Molecular Dynamics - review

Molecular Dynamics
University of Tehran
College of Engineering
School of Chemical Engineering
Student :
Hamed Hoorijani
Spring 2019
1hoorijani@ut.ac.ir

Outlines
• Introduction
• Principles
• Potential Functions
• Periodic Boundary Conditions
• Cutoff methods
• Integration Algorithms
• Challenges
• Softwares
• Example
• Books
2hoorijani@ut.ac.ir

Introduction
• Why we do Simulation?
in some cases, Experiment is :
1. impossible inside of star ,weather
forecast
2. too dangerous Flight Simulation, Explosion
Simulation
3. Expensive High Pressure Simulation,
Wind channel Simulation
4.blind some properties can’t be observe
on very short time- scales and very
small space-scales
Examples
3hoorijani@ut.ac.ir

Introduction
• Simulation is a useful complement, because it can:
• Replace Experiment
• Provoke Experiment
• Explain Experiment
• Aid in establishing intellectual property
4hoorijani@ut.ac.ir

What is Molecular Dynamics ?
• Connection between microscopic and macroscopic behavior of
the physical system with description of the atomic and
molecular interactions
5hoorijani@ut.ac.ir

Computational Tools
• Quantum Mechanics(QM)
Electronic Structure (Schrӧdinger)
• Accurate
• Expensive small system
• Classical Molecular Mechanics(MM)
Empirical Forces(Newton)
• Less Accurate
• Fast
• Mixed QM/MM
6hoorijani@ut.ac.ir

Procedure
• Calculate how a system of particles evolves in time
• Consider a set of atoms with positions /velocities and the
potential energy function of the system
• Predict the next positions of particles over some short time
interval by solving Newtonian mechanics
7hoorijani@ut.ac.ir

Basic MD Algorithm
Set initial conditions and
Get new forces
Solve the equations of motion
numerically over a short step
Is ?
Calculate results and finish
)( 0tir )( 0tiv
)( ii rF
)()( ttt ii  rr
)()( ttt ii  vv
t
ttt 
maxtt 
8hoorijani@ut.ac.ir

Principles
• In molecular Dynamic we need
• Position (r)
• Momentum (m)
• Charge (q)
• Bond Information (Which Atoms, bond angles, etc.)
For each atom in every molecule
i
jj’
rcut
L
9hoorijani@ut.ac.ir

Principles
• The base of Molecular Dynamic is the second law of newton’s
• Using the gradient of the potential energy function the main
algorithm of the simulation is presented
i i iF m a
i iF V 
2
2
i
i
i
d rdV
m
dr dt
 
10hoorijani@ut.ac.ir

Potential Functions
• A single atom will be affected by the potential energy functions
of every atom in the system:
• Bonded Neighbors
• Non-Bonded Atoms (either other atoms in the same molecule, or atoms from
different molecules)
non bonded van der Waals electrostaticE E E   
bonded bond stretch angle bend rotate along bondE E E E     
( ) bonded non bondedV R E E  

Non-Bonded Potential
• Van-der walls Potential
one of the most widely used functions for the van der waals
potential in the Lennard-Jones
12 6
ik ik
Lennard Jones
nonbonded ik ik
pairs
A C
E
r r

 
  
 

A,C depends on the atom
types, derived from
experimental data

Non-Bonded Potential
• Electrostatic Potential
opposite charges attract
• The force of the attraction is inversely proportional to the
square of the distance
i k
electrostatic
nonbonded ik
pairs
q q
E
Dr
 

Bonded Potential
• 3 types of interaction between bonded atoms:
• Stretching along the bond
• Bending between bonds
• Rotating around bonds
bonded bond stretch angle bend rotate along bondE E E E     

Boned Potential
Bond length Potentials
2
0
1,2
( )bond stretch b
pairs
E K b b  
• Both the spring constant and the ideal bond length are dependent
on the atoms involved

Bonded Potentials
Bond Angle Potentials
• The spring constant and the ideal angle are also dependent on
the chemical type of the atoms.
2
0( )bond bend
angles
E K    

Boned Potential
Torsional Potentials
• Described by a dihedral angle and coefficient of symmetry
(n=1,2,3), around the middle bond.
1,4
(1 cos( ))rotate along bond
pairs
E K n    

Some Simplified Potential Functions for
Specific cases
• Morse Potential
• For pair atomic molecules
• For modeling some metals such as copper
• For modeling structures with covalent bonds

Specific cases
• Sterlinger-weber
• For modeling semi-conductive material

Specific cases
• Tersoff Potential
• Modeling carbonic and silicone structure

Specific cases
• AIREBO Potential
• Modeling carbon-hydrogen systems

Specific cases
• EAM Potential
• Modeling metallic and different types of alloys

Periodic Boundary Condition
• Simulate a segment of molecules in a larger solution by having
repeatable regions
• When an atom moves off the edge, it reappears on the other
side (like in asteroids)
• In molecular dynamics simulation, PBC are usually applied to
calculate bulk gasses, liquids, crystals or mixtures.

