Hello. In this video, we're
going to talk a little bit about molecular dynamics because molecular dynamics is
one of the few ways to study liquids from a
theoretical point of view, or a computational point
of view, I should say. Molecular dynamics
involves following microscopic behavior at
the atomistic level, by making the Born-Oppenheimer
approximation, and solving F equals MA for
all the nuclei present. One can calculate the
forces on the nuclei using either empirical
or quantum methods. But empirical methods are
almost always used for reasons of computational
efficiency. Now, it's doing a
quantum MD calculation that's really, really slow. Empirical force fields are the subject of
molecular mechanics. That's just a term that is
used, molecular mechanics, it means for an
empirical force field. We'll talk about a force
field in a minute. In molecular dynamics,
we account for the thermal motion of all atoms, and therefore, of all molecules. Although there are methods
for lumping atoms together, and we'll not get into that, it's a subtlety of the field. So the direct output of a
calculation, of course, is the position and velocity of every atom has a
function of time, and of course, something
that's a function of position, and velocity, and time, that space is known
as Phase Space. Such a calculation
provides information about the possible conformations, thermodynamic properties, and dynamic behaviors
of molecules. So we start with Newton's
law, F equals MA. The force is the negative of the spatial derivative
of the potential energy. We can rewrite the
acceleration as force divided by mass
from Newton's law, and acceleration as
the second derivative of position with time. So d squared x_i, dt squared is equal
to f_i, over m, where the index i refers to the one of the
coordinate directions. So we would get a
solution in which we knew the position and the velocity in all three coordinate directions
as a function of time. So there's some
fundamental assumptions that go into calculating forces. The first is that
Born-Oppenheimer applies, I say, as always, that almost always as always. Force constants and bond lengths, one assumes are largely the same from one
molecule to the next. So most C-C bonds look the same, C-H bonds look the same, and that's called
transferability. You don't have to do it that way, but the more detail you put in, and the more special cases, the bigger the force field gets, and the more complicated
calculating forces becomes. The bonds and other interactions can be characterized
algebraically. That's the result of taking
an empirical approach, and we've already
done some of that. For example, with the
harmonic oscillator. So the terms include
bonds, angles, dihedral, non-bonded, and here's an example of one form of an
empirical force field. The first term, k, r minus r equilibrium squared, that's just a
harmonic oscillator. Potential energy term, it's exactly the same thing
we used when we did the harmonic oscillator back in the quantum mechanics
section of the course. The next one is angles. If you have three atoms, they form an angle. So this is a harmonic motion
in the plane of the angle. Dihedrals are when you
have four molecules, so that they can
bend out of plane. Then the last term includes both what are
called Van der Waals forces, and they're typically
represented with a Lennard-Jones 6-12 potential. These are for the mainly
attractive forces between atoms in
the same molecule that are a fair distance apart, and between molecules,
or between atoms, I should say, that aren't
connected by any bond at all. Then the last term,
q_iq_j over r_ij, that is electrostatic force term. So this is the minimal
set of terms that one would find in a
molecular dynamic situation. They can be made
more complicated, one can get into dipole effects, shape affects and so on. Like I said, you can have any degree of
complexity you want. So how do people go about
developing a force field? Well, it's a curve
sitting procedure, and the proof is in the pudding. When you're doing
these empirical forms, you have to assume some
functional form for each of the terms in the potential
energy expression. Then you have to run simulations and compare the simulations
with experiments, and then one does some kind of least squares minimization, or statistical optimization to get the best values
of the parameters. For example, the force constant
for harmonic oscillation, such that the data is best fit. This, for example, is
a penalty function, a least-squares penalty function. We're you're just summing
over calculated minus experimental values of
the various observables, divided by some weighting
function that you might give, in order to develop a force field that's appropriate for your
particular situation. There are many
published force fields. Many, many, many. Lots of PhD thesis have been devoted to
developing force fields, each have been developed
for a specific purpose. Some of the more common
ones include the MM series, Amber, Charmm and OPLS. If you go online and just search molecular
dynamic force field, you'll pop up lots
and lots of choices. So that's it for this video. Thanks for listening.
Have a great day.