Molecular Dynamics Simulations
An Introduction
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Pingwen Zhang
Molecular Dynamics
•
Definitions, Motivations
•
Force fields
•
Algorithms and computations
•
Analysis of Data
•
M
olecular
dynamics
(MD)
is
a
computer
simulation
technique
:
the
time
evolution
of
interacting
atoms
is
followed
by
integrating
their
equations
of
motion
.
•
W
e follow the laws of classical mechanics,
and most notably Newton's law:
Molecular dynamics

Introduction
Hy = Ey
F = MA
exp(

D
E/kT)
domain
quantum
chemistry
molecular
dynamics
Monte Carlo
mesoscale
continuum
Length Scale
10

10
M
10

8
M
10

6
M
10

4
M
10

12
S
10

8
S
10

6
S
Scale in Simulations
•
Modeling the motion of a complex molecule
by solving the wave functions of the various
subatomic particles would be accurate…
•
But it would also be
very
hard to program
and take more computing power than
anyone has!
Why Not Quantum Mechanics?
•
Given
an initial set of positions and velocities, the
subsequent time evolution is
in principle
completely
determined.
•
A
toms
and molecules
will
‘
move
’
in the computer,
bumping into each other,
vibrating about a mean
position (if constrained), or wandering
around (if the
system is fluid), oscillating in waves in concert with
their neighbours, perhaps evaporating away from the
system if there is a free surface, and so on, in a way
similar to what
real
atoms
and molecules
would do.
Molecular dynamics

Introduction
•
T
he
computer experiment
.
•
In a computer experiment, a model is still provided
by theorists, but the calculations are carried out by
the machine by following a recipe (the
algorithm
,
implemented in a suitable
programming language
).
•
In this way, complexity can be introduced
(
with
caution!) and more realistic systems can be
investigated, opening a road towards a better
understanding of real experiments.
Molecular dynamics

Motivation
Molecular dynamics

Motivation
•
The computer calculates a trajectory
of the system
•
6
N

dimensional phase space (3
N
positions and 3
N
momenta
).
•
A trajectory obtained by molecular dynamics provides
a set of co
nformations of the molecule,
•
They are accessible without any great expenditure of
energy (e.g. breaking bonds)
•
MD also used as an efficient tool for optimisation of
structures (
simulated annealing
).
Molecular dynamics

Motivation
•
MD
allows
to
study
the
dynamics
of
large
macromolecules
•
Dynamical
events
control
processes
which
affect
functional
properties
of
the
bio

molecule
(e
.
g
.
protein
folding)
.
•
Drug
design
is
used
in
the
pharmaceutical
industry
to
test
properties
of
a
molecule
at
the
computer
without
the
need
to
synthesize
it
.
Molecular dynamics
–
Time Limitations
•
Typical
MD
simulations
are
performed
on
systems
containing
millions
of
atoms
•
Simulation
times: picoseconds to nanoseconds
.
•
A simulation is
reliable
when the simulation time
is
much longer than the relaxation time of the
quantities we are interested in
.
Historical Perspective on MD
Procedure of MD
Initialization
•
Position
–
X

ray, NMR, simulation or analytical calculation
•
Velocity
–
T
he initial velocities are assigned taking them from a
Maxwell distribution at a certain temperature
T
•
Another possibility is to take the initial positions
and velocities to be the final positions and
velocities of a previous MD run
Choosing the Time Step
d
t
•
No
MD
follows
the
true
trajectories
for
very
many
time
steps
–
errors
always
accumulate
.
The
time
over
which
the
MD
trajectory
=
true
trajectory
is
called
correlation
time
.
•
No
MD
truly
conserves
energy
since
there
are
always
errors
.
The
goal
is
to
have
a
constant
average
E
with
fluctuations
as
small
as
possible
•
Time
step
d
t
should
be
as
large
as
possible
to
still
get
accurate
trajectories
(on
the
time
scale
needed)
and
conserve
of
energy
•
In
general,
d
t
should
be
≈
0
.
01
x
the
fastest
behavior
of
your
system
(E
.
g
.
,
atoms
oscillate
about
once
every
10

12
s
in
a
solid
MD
time
steps
are
≈
10

14
s
in
simulations
of
solids
Potentials
•
ab initio potential
–
Quantum calculation, DFT, OFDFT, Tight

binding, etc.
•
Empirical potential
–
Comes from quantum
–
Consistent with continuum
–
Fit the database: elastic moduli, surface energy, etc.
•
Connection with continuum
–
Constitutive relation
The Lennard

