populate your phase space
with what we call walkers.

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So these are sort of
dynamical systems.

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So those are the
points that represent

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those dynamical systems.

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You give them some coordinates.

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That is they are going to
have some potential energy.

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And then you give
them some temperature.

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So they are going to
have some kinetic energy.

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You can think this
as a swarm of skiers

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sort of going around
your phase space.

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And if you have a lot
of them and if these

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have a lot of temperature, they
will move all over phase space.

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But by the time you start
sort of very slowly cooling

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this dynamical systems down,
you start removing temperature

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for them, they are going
to condensate really

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in whichever local
minima they find.

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So some of them may
end up, say, here.

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And you cool them
down, and slowly they

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find themselves here.

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But if you cool them
sufficiently slow--

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and really in order to make
this into an exact schema,

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you have to pull them
almost infinitely slowly.

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So it's never going to be
an exact approach, but just

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a stochastic approach.

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But, you know, with some luck,
you'll find that some of them

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start condensing in
interesting minima.

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And then you choose basically
your lowest minima possible.

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And this is actually a very
sort of practical and useful

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approach.

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And so it's sort
of fairly widely

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used in optimization problems.

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OK, so this sort of
concludes the sort

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of set of applications
of molecular dynamics.

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Now, I wanted to
sort of hint at some

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of the advanced statistical
mechanics techniques

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that you'll actually see later
in the class that, generally

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speaking, go under the umbrella
name of Green-Kubo techniques.

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And sort of I'm taking
one of the simplest

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examples in which, actually,
Albert Einstein was involved.

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So it's an interesting approach.

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Well, this has to do on starting
a macroscopic property that

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is really diffusion in a solid.

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So you could think
of a silicon crystal.

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And we are doping that silicon
with gallium or phosphorus,

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and so you are
interested in studying

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how the impurities in
this system move around.

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And so if you
want, what you have

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is that you have a perturbation.

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You put a sort of gallium
on the surface of silicon,

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and then this diffuses
into the solid.

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And sort of the general
Green-Kubo approach

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is a formalism that relates
some of these macroscopic

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properties, like a
diffusion coefficient,

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a transpose property, to
microscopic properties.

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And you'll see that as sort of
fluctuations of the equilibrium

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distribution.

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OK.

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So suppose that we are looking
at the diffusion of, again,

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something like
gallium and silicon.

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What you would define as your
sort of fundamental variable

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is the concentration
of impurities.

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So that's a sort of
four-dimensional function is

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a concentration function
of the [INAUDIBLE] space

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you are interested
and function of time.

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And then, obviously,
there is sort

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of the diffusion
law for this that

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says that basically
the current is

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really proportional to the
gradient of the concentration.

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And then you put this together
with a sort of continuity

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equation that tells you that the
derivative of the concentration

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with respect to time plus the
divergence of your current--

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that is your flux--

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needs to be constant.

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That equals-- let's say
it's going to be equal to 0.

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I mean, this very
simply says that,

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if an infinitesimal volume
the concentration changes,

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it's that because there
is a current going out

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of that infinitesimal volume.

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And so the divergence of
the current measures this.

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And with fixed diffusion law,
we can put the 2 and 2 together

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and basically obtain this
relation, this differential

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equation, for the
concentration profile in time.

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So if you want,
these are starting

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macroscopic relationship.

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And then what
becomes interesting

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is the derivation that
Einstein did in order

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to recover the connection
with the microscopic dynamics.

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And this is just a little
bit of algebra, which

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I will go over fairly quickly.

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But, basically, what
we are doing here,

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we are actually
multiplying the left term

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and the right-hand
terms by r square

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and integrating
it over all space.

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And once we do
that, well, what we

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have is that on the
right-hand side,

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this requires some calculus.

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So I won't go really
into the derivation.

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But by sort of
integration by parts,

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you can actually show that
this integral provided,

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say, the concentration
is normalized to 1

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when integrated in all
space, what we really obtain

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is just 2 times the
dimensionality of your problem.

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That is, if you are in one
dimension, sort of small d

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is going to be equal to 1.

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If you are in two
dimensions, it's

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going to be equal to 2, to
3, and so on and so forth.

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So this integral
really just gives us

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the dimension of the space.

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And again, we won't go
into how we prove this.

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What we have on the left-hand
side, on the contrary,

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is really, you see, the
average value of r square.

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What we mean by this
bracket that you see

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is the integral
over all space of r

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square with the probability.

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If you want the
concentration, it's

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identical to the
probability of having

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a certain particle in a certain
position in a certain time.

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So what this integral
means is really

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the average value over
all space of r square,

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how much your
particles has moved.

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And we have the
derivative and the time.

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And so you see we
start seeing somehow

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a connection between a
diffusion coefficient

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that's the macroscopic property
and the microscopic properties,

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the dynamic piece, how much
the particles are moving

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once you put them in somewhere.

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Let me actually
give you an example

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of what would this
quantity look like

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and how we would calculate it in
a molecular dynamic simulation.

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Again, you can think
of your gallium atoms

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being at the surface.

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At the beginning, you
put everything in motion.

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And they are going to move
with a sort of random walk

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towards the inside.

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And if you want to calculate
this sort of average value

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of what is called the
mean square displacement,

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the average value
of r square, well,

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you need really to do the
average over all your particle

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at a certain time
of delta r square.

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Delta r square is really how far
all these particles have moved.

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And you know, this
would look at something

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like this in a
typical, say, solid.

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Actually, the example
that I have here

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is slightly more exotic, but
it was a sort of good example

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to show you the difference
between a liquid and a solid.

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And it's the case
of silver iodide

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that is, in particular,
an ionic crystal

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in which the blue iodine
atoms sit on a BCC lattice.

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And for each iodine,
there is a silver atom.

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But above, say, 420
Kelvin, this system

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is in a very peculiar
state of matter, what

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is called a super ionic state
in which this iodine atom

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oscillates around an
equilibrium position exactly

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as atoms do in a solid, while
the silver atoms sort of move

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around exactly like a liquid.

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So it's a system
that is really solid.

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It's crystalline.

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But one sublattice,
the silver sublattice,

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is sort of moving
around as a liquid.

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And so if you calculate what
are the average mean square

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displacements-- that
is if you average, say,

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all the iodine atoms, what is
their average delta r squared,

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how much they move around--

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well, you'll discover
that really, in a sense,

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that they are a
crystalline solid.

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They are just going to oscillate
around their equilibrium

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position.

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And so you calculate
that average.

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During a molecular
dynamic simulation,

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you see there is a sort
of first very short period

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of thermalization.

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The atoms start
moving, but then they

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are just oscillating around.

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They are not going anywhere.

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So their instantaneous
mean squared displacement

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as a value that is
just a constant.

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So this is sort of typical
of a crystalline solid.

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If you do the same
thing for the silvers,

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you see that, as a function
of increase in temperature,

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you see, at 250 Kelvin below
the superionic transition, also

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the silvers sort of look
more or less flatter.

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But when you increase
the temperature,

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their mean square
displacement starts becoming

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a linear function of time.

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So the silvers are
really moving around.

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They are diffusing
away as a liquid.

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And so in the mean
square displacement,

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you can see the
signature of a liquid

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when they increase
linearly with time.

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Or you can see the
signature of a solid

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where they just are
not going anywhere.

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They just vibrate around
the equilibrium position.

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And so this is the
quantity that we

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could calculate in a
molecular dynamic simulation.

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The average of the mean
square displacement,

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I have written it here.

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Remember, this is going to be
equal to 2 times the diffusion

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coefficient times the
dimensionality of your system.

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And now, we sort of work
a little bit with this.

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And in particular, we
introduce this sort

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of alternative definition of
the displacement of an atom.

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This is actually very useful.

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If you are sort of working in
periodic boundary conditions,

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you need to be careful.

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Often, your molecular dynamics
codes, for simplicity,

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always show you the
coordinates of the ions

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as they were sitting in
your first unit cells.

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If an atom is
diffusing as a liquid

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and it sort of moves
around and from one cell

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moves in the second unit
cell, your code usually

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will actually sort
of bring it back

250
00:13:36,040 --> 00:13:38,890
by a lattice translation
in the first unit cell.

251
00:13:38,890 --> 00:13:43,480
So a universal way to
sort of take into account

252
00:13:43,480 --> 00:13:46,360
what the actual
position of an atom is

253
00:13:46,360 --> 00:13:50,890
is actually write the
position that is usually

254
00:13:50,890 --> 00:13:53,680
a vector that satisfies this
periodic boundary condition

255
00:13:53,680 --> 00:13:55,760
as an integral of the velocity.

256
00:13:55,760 --> 00:13:59,890
So in that sense, the
position of your particle

257
00:13:59,890 --> 00:14:04,150
is always taking correctly
into account all the distance

258
00:14:04,150 --> 00:14:05,650
that has been traversed.

259
00:14:05,650 --> 00:14:08,140
So if you have a
unit cell really

260
00:14:08,140 --> 00:14:11,590
and your particle
is moving, you'll

261
00:14:11,590 --> 00:14:14,680
sort of integrate
correctly your trajectory

262
00:14:14,680 --> 00:14:17,830
by doing the integral of
the velocity, while sort

263
00:14:17,830 --> 00:14:23,200
of your algorithm would actually
bring back the particle once it

264
00:14:23,200 --> 00:14:24,220
crosses a boundary.

265
00:14:24,220 --> 00:14:24,790
OK.

266
00:14:24,790 --> 00:14:26,320
Nothing deep here,
we are just sort

267
00:14:26,320 --> 00:14:30,670
of writing the position as
the integral of the velocity.

268
00:14:30,670 --> 00:14:33,580
But with this definition,
we can actually

269
00:14:33,580 --> 00:14:37,210
go back and look at
what is the expression

270
00:14:37,210 --> 00:14:41,220
for the average of the
mean square displacement.

271
00:14:41,220 --> 00:14:42,760
That is the delta x square.

272
00:14:42,760 --> 00:14:46,090
And now, we write it
actually as the square

273
00:14:46,090 --> 00:14:49,910
of the integral of the
velocity at the instant of t.

274
00:14:49,910 --> 00:14:53,830
And so I'm just sort of writing
this integral out explicitly

275
00:14:53,830 --> 00:14:56,890
when I sort of
expand the square.

276
00:14:56,890 --> 00:14:58,810
I have an integral
in t and t prime.

277
00:14:58,810 --> 00:15:03,460
And again, the bracket means
just the ensemble average

278
00:15:03,460 --> 00:15:06,370
on all your particles, so it
commutes with the integral.

279
00:15:06,370 --> 00:15:09,130
And so what we are
calculating here

280
00:15:09,130 --> 00:15:16,210
is really the sum over all the
particles and averaging that.

281
00:15:16,210 --> 00:15:20,260
That is dividing by the
total number of particles

282
00:15:20,260 --> 00:15:24,970
of the velocity at certain
instant t prime times

283
00:15:24,970 --> 00:15:30,570
the velocity of a
certain instant t2 prime.

284
00:15:30,570 --> 00:15:34,870
And we can actually, for
computational convenience,

285
00:15:34,870 --> 00:15:37,200
this is really an
integral on a square,

286
00:15:37,200 --> 00:15:40,920
integral from 0 to t in one
dimension and the integral

287
00:15:40,920 --> 00:15:42,780
from 0 to t in the
other dimension.

288
00:15:42,780 --> 00:15:44,580
But this expression
is symmetrical.

289
00:15:44,580 --> 00:15:48,540
So we can actually integrate
this only on half the square.

290
00:15:48,540 --> 00:15:51,420
We are integrating it
only on the triangle.

291
00:15:51,420 --> 00:15:58,680
If this is were t and t prime
and our integration interval

292
00:15:58,680 --> 00:16:01,740
in this, we can just
sort, for simplicity,

293
00:16:01,740 --> 00:16:04,960
integrate into half of this.

294
00:16:04,960 --> 00:16:06,150
OK.

295
00:16:06,150 --> 00:16:09,240
So what we have
achieved here is we

296
00:16:09,240 --> 00:16:12,450
have written this
average mean squared

297
00:16:12,450 --> 00:16:18,540
displacement as an integral
of the average product

298
00:16:18,540 --> 00:16:22,260
between the velocity
at a certain instant

299
00:16:22,260 --> 00:16:25,290
and the velocity at
a different instant.

300
00:16:25,290 --> 00:16:29,230
And this is sort of where our
connection with the equilibrium

301
00:16:29,230 --> 00:16:31,050
properties is
starting to emerge.

302
00:16:31,050 --> 00:16:32,520
And you'll see this in a moment.

303
00:16:32,520 --> 00:16:37,140
This is called velocity
autocorrelation function.

304
00:16:37,140 --> 00:16:40,740
And I'm writing it
more explicitly.

305
00:16:40,740 --> 00:16:44,160
All of this has been done, again
to keep the algebra simple,

306
00:16:44,160 --> 00:16:45,730
in two dimensions.

307
00:16:45,730 --> 00:16:48,930
So remember, the
Einstein relation.

308
00:16:48,930 --> 00:16:50,890
Sorry, we are in one dimension.

309
00:16:50,890 --> 00:16:54,000
So remember, the
Einstein relation.

310
00:16:54,000 --> 00:16:57,090
The small d, the dimensionality
of your system, is 1.

311
00:16:57,090 --> 00:17:01,620
So we have the 2 times
the diffusion coefficient

312
00:17:01,620 --> 00:17:05,880
is equal to the derivative of
the mean square displacement.

313
00:17:05,880 --> 00:17:09,630
And mean square
displacement in itself

314
00:17:09,630 --> 00:17:12,420
has been written as
the double integral.

315
00:17:12,420 --> 00:17:14,700
But when you take the
derivative with respect to t,

316
00:17:14,700 --> 00:17:17,849
one of the integral cancels out.

317
00:17:17,849 --> 00:17:20,920
So if you want, this
is our final relation.

318
00:17:20,920 --> 00:17:25,020
Let me actually sort
of write it over here

319
00:17:25,020 --> 00:17:27,780
and remove the factor of 2.

320
00:17:27,780 --> 00:17:31,860
We have the diffusion
coefficient as

321
00:17:31,860 --> 00:17:33,390
written as this integral.

322
00:17:33,390 --> 00:17:36,930
We can sort of exploit the
translational invariance

323
00:17:36,930 --> 00:17:37,860
in time.

324
00:17:37,860 --> 00:17:40,170
Really, if we are
looking at what

325
00:17:40,170 --> 00:17:44,730
is the average value of
the product of the velocity

326
00:17:44,730 --> 00:17:47,490
at a certain instant, d
prime, times the velocity

327
00:17:47,490 --> 00:17:53,160
of another instant d2 prime,
well, that average product

328
00:17:53,160 --> 00:17:57,330
is not going to be different
if we translate it in time.

329
00:17:57,330 --> 00:18:01,540
So we can refer it to
an arbitrary origin.

330
00:18:01,540 --> 00:18:04,550
And so we just do a
translation in which we shift

331
00:18:04,550 --> 00:18:07,930
d2 prime into 0 and so on.

332
00:18:07,930 --> 00:18:10,770
So this is sort of
our final expression.

