Molecular Dynamics

Molecular Dynamics (Part 1)

Sparisoma Viridi*

Nuclear Physics and Biophysics Research Division
Institut Teknologi Bandung, Jalan Ganesha 10, Bandung 40132, Indonesia
*dudung@gmail.com

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Kapita Selekta Sains Komputasi

Outline
• Short history of molecular dynamics (MD)
• Introduction to MD
• MD Algorithm
• Example
• Assignments

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Short history

Molecular dynamics
Contribution from some works

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History of MD
• It was first introduced in studying the interact-
ions of hard spheres which exhibits phase
transitions (Alder et. al, 1957)
• Then, a series of paper led by Alder is then pu-
blished during 1959-1980 investigating this
method

B. J. Alder and T. E. Wainwright, “Phase Transition for a Hard Sphere System”, Journal of Chemical Physics 27 (5)
1208-1209 (1957)

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History of MD (cont.)
• Studies of Alder and Wainwright in 1957 and
1959 induced other studies concerning beha-
ior of simple liquids
• Realistic potential for liquid argon is then used
(Rahman, 1964)
• Simulation of realistic system is conducted for
the first time for water (Stillinger et. al, 1974)
A. Rahman, “Correlations in the Motion of Atoms in Liquid Argon”, Physical Review 136 (2A), A405-A411 (1964)
F. H. Stillinger and A. Rahman, “Improved Simulation of Liquid Water by Molecular Dynamics”, Journal of Chemical
Physics 60 (4), 1545-1557 (1974)

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History of MD (cont.)
• The dynamics of folded of globular protein
(bovine pancreatic trypsin inhibitor) is the first
protein simulation (McCammon et. al, 1977)
• Many program and code are released, e.g.
Chemistry HARvard Molecular Mechanics
(CHARMM) (Stote et. al, 1999)

J. A. McCammon, B. R. Gelin, and M. Karplus, “Dynamics of Folded Proteins”, Nature 267 (5612) 585-590 (1977)
R. Stote, A. Dejaegere, D. Kuznetsov, and L. Falquet, “Theory of Molecular Dynamcis Simulation ” in Tutori@l
Molecular Dynamics Simulation CHARMM, URI http://www.ch .embnet.org/MD_tutorial /pages/MD.Part1. html
[2012.02.13]
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Alder’s papers
• This series of papers published during 1959-
1980, a lot of time of consistency of studying
something
– I. General Method (1959)
– IV. Behavior of a Small Number of Elastic Spheres
(1960)
– III. A Mixture of Hard Spheres (1964)
– IV. The Pressure, Collision Rate, and Their Number
Dependence for Hard Disks (1967)
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Alder’s papers (cont.)
– V. High-Density Equation of State and Entropy for
Hard Disks and Spheres
– VI. Free-Path Distributions and Collision Rates for
Hard-Sphere and Square-Well Molecules (1968)
– VII. Hard-Sphere Distribution Functions and an
Augmented van der Waals Theory (1969)
– VIII. The Transport Coefficients for a Hard-Sphere
Fluid (1970)
– IX. Vacancies in Hard Sphere Crystals (1971)
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– X. Corrections to the Augmented van der Waals
Theory for the Square Well Fluid (1972)
– XI. Correlation Functions of a Hard-Sphere Test
Particle (1972)
– XII. Band Shape of the Depolarized Light Scattered
from Atomic Fluids (1973)
– XIII. Singlet and Pair Distribution Functions for
Hard-Disk and Hard-Sphere Solids (1974)

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– XIV. Mass and Size Dependence of the Binary
Diffusion Coefficient (1974)
– XV. High Temperature Description of the
Transport Coefficients (1975)
– XVI. Fluctuation Driven Resonance (1977)
– XVII. Phase diagrams for ’’step’’ potentials in two
and three dimensions (1979)
– XVIII. The square-well phase diagram (1980)

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Introduction

Molecular dynamics
Definitions, use, and limitations

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Molecular dynamics
• Molecular dynamics (MD) is a computer
simulation of physical movements of atoms
and molecules (Wikipedia, 2011)
• MD simulation consists of the numerical, step-
by-step, solution of classical equation of
motion (Allen, 2004)