Cutoff Methods
• Ideally, every atom in the system should interact with every
other atom which leads to a force calculation algorithm of
quadratic order
• The cutoff methods explains different approaches to ignore
atoms at large distances from each other without loosing too
much accuracy

Integration Algorithms
• Using conventional algorithms to solve the motion equation in MD is
not efficient cause:
• the forces are very rapidly changing non-linear functions
• The RK in some cases is justifiable it allows you to take larger time steps but
requires multiple force calculations per each timestep
So an algorithm was needed to provide the stability benefits of RK without the
cost of extra force calculations!
In short :
Numerical approximation of the integral over time

• Different Algorithms have been suggested :
• Verlet Algorithm
• Leap-frog Algorithm
• Velocity Verlet Algorithm
• Boeman’s Algorithm
• In choosing the right algorithm we should consider:
• it should be computational efficient
• it should conserve energy and momentum
• it should permit a long time step for integration

• They are all assume that position, velocities and acceleration
can be approximate based on a Taylor series expansion

Verlet Algorithm
• It uses the position and acceleration at time step (t) and position at 𝑡 − 𝑑𝑡
))((
1
)( t
m
rFra 
2
)(
2
1
)()()( tttttt  ravrr
ttttt  )(
2
1
)()2/( ravv
))((
1
)( tt
m
tt  raa
ttttttt  )(
2
1
)2/()( avv
From the initial ,)(tir )(tiv
tt 
28
Obtain the positions and velocities at
hoorijani@ut.ac.ir

Leap-Frog Algorithm
• In this algorithm using the velocity at time (𝑡 +
𝑑𝑡
2
) and position
at time (t), calculates the position at t+dt

Velocity Verlet Algorithm
• More accuracy than the Verlet
• Using the position, velocity, acceleration at time (t) the velocity
and position at time (𝑡 + 𝑑𝑡)

Boeman’s Algorithm
• More accuracy than Verlet but computational expensive

MD time Step
1/20 of the nearest atom distance tr/
In practice fs (femto-second).4t
MD is limited to <~100 ns
If Too long : energy is not conservedt

Calculating physical Properties
• Thermodynamic Properties
• Kinetic Energy:
• Temperature:
K E m vi i
i
N
. .  
1
2
2
T
Nk
K E
B

2
3
. .

• Configuration Energy:
• Pressure:
• Specific Heat
U V rc ij
j i
N
i


 ( )


 

1
13
1 N
i
N
ij
ijijB frTNkPV

( ) ( )U Nk T
Nk
C
c NVE B
B
v
2 2 23
2
1
3
2
 

• Structural Properties
• Pair correlation (Radial Distribution Function):
• Structure factor:
Note: S(k) available from x-ray diffraction
g r
n r
r r
V
N
r rij
j i
N
i
( )
( )
( )  

4 2 2



  drrrg
kr
kr
kS 2
0
1)(
)sin(
41)(  



Radial Distribution Function

Radial Distribution Function obtained of
Lennard-Jones System

Limitations of classical MD
Problems
1.Fixed set of atom types
2.No electronic Polarization
-fixed partial charges allow
for conformational polarization
but not electronic polarization
3.Parameters are imperfect

Softwares
Package Name Supported Force Fields Website Developer Team
CHARMM CHARMM(E / I; AA / UA) www.charm.org Harvard
Amber Amber(E / I;AA) Amber.scripps.edu San Francisco
GROMOS Gromos (E / vacuum ; UA) Igc.ethz.ch/GROMOS Zurich
Gromacs Amber,Gromos,OPLS – (all
E)
Gromacs.org Groningen
NAMD CHARMM, Amber
,Gromos
Ks.uiuc.edu/Research/na
md
Orban, USA
Lammps CHARMM, AMBER,
COMPASS, and DREIDING
Lammps.sandia.gov USA

LAMMPS Example Case
• Uniaxial compressive loading of an cupper single crystal

LAMMPS Example Case

LAMMPS Example Case
You can see a clip of the results available on the link below:
https://youtu.be/FYhw5FVotKI

LAMMPS Example Case

LAMMPS Example Case
• a reactive deformation of a single polyethylene chain

LAMMPS Example Case

LAMMPS Example Case
You can see a clip of the results available on the link below:
: https://youtu.be/q340EYNh5XE

Thank you for your attention 

SUMMARY
• Review Molecular Dynamics
• MD simulation for ax uniaxial compressive loading of an cupper
single crystal
• MD simulation for a reactive deformation of a single
polyethylene chain

Molecular Dynamics - review

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