333
00:18:10,770 --> 00:18:14,340
And this is what is
called a velocity-velocity

334
00:18:14,340 --> 00:18:16,330
autocorrelation function.

335
00:18:16,330 --> 00:18:21,030
So you see, what we have is
that the macroscopic property,

336
00:18:21,030 --> 00:18:23,880
the diffusion coefficient,
that is really

337
00:18:23,880 --> 00:18:27,990
telling you how a system
responded to a perturbation.

338
00:18:27,990 --> 00:18:31,470
You put some gallium on
your surface of silicon,

339
00:18:31,470 --> 00:18:36,390
and then there is this
macroscopic diffusion transfer

340
00:18:36,390 --> 00:18:37,440
process.

341
00:18:37,440 --> 00:18:40,500
That is there is a perturbation,
and the system evolves.

342
00:18:40,500 --> 00:18:44,190
It can actually be
related to an equilibrium

343
00:18:44,190 --> 00:18:46,170
property of the system.

344
00:18:46,170 --> 00:18:47,970
That is really,
ultimately, what are

345
00:18:47,970 --> 00:18:52,560
the mean square displacements,
how atom microscopically move

346
00:18:52,560 --> 00:18:53,400
around.

347
00:18:53,400 --> 00:18:57,810
And in particular, the
quantity that is playing here

348
00:18:57,810 --> 00:19:00,510
is this velocity-velocity
autocorrelation.

349
00:19:00,510 --> 00:19:05,370
You see, what this is
looking at is suppose

350
00:19:05,370 --> 00:19:09,360
that you have a velocity
at a certain instant.

351
00:19:09,360 --> 00:19:11,870
And then you look
at that instant

352
00:19:11,870 --> 00:19:18,150
that is sort of tau away in time
from sort of your instant 0.

353
00:19:18,150 --> 00:19:22,350
And you look at the product
of these two velocities.

354
00:19:22,350 --> 00:19:27,270
Now, if tau is very
small, the velocity

355
00:19:27,270 --> 00:19:29,990
is not going to have
changed very much.

356
00:19:29,990 --> 00:19:34,920
So in the limit of small
tau, your velocity at 0

357
00:19:34,920 --> 00:19:38,223
and your velocity at tau are
going to be very similar.

358
00:19:38,223 --> 00:19:40,140
So if you want, there
is a lot of correlation.

359
00:19:40,140 --> 00:19:42,690
When you take this
product, it's going

360
00:19:42,690 --> 00:19:46,440
to look a lot like v0 squared.

361
00:19:46,440 --> 00:19:48,690
And then all these things
are normalized properly.

362
00:19:48,690 --> 00:19:51,510
So it could look like 1.

363
00:19:51,510 --> 00:19:54,270
But as you sort of
move away in time,

364
00:19:54,270 --> 00:19:57,540
you go towards longer
and longer times,

365
00:19:57,540 --> 00:19:59,610
there is going to
be no correlation.

366
00:19:59,610 --> 00:20:02,580
Your velocity at 0
and your velocity

367
00:20:02,580 --> 00:20:05,580
at an instant that is
very far away in time

368
00:20:05,580 --> 00:20:07,320
is not going to be correlated.

369
00:20:07,320 --> 00:20:12,960
And so the product of this can
be, if you want, any number.

370
00:20:12,960 --> 00:20:14,910
And when you take
the average, you

371
00:20:14,910 --> 00:20:17,730
see the sort of ensemble
average, the bracket.

372
00:20:17,730 --> 00:20:21,510
When you average this
quantity on all your particles

373
00:20:21,510 --> 00:20:23,640
in the system,
this is just going

374
00:20:23,640 --> 00:20:26,910
to be 0 because
this thing can have

375
00:20:26,910 --> 00:20:29,010
any positive or negative value.

376
00:20:29,010 --> 00:20:31,820
Because there is no correlation
between the velocity

377
00:20:31,820 --> 00:20:33,050
at the large tau.

378
00:20:33,050 --> 00:20:35,990
And so this thing
is going to be 0.

379
00:20:35,990 --> 00:20:36,530
OK.

380
00:20:36,530 --> 00:20:39,860
So the limit of this
velocity-velocity

381
00:20:39,860 --> 00:20:44,690
autocorrelation for very large
tau is going to be equal to 0.

382
00:20:44,690 --> 00:20:49,910
The limit for sort of very small
tau is going to be equal to 1

383
00:20:49,910 --> 00:20:52,670
or whatever your
normalization factor is.

384
00:20:52,670 --> 00:20:55,610
And then there should be
some interesting structure

385
00:20:55,610 --> 00:20:57,680
at a certain time.

386
00:20:57,680 --> 00:21:00,110
And we'll see it
in a moment that

387
00:21:00,110 --> 00:21:03,500
suppose your system is
actually sort of oscillating.

388
00:21:03,500 --> 00:21:05,660
You are looking not
at the liquid part

389
00:21:05,660 --> 00:21:08,210
of your silver iodide,
but you are looking just

390
00:21:08,210 --> 00:21:11,090
at the sort of
crystalline iodine

391
00:21:11,090 --> 00:21:13,130
sort of oscillating around.

392
00:21:13,130 --> 00:21:17,420
Well, suppose that your iodine
atom has a certain period

393
00:21:17,420 --> 00:21:18,390
of oscillation.

394
00:21:18,390 --> 00:21:20,900
So it sort of keeps
going back and forth.

395
00:21:20,900 --> 00:21:23,570
Well, then there will
be a lot of correlation

396
00:21:23,570 --> 00:21:26,840
if you look at your
velocity at a time 0

397
00:21:26,840 --> 00:21:29,930
and your velocity
at an instant that

398
00:21:29,930 --> 00:21:32,360
is roughly equal to a
period of oscillation.

399
00:21:32,360 --> 00:21:35,660
Because if this thing
were to oscillate exactly

400
00:21:35,660 --> 00:21:39,150
around this equilibrium,
every sort of period

401
00:21:39,150 --> 00:21:41,700
you would have that the
velocity has become the same.

402
00:21:41,700 --> 00:21:42,320
OK.

403
00:21:42,320 --> 00:21:45,710
So you will see a very
definite structure

404
00:21:45,710 --> 00:21:48,080
in this correlation function.

405
00:21:48,080 --> 00:21:50,630
If your system is not a
crystal, if your system is

406
00:21:50,630 --> 00:21:55,430
sort of diffusing away as a
liquid, as your time increases,

407
00:21:55,430 --> 00:21:57,800
the correlation becomes 0.

408
00:21:57,800 --> 00:22:01,010
And even if it's a crystal,
but it's not really

409
00:22:01,010 --> 00:22:05,750
oscillating perfectly,
like a harmonic

410
00:22:05,750 --> 00:22:08,300
oscillator around the
period, when you really

411
00:22:08,300 --> 00:22:10,730
got to very, very
long time away,

412
00:22:10,730 --> 00:22:14,930
you start losing this
correlation that sort of takes

413
00:22:14,930 --> 00:22:16,580
place at every beta.

414
00:22:16,580 --> 00:22:20,510
And that average quantity
starts in itself to be 0.

415
00:22:20,510 --> 00:22:23,940
So how does our
velocity-velocity

416
00:22:23,940 --> 00:22:26,110
autocorrelation
function looks like?

417
00:22:29,020 --> 00:22:31,120
Well, it looks like
something like this.

418
00:22:31,120 --> 00:22:33,950
I need to sort of graph it here.

419
00:22:33,950 --> 00:22:38,410
So again, when you are really
sort of far away in time--

420
00:22:38,410 --> 00:22:40,270
so, you know, what
we are plotting

421
00:22:40,270 --> 00:22:47,260
is the ensemble average
of vdt times vf0.

422
00:22:47,260 --> 00:22:51,340
So very, very far away in
time, it's going to be 0.

423
00:22:51,340 --> 00:22:55,930
There is no correlation between
the velocity of the atoms.

424
00:22:55,930 --> 00:23:02,680
At very, very small times, there
is really maximum correlation.

425
00:23:02,680 --> 00:23:05,800
And then what is
in between really

426
00:23:05,800 --> 00:23:08,500
depends on the
dynamics of the system.

427
00:23:08,500 --> 00:23:14,110
And it can look very different
in different systems.

428
00:23:14,110 --> 00:23:18,250
But Einstein relation and
the Green-Kubo formula

429
00:23:18,250 --> 00:23:20,170
that we have seen
before actually

430
00:23:20,170 --> 00:23:24,520
relates, first of all,
the integral from 0

431
00:23:24,520 --> 00:23:29,120
to infinity of the expansion
to the diffusion coefficient.

432
00:23:29,120 --> 00:23:31,720
So again, these functions
really represent

433
00:23:31,720 --> 00:23:34,510
a [INAUDIBLE]
microscopic fluctuation

434
00:23:34,510 --> 00:23:38,290
at equilibrium, sort of how the
velocities are correlated, so

435
00:23:38,290 --> 00:23:40,870
how your vibrate around.

436
00:23:40,870 --> 00:23:43,780
But all the sort of
algebra we have seem before

437
00:23:43,780 --> 00:23:47,350
shows also that the
integral of this quantity

438
00:23:47,350 --> 00:23:50,780
gives you the diffusion
coefficient [INAUDIBLE]..

439
00:23:50,780 --> 00:23:53,270
And that's one way of
calculating the diffusion

440
00:23:53,270 --> 00:23:54,410
coefficient.

441
00:23:54,410 --> 00:23:57,500
The other way, if you
go back to your slides,

442
00:23:57,500 --> 00:24:01,350
would be just calculating
the derivative with respect

443
00:24:01,350 --> 00:24:03,740
to time of the mean
squared displacement.

444
00:24:03,740 --> 00:24:05,750
And so in a liquid,
you have seen

445
00:24:05,750 --> 00:24:09,080
that your mean square
displacements sort of become

446
00:24:09,080 --> 00:24:12,350
linear if you sort of weighed
enough time with respect

447
00:24:12,350 --> 00:24:13,080
to time.

448
00:24:13,080 --> 00:24:17,150
And so the diffusion coefficient
is also equivalently given

449
00:24:17,150 --> 00:24:20,000
by the slope of this system.

450
00:24:20,000 --> 00:24:22,850
One of the interesting
things that you

451
00:24:22,850 --> 00:24:26,510
obtain from the sort
of velocity-velocity

452
00:24:26,510 --> 00:24:30,020
autocorrelation function
that you don't obtain

453
00:24:30,020 --> 00:24:33,380
from the slope of your
mean square displacement

454
00:24:33,380 --> 00:24:36,980
is that you can obtain what
we call the power spectrum.

455
00:24:36,980 --> 00:24:40,670
That is you can look at
a system like a liquid

456
00:24:40,670 --> 00:24:44,870
and figure out what are
the typical vibrations

457
00:24:44,870 --> 00:24:46,130
in your system.

458
00:24:46,130 --> 00:24:49,400
Again, this is because, you
know, what we have said.

459
00:24:49,400 --> 00:24:54,150
If a system, if an atom,
is oscillating around, say,

460
00:24:54,150 --> 00:24:56,300
an equilibrium
position, there is going

461
00:24:56,300 --> 00:24:58,460
to be a lot of correlation.

462
00:24:58,460 --> 00:25:01,860
That is, if we look at time 0
and if you look after a period

463
00:25:01,860 --> 00:25:05,900
t, the velocities are, again,
going to be very similar.

464
00:25:05,900 --> 00:25:08,550
If you look again at
time 2t, they are, again,

465
00:25:08,550 --> 00:25:09,560
going to be similar.

466
00:25:09,560 --> 00:25:11,400
If you look at the time
3t, they are, again,

467
00:25:11,400 --> 00:25:12,690
going to be similar.

468
00:25:12,690 --> 00:25:16,250
So in the velocity-velocity
autocorrelation function,

469
00:25:16,250 --> 00:25:19,820
actually you will see not
only a peak at 0, but a peak

470
00:25:19,820 --> 00:25:24,620
at t, at 2t, and 3t and so on,
and then sort of slowly decay.

471
00:25:24,620 --> 00:25:26,900
But so that means that
the velocity-velocity

472
00:25:26,900 --> 00:25:32,000
autocorrelation function
will sort of show somehow

473
00:25:32,000 --> 00:25:34,940
some periodic features
that are related

474
00:25:34,940 --> 00:25:38,970
to what is the typical
dynamics of your particles.

475
00:25:38,970 --> 00:25:41,900
And so if you do the Fourier
transform of that, the Fourier

476
00:25:41,900 --> 00:25:44,340
transform of a
function, it picks up

477
00:25:44,340 --> 00:25:47,640
what are the relevant
frequencies in that function.

478
00:25:47,640 --> 00:25:49,580
And so if you do
that, you actually

479
00:25:49,580 --> 00:25:52,760
find out that what is
the vibrational density

480
00:25:52,760 --> 00:25:55,700
of states, what are the typical
frequencies of your system.

481
00:25:55,700 --> 00:25:58,640
And this is sort
of an example that

482
00:25:58,640 --> 00:26:02,580
comes from a molecular
dynamics simulation of water.

483
00:26:02,580 --> 00:26:05,570
So what you have is you
have your liquid water.

484
00:26:05,570 --> 00:26:09,060
Remember, that in a
system like liquid water,

485
00:26:09,060 --> 00:26:13,760
you actually need only 30, 40,
50 molecules in a unit cell

486
00:26:13,760 --> 00:26:17,150
to basically simulate the
infinite system as long as you

487
00:26:17,150 --> 00:26:21,020
are sort of enough far away
from the melting or freezing

488
00:26:21,020 --> 00:26:22,100
transition.

489
00:26:22,100 --> 00:26:25,040
So you take the system,
you let it evolve.

490
00:26:25,040 --> 00:26:29,210
And then you calculate at
every instant in time t

491
00:26:29,210 --> 00:26:31,220
the product of the velocity.

492
00:26:31,220 --> 00:26:33,080
You average that on
all the molecules,

493
00:26:33,080 --> 00:26:34,950
and you have the
velocity-velocity

494
00:26:34,950 --> 00:26:36,590
autocorrelation function.

495
00:26:36,590 --> 00:26:40,310
And that, as a
function of time, will

496
00:26:40,310 --> 00:26:44,960
show typical beatings that
have to do with the frequencies

497
00:26:44,960 --> 00:26:46,160
in your system.

498
00:26:46,160 --> 00:26:48,770
You do the Fourier
transform of this,

499
00:26:48,770 --> 00:26:50,930
and you find a power spectrum.

500
00:26:50,930 --> 00:26:55,430
You find a vibrational spectrum
of your liquid in which you see

501
00:26:55,430 --> 00:26:58,465
you can identify
a very large peak,

502
00:26:58,465 --> 00:27:02,990
a very sort of typical set of
vibrations that have really

503
00:27:02,990 --> 00:27:07,340
to do-- this is actually heavy
water, so it's got deuterium--

504
00:27:07,340 --> 00:27:12,710
with the stretching mode of
the deuterium oxygen distance

505
00:27:12,710 --> 00:27:16,430
in the water molecule, so the
modes in which the two atoms

506
00:27:16,430 --> 00:27:18,720
vibrate one against the other.