Wikipedia contributors, “Molecular dynamics”, Wikipedia, The Free Encyclopedia, 5 September 2011, 15:49 UTC,
oldid:448597141 [2011.09.21]
M. P. Allen, “Introduction to Molecular Dynamics Simulation”, in Computational Soft Matter: From Synthetic Polymers
to Proteins, Lecture Notes, Norberg Attig, Kurt Binder, Helmut Grubmüller, Kurt Kremer (Eds.), John von Nuemann
Institut for Computing, Jülich, NIC Series, Vol. 23, pp. 1-28, 2004
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Molecular dynamics (cont.)
• It is a computer simulation technique where
the time evolution of a set of interacting
atoms is followed by integrating their
equations of motion (Ercolessi, 1997)
• MD simulations can provide the ultimate
detail concerning individual motions as a
function of time (Karplus et. al, 2002)
F. Ercolessi, “A Molecular Dynamics Primer”, Spring College in Computational Physics, ICTP, Trieste, 9/10/1997 URI
http://www.fisica.uniud.it/~ercolessi/md /md/node6.html [2011.09.21]
M. Karplus and J. A. McCammon, “Molecular Dynamics Simulations of Biomolecules”, Nature Structural Biology 9 (9),
646-653 (2002)
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Range of use
• It is used from atomic until planetoid scale
• Calculation of electronic ground state as
function of time of liquid metal (Kresse et. al,
1993)
• Motion of n-Alkanes molecules (Ryckaert et.
al, 1977)

G. Kresse and J. Hafner, “Ab Initio Molecular Dynamics for Liquid Metals”, Physical Review B 47 (1), 558-561 (1993)
J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, “Numerical Integration of the Cartesian Equations of Motion of a
System with Constraints: Molecular Dynamics of n-Alkanes”, Journal of Computational Physics 23 (3), 327-341 (1977)

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Range of use (cont.)
• Nanodroplet on a surface (Sedighi et. al, 2010)
• Grains in mm and cm size (Gallas et. al, 1996)
• Simulation of asteroids movement (Jaffé et. al,
2002)

N, Sedighi, S. Murad, and S. K. Aggarwal, “Molecular Dynamics Simulations of Nanodroplet Spreading on Solid
Surfaces, Effect of Droplet Size”, Fluid Dynamics Research 42 (3), 035501 (2010)
J. A. C. Gallas, H. J. Herrmann, T. Pöschel, and Stefan Sokolowski, “Molecular Dynamics Simulation of Size
Segregation in Three Dimensions”, Journal of Statistical Physics 82 (1-2), 443-450 (1996)
C. Jaffé, S. D. Ross, M. W. Lo, J. Marsden, D. Farrelly, and T. Uzer, “Statistical Theory of Asteroid Escape Rates”,
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Use of MD
• There are three main scenarios for the use of
MD (Fedman, 2006)
• In the first scenario the simulated properties
are compared with experimental results, and
when the two agree it is reasonable to claim
that the experimental results can be explained
by the simulation model.
F. Hedman, “Algorithms for Molecular Dynamics Simulations: Advancing the Computational Horizon”, Ph.D. Thesis,
Avdelningen för fysikalisk kemi, Arrheniuslaboratoriet, Stockholms Universitet, Stockholm, 2006

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Use of MD (cont.)
• In the second scenario, MD simulations are
used to interpret experimental results. In a
sense the second scenario is the inverse of the
first.
• In the third scenario, simulations are used as
an exploratory tool to help gain an initial
understanding of a problem and give guidance
among possible lines of investigation, be it
theoretical or experimental.
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Experiment using simulation

M. P. Allen, “Introduction to Molecular Dynamics Simulation”, in Computational Soft Matter: From Synthetic Polymers
to Proteins, Lecture Notes, Norberg Attig, Kurt Binder, Helmut Grubmüller, Kurt Kremer (Eds.), John von Nuemann
Institut for Computing, Jülich, NIC Series, Vol. 23, pp. 1-28, 2004

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Experiment .. simulation (cont.)
• It is a bridge between microscopic and
macroscopic
• It is also a bridge between theory and
experiment
• Do the experiment using simulation is a smart
way to reduce the financial problem
• Even all considered nature laws are input to
the system, it could give the unexpected
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Is MD so perfect?
• Unfortunately not
• It has problem even all forces are already
considered
• It can produce unreported results or
unexpected (wrong) results
• It has problem in time stamp

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Time stamp problem
• Nanodroplet

N, Sedighi, S. Murad, and S. K. Aggarwal, “Molecular Dynamics Simulations of Nanodroplet Spreading on Solid
Surfaces, Effect of Droplet Size”, Fluid Dynamics Research 42 (3), 035501 (2010)

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Time stamp problem (cont.)