Jones Potential
•
The truncated and shifted Lennard

Jones potential
•
The truncated Lennard

Jones potential
•
The Lennard

Jones potential
FENE potential
•
FENE stands for:
Finitely extensible nonlinear elastic
EAM potential
•
Embedded Atom Method
works for metallic solids
•
Two contributions
•
nuclear

nuclear interaction
•
embedding an atom to the electron cloud
Potential for Covalent Carbon
•
The Stillinger

Weber potential
•
The Tersoff Potential
Not
only
accounts
for
the
contribution
of
bond
lengths,
but
also
for
the
bond
angles
Non

Bonded Atoms
There are two potential functions we need to
be concerned about between non

bonded
atoms:
•
van der Waals Potential
•
Electrostatic Potential
The van der Waals Potential
•
Atoms with no net electrostatic charge will still
tend to attract each other at short distances
•
Atoms tend to repel when they get too close
The
Constants
A
and
C
depend
on
the
atom
types,
and
are
derived
from
experimental
data
The Electrostatic Potential:
Coulomb’s Law
•
Opposite Charges Attract
•
Like Charges Repel
•
The force of the attraction is inversely
proportional to the square of the distance
Bonded Atoms
There are three types of
interaction between
bonded atoms:
•
Stretching along the
bond
•
Bending between bonds
•
Rotating around bonds
Bond Length Potentials
Both the spring constant
and the ideal bond
length are dependent on
the atoms involved.
Bond Angle Potentials
The spring constant and
the ideal angle are also
dependent on the
chemical type of the
atoms.
Torsional Potentials
Described by a dihedral
angle and coefficient
of symmetry
(
n=1,2,3
), around the
middle bond.
Effects of solvents
•
Implicit models
–
“Generalized Born” solvent Model
coarse

graining the effects of
solvent by approximately solving
the Poisson equation
•
Explicit models
–
Explicitly adding the water
molecules which are regarded as
rigid bodies
Integrator: Verlet Algorithm
Start with {r(t), v(t)}, integrate it to {r(t+
D
t),=瘨tH
D
t)}:
{r(t), v(t)}
{r(t+
D
t), v(t+
D
t)}
The new position at t+
D
t:
Similarly, the old position at t

D
t:
(1)
(2)
Add
(
1
)
and
(
2
)
:
Thus the velocity at t is:
(3)
(4)
Verlet Scheme
•
Is time reversible
•
Does conserve volume in phase space
•
(Is symplectic)
•
Doe not suffer from energy drift
Velocity Verlet scheme
•
Velocity calculated explicitly
•
Possible to control the temperature
•
Stable in long time simulation
•
Most commonly used algorithm
Central
Simulation
box
r
c
Periodic Boundary Conditions
Minimum Image
Ewald Sum: split into two teems, real space
sum and k

space sum
Periodic Boundary Conditions
Ewald Method
Saving CPU time
Cell list
Verlet list
Ensembles
•
NVE
–
micro

canonical
ensemble
•
NVT
–
canonical ensemble
•
NPT
–
grand

canonical
ensemble
•
Temperature control
–
Berendsen thermostat (velocity
rescaling)
–
Andersen thermostat (velocity
resampling)
–
Nose

Hoover chain
•
Pressure control
–
Berendsen volume rescaling
–
Andersen piston
MD as Optimization tool
Simulated Annealing
•
Most popular global optimization
algorithm
•
Start at high T, decrease T in small
steps (cooling schedule)
•
Easy to understand & implement
•
Drawback: might be easily trapped in
local minima
Cooling Schedules
:
Molecular dynamics
–
Analyses
•
The simplest
way of
analyzing
the system during (or
after) its dynamic motion
is
looking at it
.
•
One can assign a radius to the atoms
,
represent the
atoms as balls having that radius, and have a
computer
program construct a
‘
photograph
’
of the
system
.
•
We may also color the atoms according to its
properties (charge, displacement, ‘temperature’…)
Molecular dynamics
–
Analyses
•
We also can
measure
instantaneous and time
averages of various physically important quantities
•
To measure time averages: If the instantaneous
values of some property A at time t is
then its average is
where
N
T
is the number of steps in the trajectory
Softwares
•
AMBER
•
CHARMM
•
VASP (DFT)
•
XMD
•
CPMD(DFT) (Car

Parrinello MD)
References
•
M. P. Allen, D. J. Tildesley (1989) Computer simulation
of liquids. Oxford University Press.
•
Frenkel Daan; Smit, Berend [2001].
Understanding
Molecular Simulation
: from algorithms to applications
.
Academic Press. D. C.
•
Rapaport (1996) The Art of Molecular Dynamics
Simulation.
•
Tamar Schlick (2002) Molecular Modeling and
Simulation. Springer.
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