Table Of Content5
Texts in Computational Science
and Engineering
Editors
TimothyJ.Barth
MichaelGriebel
DavidE.Keyes
RistoM.Nieminen
DirkRoose
TamarSchlick
Michael Griebel
Stephan Knapek
Gerhard Zumbusch
Numerical Simulation
in Molecular Dynamics
Numerics, Algorithms,
Parallelization, Applications
With180Figures,43inColor,and63Tables
123
MichaelGriebel StephanKnapek
InstitutfürNumerischeSimulation e-mail:[email protected]
UniversitätBonn
Wegelerstr.6
53115Bonn,Germany
e-mail:[email protected]
GerhardZumbusch
InstitutfürAngewandteMathematik
Friedrich-Schiller-UniversitätJena
Ernst-Abbe-Platz2
07743Jena,Germany
e-mail:[email protected]
LibraryofCongressControlNumber:2007928345
Mathematics Subject Classification (2000): 68U20, 70F10, 70H05, 83C10, 65P10, 31C20,
65N06,65T40,65Y05,68W10,65Y20
ISSN 1611-0994
ISBN 978-3-540-68094-9 SpringerBerlinHeidelbergNewYork
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Preface
The rapid development of parallel computing systems made it possible to
recreate and predict physical processes on computers. Nowadays, computer
simulations complement and even substitute experiments. Moreover, simu-
lations allow to study processes that cannot be analyzed directly through
experiments. This accelerates the development of new products since costly
physical experiments can be avoided. Additionally, the quality of products
canbeimprovedbytheinvestigationofphenomenanotpreviouslyaccessible.
Hence, computer simulation plays a decisive role especially in the develop-
ment of new substances in the material sciences as well as in biotechnology
and nanotechnology.
Many interesting processes cannot be described nor understood in their
entirety in a continuum model, but have to be studied with molecular or
atomic models. The numerical simulationofmodels onthis lengthscale usu-
ally relies on particle methods and other methods of molecular dynamics.
Areas of application range from physics, biology, and chemistry to modern
material sciences.
The fundamental mathematical model here is Newton’s second law. It is
asystemofordinarydifferentialequationsofsecondorder.Thelawdescribes
therelationshipbetweentheforceactingonaparticleandtheresultingaccel-
erationoftheparticle.Theforceoneachparticleiscausedbytheinteraction
of that particle with all other particles. The resulting system of differential
equations has to be numerically approximated in an efficient way. After an
appropriatetimediscretization,forcesonallparticleshavetobecomputedin
each time step. Different fast and memory-efficient numerical methods exist
which are tailored to the short-range or long-range force fields and poten-
tials thatare used.Here,especiallythe linkedcellmethod, theparticle mesh
method, the P3M method and its variants, as well as several tree methods
suchas the Barnes-Hutmethod or the fast multipole method are to be men-
tioned. The execution times of these methods can be substantially reduced
by a parallel implementation on modern supercomputers. Such an approach
is also of fundamental importance to reach the very large numbers of parti-
cles and long simulation times necessary for some problems. The numerical
methods mentioned are already used with great success in many different
implementations by physicists, chemists, and material scientists. However,
VI Preface
without a deeper understanding of the particular numerical method applied
it is hard to make changes or modifications, or to parallelize or otherwise
optimize the available programs.
The aim of this book is to present the necessary numerical techniques
of molecular dynamics in a compact form, to enable the reader to write a
molecular dynamics programin the programming language C, to implement
thisprogramwithMPIonaparallelcomputerwithdistributedmemory,and
to motivate the reader to set up his own experiments. For this purpose we
present in all chapters a precise description of the algorithms used, we give
additional hints for the implementation on modern computers and present
numerical experiments in which these algorithms are employed. Further in-
formation and some programs can also be found on the internet. They are
available on the web page
http://www.ins.uni-bonn.de/info/md.
Afterashortintroductiontonumericalsimulationinchapter1,wederive
in chapter 2 the classical molecular dynamics of particle systems from the
principlesofquantummechanics.Inchapter3weintroducethebasicmodules
of molecular dynamics methods for short-range potentials and force fields
(linkedcellimplementation,Verlettimeintegration).Additionally,wepresent
afirstsetofexamplesoftheiruse.Here,thetemperatureistakenintoaccount
usingstatisticalmechanicsinthesettingofanNVTensemble.TheParrinello-
Rahman method for the NPT ensemble is also reviewed. Subsequently we
discuss in detail the parallel implementation of the linked cell method in
chapter 4 and give a set of further examples. In chapter 5 we extend our
methodstomolecularsystemsandmorecomplexpotentials.Furthermore,in
chapter 6 we give an overview of methods for time integration.