507
00:27:18,720 --> 00:27:22,400
So this would be the optical
intermolecular modes.

508
00:27:22,400 --> 00:27:24,530
And then you see
another peak that

509
00:27:24,530 --> 00:27:27,020
has to do with the
vibration modes, the fact

510
00:27:27,020 --> 00:27:29,990
that the water molecules
act as a scissors.

511
00:27:29,990 --> 00:27:34,230
And then you find a lot
of lower energy modes.

512
00:27:34,230 --> 00:27:36,260
But again, just the
Fourier transform

513
00:27:36,260 --> 00:27:38,840
of this velocity-velocity
autocorrelation function

514
00:27:38,840 --> 00:27:43,040
gives you right away what is the
vibrational density of states

515
00:27:43,040 --> 00:27:46,490
and, again, sort of
a snapshot of what

516
00:27:46,490 --> 00:27:48,740
are the important
vibrations in the system.

517
00:27:48,740 --> 00:27:51,710
And a lot of
spectroscopic properties

518
00:27:51,710 --> 00:27:55,070
would be correlated with this
vibrational density of states.

519
00:27:55,070 --> 00:27:57,740
Because suppose that
some of these states

520
00:27:57,740 --> 00:28:00,590
are, say, what we
call infrared active.

521
00:28:00,590 --> 00:28:02,810
That is, when the
atoms move around,

522
00:28:02,810 --> 00:28:05,420
they create a
little polarization.

523
00:28:05,420 --> 00:28:08,730
They create a little
local electric field.

524
00:28:08,730 --> 00:28:12,290
Well, then those
modes would interact

525
00:28:12,290 --> 00:28:14,720
and couple very
strongly with sort

526
00:28:14,720 --> 00:28:17,190
of electromagnetic radiation.

527
00:28:17,190 --> 00:28:18,890
And so depending
on your frequency

528
00:28:18,890 --> 00:28:21,150
of your electromagnetic
radiation,

529
00:28:21,150 --> 00:28:24,170
you would couple
very strongly with

530
00:28:24,170 --> 00:28:27,570
the appropriate frequencies
of your liquid system.

531
00:28:27,570 --> 00:28:30,650
And so this is, again, one
of the sort of very important

532
00:28:30,650 --> 00:28:33,320
quantities that you
might want to extract

533
00:28:33,320 --> 00:28:35,910
from a molecular
dynamics simulation.

534
00:28:35,910 --> 00:28:39,680
And if, instead of having a
liquid, you had a solid, again,

535
00:28:39,680 --> 00:28:42,590
that would be the sort
of vibrational density

536
00:28:42,590 --> 00:28:45,930
of states of your phonon modes.

537
00:28:45,930 --> 00:28:46,430
OK.

538
00:28:46,430 --> 00:28:48,950
This sort of concludes
this parenthesis

539
00:28:48,950 --> 00:28:51,450
on Green-Kubo approach.

540
00:28:51,450 --> 00:28:55,660
You'll see more of this in
one of the later classes.

541
00:28:55,660 --> 00:28:58,140
But again, if there is one
thing that you need to remember,

542
00:28:58,140 --> 00:29:03,090
it's this, that this general
sort of set of relation

543
00:29:03,090 --> 00:29:06,810
make a connection between
a macroscopic property

544
00:29:06,810 --> 00:29:10,320
and, in particular, a response
property of the system.

545
00:29:10,320 --> 00:29:13,200
The diffusion coefficient
is a response property

546
00:29:13,200 --> 00:29:17,730
of the system to a
concentration inhomogeneity.

547
00:29:17,730 --> 00:29:21,600
And these macroscopic property,
these response properties,

548
00:29:21,600 --> 00:29:25,950
can be connected to equilibrium
fluctuations or something

549
00:29:25,950 --> 00:29:29,700
like the velocity-velocity
autocorrelation function.

550
00:29:29,700 --> 00:29:32,310
And there are a lot of
interesting quantities

551
00:29:32,310 --> 00:29:34,530
that you can obtain from
Green-Kubo relations

552
00:29:34,530 --> 00:29:37,300
besides the diffusion
coefficient.

553
00:29:37,300 --> 00:29:39,780
So in particular,
you could find out

554
00:29:39,780 --> 00:29:41,670
the viscosity of your system.

555
00:29:41,670 --> 00:29:44,370
Suppose that you want to study
liquid iron because you want

556
00:29:44,370 --> 00:29:47,190
to understand how
seismic waves propagate

557
00:29:47,190 --> 00:29:50,210
in this sort of liquid
inner core of Earth.

558
00:29:50,210 --> 00:29:53,430
Well, you are very interested
in the viscosity of the system

559
00:29:53,430 --> 00:29:56,130
because you want to know how
sort of shear propagation

560
00:29:56,130 --> 00:29:57,210
takes place.

561
00:29:57,210 --> 00:30:02,460
And just from the fluctuations
in the stress tensor,

562
00:30:02,460 --> 00:30:05,910
if you have your unit cell
with all the iron liquid atoms

563
00:30:05,910 --> 00:30:09,210
moving around, the
fluctuations instantaneous

564
00:30:09,210 --> 00:30:11,985
due to the thermal motion
in the stress tensor

565
00:30:11,985 --> 00:30:14,820
for that system give
you the sheer viscosity.

566
00:30:14,820 --> 00:30:17,970
Or say you want to
study how, say, one

567
00:30:17,970 --> 00:30:20,640
of the systems like
water would couple

568
00:30:20,640 --> 00:30:22,920
and what would be its
infrared absorption.

569
00:30:22,920 --> 00:30:27,690
Well, you can actually look at
the instantaneous fluctuations

570
00:30:27,690 --> 00:30:30,150
in the total polarization
in the system.

571
00:30:30,150 --> 00:30:33,960
Because what is the microscopic
local electric field there

572
00:30:33,960 --> 00:30:35,330
and how it couples?

573
00:30:35,330 --> 00:30:39,060
So again, you can sort of find
out macroscopic properties

574
00:30:39,060 --> 00:30:41,550
from microscopic
fluctuation or sort

575
00:30:41,550 --> 00:30:44,790
of electrical thermal
conductivities.

576
00:30:44,790 --> 00:30:47,610
Again, sort of macroscopic
transfer properties

577
00:30:47,610 --> 00:30:50,700
can be found out
from fluctuations

578
00:30:50,700 --> 00:30:54,150
in the autocorrelation
functions for the sort

579
00:30:54,150 --> 00:31:01,320
of electrical charge or a
thermal carriers in a system.

580
00:31:01,320 --> 00:31:02,460
OK.

581
00:31:02,460 --> 00:31:07,480
This basically concludes the
classical molecular dynamic

582
00:31:07,480 --> 00:31:08,490
part.

583
00:31:08,490 --> 00:31:11,220
And what I wanted
to show you next

584
00:31:11,220 --> 00:31:15,640
is how we actually do first
principle molecular dynamics.

585
00:31:15,640 --> 00:31:19,260
That is how we sort of
evolve atoms in time

586
00:31:19,260 --> 00:31:22,800
not using a classic
field, but using

587
00:31:22,800 --> 00:31:25,080
our favorite electronic
structure methods.

588
00:31:25,080 --> 00:31:26,940
That, actually for
most of this class,

589
00:31:26,940 --> 00:31:28,620
has been density
functional theory,

590
00:31:28,620 --> 00:31:31,230
but doesn't really have to
be density functional theory.

591
00:31:31,230 --> 00:31:35,102
Any of the electronic
structure methods would work.

592
00:31:35,102 --> 00:31:37,560
Density functional theory tends
to be the simplest and most

593
00:31:37,560 --> 00:31:40,110
efficient to implement.

594
00:31:40,110 --> 00:31:42,360
Sadly, in order
to do this, I need

595
00:31:42,360 --> 00:31:49,740
to give you some other reminders
of formal classical mechanics.

596
00:31:49,740 --> 00:31:53,310
Because especially in first
principle molecular dynamics,

597
00:31:53,310 --> 00:31:57,750
we use a lot of the concept
of extended Lagrangian and

598
00:31:57,750 --> 00:31:58,980
extended Hamiltonians.

599
00:31:58,980 --> 00:32:02,400
That is we'll derive
the equation of motion

600
00:32:02,400 --> 00:32:06,540
from an appropriate functional
that includes sometimes very

601
00:32:06,540 --> 00:32:08,530
exotic degrees of freedom.

602
00:32:08,530 --> 00:32:10,890
So let me remind
you how you have

603
00:32:10,890 --> 00:32:14,280
seen this in some of your
physics or mechanics class.

604
00:32:14,280 --> 00:32:17,730
But let me show you
how, in general, one

605
00:32:17,730 --> 00:32:21,810
can think at evolution
in phase space

606
00:32:21,810 --> 00:32:23,700
and sort of find
out the equations

607
00:32:23,700 --> 00:32:25,860
that integrated the trajectory.

608
00:32:25,860 --> 00:32:28,530
And up to now, we
have really just seen

609
00:32:28,530 --> 00:32:30,840
Newton equation
of motion, forces

610
00:32:30,840 --> 00:32:33,270
equal to the mass
times acceleration.

611
00:32:33,270 --> 00:32:36,240
But there is a sort of more
complex and, if you want,

612
00:32:36,240 --> 00:32:41,190
more elegant formalism to
derive the equation of motion

613
00:32:41,190 --> 00:32:42,240
for a system.

614
00:32:42,240 --> 00:32:44,160
And in particular,
what I'm showing here

615
00:32:44,160 --> 00:32:47,640
is sort of what is called
Lagrangian dynamics.

616
00:32:47,640 --> 00:32:50,230
In fact, this is actually
very, very simple.

617
00:32:50,230 --> 00:32:54,490
And the way Lagrangian
dynamics works is this.

618
00:32:54,490 --> 00:32:58,920
First of all, you have to
construct your Lagrangian.

619
00:32:58,920 --> 00:33:02,640
That is the
functional that drives

620
00:33:02,640 --> 00:33:05,580
the evolution of your systems.

621
00:33:05,580 --> 00:33:10,560
And there are various ways
in which one of constructs

622
00:33:10,560 --> 00:33:12,090
these functionals.

623
00:33:12,090 --> 00:33:14,160
And there are even sort
of equivalent ways.

624
00:33:14,160 --> 00:33:16,950
One can construct
different Lagrangians

625
00:33:16,950 --> 00:33:19,380
that give you the same
equation of motion

626
00:33:19,380 --> 00:33:20,910
of the same trajectories.

627
00:33:20,910 --> 00:33:24,780
But, see, what is important
here is the standard way.

628
00:33:24,780 --> 00:33:27,340
That is the way we
construct this functional,

629
00:33:27,340 --> 00:33:29,400
it's not very different
from thermodynamics.

630
00:33:29,400 --> 00:33:32,040
We just take the kinetic
energy of the system,

631
00:33:32,040 --> 00:33:36,070
T. We subtract the potential
energy of the system.

632
00:33:36,070 --> 00:33:39,930
And this is sort of our
Lagrangian, T minus V.

633
00:33:39,930 --> 00:33:45,090
In general, the potential
energy, as you have seen it,

634
00:33:45,090 --> 00:33:49,300
tends to be a function
of position only.

635
00:33:49,300 --> 00:33:51,970
So usually, we
written as a function

636
00:33:51,970 --> 00:33:55,720
of, if we have n
particles, r1 to rn.

637
00:33:55,720 --> 00:33:59,230
So you know, this is what is
called a conservative field.

638
00:33:59,230 --> 00:34:01,270
If you are sort of
a particle living

639
00:34:01,270 --> 00:34:03,430
in a gravitational
field, well, you

640
00:34:03,430 --> 00:34:05,230
are going to be in
a certain position.

641
00:34:05,230 --> 00:34:07,300
You are going to feel
a certain potential,

642
00:34:07,300 --> 00:34:09,070
or you are going to
feel a certain force.

643
00:34:09,070 --> 00:34:11,020
That's the gradient
of that potential.

644
00:34:11,020 --> 00:34:13,420
You go somewhere else, you'll
feel a different potential.

645
00:34:13,420 --> 00:34:14,920
You see a different force.

646
00:34:14,920 --> 00:34:18,159
And the work that you sort of
make in going from one place

647
00:34:18,159 --> 00:34:20,889
to the other, it's just
the integral of that force.

648
00:34:20,889 --> 00:34:23,300
And it's independent
of the trajectory.

649
00:34:23,300 --> 00:34:25,630
So this is sort of, you
know, a very general sort

650
00:34:25,630 --> 00:34:27,820
of potential function
that you have seen.

651
00:34:27,820 --> 00:34:30,710
And you know, again, the
kinetic energy, you have seen it

652
00:34:30,710 --> 00:34:37,245
and tends to be a function of
the square of the velocities.

653
00:34:37,245 --> 00:34:37,745
OK.

654
00:34:42,040 --> 00:34:45,280
So again, if you have
only one particle,

655
00:34:45,280 --> 00:34:48,250
its kinetic energy is
going to be 1/2 times

656
00:34:48,250 --> 00:34:52,389
the mass times the
square velocity.

657
00:34:52,389 --> 00:34:55,600
We usually, in the
Lagrangian formulation,

658
00:34:55,600 --> 00:35:01,420
don't use the sort of notation
for the positions r1, rn.

659
00:35:01,420 --> 00:35:02,860
But instead, say
in particular, we

660
00:35:02,860 --> 00:35:06,460
use the other notation in
which the coordinates are

661
00:35:06,460 --> 00:35:09,460
given by q1, q2, qn.

662
00:35:09,460 --> 00:35:13,780
And then the velocities we
just indicate them as q dot.

663
00:35:13,780 --> 00:35:17,950
And the reason we call them
q is that what you sometimes

664
00:35:17,950 --> 00:35:22,990
want to do is not use
your regular coordinates

665
00:35:22,990 --> 00:35:26,350
as the description
of your position

666
00:35:26,350 --> 00:35:28,810
for your dynamical
system, but you might want

667
00:35:28,810 --> 00:35:31,420
to use generalized coordinates.

668
00:35:31,420 --> 00:35:34,870
Say you study water molecules.

669
00:35:34,870 --> 00:35:38,920
And all of a sudden you want to
describe this liquid of water

670
00:35:38,920 --> 00:35:41,720
molecules as rigid molecules.

671
00:35:41,720 --> 00:35:45,130
So you want to
say that the angle

672
00:35:45,130 --> 00:35:48,700
between the hydrogen,
oxygen, and the hydrogen

673
00:35:48,700 --> 00:35:49,670
doesn't change.

674
00:35:49,670 --> 00:35:53,140
And if you want to say that the
distance between the hydrogen

675
00:35:53,140 --> 00:35:55,270
and oxygen doesn't
change, well, then you

676
00:35:55,270 --> 00:35:59,440
want to sort of develop
a dynamic in which what

677
00:35:59,440 --> 00:36:03,200
you really move around are
not the position of the atoms,

678
00:36:03,200 --> 00:36:05,740
but you move around
the center of mass

679
00:36:05,740 --> 00:36:07,360
of your water molecules.

680
00:36:07,360 --> 00:36:10,197
And you move around
the orientation.

681
00:36:10,197 --> 00:36:11,530
This is actually very important.