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• Granular oscillation

K. -C. Chen, C. -H. Lin, C. -C. Li, and J. -J. Li, “Dual Granular Temperature Oscillation of a Compartmentalized
Bidisperse Granular Gas”, Journal of the Physical Society of Japan 78 (4), 044401 (2009)

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MD Algorithm

Trajectory of a particle
Potentials, the forms, and their physical meaning

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Algorithms
• It is uses Newton’s second law of motion to
get the acceleration a
• It using numerical integration to get the
equation of motion, use the simple method
i.e. original Euler method
• New motion parameters will cause new value
of all forces
• Calculate the new forces to get new a
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Algorithms (cont.)
• Newton’s second law of motion
 
∑ F = ma

• Left side consists of all considered forces

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Algorithms (cont.)
• Euler method:
  
vi +1 = vi + ai ∆t
  
ri + 1 = ri + vi ∆t

t i +1 = t i + ∆t

• Particle position is given by ri at time t i

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Algorithms (cont.)
• You must pay attention to influence from out-
side of the system that changed with order of
magnitude of chosen Δt
• Normally it is chosen that Δt must be 100
times smaller than that change

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Potentials
• Lennard-Jones potential
 σ 12  σ  6 
U LJ ( r ) = 4ε   −   
 r 
 r  

• Coulomb potential
q1q 2 1
UC (r) =
4πε 0 r
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Potentials (cont.)
• Gravitation potensial near large object
U G ( r ) = −mgr

• Gravitation potensial
1
U G ( r ) = −Gm1m2
r
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Potentials (cont.)
• Morse potential

U M ( r ) = De 1 − e [ ]
−α ( r − re ) 2

• Yukawa potential
− kmr
e
U Y ( r) = −g 2

r
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Potentials (cont.)
• Harmonic oscillator potential
1
U HO ( r ) = k ( r − r0 )
2

2

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Force
• Force can be obtained from potential through
 
F = −∇U

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Granular memory device

Example
Sequence of particle under gravitation potential

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System
• Granular device (D), sensor (S), particle
sequence (P)

• P moves with constant initial velocity before
entering D
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Typical states
• Observed states are:
– s10w0r0 (two configurations)
– s10w1r0
– s10w1r1

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Typical states (cont.)
• Writing zero particle and relecting none from
ten particles sequence (s10w0r0)

• gn = 0, gp = 0, b = 0, v0 = 4

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• Writing zero particle and relecting none from

• gn = 1, gp = -2, b = 2, v0 = 6

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• Writing one particle and relecting none from

• gn = 1, gp = -2, b = 2, v0 = 5

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• Writing one particle and relecting another one
from ten particles sequence (s10w1r1)

• gn = 1, gp = -3, b = 2, v0 = 4

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Assignments

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Assignments
• Make six groups of 2-3 students
• Each group collects only one answer file
• Answer file should be sent to
dudung@gmail.com with subject [SK6202] MD
Assignment 1
• The file sould be received before 11 February
2013

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Assignments (cont.)
• Question 1. Derive force formulation for
following potential:
(a) harmonic oscillator,
(b) Coulomb,
(c) gravitation,
(d) Lennard-Jones,
(e) Morse,
(f) Yukawa
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Assignments (cont.)
• Question 2. Describe the physical meaning of
parameters used in each force or potential
formulation
• Question 3. Tell the difference between
molecular dynamics and molecular mechanics
• Question 4. Find the Euler, Verlet, Gear pre-
dictor-corrector, Rattle, and Shake algorithm
• Question 5. Find a topic to be solved using
molecular dynamics and explain
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Thank you

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Molecular Dynamics

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