Different numerical methods for the efficient computation of long-range
forcefields arediscussedinthe followingchapters7 and8.The P3Mmethod
approximates long-rangepotentials on an auxiliary grid. Using this method,
further examples can be studied that involve in particular Coulomb forces
or, on a different length scale, also gravitational forces. We review both the
sequential and the parallel implementation of the SPME technique, which
is a variant of the P3M method. In chapter 8 we introduce and discuss tree
methods. Here, the emphasis is on the Barnes-Hut method, its extension to
higher order, and on a method from the family of fast multipole methods.
Both sequential and parallel implementations using space filling curves are
presented. In chapter 9 we give examples from biochemistry that require a
combination of the methods introduced before.
We thank the SFB 611 (Collaborative ResearchCenter sponsored by the
DFG- the GermanResearchAssociation)“SingularPhenomenaandScaling
in Mathematical Models” at the University of Bonnfor its support, Barbara
Hellriegel and Sebastian Reich for valuable hints and references, our col-
Preface VII
leagues and coworkers Marcel Arndt, Attila Caglar, Thomas Gerstner, Jan
Hamaekers, Lukas Jager, Marc Alexander Schweitzer and Ralf Wildenhues
for numerous discussions, their support in the implementation of the algo-
rithms as well as for the programming and computation for various model
problems andapplications.Inparticular we thank Attila andalso Alex,Jan,
Lukas, Marcel, Ralf and Thomas for their active help. We also thank Bern-
hard Hientzsch for the efforts he put in translating the German version1 of
this book.
Bonn, Michael Griebel
April 2007 Stephan Knapek
Gerhard Zumbusch
1 Numerische Simulation in der Moleku¨ldynamik, Numerik, Algorithmen, Paral-
lelisierung, Anwendungen, SpringerVerlag, Heidelberg, 2004.
Table of Contents
1 Computer Simulation – a Key Technology ................ 1
2 From the Schro¨dinger Equation to Molecular Dynamics .. 17
2.1 The Schro¨dinger Equation ............................... 17
2.2 A Derivation of Classical Molecular Dynamics.............. 21
2.2.1 TDSCF Approach and Ehrenfest Molecular Dynamics. 21
2.2.2 Expansion in the Adiabatic Basis................... 24
2.2.3 Restriction to the Ground State.................... 26
2.2.4 Approximation of the Potential Energy Hypersurface . 26
2.3 An Outlook on Methods of Ab Initio Molecular Dynamics ... 30
3 The Linked Cell Method for Short-Range Potentials...... 37
3.1 Time Discretization – the Sto¨rmer-Verlet Method........... 40
3.2 Implementation of the Basic Algorithm.................... 46
3.3 The Cutoff Radius...................................... 53
3.4 The Linked Cell Method ................................ 56
3.5 Implementation of the Linked Cell Method ................ 58
3.6 First Application Examples and Extensions................ 64
3.6.1 Collision of Two Bodies I.......................... 66
3.6.2 Collision of Two Bodies II ......................... 68
3.6.3 Density Gradient................................. 72
3.6.4 Rayleigh-TaylorInstability ........................ 73
3.6.5 Rayleigh-B´enardConvection ....................... 79
3.6.6 Surface Waves in Granular Materials................ 82
3.7 Thermostats, Ensembles, and Applications................. 86
3.7.1 Thermostats and Equilibration..................... 87
3.7.2 Statistical Mechanics and Thermodynamic Quantities. 93
3.7.3 Phase Transition of Argon in the NVT Ensemble ..... 96
3.7.4 The Parrinello-RahmanMethod.................... 104
3.7.5 Phase Transition of Argon in the NPT Ensemble ..... 107
4 Parallelization ............................................ 113
4.1 ParallelComputers and ParallelizationStrategies........... 113
4.2 Domain Decomposition for the Linked Cell Method ......... 122
4.3 Implementation ........................................ 128
X Table of Contents
4.4 Performance Measurements and Benchmarks............... 139
4.5 Application Examples................................... 146
4.5.1 Collision of Two Bodies ........................... 146
4.5.2 Rayleigh-TaylorInstability ........................ 148
5 Extensions to More Complex Potentials and Molecules ... 151
5.1 Many-Body Potentials .................................. 151
5.1.1 Cracks in Metals – Finnis-Sinclair Potential.......... 152