682
00:36:11,530 --> 00:36:14,080
If you remember in
sort of last class,

683
00:36:14,080 --> 00:36:17,560
I've told you that, when
we study water, actually

684
00:36:17,560 --> 00:36:20,770
because water at
regular temperature

685
00:36:20,770 --> 00:36:22,780
is still a quantum
system, is still

686
00:36:22,780 --> 00:36:27,220
a system that has most
of its vibrational states

687
00:36:27,220 --> 00:36:31,150
frozen in the sort of 0
point motion quantum state,

688
00:36:31,150 --> 00:36:34,870
you actually tend to
describe better liquid water

689
00:36:34,870 --> 00:36:39,940
if you describe it as a set of
rigid molecules moving around.

690
00:36:39,940 --> 00:36:41,990
This is, again,
an approximation.

691
00:36:41,990 --> 00:36:43,960
But it's actually a
better approximation

692
00:36:43,960 --> 00:36:47,210
of the true dynamics of the
system than an approximation

693
00:36:47,210 --> 00:36:50,270
which you let also the internal
degrees of freedom change.

694
00:36:50,270 --> 00:36:52,030
So suppose you want
to sort of simulate

695
00:36:52,030 --> 00:36:53,350
the rigid water around.

696
00:36:53,350 --> 00:36:56,500
You need to find out what
are the equation of motion

697
00:36:56,500 --> 00:36:59,620
for this generalized
set of coordinates

698
00:36:59,620 --> 00:37:02,680
in which what you really move
around when you move water

699
00:37:02,680 --> 00:37:05,990
is the center of mass
and their orientation.

700
00:37:05,990 --> 00:37:08,050
And you know, it would
be very difficult to do

701
00:37:08,050 --> 00:37:11,860
sort of using Newton's equation
of motion with a constraint.

702
00:37:11,860 --> 00:37:15,550
And so what you do, you use
your Lagrangian formulation

703
00:37:15,550 --> 00:37:19,450
in a generalized formalism
of generalized coordinates q

704
00:37:19,450 --> 00:37:22,270
and generalized velocity q dot.

705
00:37:22,270 --> 00:37:24,580
So what Lagrangian
dynamics tell us

706
00:37:24,580 --> 00:37:27,790
is that we construct our
Lagrangian function T

707
00:37:27,790 --> 00:37:32,410
minus V. And then the equation
of motion are given by these.

708
00:37:32,410 --> 00:37:33,880
These are the
Lagrangian equation.

709
00:37:33,880 --> 00:37:37,810
We want to derive them, but
this is how they are written.

710
00:37:37,810 --> 00:37:40,975
The derivative with respect to
time of the partial derivatives

711
00:37:40,975 --> 00:37:44,540
of the Lagrangian with respect
to the generalized velocity q

712
00:37:44,540 --> 00:37:47,530
dot minus the partial
derivatives of the Lagrangian

713
00:37:47,530 --> 00:37:49,480
with respect to the
generalized coordinates

714
00:37:49,480 --> 00:37:51,550
needs to be equal to 0.

715
00:37:51,550 --> 00:37:54,490
And you know, this
also focuses us

716
00:37:54,490 --> 00:37:58,210
on just constructing
the two scalar

717
00:37:58,210 --> 00:38:04,038
function, kinetic energy and
potential energy, for a system.

718
00:38:04,038 --> 00:38:05,830
And that is just, you
know, straightforward

719
00:38:05,830 --> 00:38:08,440
algebra to derive this equation.

720
00:38:08,440 --> 00:38:10,660
And I've actually
done the derivation

721
00:38:10,660 --> 00:38:15,550
for the sort of simple
case of Newtonian dynamics.

722
00:38:15,550 --> 00:38:20,680
You actually see how trivially
the Lagrange equation

723
00:38:20,680 --> 00:38:25,330
that is written here turns into
Newton's equation of motion

724
00:38:25,330 --> 00:38:29,950
when you plug in for your
Lagrangian 1/2 nV squared

725
00:38:29,950 --> 00:38:32,480
minus your potential energy.

726
00:38:32,480 --> 00:38:34,810
And you see, when you take
the partial derivatives

727
00:38:34,810 --> 00:38:38,410
of the Lagrangian with respect
to the generalized velocity,

728
00:38:38,410 --> 00:38:40,510
since it is a
partial derivative,

729
00:38:40,510 --> 00:38:42,790
you only need to
take the derivative

730
00:38:42,790 --> 00:38:46,030
of the kinetic energy with
respect to the velocity,

731
00:38:46,030 --> 00:38:47,710
with respect to x dot.

732
00:38:47,710 --> 00:38:50,890
And then you need to
take partial derivatives

733
00:38:50,890 --> 00:38:52,330
with respect to the position.

734
00:38:52,330 --> 00:38:55,190
And there is no position in
the kinetic energy functional.

735
00:38:55,190 --> 00:38:58,760
So there is only the position
in that potential energy.

736
00:38:58,760 --> 00:39:02,155
So what you see is that,
from the left-hand term,

737
00:39:02,155 --> 00:39:03,760
you have the
derivative with respect

738
00:39:03,760 --> 00:39:07,850
to time of mass times velocity.

739
00:39:07,850 --> 00:39:11,260
And so that's nothing else
than mass times acceleration.

740
00:39:11,260 --> 00:39:14,710
And then on the right,
the right-hand term,

741
00:39:14,710 --> 00:39:20,140
you have minus the gradient
of your conservative potential

742
00:39:20,140 --> 00:39:20,800
field.

743
00:39:20,800 --> 00:39:22,970
And so this is nothing
else than the force.

744
00:39:22,970 --> 00:39:27,430
So we have force equal to
mass times acceleration.

745
00:39:27,430 --> 00:39:29,380
And we have recovered.

746
00:39:29,380 --> 00:39:33,910
Just by applying
Lagrange equations

747
00:39:33,910 --> 00:39:38,110
to the Lagrangian of
classical dynamics, kinetics

748
00:39:38,110 --> 00:39:42,710
minus potential, we have derived
Newton's equation of motion.

749
00:39:42,710 --> 00:39:48,950
So this is one way of deriving
equation of motion, but, again,

750
00:39:48,950 --> 00:39:52,060
are going to be
second-order differential

751
00:39:52,060 --> 00:39:54,190
equation with respect to time.

752
00:39:54,190 --> 00:39:56,560
Because, if you
think, you are taking

753
00:39:56,560 --> 00:40:00,430
the derivative of a kinetic
energy with respect to q dot.

754
00:40:00,430 --> 00:40:03,310
There is a second formulation
of classical mechanics--

755
00:40:03,310 --> 00:40:05,920
and I also need to sort
of present this here--

756
00:40:05,920 --> 00:40:08,920
that is called the
Hamiltonian formulation.

757
00:40:08,920 --> 00:40:12,370
And it's sort of,
again, very easy

758
00:40:12,370 --> 00:40:16,630
to see this if you think
of a thermodynamic analogy.

759
00:40:16,630 --> 00:40:20,320
Say, when you're sort of
looking at thermodynamics,

760
00:40:20,320 --> 00:40:23,230
suppose that you are in the
microcanonical ensemble.

761
00:40:23,230 --> 00:40:26,090
Your thermodynamical
functional is the energy.

762
00:40:26,090 --> 00:40:30,850
And so when you sort of look at
that thermodynamical ensemble,

763
00:40:30,850 --> 00:40:33,100
it means that, say, you are
looking at the system that

764
00:40:33,100 --> 00:40:35,980
has a constant energy,
constant number of particle,

765
00:40:35,980 --> 00:40:37,600
and constant volume.

766
00:40:37,600 --> 00:40:41,500
But many times it becomes
appropriate to look

767
00:40:41,500 --> 00:40:43,690
at, actually, sort
of a system that

768
00:40:43,690 --> 00:40:47,170
lives at constant pressure
or constant temperature.

769
00:40:47,170 --> 00:40:50,020
And so you transform your
thermodynamical functional

770
00:40:50,020 --> 00:40:53,020
from the energy to one, say,
of the Helmholtz or Gibbs

771
00:40:53,020 --> 00:40:53,890
free energies.

772
00:40:53,890 --> 00:40:58,030
You do e minus ds to obtain a
thermodynamic functional that

773
00:40:58,030 --> 00:41:01,090
depends on temperature instead
of depending on entropy.

774
00:41:01,090 --> 00:41:04,720
Or you do e plus pv to
obtain the enthalpy that

775
00:41:04,720 --> 00:41:07,900
sort of depends on
pressure and not on volume.

776
00:41:07,900 --> 00:41:12,860
And this is this general concept
of Legendre transformation.

777
00:41:12,860 --> 00:41:15,010
If you have a
function, let's say,

778
00:41:15,010 --> 00:41:20,200
for the Lagrangian was a
functional of q and q dot,

779
00:41:20,200 --> 00:41:27,580
you can construct a new one that
doesn't depend, say, on q dot,

780
00:41:27,580 --> 00:41:31,060
but depends only
on a new variable

781
00:41:31,060 --> 00:41:35,050
that we call it the
conjugate variable 2q dot.

782
00:41:35,050 --> 00:41:39,760
So pressure and volume,
temperature and entropy,

783
00:41:39,760 --> 00:41:42,670
chemical potential and
number of particles

784
00:41:42,670 --> 00:41:44,740
are all conjugate variables.

785
00:41:44,740 --> 00:41:49,750
And so, say, if you take the
Lagrangian and you derive it

786
00:41:49,750 --> 00:41:51,470
with respect to
q dot-- remember,

787
00:41:51,470 --> 00:41:53,680
the Lagrangian is a
function of q dot--

788
00:41:53,680 --> 00:41:56,710
what you obtain is
at conjugate variable

789
00:41:56,710 --> 00:42:00,010
that we call a
conjugate momentum.

790
00:42:00,010 --> 00:42:03,220
And then if you do this
operation, this Legendre

791
00:42:03,220 --> 00:42:06,070
transform in which you take--

792
00:42:06,070 --> 00:42:08,450
the sign doesn't really matter.

793
00:42:08,450 --> 00:42:10,330
In this case, you
take a minus sign.

794
00:42:10,330 --> 00:42:14,710
And you sum the product
of the conjugate variable

795
00:42:14,710 --> 00:42:16,690
times your original variable.

796
00:42:16,690 --> 00:42:20,650
You get a new function
that doesn't depend

797
00:42:20,650 --> 00:42:24,580
on your original q dot
variable, but depends only

798
00:42:24,580 --> 00:42:26,530
on its conjugate variable p.

799
00:42:26,530 --> 00:42:29,620
That's how you remove the
dependence, say, on the volume

800
00:42:29,620 --> 00:42:32,080
and you put in the
dependence on the pressure,

801
00:42:32,080 --> 00:42:33,700
making the Legendre
transformation

802
00:42:33,700 --> 00:42:36,370
which you have pv, or that's
how you sort of move on

803
00:42:36,370 --> 00:42:40,720
from entropy to temperature
in the Helmholtz free energy.

804
00:42:40,720 --> 00:42:42,760
And this is just
very simple to do

805
00:42:42,760 --> 00:42:47,440
when you take the
differential of this quantity.

806
00:42:47,440 --> 00:42:49,600
Because you are going
to see that really,

807
00:42:49,600 --> 00:42:54,220
because of this relation,
you remove all the dependency

808
00:42:54,220 --> 00:42:54,910
in q dot.

809
00:42:54,910 --> 00:42:59,020
And you put into your
system an independence on p.

810
00:42:59,020 --> 00:43:00,800
Well, why do we do this?

811
00:43:00,800 --> 00:43:03,350
Well, because, again
with some algebra,

812
00:43:03,350 --> 00:43:07,750
we can find that, for this
new function that is now

813
00:43:07,750 --> 00:43:11,500
called the Hamiltonian,
another alternative

814
00:43:11,500 --> 00:43:13,780
sets of equation of motions.

815
00:43:13,780 --> 00:43:16,120
Remember, from
the Lagrangian, we

816
00:43:16,120 --> 00:43:20,080
had obtained equation of motion
basically for q dot and q.

817
00:43:20,080 --> 00:43:22,930
And from the
Hamiltonian formulation,

818
00:43:22,930 --> 00:43:26,860
if we work this out, we
obtain equation of motion

819
00:43:26,860 --> 00:43:29,470
for q and for p.

820
00:43:29,470 --> 00:43:31,570
And sort of the
slight difference

821
00:43:31,570 --> 00:43:37,300
is that instead of having
basically a second order

822
00:43:37,300 --> 00:43:40,990
differential equation from the
Lagrangian formulation, now

823
00:43:40,990 --> 00:43:45,430
we have a double
set of differential

824
00:43:45,430 --> 00:43:47,710
that are only first-order.

825
00:43:47,710 --> 00:43:49,840
So depending on
the your problem,

826
00:43:49,840 --> 00:43:52,180
they can actually
be easier to solve.

827
00:43:52,180 --> 00:43:54,340
They can actually be
different, but they could

828
00:43:54,340 --> 00:43:56,450
lead to the same trajectories.

829
00:43:56,450 --> 00:44:00,430
So again, all these formalisms
of Lagrangian and Hamiltonian

830
00:44:00,430 --> 00:44:04,870
is that's the very general
way to construct functions,

831
00:44:04,870 --> 00:44:09,370
either the Lagrangian T
minus V or the Hamiltonian

832
00:44:09,370 --> 00:44:12,670
via the Legendre
transform that give us

833
00:44:12,670 --> 00:44:15,640
equation of motion
in q and q dot

834
00:44:15,640 --> 00:44:19,390
for the case of the
Lagrangian and in q and p

835
00:44:19,390 --> 00:44:21,880
for the case of the Hamiltonian.

836
00:44:21,880 --> 00:44:23,530
And if you actually
work this out

837
00:44:23,530 --> 00:44:27,460
for the standard case
of Newtonian dynamics,

838
00:44:27,460 --> 00:44:30,850
you find out that your
Hamiltonian is, again,

839
00:44:30,850 --> 00:44:32,950
something very trivial.

840
00:44:32,950 --> 00:44:37,010
It's just the kinetic energy
plus the potential energy.

841
00:44:37,010 --> 00:44:40,480
So at the end of all of
this, for all the cases

842
00:44:40,480 --> 00:44:43,260
of interest to you, you see
that either you construct

843
00:44:43,260 --> 00:44:46,550
a Lagrangian T
minus V and you have

844
00:44:46,550 --> 00:44:48,610
the Lagrangian
equation of motion,

845
00:44:48,610 --> 00:44:52,750
or your construct your
Hamiltonian, p plus v.

846
00:44:52,750 --> 00:44:56,680
And you have the sort of
Hamiltonian equation of motion

847
00:44:56,680 --> 00:44:59,380
that I have written here.

848
00:44:59,380 --> 00:45:06,380
And one of the reasons
that we do this--

849
00:45:06,380 --> 00:45:15,070
so this is-- is that,
often, we want actually

850
00:45:15,070 --> 00:45:18,620
to simulate the
microscopic dynamics

851
00:45:18,620 --> 00:45:21,200
not in the
microcanonical ensemble,

852
00:45:21,200 --> 00:45:23,930
but in different
thermodynamical ensembles,

853
00:45:23,930 --> 00:45:26,510
say, in which we
control the temperature.

