Introduction to MD

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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.