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A Molecular Dynamics Primer - Furio Ercolessi

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A molecular dynamics primer
Furio Ercolessi
International School for Advanced Studies (SISSA-ISAS)
Trieste, Italy
Email: furio@sissa.it
WWW: http://www.sissa.it/furio/
Spring College in Computational Physics, ICTP, Trieste, June 1997.
Contents
Preface 1
1 Introduction 2
1.1 The role of computer experiments . . . . . . . . . . . . . . . . . . 2
1.1.1 But is it theory or experiment? . . . . . . . . . . . . . . . 3
1.2 What is molecular dynamics? . . . . . . . . . . . . . . . . . . . . 4
1.3 Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Today's role of molecular dynamics . . . . . . . . . . . . . . . . . 6
1.5 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5.1 Use of classical forces . . . . . . . . . . . . . . . . . . . . 7
1.5.2 Realism of forces . . . . . . . . . . . . . . . . . . . . . . . 8
1.5.3 Time and size limitations . . . . . . . . . . . . . . . . . . 8
2 The basic machinery 10
2.1 Modeling the physical system . . . . . . . . . . . . . . . . . . . . 10
2.1.1 The Lennard-Jones potential . . . . . . . . . . . . . . . . 10
2.1.2 Potential truncation and long-range corrections . . . . . . 11
2.2 Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . 12
2.2.1 The minimum image criterion . . . . . . . . . . . . . . . . 13
2.2.2 Geometries with surfaces . . . . . . . . . . . . . . . . . . 14
2.3 Time integration algorithm . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 The Verlet algorithm . . . . . . . . . . . . . . . . . . . . . 15
2.3.2 Predictor-corrector algorithm . . . . . . . . . . . . . . . . 16
3 Running, measuring, analyzing 18
3.1 Starting a simulation . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.1 Starting from scratch . . . . . . . . . . . . . . . . . . . . 18
3.1.2 Continuing a simulation . . . . . . . . . . . . . . . . . . . 19
3.2 Controlling the system . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Equilibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 Looking at the atoms . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5 Simple statistical quantities to measure . . . . . . . . . . . . . . 21
3.5.1 Potential energy . . . . . . . . . . . . . . . . . . . . . . . 22
3.5.2 Kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5.3 Total energy . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5.4 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 23
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3.5.5 The caloric curve . . . . . . . . . . . . . . . . . . . . . . . 23
3.5.6 Mean square displacement . . . . . . . . . . . . . . . . . . 24
3.5.7 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.6 Measuring the melting temperature . . . . . . . . . . . . . . . . . 25
3.7 Real space correlations . . . . . . . . . . . . . . . . . . . . . . . . 26
3.8 Reciprocal space correlations . . . . . . . . . . . . . . . . . . . . 27
3.9 Dynamical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.10 Annealing and quenching: MD as an optimization tool . . . . . . 29
3.11 Other statistical ensembles . . . . . . . . . . . . . . . . . . . . . 30
4 Interatomic potentials 32
4.1 The Born-Oppenheimer approximation . . . . . . . . . . . . . . . 32
4.2 The design of potentials . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 The problems with two-body potentials . . . . . . . . . . . . . . 34
4.4 Many-body potentials for metals . . . . . . . . . . . . . . . . . . 35
4.5 Many-body potentials for semiconductors . . . . . . . . . . . . . 37
4.5.1 The Stillinger-Weber potential . . . . . . . . . . . . . . . 37
4.5.2 The Terso� potential . . . . . . . . . . . . . . . . . . . . . 38
4.6 Long-range forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.7 But do we really need potentials? . . . . . . . . . . . . . . . . . . 39
4.8 Fitting to ab initio data by \force matching" . . . . . . . . . . . 40
A Choosing hardware and software 42
A.1 What kind of computer? . . . . . . . . . . . . . . . . . . . . . . . 42
A.1.1 Selecting a computer for molecular dynamics . . . . . . . 42
A.1.2 Storage space . . . . . . . . . . . . . . . . . . . . . . . . . 44
A.2 What language? . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
A.2.1 High level languages . . . . . . . . . . . . . . . . . . . . . 44
A.2.2 Fortran or C? . . . . . . . . . . . . . . . . . . . . . . . . . 45
Bibliography 47
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Preface
These notes constitute the skeleton for a short hands-on course on molecular
dynamics. It is assumed that the reader has basic notions of classical mechanics,
statistical mechanics and thermodynamics, but knows very little|if anything
at all|on methods for atomistic computer simulation. The basics of molecular
dynamics are explained, together with a few tricks of the trade.
No attempt is made to address \advanced topics" in molecular dynamics,
such as calculation of free energies, non equilibrium simulations, etc., and many
important subjects are treated only super�cially. This document is meant to be
just a starting point. Many excellent books devoted to computer simulations
in condensed matter physics are available [1, 2, 3, 4, 5, 6, 7], and the reader is
advised to make use of them to dig into more speci�c subjects.
These note are accompanied by example programs written in
Fortran 90 and available on the World Wide Web at the URL
http://www.sissa.it/furio/md/f90. While still relatively uncommon
among scientists, Fortran 90 is bound to become one of the most important
languages for scienti�c programming in the years to come. Fortran 90
incorporates Fortran 77|allowing it to inherit smoothly the legacy of the
immense body of existing codes|but also features elements now recognized to
be essential in a language for large projects in scienti�c computing.
I hope that the examples contained here will be of help to young compu-
tational physicists to get familiar with Fortran 90, and to people grown with
Fortran 77 (or 66!) to migrate towards this new, more modern environment.
Any comment or suggestion to improve this document, which will also
be kept online at the URL http://www.sissa.it/furio/md/, will be wel-
come and appreciated. The best way to contact the author is by email at
furio@sissa.it.
1
Chapter 1
Introduction
1.1 The role of computer experiments
Computer experiments play a very important role in science today. In the past,
physical sciences were characterized by an interplay between experiment and
theory. In experiment, a system is subjected to measurements, and results,
expressed in numeric form, are obtained. In theory, a model of the system is
constructed, usually in the form of a set of mathematical equations. The model
is then validated by its ability to describe the system behavior in a few selected
cases, simple enough to allow a solution to be computed from the equations.
In many cases, this implies a considerable amount of simpli�cation in order to
eliminate all the complexities invariably associated with real world problems,
and make the problem solvable.
In the past, theoretical models could be easily tested only in a few simple
\special circumstances". So, for instance, in condensed matter physics a model
for intermolecular forces in a speci�c material could be veri�ed in a diatomic
molecule, or in a perfect, in�nite crystal. Even then, approximations were often
required to carry out the calculation. Unfortunately, many physical problems of
extreme interest (both academic and practical) fall outside the realm of these
\special circumstances". Among them, one could mention the physics and
chemistry of defects, surfaces, clusters of atoms, organic molecules, involving
a large amount