854
00:45:26,510 --> 00:45:28,940
Or we control the
pressure, or maybe we

855
00:45:28,940 --> 00:45:31,220
control the number of particles.

856
00:45:31,220 --> 00:45:34,110
Or maybe we control
the chemical potential.

857
00:45:34,110 --> 00:45:38,690
And so we need to have some
of this sort of formalism

858
00:45:38,690 --> 00:45:40,430
to do this effectively.

859
00:45:40,430 --> 00:45:44,000
In particular, remember when
we have discussed the sort

860
00:45:44,000 --> 00:45:45,710
of control of temperature.

861
00:45:45,710 --> 00:45:47,240
I've told you that
there are sort

862
00:45:47,240 --> 00:45:49,520
of three different
approaches in which you

863
00:45:49,520 --> 00:45:52,580
can do a canonical simulation,
which you can control

864
00:45:52,580 --> 00:45:54,050
the temperature in your system.

865
00:45:54,050 --> 00:45:56,300
And you could have
Langevin dynamics.

866
00:45:56,300 --> 00:45:59,510
You could have a stochastic
approach in which you randomly

867
00:45:59,510 --> 00:46:02,480
pick atoms in order
to accelerate them

868
00:46:02,480 --> 00:46:04,040
or to slow them down.

869
00:46:04,040 --> 00:46:06,710
So that in analogy
with a thermal bath,

870
00:46:06,710 --> 00:46:10,790
they sort of, on average,
have the right kinetic target,

871
00:46:10,790 --> 00:46:13,280
kinetic energy for your problem.

872
00:46:13,280 --> 00:46:16,940
Or you could do something that
is probably even more brutal,

873
00:46:16,940 --> 00:46:20,870
although it tends to be very
efficient in thermalizing,

874
00:46:20,870 --> 00:46:22,370
effectively, your system.

875
00:46:22,370 --> 00:46:25,490
You could actually do
a dynamics in which,

876
00:46:25,490 --> 00:46:28,250
every time you have
got new position,

877
00:46:28,250 --> 00:46:32,150
you renormalize those new
position by renormalizing

878
00:46:32,150 --> 00:46:35,540
the velocity of the
particle, so that the sum

879
00:46:35,540 --> 00:46:38,670
of the kinetic energy
is a actually constant.

880
00:46:38,670 --> 00:46:41,390
So a constraint
method would actually

881
00:46:41,390 --> 00:46:45,410
keep, strictly speaking, the
temperature of your system

882
00:46:45,410 --> 00:46:47,990
sort of constant to
your target value.

883
00:46:47,990 --> 00:46:51,290
That can be actually
very effective

884
00:46:51,290 --> 00:46:53,700
to thermalize your
system, to bring

885
00:46:53,700 --> 00:46:58,520
it really very close to your
equilibrium distribution.

886
00:46:58,520 --> 00:47:02,660
But it does have
some counter-effects.

887
00:47:02,660 --> 00:47:05,030
That is, if you
actually look at what

888
00:47:05,030 --> 00:47:09,080
is going to be your equilibrium
distribution in positions,

889
00:47:09,080 --> 00:47:12,410
it's really going to sort of
be a canonical distribution

890
00:47:12,410 --> 00:47:14,840
according to the Boltzmann
canonical ensemble.

891
00:47:14,840 --> 00:47:17,690
But if you look at your
distribution of velocity,

892
00:47:17,690 --> 00:47:19,710
it's only pseudo-canonical.

893
00:47:19,710 --> 00:47:21,470
So often, people,
especially those

894
00:47:21,470 --> 00:47:25,460
that do very complex and
long molecular dynamics,

895
00:47:25,460 --> 00:47:29,120
use the sort of most elegant
and most accurate approach.

896
00:47:29,120 --> 00:47:32,480
That is really
coupling your system

897
00:47:32,480 --> 00:47:35,180
to an additional
dynamical variable

898
00:47:35,180 --> 00:47:39,380
using an extended Lagrangian
or an extended Hamiltonian.

899
00:47:39,380 --> 00:47:43,160
So we have written our
Lagrangian or Hamiltonian

900
00:47:43,160 --> 00:47:45,470
in terms of generalized
coordinates.

901
00:47:45,470 --> 00:47:49,510
All of a sudden, you can add
one more generalized coordinate.

902
00:47:49,510 --> 00:47:52,490
So you can add, if you
want, a pseudo-particle

903
00:47:52,490 --> 00:47:56,510
in your system with its
own sort of kinetic energy

904
00:47:56,510 --> 00:47:58,940
and with its own
potential energy.

905
00:47:58,940 --> 00:48:01,370
And you can construct
the kinetic energy.

906
00:48:01,370 --> 00:48:04,230
And in particular, you can
construct the potential energy,

907
00:48:04,230 --> 00:48:08,390
so that this additional
dynamical system,

908
00:48:08,390 --> 00:48:11,540
this additional dynamical
variable, interacts

909
00:48:11,540 --> 00:48:13,820
with the other
dynamical variables,

910
00:48:13,820 --> 00:48:17,400
basically exchanging
temperature with it

911
00:48:17,400 --> 00:48:20,690
to bring the sort of
average temperature

912
00:48:20,690 --> 00:48:24,770
of the real classical
particles to the equilibrium

913
00:48:24,770 --> 00:48:25,868
distribution.

914
00:48:25,868 --> 00:48:27,410
And so this is
actually how you would

915
00:48:27,410 --> 00:48:31,460
write the extended
Lagrangian for the case

916
00:48:31,460 --> 00:48:34,130
of a canonical
simulation, something

917
00:48:34,130 --> 00:48:37,460
in which we want to keep
the temperature constant.

918
00:48:37,460 --> 00:48:40,640
And so you see that's
where the power of all

919
00:48:40,640 --> 00:48:43,680
this generalized
system comes into play.

920
00:48:43,680 --> 00:48:48,230
And again, you sort of write
out your kinetic energy.

921
00:48:48,230 --> 00:48:50,020
And note that there
is this sort of s

922
00:48:50,020 --> 00:48:54,440
squared term that couples
the kinetic energy

923
00:48:54,440 --> 00:48:58,670
of your real particles
with the kinetic energy

924
00:48:58,670 --> 00:49:02,160
of this new thermostat,
as we call it.

925
00:49:02,160 --> 00:49:05,730
So we have kinetic energy
minus potential energy.

926
00:49:05,730 --> 00:49:07,700
And then this is
the new particle.

927
00:49:07,700 --> 00:49:10,610
This is the new
extended coordinate

928
00:49:10,610 --> 00:49:12,860
that is described by
a generalized position

929
00:49:12,860 --> 00:49:17,720
s and the generalized
velocity s dot.

930
00:49:17,720 --> 00:49:20,600
And so, if you want,
its kinetic energy

931
00:49:20,600 --> 00:49:24,755
is written in a trivial way,
1/2 our generalized mass times

932
00:49:24,755 --> 00:49:26,060
the square velocity.

933
00:49:26,060 --> 00:49:30,500
And this is actually how Nosé
figured out the potential

934
00:49:30,500 --> 00:49:35,840
energy should look like for a
system of n classical particles

935
00:49:35,840 --> 00:49:38,890
that you want to keep at an
inverse temperature beta.

936
00:49:38,890 --> 00:49:42,260
Remember, beta is just 1 over
the Boltzmann constant times

937
00:49:42,260 --> 00:49:43,250
the temperature.

938
00:49:43,250 --> 00:49:45,710
And so it's a very
exotic potential energy.

939
00:49:45,710 --> 00:49:50,840
And it's sort of the logarithm
of this generalized coordinate.

940
00:49:50,840 --> 00:49:53,450
And so if you use
these Lagrangian,

941
00:49:53,450 --> 00:49:57,210
you can construct the
Lagrangian equation of motion.

942
00:49:57,210 --> 00:50:01,790
So you will have equation of
motion for your 3N particle

943
00:50:01,790 --> 00:50:04,580
that will be very similar
to your standard equation

944
00:50:04,580 --> 00:50:08,030
of motion, but we'll
have in there also terms

945
00:50:08,030 --> 00:50:09,140
that contain s.

946
00:50:09,140 --> 00:50:11,870
And then you will have a
sort of an equation of motion

947
00:50:11,870 --> 00:50:16,430
for your Nosé or Nosé-Hoover
variable that is equation

948
00:50:16,430 --> 00:50:18,020
of motion for s.

949
00:50:18,020 --> 00:50:20,810
And you let all
this system evolve.

950
00:50:20,810 --> 00:50:26,360
And what will happen is
that the kinetic energies

951
00:50:26,360 --> 00:50:32,240
of your classical particles
will start to sort of distribute

952
00:50:32,240 --> 00:50:36,020
themselves according to a
Maxwell-Boltzmann distribution.

953
00:50:36,020 --> 00:50:39,020
That is according to
the canonical ensemble.

954
00:50:39,020 --> 00:50:42,950
And people have worked out
all the statistical mechanics

955
00:50:42,950 --> 00:50:45,560
of your problem
and so worked out

956
00:50:45,560 --> 00:50:47,120
the appropriate coordination.

957
00:50:47,120 --> 00:50:50,720
You would need to have a sort of
Maxwell-Boltzmann distribution.

958
00:50:50,720 --> 00:50:52,610
This is in the moduli
of your system.

959
00:50:52,610 --> 00:50:54,200
And really, if
you do all of this

960
00:50:54,200 --> 00:50:58,070
and you sort of take an average
from your molecular dynamic

961
00:50:58,070 --> 00:51:01,880
simulation, you really
see that both the position

962
00:51:01,880 --> 00:51:05,420
and the velocity of
your classical particles

963
00:51:05,420 --> 00:51:08,460
are distributed in
the appropriate way.

964
00:51:08,460 --> 00:51:12,710
So it's very sort of useful
to calculate, say, response

965
00:51:12,710 --> 00:51:14,900
suppose property of the system.

966
00:51:14,900 --> 00:51:17,000
Say, again, if you want
to calculate the diffusion

967
00:51:17,000 --> 00:51:19,310
coefficient from the
velocity-velocity

968
00:51:19,310 --> 00:51:21,200
autocorrelation
function, you really

969
00:51:21,200 --> 00:51:23,870
need to have your
velocity distributed

970
00:51:23,870 --> 00:51:27,530
according to the appropriate
thermodynamical ensemble.

971
00:51:27,530 --> 00:51:32,000
So it's sort of the, if you
want, most accurate and careful

972
00:51:32,000 --> 00:51:35,660
way of doing the dynamics.

973
00:51:35,660 --> 00:51:39,170
Nosé-Hoover, so it's
probably the most common way

974
00:51:39,170 --> 00:51:41,480
of thermostarting your problem.

975
00:51:41,480 --> 00:51:44,465
It's not the most robust.

976
00:51:44,465 --> 00:51:50,030
It's the approach that
gives you the long time

977
00:51:50,030 --> 00:51:52,610
thermodynamical
properties correctly.

978
00:51:52,610 --> 00:51:57,350
But it tends to be
very poorly, say,

979
00:51:57,350 --> 00:52:01,040
for a system that
is very harmonic.

980
00:52:01,040 --> 00:52:01,670
OK.

981
00:52:01,670 --> 00:52:05,960
A system is very harmonic when
its potential energy is really

982
00:52:05,960 --> 00:52:09,420
a quadratic function
of its coordinate.

983
00:52:09,420 --> 00:52:11,930
So if you have, say,
a single particle

984
00:52:11,930 --> 00:52:14,090
that sits in a
parabolic well, what

985
00:52:14,090 --> 00:52:16,480
we call a harmonic
oscillator also

986
00:52:16,480 --> 00:52:21,120
from a standard pendulum, that
is a perfect harmonic system.

987
00:52:21,120 --> 00:52:24,290
A solid at very,
very low temperature

988
00:52:24,290 --> 00:52:25,790
is also very harmonic.

989
00:52:25,790 --> 00:52:29,000
If you want the potential
energy of each particle,

990
00:52:29,000 --> 00:52:32,600
it's just a quadratic function
of the [INAUDIBLE] displacement

991
00:52:32,600 --> 00:52:34,590
from its equilibrium position.

992
00:52:34,590 --> 00:52:39,230
And so what are the trajectories
in a harmonic oscillator?

993
00:52:39,230 --> 00:52:41,660
Well, they look
something like this.

994
00:52:41,660 --> 00:52:46,600
If you look now, if you go back
to a one-dimensional position

995
00:52:46,600 --> 00:52:50,440
and momentum
representation, well,

996
00:52:50,440 --> 00:52:54,960
you know, a pendulum does
really this over and over again.

997
00:52:58,570 --> 00:53:02,410
So this would be the
trajectory moving in time.

998
00:53:02,410 --> 00:53:05,770
So the position sort of
oscillates back and forth.

999
00:53:05,770 --> 00:53:09,250
And the momentum or the velocity
oscillate back and forth.

1000
00:53:09,250 --> 00:53:10,720
And they are in a
positional phase.

1001
00:53:10,720 --> 00:53:13,630
When the elongation is
maximum, your momentum

1002
00:53:13,630 --> 00:53:15,130
is linear and so on.

1003
00:53:15,130 --> 00:53:22,000
I actually sort of made,
I guess, incorrect axis.

1004
00:53:22,000 --> 00:53:27,100
But thanks to the power
of the [INAUDIBLE] tablet,

1005
00:53:27,100 --> 00:53:31,650
now this looks much better.

1006
00:53:31,650 --> 00:53:32,320
OK.

1007
00:53:32,320 --> 00:53:36,730
Now, suppose that
you sort of studying

1008
00:53:36,730 --> 00:53:38,950
not a single
oscillator, but you are

1009
00:53:38,950 --> 00:53:42,850
studying a solid at
very low temperature.

1010
00:53:42,850 --> 00:53:47,380
What your Nosé-Hoover thermostat
would give you is something

1011
00:53:47,380 --> 00:53:50,900
that is very different from
the equilibrium distribution.

1012
00:53:50,900 --> 00:53:54,490
So the sort of equilibrium
canonical distribution

1013
00:53:54,490 --> 00:53:57,870
is going to look
something in which there's

1014
00:53:57,870 --> 00:54:03,520
of momenta and velocity of
the particle are distributed.

1015
00:54:03,520 --> 00:54:06,430
Because, now,
there is obviously,

1016
00:54:06,430 --> 00:54:10,280
if we have a real solid, a
small amount of interaction.

1017
00:54:10,280 --> 00:54:13,240
So each atom, each
harmonic oscillator,

1018
00:54:13,240 --> 00:54:16,880
is going to be talking
with its neighbors.

1019
00:54:16,880 --> 00:54:20,980
And so they are not going to
be all in phase and perpetually

1020
00:54:20,980 --> 00:54:24,100
in phase, but they are going
to sort of exchange energy

1021
00:54:24,100 --> 00:54:26,060
between one and the other.