5.1.2 Phase Transition in Metals – EAM Potential......... 160
5.1.3 Fullerenes and Nanotubes – Brenner Potential ....... 167
5.2 Potentials with Fixed Bond Structures .................... 181
5.2.1 Membranes and Minimal Surfaces .................. 181
5.2.2 Systems of Linear Molecules ....................... 186
5.2.3 Outlook to More Complex Molecules................ 202
6 Time Integration Methods................................ 211
6.1 Errors of the Time Integration ........................... 212
6.2 Symplectic Methods .................................... 221
6.3 Multiple Time Step Methods – the Impulse Method......... 226
6.4 Constraints – the RATTLE Algorithm .................... 230
7 Mesh-Based Methods for Long-Range Potentials.......... 239
7.1 Solution of the Potential Equation........................ 243
7.1.1 Boundary Conditions ............................. 243
7.1.2 Potential Equation and Potential Decomposition ..... 244
7.1.3 Decomposition of the Potential Energy and of the
Forces .......................................... 248
7.2 Short-Range and Long-Range Energy and Force Terms...... 250
7.2.1 Short-Range Terms – Linked Cell Method ........... 250
7.2.2 Long-Range Terms – Fast Poisson Solvers ........... 252
7.2.3 Some Variants ................................... 258
7.3 Smooth Particle-MeshEwald Method (SPME) ............. 260
7.3.1 Short-Range Terms ............................... 261
7.3.2 Long-Range Terms ............................... 263
7.3.3 Implementation of the SPME method............... 273
7.4 Application Examples and Extensions..................... 281
7.4.1 Rayleigh-TaylorInstability with Coulomb Potential... 281
7.4.2 Phase Transition in Ionic Microcrystals ............. 284
7.4.3 Water as a Molecular System ...................... 287
7.5 Parallelization ......................................... 294
7.5.1 Parallelizationof the SPME Method................ 294
7.5.2 Implementation .................................. 299
7.5.3 PerformanceMeasurements and Benchmarks......... 302
7.6 Example Application: Structure of the Universe ............ 306
Table of Contents XI
8 Tree Algorithms for Long-Range Potentials............... 313
8.1 Series Expansion of the Potential......................... 314
8.2 Tree Structures for the Decomposition of the Far Field ...... 320
8.3 Particle-Cluster Interactions and the Barnes-Hut Method.... 325
8.3.1 Method ......................................... 326
8.3.2 Implementation .................................. 328
8.3.3 Applications from Astrophysics .................... 339
8.4 ParallelTree Methods................................... 341
8.4.1 An Implementation with Keys ..................... 343
8.4.2 Dynamical Load Balancing ........................ 357
8.4.3 Data Distribution with Space-Filling Curves ......... 359
8.4.4 Applications ..................................... 366
8.5 Methods of Higher Order................................ 370
8.5.1 Implementation .................................. 371
8.5.2 Parallelization ................................... 376
8.6 Cluster-Cluster Interactions and the Fast Multipole Method . 377
8.6.1 Method ......................................... 377
8.6.2 Implementation .................................. 382
8.6.3 Error Estimate................................... 385
8.6.4 Parallelization ................................... 386
8.7 Comparisons and Outlook ............................... 387
9 Applications from Biochemistry and Biophysics........... 391
9.1 Bovine Pancreatic Trypsin Inhibitor ...................... 392
9.2 Membranes ............................................ 394
9.3 Peptides and Proteins................................... 398
9.4 Protein-LigandComplex and Bonding..................... 408
10 Prospects................................................. 413
A Appendix................................................. 417
A.1 Newton’s, Hamilton’s, and Euler-Lagrange’sEquations ...... 417
A.2 Suggestions for Coding and Visualization.................. 418
A.3 Parallelizationby MPI .................................. 421
A.4 Maxwell-Boltzmann Distribution ......................... 425
A.5 Parameters ............................................ 428
References.................................................... 431
Index......................................................... 467