1022
00:54:26,060 --> 00:54:30,940
And so a large harmonic
solid, but slightly

1023
00:54:30,940 --> 00:54:33,160
sort of different
from 0 temperature,

1024
00:54:33,160 --> 00:54:35,620
will have a distribution
of velocity and momentum

1025
00:54:35,620 --> 00:54:37,960
that doesn't sit in
a perfect circle,

1026
00:54:37,960 --> 00:54:40,330
but is sort of
distributed around.

1027
00:54:40,330 --> 00:54:43,750
If you do this with a
Nosé-Hoover thermostat,

1028
00:54:43,750 --> 00:54:47,830
sadly there will be
no exchange of energy.

1029
00:54:47,830 --> 00:54:51,250
There will be no thermalization
between all different atoms.

1030
00:54:51,250 --> 00:54:54,130
They are going to
be doing perpetually

1031
00:54:54,130 --> 00:54:56,020
their own thing in synchrony.

1032
00:54:56,020 --> 00:55:01,120
And so the Nosé-Hoover
thermostat works very poorly

1033
00:55:01,120 --> 00:55:03,990
for systems.

1034
00:55:03,990 --> 00:55:06,130
The more harmonic
your system is,

1035
00:55:06,130 --> 00:55:08,950
the poorer the equilibration
that comes from the Nosé

1036
00:55:08,950 --> 00:55:10,760
thermostat is.

1037
00:55:10,760 --> 00:55:13,480
And so you have to
pay a lot of attention

1038
00:55:13,480 --> 00:55:15,880
to system at low temperature.

1039
00:55:15,880 --> 00:55:19,180
And sort of one of the usual
solutions that people mention

1040
00:55:19,180 --> 00:55:23,650
is using Nosé-Hoover chains
in which you actually have

1041
00:55:23,650 --> 00:55:25,250
your dynamical system.

1042
00:55:25,250 --> 00:55:26,740
You have your thermostat.

1043
00:55:26,740 --> 00:55:28,810
And then you have
another thermostat,

1044
00:55:28,810 --> 00:55:30,735
thermostarting your
thermostart to thermostart

1045
00:55:30,735 --> 00:55:31,870
for your particle.

1046
00:55:31,870 --> 00:55:33,370
And then you have
a third thermostat

1047
00:55:33,370 --> 00:55:35,170
that thermostarts the
second thermostat.

1048
00:55:35,170 --> 00:55:36,730
And suddenly, you
have to do this.

1049
00:55:39,460 --> 00:55:42,890
Harmonic solids are
actually very important.

1050
00:55:42,890 --> 00:55:45,520
So there is a reason
why a lot of effort

1051
00:55:45,520 --> 00:55:46,998
has been put into this.

1052
00:55:46,998 --> 00:55:48,790
Because in particular--
and this is, again,

1053
00:55:48,790 --> 00:55:51,490
something that we'll see
in one of the last classes.

1054
00:55:51,490 --> 00:55:55,990
In order to calculate the
free energy of a system

1055
00:55:55,990 --> 00:55:58,420
at finite temperature,
often you need

1056
00:55:58,420 --> 00:56:01,720
to do an integration
from 0 temperature

1057
00:56:01,720 --> 00:56:03,430
to the present temperature.

1058
00:56:03,430 --> 00:56:07,180
And so you really need to
start from a harmonic solid

1059
00:56:07,180 --> 00:56:09,130
that you need to
describe correctly.

1060
00:56:09,130 --> 00:56:11,050
And if you don't
do that, so if you

1061
00:56:11,050 --> 00:56:14,590
don't use your proper
thermostating techniques,

1062
00:56:14,590 --> 00:56:16,510
this is not going to work out.

1063
00:56:16,510 --> 00:56:18,940
And there is really a
lot of dark [INAUDIBLE]

1064
00:56:18,940 --> 00:56:19,960
in all of this.

1065
00:56:19,960 --> 00:56:23,920
You really need to be careful
because the temperature is

1066
00:56:23,920 --> 00:56:25,600
a global quantity.

1067
00:56:25,600 --> 00:56:29,920
But if you have, say, a
system with 100 particles,

1068
00:56:29,920 --> 00:56:32,320
you can have a
temperature of 300 Kelvin

1069
00:56:32,320 --> 00:56:35,410
by having sort of all
particles distributed

1070
00:56:35,410 --> 00:56:37,100
along the same temperature.

1071
00:56:37,100 --> 00:56:39,160
But you can have also
the same temperature

1072
00:56:39,160 --> 00:56:42,700
if part of your system is very
cold and part of your system

1073
00:56:42,700 --> 00:56:43,910
is very hot.

1074
00:56:43,910 --> 00:56:45,880
And if you wanted,
the thermostat

1075
00:56:45,880 --> 00:56:50,380
has this role of making sure
that the energy flow goes

1076
00:56:50,380 --> 00:56:53,350
around, so that all
part of your system

1077
00:56:53,350 --> 00:56:54,970
are really at the equilibrium.

1078
00:56:54,970 --> 00:56:59,590
And you know, this can be
actually a trickier thing

1079
00:56:59,590 --> 00:57:02,320
than sort of it actually seems.

1080
00:57:02,320 --> 00:57:04,890
Obviously, all these
dots and [INAUDIBLE]..

1081
00:57:08,700 --> 00:57:09,200
OK.

1082
00:57:09,200 --> 00:57:12,490
These are sort of a
summary of all the books

1083
00:57:12,490 --> 00:57:15,980
in which you can find a
description of these concepts.

1084
00:57:15,980 --> 00:57:18,730
If you have to read
one for this class--

1085
00:57:18,730 --> 00:57:20,780
the sort of free primer
that Furio Ercolessi

1086
00:57:20,780 --> 00:57:24,320
has put on his website.

1087
00:57:24,320 --> 00:57:27,190
I'm actually very proud of this
because Furio moved recently

1088
00:57:27,190 --> 00:57:29,290
to this very small
town in northern Italy

1089
00:57:29,290 --> 00:57:30,790
that is sort of my hometown.

1090
00:57:30,790 --> 00:57:33,280
But anyhow, this is
available and free.

1091
00:57:33,280 --> 00:57:34,780
It's 30 or 40 pages.

1092
00:57:34,780 --> 00:57:37,910
I posted it also on
the Stella website.

1093
00:57:37,910 --> 00:57:40,810
And it gives you a
very clean introduction

1094
00:57:40,810 --> 00:57:42,490
to molecular dynamics.

1095
00:57:42,490 --> 00:57:46,030
And then I think this is a
field in which there are,

1096
00:57:46,030 --> 00:57:49,180
in particular, two
exemplary books.

1097
00:57:49,180 --> 00:57:52,420
So if you really want
to do this for a living,

1098
00:57:52,420 --> 00:57:55,660
there are a couple of books
that are very, very good--

1099
00:57:55,660 --> 00:57:58,600
the Allen and Tildesley book
that, by now, it's very old.

1100
00:57:58,600 --> 00:58:01,030
It's sort of really
written at the beginning

1101
00:58:01,030 --> 00:58:05,020
of the '80s when you can imagine
computer capabilities were

1102
00:58:05,020 --> 00:58:06,310
very different.

1103
00:58:06,310 --> 00:58:09,370
And it's actually
remarkable somehow how

1104
00:58:09,370 --> 00:58:13,300
the clarity that you develop
by dealing with systems that

1105
00:58:13,300 --> 00:58:16,060
are very primitive from the
computational point of view

1106
00:58:16,060 --> 00:58:19,100
actually develops your
intellectual skills.

1107
00:58:19,100 --> 00:58:21,370
So this is really
very, very good.

1108
00:58:21,370 --> 00:58:23,860
And equally good is a
much more recent book

1109
00:58:23,860 --> 00:58:27,925
that is in its third edition
from last year, the Frenkel

1110
00:58:27,925 --> 00:58:33,040
and Smit that also has
many more advanced topics.

1111
00:58:33,040 --> 00:58:35,620
But either of these
are really sort

1112
00:58:35,620 --> 00:58:37,870
of very, very good references.

1113
00:58:37,870 --> 00:58:40,690
And now, in the last
20 minutes, I'll

1114
00:58:40,690 --> 00:58:43,930
give you really the flavor
on how we do first principle

1115
00:58:43,930 --> 00:58:45,460
molecular dynamics.

1116
00:58:45,460 --> 00:58:49,720
That is really nothing else than
a classical molecular dynamics

1117
00:58:49,720 --> 00:58:53,390
with a lot of
additional variables.

1118
00:58:53,390 --> 00:58:57,640
So we use this concept
of extended Hamiltonians.

1119
00:58:57,640 --> 00:59:01,070
And you know, somehow it's,
by now, a little bit old.

1120
00:59:01,070 --> 00:59:04,210
I had sort of once
downloaded this.

1121
00:59:04,210 --> 00:59:09,490
This is actually the citations
of first principle molecular

1122
00:59:09,490 --> 00:59:13,060
dynamics, of initial molecular
dynamics or the citation

1123
00:59:13,060 --> 00:59:15,820
of the first paper that
developed this concept that

1124
00:59:15,820 --> 00:59:18,840
goes under the name of the
[INAUDIBLE] techniques that

1125
00:59:18,840 --> 00:59:20,497
is from 1985.

1126
00:59:20,497 --> 00:59:22,330
And you see we are sort
of very pleased with

1127
00:59:22,330 --> 00:59:24,970
this exponential explosion.

1128
00:59:24,970 --> 00:59:28,330
It means that basically there is
a lot of useless literature out

1129
00:59:28,330 --> 00:59:31,060
there that you can look at.

1130
00:59:31,060 --> 00:59:33,700
So in order to do first
principle molecular dynamics,

1131
00:59:33,700 --> 00:59:37,900
I need to give you a reminder
of your favorite topic.

1132
00:59:37,900 --> 00:59:41,590
That is the expansion in plane
waves of the total energy

1133
00:59:41,590 --> 00:59:42,790
of a system.

1134
00:59:42,790 --> 00:59:46,270
And hopefully, you still
remind something about this.

1135
00:59:46,270 --> 00:59:48,400
But if you have a
crystal that's going

1136
00:59:48,400 --> 00:59:53,410
to be defined by the three
primitive direct lattice

1137
00:59:53,410 --> 00:59:54,030
vectors--

1138
00:59:54,030 --> 00:59:59,740
so you could have an FCC crystal
and they point to 1/2, 1/2, 0,

1139
00:59:59,740 --> 01:00:02,080
0, 0, and the third one.

1140
01:00:02,080 --> 01:00:05,810
And then we can define
reciprocal space

1141
01:00:05,810 --> 01:00:11,080
and the reciprocal lattice whose
three primitive vectors, this g

1142
01:00:11,080 --> 01:00:16,460
vectors, are just defined as
the duals of your direct lattice

1143
01:00:16,460 --> 01:00:16,960
vector.

1144
01:00:16,960 --> 01:00:19,960
So the scalar total
between the reciprocals

1145
01:00:19,960 --> 01:00:24,380
and the direct lattice vectors
needs to be equal to 0 or 2 pi.

1146
01:00:24,380 --> 01:00:24,910
OK.

1147
01:00:24,910 --> 01:00:26,470
So this is the definition.

1148
01:00:26,470 --> 01:00:30,550
Once we have this
reciprocal lattice vectors

1149
01:00:30,550 --> 01:00:33,160
that, for simplicity,
I'll draw here

1150
01:00:33,160 --> 01:00:38,530
as being two-dimensional
and square, what we have is,

1151
01:00:38,530 --> 01:00:46,750
again, sort of these points here
represents all possible integer

1152
01:00:46,750 --> 01:00:52,030
combination of reciprocal
lattice vectors.

1153
01:00:52,030 --> 01:00:55,750
That, for my problem
at hand here,

1154
01:00:55,750 --> 01:00:57,730
say, this would be
the Brillouin zone.

1155
01:00:57,730 --> 01:01:01,210
This would be the unit cell
of the reciprocal lattice.

1156
01:01:01,210 --> 01:01:04,810
And so all these
g vectors in blue,

1157
01:01:04,810 --> 01:01:10,510
remember, are such that
the complex function e

1158
01:01:10,510 --> 01:01:15,640
to the iG times
r, where G is any

1159
01:01:15,640 --> 01:01:19,380
of these vectors represented
by one of these point,

1160
01:01:19,380 --> 01:01:23,170
this function in real
space is compatible

1161
01:01:23,170 --> 01:01:25,480
with your periodic
boundary condition.

1162
01:01:25,480 --> 01:01:28,930
It's going to have either one
oscillation inside your unit

1163
01:01:28,930 --> 01:01:32,590
cell, 2 oscillation, 3
oscillation, 10 oscillations.

1164
01:01:32,590 --> 01:01:36,320
And it's going to be, generally
speaking, in three dimensions.

1165
01:01:36,320 --> 01:01:40,240
And so if you remember, we
expand our wave function

1166
01:01:40,240 --> 01:01:42,910
into linear combination
of these plane waves

1167
01:01:42,910 --> 01:01:46,780
because these plane waves
are all the possible--

1168
01:01:46,780 --> 01:01:49,060
it's a complete set
of functions that

1169
01:01:49,060 --> 01:01:52,360
describe orbitals that
have the periodicity

1170
01:01:52,360 --> 01:01:53,390
of the direct lattice.

1171
01:02:01,720 --> 01:02:02,560
OK.

1172
01:02:02,560 --> 01:02:05,810
So what was our quantum
mechanical system?

1173
01:02:05,810 --> 01:02:07,070
We keep it simple.

1174
01:02:07,070 --> 01:02:09,070
We won't go into density
functional theory.

1175
01:02:09,070 --> 01:02:12,190
We'll just think
again at the operator.

1176
01:02:12,190 --> 01:02:15,680
You see the Hamiltonian
raises its head again.

1177
01:02:15,680 --> 01:02:18,220
It's just going to be the
quantum kinetic energy

1178
01:02:18,220 --> 01:02:22,510
minus 1/2 the Laplacian
plus the potential energy

1179
01:02:22,510 --> 01:02:25,870
and with the caveat
that we develop,

1180
01:02:25,870 --> 01:02:29,800
we expand, the periodic
part of your wave function,

1181
01:02:29,800 --> 01:02:34,300
what we call sort of, via the
Bloch theorem, the periodic.

1182
01:02:34,300 --> 01:02:36,250
Sometimes I call it u.

1183
01:02:36,250 --> 01:02:40,120
But what is the
periodic component?

1184
01:02:40,120 --> 01:02:41,690
We expand it.

1185
01:02:41,690 --> 01:02:46,420
We write it as a linear
combination of plane waves

1186
01:02:46,420 --> 01:02:48,800
with appropriate coefficients.

1187
01:02:48,800 --> 01:02:53,230
So if you want, your
quantum mechanical problem

1188
01:02:53,230 --> 01:02:56,380
is, yet again, nothing
else than trying

1189
01:02:56,380 --> 01:02:58,360
to find that these numbers.

1190
01:02:58,360 --> 01:03:01,180
Once you have defined
your basis sector,

1191
01:03:01,180 --> 01:03:03,220
then you have
written an expansion.

1192
01:03:03,220 --> 01:03:06,760
And then all your algebra and
all your computational problem

1193
01:03:06,760 --> 01:03:10,600
is finding out what this
coefficient should be.

1194
01:03:10,600 --> 01:03:12,700
And this tends to be a
very good basis sector

1195
01:03:12,700 --> 01:03:16,000
because it can be made more
and more accurate that by

1196
01:03:16,000 --> 01:03:19,570
including G vectors of
larger and larger models

1197
01:03:19,570 --> 01:03:22,420
that correspond to
plane waves of higher

1198
01:03:22,420 --> 01:03:24,700
and higher resolution.

1199
01:03:24,700 --> 01:03:29,380
Not only that, but it's actually
a very manageable basis sector

1200
01:03:29,380 --> 01:03:30,760
to use.

1201
01:03:30,760 --> 01:03:34,910
Because it's very easy to
take first derivatives,

1202
01:03:34,910 --> 01:03:38,590
second derivatives of a wave
function of square moduli.

1203
01:03:38,590 --> 01:03:43,090
Because, say, if we take a first
derivative of psi with respect

1204
01:03:43,090 --> 01:03:47,170
to r, the only thing that we
have is that each term in G

1205
01:03:47,170 --> 01:03:51,100
gets down from here
a factor i times G.

1206
01:03:51,100 --> 01:03:52,690
And the second
derivative gives us

1207
01:03:52,690 --> 01:03:55,960
a factor iG scalar product iG.

1208
01:03:55,960 --> 01:03:57,880
That is minus G squared.

1209
01:03:57,880 --> 01:03:58,690
OK.

1210
01:03:58,690 --> 01:04:02,140
So our quantum
mechanical problem basis

1211
01:04:02,140 --> 01:04:04,480
for our first principle
molecular dynamics

1212
01:04:04,480 --> 01:04:08,590
has really nothing else
to do than finding what

1213
01:04:08,590 --> 01:04:10,840
is the ground state energy.

1214
01:04:10,840 --> 01:04:14,145
That, not really in a
density functional formalism,

1215
01:04:14,145 --> 01:04:16,750
but in a sort of
simplified formalism,

1216
01:04:16,750 --> 01:04:21,610
I write out just as a
sum of the eigenvalues

1217
01:04:21,610 --> 01:04:23,890
of our single particle orbitals.

1218
01:04:23,890 --> 01:04:28,000
That is just the sum of all the
electrons of this expectation

1219
01:04:28,000 --> 01:04:28,663
value.

1220
01:04:28,663 --> 01:04:30,580
If you remember, in
density functional theory,

1221
01:04:30,580 --> 01:04:34,840
the energy is slightly different
because the Hamiltonian

1222
01:04:34,840 --> 01:04:39,640
in that approach has actually
become self-consistent.

1223
01:04:39,640 --> 01:04:42,980
The Hamiltonian depends on
the charge density itself.

1224
01:04:42,980 --> 01:04:45,790
And so there are some
additional collective terms.

1225
01:04:45,790 --> 01:04:49,430
But you know, from the point of
view of our sort of [INAUDIBLE]

1226
01:04:49,430 --> 01:04:51,640
here, it doesn't
really matter that we

1227
01:04:51,640 --> 01:04:53,830
include these additional terms.

1228
01:04:53,830 --> 01:04:57,220
And so, you know, what is that
we have to deal with here?

1229
01:04:57,220 --> 01:05:00,640
Well, we have, again, a
quantum kinetic energy term

1230
01:05:00,640 --> 01:05:02,950
and a potential energy term.

1231
01:05:02,950 --> 01:05:05,440
And so the quantum
kinetic energy

1232
01:05:05,440 --> 01:05:08,860
is going to be the sum of the
kinetic energy of each orbital,

1233
01:05:08,860 --> 01:05:10,360
as written here.

1234
01:05:10,360 --> 01:05:12,040
And you know what I
was saying before.

1235
01:05:12,040 --> 01:05:16,450
It's actually trivial to
calculate the kinetic energy

1236
01:05:16,450 --> 01:05:20,800
if you have a wave function
that is expanded in plane waves

1237
01:05:20,800 --> 01:05:22,180
because this is it.

1238
01:05:22,180 --> 01:05:23,830
This is the expectation value.

1239
01:05:23,830 --> 01:05:27,970
Remember, it's the integral
over all your Hilbert space.

1240
01:05:27,970 --> 01:05:31,230
That is the integral
in space of this.

1241
01:05:31,230 --> 01:05:32,230
This is the [INAUDIBLE].

1242
01:05:32,230 --> 01:05:36,550
So this is the complex
conjugate of the plane waves

1243
01:05:36,550 --> 01:05:37,850
e to the iGr.

1244
01:05:37,850 --> 01:05:42,490
So it's e to the minus
iGr times the action

1245
01:05:42,490 --> 01:05:47,090
of this operator to the plane
wave e to the iG prime r.

1246
01:05:47,090 --> 01:05:50,280
And so, you know, this
second derivative,

1247
01:05:50,280 --> 01:05:53,020
so with respect to
r, gives us an iG

1248
01:05:53,020 --> 01:05:56,800
prime times iG prime
that is a minus G square.

1249
01:05:56,800 --> 01:05:59,150
The minus sign cancels out.

1250
01:05:59,150 --> 01:06:03,700
And so we have 1/2 g square
times the integral with respect

1251
01:06:03,700 --> 01:06:09,670
to space of e to the minus
iGr times e to the iG prime r.

1252
01:06:09,670 --> 01:06:12,580
And actually, if G
is equal to G prime,

1253
01:06:12,580 --> 01:06:15,130
that is just going to be 1.

1254
01:06:15,130 --> 01:06:16,810
And so the integral,
it's actually--

1255
01:06:16,810 --> 01:06:18,730
I sort of skipped all
the normalization.

1256
01:06:18,730 --> 01:06:20,447
So the integral
is going to be 1.

1257
01:06:20,447 --> 01:06:21,655
You can work out the algebra.

1258
01:06:21,655 --> 01:06:25,090
If G is different from G
prime, it's going to be 0.

1259
01:06:25,090 --> 01:06:29,950
So this expectation value of
the kinetic energy operator

1260
01:06:29,950 --> 01:06:32,290
between two plane
waves is just going

1261
01:06:32,290 --> 01:06:36,790
to be 1/2 G square between
two identical plane waves.

1262
01:06:36,790 --> 01:06:40,480
And it's going to be 0 if the
plane waves are different.

1263
01:06:40,480 --> 01:06:43,960
This is what we say when we
say that the kinetic energy

1264
01:06:43,960 --> 01:06:48,370
operator in a matrix
representation in which

1265
01:06:48,370 --> 01:06:53,860
the rows and the columns are
the different plane waves

1266
01:06:53,860 --> 01:06:55,690
is actually diagonal.

1267
01:06:55,690 --> 01:07:01,260
That is, if you calculate this,
so you would have, say, G here

1268
01:07:01,260 --> 01:07:06,210
and G prime here, all the
terms in this matrix function

1269
01:07:06,210 --> 01:07:09,720
of G and G prime has 0
outside the diagonal.

1270
01:07:09,720 --> 01:07:13,860
And the diagonal is
1/2 G square, so very

1271
01:07:13,860 --> 01:07:17,520
trivial to do in plane waves.

1272
01:07:17,520 --> 01:07:21,280
If we want to calculate the
second term in the energy,

1273
01:07:21,280 --> 01:07:24,255
we need to calculate
the potential energy.

1274
01:07:32,770 --> 01:07:35,470
And the potential
energy is going to be,

1275
01:07:35,470 --> 01:07:40,205
again, sort of the sum over
all the single particle

1276
01:07:40,205 --> 01:07:42,520
terms in the potential energy.

1277
01:07:42,520 --> 01:07:44,570
And I'm doing the
same thing here.

1278
01:07:44,570 --> 01:07:48,520
I'm writing this expectation
value between two plane waves.

1279
01:07:48,520 --> 01:07:51,400
And so I have same as
before, the exponential

1280
01:07:51,400 --> 01:07:55,090
of the complex conjugate
minus iGr times e of r

1281
01:07:55,090 --> 01:07:57,590
times e to the iG prime r.

1282
01:07:57,590 --> 01:08:01,090
And if you look at this,
this is nothing else

1283
01:08:01,090 --> 01:08:03,940
than the definition
of the Fourier

1284
01:08:03,940 --> 01:08:08,080
transform coefficient of your
potential energy with respect

1285
01:08:08,080 --> 01:08:11,230
to the wave vector
G minus G prime.

1286
01:08:11,230 --> 01:08:17,410
So now, this part, the potential
energy in a plane wave basis

1287
01:08:17,410 --> 01:08:19,784
sector, is not anymore diagonal.

1288
01:08:22,330 --> 01:08:25,359
It's going to have
off-diagonal terms.

1289
01:08:25,359 --> 01:08:28,990
That is, if you look in
the previous sort of matrix

1290
01:08:28,990 --> 01:08:34,390
representation in G and G prime,
what we have along the diagonal

1291
01:08:34,390 --> 01:08:39,310
is just the kinetic energy
terms T that are 1/2 G squared.

1292
01:08:39,310 --> 01:08:44,470
But we are going to have
a number of diagonal terms

1293
01:08:44,470 --> 01:08:49,540
and also on-diagonal diagonal
terms from the potential.

1294
01:08:49,540 --> 01:08:53,950
Obviously, the more you
go outside the diagonal,

1295
01:08:53,950 --> 01:08:58,399
the more G minus G prime
is going to be large.

1296
01:08:58,399 --> 01:09:01,450
So the more you're going
to look at high frequency

1297
01:09:01,450 --> 01:09:03,580
components of your potential.

1298
01:09:03,580 --> 01:09:06,220
And in general,
unless your potential

1299
01:09:06,220 --> 01:09:09,819
is really ill-defined, the
high frequency components

1300
01:09:09,819 --> 01:09:11,899
are going to be
smaller and smaller.

1301
01:09:11,899 --> 01:09:15,430
So again, sort of the
overall Hamiltonian matrix

1302
01:09:15,430 --> 01:09:18,490
kinetic plus potential
in a plane wave basis

1303
01:09:18,490 --> 01:09:20,229
tends to be diagonally dominant.

1304
01:09:20,229 --> 01:09:21,939
As we move farther
and farther away,

1305
01:09:21,939 --> 01:09:24,560
it tends to be
smaller and smaller.

1306
01:09:24,560 --> 01:09:26,859
So we have actually
written all the terms.

1307
01:09:26,859 --> 01:09:29,710
And then we can take the sum
over all these expectation

1308
01:09:29,710 --> 01:09:30,340
value.

1309
01:09:30,340 --> 01:09:33,819
And so we have here
sort of a [INAUDIBLE],,

1310
01:09:33,819 --> 01:09:39,700
if you want, the total energy as
a function of this coefficient

1311
01:09:39,700 --> 01:09:41,109
of the plane waves.

1312
01:09:41,109 --> 01:09:45,310
So we have the kinetic energy
term and the potential energy

1313
01:09:45,310 --> 01:09:46,210
term.

1314
01:09:46,210 --> 01:09:50,770
And the fundamental approach
that we take in first principle

1315
01:09:50,770 --> 01:09:55,720
calculation is not trying to
diagonalize this Hamiltonian

1316
01:09:55,720 --> 01:09:58,300
to find eigenvalues
and eigenvectors

1317
01:09:58,300 --> 01:10:00,820
because that
operation would scale

1318
01:10:00,820 --> 01:10:03,820
as the cube of the
number of plane waves.

1319
01:10:03,820 --> 01:10:08,620
And it gives us not only the
10 or 20 lowest eigenvalues

1320
01:10:08,620 --> 01:10:10,360
that have the occupied states.

1321
01:10:10,360 --> 01:10:13,720
But, say, if we have a simple
system like the silicon atom--

1322
01:10:13,720 --> 01:10:16,240
remember, you have seen
semiconductors in your lab

1323
01:10:16,240 --> 01:10:18,790
two, where, at the
end, what you need

1324
01:10:18,790 --> 01:10:23,830
is really the four lowest
eigenvalues and the four lowest

1325
01:10:23,830 --> 01:10:25,000
eigenvectors.

1326
01:10:25,000 --> 01:10:27,430
Well, if you diagonalize
this Hamiltonian

1327
01:10:27,430 --> 01:10:31,090
and you have plane wave basis
set with 1,000 elements,

1328
01:10:31,090 --> 01:10:34,900
you get all 1,000
eigenvalues and eigenvectors.

1329
01:10:34,900 --> 01:10:35,710
We don't need that.

1330
01:10:35,710 --> 01:10:36,710
It's too expensive.

1331
01:10:36,710 --> 01:10:39,540
We need only 4 out of 1,000.

1332
01:10:39,540 --> 01:10:40,090
OK.

1333
01:10:40,090 --> 01:10:43,960
So the approach that most
or all electronic structure

1334
01:10:43,960 --> 01:10:46,630
approaches take is
actually an approach

1335
01:10:46,630 --> 01:10:51,580
in which we find what
are these coefficients

1336
01:10:51,580 --> 01:10:55,180
of the relevant
occupied orbitals.

1337
01:10:55,180 --> 01:10:57,358
You see, here, the
total energy, that's

1338
01:10:57,358 --> 01:10:58,900
the function of the
occupied orbital.

1339
01:10:58,900 --> 01:11:01,900
So for silicon,
we would sum only

1340
01:11:01,900 --> 01:11:04,240
on the four lowest orbitals.

1341
01:11:04,240 --> 01:11:08,110
And by minimizing
this quantity, we

1342
01:11:08,110 --> 01:11:10,610
find what are the coefficients.

1343
01:11:10,610 --> 01:11:13,540
So the electronic structure
problem, again, it's

1344
01:11:13,540 --> 01:11:17,800
a minimum problem in which
you have an expression

1345
01:11:17,800 --> 01:11:20,320
for the energy that, in
density functional theory,

1346
01:11:20,320 --> 01:11:24,070
is slightly more complex,
but at the end depends only

1347
01:11:24,070 --> 01:11:27,760
on the coefficients of
your plane wave expansion

1348
01:11:27,760 --> 01:11:31,690
and on the matrix elements of
the potential between plane

1349
01:11:31,690 --> 01:11:35,170
waves and of the kinetic
energy between plane waves.

1350
01:11:35,170 --> 01:11:39,370
You calculate these ones for
all or, in density functional

1351
01:11:39,370 --> 01:11:40,900
theory, at every step.

1352
01:11:40,900 --> 01:11:43,750
Because your potential,
being self-consistent,

1353
01:11:43,750 --> 01:11:45,670
changes up every time step.

1354
01:11:45,670 --> 01:11:48,130
But if you have this
and you have this,

1355
01:11:48,130 --> 01:11:50,650
it's nothing else than
a problem of finding

1356
01:11:50,650 --> 01:11:53,380
the minimum with respect
to these variables.

1357
01:11:53,380 --> 01:11:56,320
Sadly, you have sort
of thousands or tens

1358
01:11:56,320 --> 01:11:59,060
of thousands or hundreds
of thousands of them.

1359
01:11:59,060 --> 01:12:04,060
So it tends to be a
very expansive problem

1360
01:12:04,060 --> 01:12:07,330
of minimization of a
non-linear function that

1361
01:12:07,330 --> 01:12:11,570
depends on basically
zillions of variables.

1362
01:12:11,570 --> 01:12:15,200
So how do we do this?

1363
01:12:15,200 --> 01:12:16,000
Well, let's see.

1364
01:12:21,920 --> 01:12:25,550
We use a molecular
dynamics idea.

1365
01:12:25,550 --> 01:12:30,230
That is we have an energy,
a function of thousands

1366
01:12:30,230 --> 01:12:31,560
of variables.

1367
01:12:31,560 --> 01:12:36,050
So we have
multi-dimensional space

1368
01:12:36,050 --> 01:12:39,560
with really thousands
of coordinates.

1369
01:12:39,560 --> 01:12:43,920
And as a function of those
thousands of coordinates,

1370
01:12:43,920 --> 01:12:51,770
these c and G variables, what we
have is a total energy surface.

1371
01:12:51,770 --> 01:12:56,600
So our problem looks
nothing else like this.

1372
01:12:56,600 --> 01:12:59,120
We have a total energy
that is this sort

1373
01:12:59,120 --> 01:13:04,010
of hyper surface in black
that is non-linear functional

1374
01:13:04,010 --> 01:13:06,110
of these things.

1375
01:13:06,110 --> 01:13:08,605
In density functional
theory, for a standard LDA,

1376
01:13:08,605 --> 01:13:13,220
GGA calculation, this system
tends to have only one minimum.

1377
01:13:13,220 --> 01:13:15,650
If you are doing a spin
polarized calculation,

1378
01:13:15,650 --> 01:13:17,330
you will have
different minima that

1379
01:13:17,330 --> 01:13:22,610
corresponds to a
non-magnetic solution or all

1380
01:13:22,610 --> 01:13:26,030
the magnetic solution with
different set of values

1381
01:13:26,030 --> 01:13:27,410
for the magnetization.

1382
01:13:27,410 --> 01:13:29,810
Or maybe have
anti-ferromagnetic solution

1383
01:13:29,810 --> 01:13:32,600
with a total magnetization
of 0, but different

1384
01:13:32,600 --> 01:13:34,380
spins and so on and so forth.

1385
01:13:34,380 --> 01:13:36,170
So this potential
energy surface can

1386
01:13:36,170 --> 01:13:39,478
start to develop some
of the complex minima

1387
01:13:39,478 --> 01:13:41,270
that I was showing you
when we were talking

1388
01:13:41,270 --> 01:13:42,710
about simulated annealing.

1389
01:13:42,710 --> 01:13:44,600
And that's why actually
finding the ground

1390
01:13:44,600 --> 01:13:49,040
state of magnetic system tend to
become, very quickly, much more

1391
01:13:49,040 --> 01:13:49,700
complex.

1392
01:13:49,700 --> 01:13:52,580
In LDA or GGA, it
has only one minimum.

1393
01:13:52,580 --> 01:13:55,310
And we need to find that
minimum as a function

1394
01:13:55,310 --> 01:13:58,190
of this very many coordinates.

1395
01:13:58,190 --> 01:14:02,630
And so we use molecular
dynamics techniques

1396
01:14:02,630 --> 01:14:04,940
and molecular dynamic analogies.

1397
01:14:04,940 --> 01:14:08,390
And you know, I'll do
this in one dimension,

1398
01:14:08,390 --> 01:14:15,170
but really what we want to do
is find an equation of motion,

1399
01:14:15,170 --> 01:14:20,870
something that evolves
our coefficients, c, G,

1400
01:14:20,870 --> 01:14:24,500
so that they move
towards the minimum

1401
01:14:24,500 --> 01:14:27,020
of that potential
energy surface.

1402
01:14:27,020 --> 01:14:30,170
And in order to do that,
we have nothing else

1403
01:14:30,170 --> 01:14:34,070
to do than to calculate the
derivative of that Bloch

1404
01:14:34,070 --> 01:14:36,770
potential energy
surface with respect

1405
01:14:36,770 --> 01:14:40,610
to each and every variable, with
respect to each and every plane

1406
01:14:40,610 --> 01:14:42,800
wave coefficient.

1407
01:14:42,800 --> 01:14:46,520
Again-- sort of lots of algebra
that I'm hitting in there.

1408
01:14:46,520 --> 01:14:49,070
But actually, and
very pleasantly,

1409
01:14:49,070 --> 01:14:50,810
it's actually very simple.

1410
01:14:50,810 --> 01:14:54,860
And we sort of write
this generalized force.

1411
01:14:54,860 --> 01:14:58,130
That is the derivative
with the minus signs,

1412
01:14:58,130 --> 01:15:00,710
the gradient of the
energy with respect

1413
01:15:00,710 --> 01:15:02,900
to each and every coefficient.

1414
01:15:02,900 --> 01:15:06,080
And it's, by chance
and nothing else than,

1415
01:15:06,080 --> 01:15:09,410
the application of
the Hamiltonian itself

1416
01:15:09,410 --> 01:15:13,100
to the wave function
coefficient by coefficient.

1417
01:15:13,100 --> 01:15:16,830
And so it's actually sort
of very easy to calculate.

1418
01:15:16,830 --> 01:15:21,110
And then once we
have that, well,

1419
01:15:21,110 --> 01:15:25,220
so if we look at this
problem in one dimension,

1420
01:15:25,220 --> 01:15:29,720
what we have is our Bloch
potential energy surface.

1421
01:15:29,720 --> 01:15:32,730
And we need to find the minimum.

1422
01:15:32,730 --> 01:15:35,660
So this is the
energy as a function

1423
01:15:35,660 --> 01:15:40,610
of all this coefficient
of plane waves.

1424
01:15:40,610 --> 01:15:46,700
And we can choose a random
value for this coefficient,

1425
01:15:46,700 --> 01:15:48,690
calculate the energy.

1426
01:15:48,690 --> 01:15:52,220
And what we will have is
a value for this energy.

1427
01:15:52,220 --> 01:15:56,930
And now, what we want
is really to evolve,

1428
01:15:56,930 --> 01:16:01,770
to move the value of each
and every coefficient.

1429
01:16:01,770 --> 01:16:05,840
So that the total energy of
our system reaches the minimum.

1430
01:16:05,840 --> 01:16:08,930
And in order to do that,
we need the gradient

1431
01:16:08,930 --> 01:16:13,070
that we have written in the
previous expression that

1432
01:16:13,070 --> 01:16:16,670
is nothing than minus the
Hamiltonian applied to the wave

1433
01:16:16,670 --> 01:16:17,540
function.

1434
01:16:17,540 --> 01:16:18,500
That is the force.

1435
01:16:18,500 --> 01:16:22,400
And it's a perfect
dynamical analogy.

1436
01:16:22,400 --> 01:16:24,620
If you are somewhere
and you know

1437
01:16:24,620 --> 01:16:28,190
what is the force that
drives you downhill,

1438
01:16:28,190 --> 01:16:31,310
well, then you have
molecular dynamics techniques

1439
01:16:31,310 --> 01:16:35,390
to get to the bottom of your
potential energy surface.

1440
01:16:35,390 --> 01:16:39,810
And this is particularly
easy if, as in LDA or GGA,

1441
01:16:39,810 --> 01:16:43,340
your potential energy
surface has only one minimum.

1442
01:16:43,340 --> 01:16:46,490
And there are actually
two different approaches

1443
01:16:46,490 --> 01:16:48,380
that you could use.

1444
01:16:48,380 --> 01:16:52,680
You could use standard
molecular dynamics.

1445
01:16:52,680 --> 01:16:56,210
So if you are here, you
put yourself in motion.

1446
01:16:56,210 --> 01:16:59,240
And what you start
doing is going down.

1447
01:16:59,240 --> 01:17:02,570
But in a pure conservative
molecular dynamics,

1448
01:17:02,570 --> 01:17:04,760
you are going to
overshoot your minimum.

1449
01:17:04,760 --> 01:17:08,300
Go back and oscillate
perennially.

1450
01:17:08,300 --> 01:17:12,260
So what you do, you
actually put some friction,

1451
01:17:12,260 --> 01:17:16,430
so that you sort of
slowly lose energy.

1452
01:17:16,430 --> 01:17:19,850
And you basically,
again, condense down

1453
01:17:19,850 --> 01:17:23,810
to your sort of lowest
energy solution.

1454
01:17:23,810 --> 01:17:28,630
And so a standard molecular
dynamics technique

1455
01:17:28,630 --> 01:17:32,380
applied on the coefficient
of the plane waves

1456
01:17:32,380 --> 01:17:37,270
with a friction term
added is one approach that

1457
01:17:37,270 --> 01:17:41,110
brings all these additional
degrees of freedom of plane

1458
01:17:41,110 --> 01:17:44,140
waves down to the ground state.

1459
01:17:44,140 --> 01:17:46,210
This would be one approach.

1460
01:17:46,210 --> 01:17:50,500
The other approach would be
actually evolving your system

1461
01:17:50,500 --> 01:17:55,870
in a way in which the velocity
is proportional to the force.

1462
01:17:55,870 --> 01:17:58,870
And so, again, you
can think that, when

1463
01:17:58,870 --> 01:18:02,290
you are at the minimum,
your force is 0.

1464
01:18:02,290 --> 01:18:05,140
And so your velocity is 0.

1465
01:18:05,140 --> 01:18:08,710
And so if instead of
accelerating your system

1466
01:18:08,710 --> 01:18:11,770
proportionally to the force
and putting a friction that

1467
01:18:11,770 --> 01:18:15,940
is something that will oscillate
and dump down to the minimum,

1468
01:18:15,940 --> 01:18:20,230
you move with a velocity that
is proportional to your force.

1469
01:18:20,230 --> 01:18:23,440
It's something that, again,
it will sort of move you

1470
01:18:23,440 --> 01:18:25,750
towards the ground state.

1471
01:18:25,750 --> 01:18:30,580
Once you are here, your
driving force is going to be 0.

1472
01:18:30,580 --> 01:18:32,620
So your velocity
is going to be 0.

1473
01:18:32,620 --> 01:18:35,140
So somehow it's a
different approach

1474
01:18:35,140 --> 01:18:37,150
that also brings you to the 0.

1475
01:18:37,150 --> 01:18:48,610
And sort of the two approaches
have a slightly different

1476
01:18:48,610 --> 01:18:51,640
performance depending
on the kind of system

1477
01:18:51,640 --> 01:18:52,660
that you are doing.

1478
01:18:52,660 --> 01:18:56,440
In this case, in the
sort of dynamical way,

1479
01:18:56,440 --> 01:19:00,520
you need to make sure that
your friction is good enough.

1480
01:19:00,520 --> 01:19:03,340
That is you don't use
too much friction,

1481
01:19:03,340 --> 01:19:06,310
so you sort of flow down
and never reach the minimum.

1482
01:19:06,310 --> 01:19:07,810
And you have to
be sure that there

1483
01:19:07,810 --> 01:19:11,410
is sort of enough friction, so
you don't oscillate forever.

1484
01:19:11,410 --> 01:19:14,050
And that's one way in
this sort of approach

1485
01:19:14,050 --> 01:19:18,100
that you need to make sure
that your system, again,

1486
01:19:18,100 --> 01:19:20,800
doesn't really
sort of stop short

1487
01:19:20,800 --> 01:19:21,970
of getting to the minimum.

1488
01:19:21,970 --> 01:19:24,400
Because the closer you
get to the minimum,

1489
01:19:24,400 --> 01:19:27,340
the closer your
velocity is to 0.

1490
01:19:27,340 --> 01:19:31,570
And so you tend to just get very
close, but not right to there.

1491
01:19:31,570 --> 01:19:33,940
And there are
advanced techniques

1492
01:19:33,940 --> 01:19:36,520
to deal with both
of these systems.

1493
01:19:36,520 --> 01:19:46,360
And with this, I think
I'll actually conclude here

1494
01:19:46,360 --> 01:19:51,280
because I wanted to sort of tell
you more about how we actually

1495
01:19:51,280 --> 01:19:56,860
evolve in time this problem
when we start moving the atoms.

1496
01:19:56,860 --> 01:20:00,370
Up to now, we were sort of
thinking at just how to get out

1497
01:20:00,370 --> 01:20:02,080
at the ground state solution.

1498
01:20:02,080 --> 01:20:04,690
But then we have the
additional problem

1499
01:20:04,690 --> 01:20:09,130
of having the atoms
move, so our potential

1500
01:20:09,130 --> 01:20:11,620
starts changing in real time.

1501
01:20:11,620 --> 01:20:14,920
And we need to sort of
follow in real time what

1502
01:20:14,920 --> 01:20:16,120
happens to the solution.

1503
01:20:16,120 --> 01:20:19,450
And again, there are sort of
two general approaches that

1504
01:20:19,450 --> 01:20:22,810
go under the name of sort of
Born-Oppenheimer molecular

1505
01:20:22,810 --> 01:20:26,140
dynamics or sort of
Car-Parrinello molecular

1506
01:20:26,140 --> 01:20:26,830
dynamics.

1507
01:20:26,830 --> 01:20:30,280
And I think we'll leave
that at this stage for one

1508
01:20:30,280 --> 01:20:31,540
of the next lectures.

1509
01:20:31,540 --> 01:20:34,150
So let me just
sort of remind you

1510
01:20:34,150 --> 01:20:36,820
that another of the
terms that you'll

1511
01:20:36,820 --> 01:20:41,140
see for this kind of dynamics
in which the velocity drives

1512
01:20:41,140 --> 01:20:45,430
the system towards the
minimum goes actually

1513
01:20:45,430 --> 01:20:49,060
under the name of steepest
descent or conjugate gradient

1514
01:20:49,060 --> 01:20:49,990
minimization.

1515
01:20:49,990 --> 01:20:51,970
And that's, again,
a sort of technique

1516
01:20:51,970 --> 01:20:54,760
that you see very often
in minimization problem

1517
01:20:54,760 --> 01:20:58,960
where sort of this is really
a molecular dynamic kind

1518
01:20:58,960 --> 01:21:00,880
of simulated [INAUDIBLE].

1519
01:21:00,880 --> 01:21:02,470
With this, I conclude.

1520
01:21:02,470 --> 01:21:06,610
We'll keep some of the sort
of ab initio molecular dynamic

1521
01:21:06,610 --> 01:21:08,860
techniques for one
of the last lectures

1522
01:21:08,860 --> 01:21:12,700
when we actually go back and
look at case studies of this.

1523
01:21:12,700 --> 01:21:15,610
And in the last transparencies,
in the last few graph

1524
01:21:15,610 --> 01:21:19,430
of your notes, you have,
again, some freely available

1525
01:21:19,430 --> 01:21:24,790
bibliography of quantum
molecular dynamics

1526
01:21:24,790 --> 01:21:27,730
primers, in particular,
the sort of primer

1527
01:21:27,730 --> 01:21:30,250
of Dominik Marx on his website.

1528
01:21:30,250 --> 01:21:34,150
The last note is something
very completely and very well

1529
01:21:34,150 --> 01:21:34,810
written.

1530
01:21:34,810 --> 01:21:36,220
With this, I conclude.

1531
01:21:36,220 --> 01:21:39,220
Have a very happy weekend,
and we'll see you on Tuesday

1532
01:21:39,220 --> 01:21:42,315
in the lab in 115.