 | Griebel M. Numerical simulation in molecular dynamics: numerics, algorithms, parallelization, applications / M.Griebel, S.Knapek, G.Zambusch. - Berlin: Springer, 2007. - xi, 470 p.: ill. - (Texts in computational science and engineering; 5). - ISSN 1611-0994; ISBN 978-3-642-48776-9
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1 Computer Simulation - a Key Technology ....................... 1
2 From the Schrödinger Equation to Molecular Dynamics ......... 17
2.1 The Schrödinger 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 Störmer-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-Taylor Instability ..................... 73
3.6.5 Rayleigh-Bénard Convection ...................... 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-Rahman Method ................... 104
3.7.5 Phase Transition of Argon in the NPT
Ensemble ....................................... 107
4 Parallelization ............................................ 113
4.1 Parallel Computers and Parallelization Strategies ..... 113
4.2 Domain Decomposition for the Linked Cell Method ....... 122
4.3 Implementation ........................................ 128
4.4 Performance Measurements and Benchmarks ............... 139
4.5 Application Examples .................................. 146
4.5.1 Collision of Two Bodies ........................ 146
4.5.2 Rayleigh-Taylor Instability .................... 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-Mesh Ewald 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-Taylor Instability 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 Parallelization of the SPME Method ............. 294
7.5.2 Implementation ................................. 299
7.5.3 Performance Measurements and Benchmarks ........ 302
7.6 Example Application: Structure of the Universe ........ 306
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 Parallel Tree 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-Ligand Complex and Bonding .................... 408
10 Prospects .................................................. 413
A Appendix ................................................... 417
A.l Newton's, Hamilton's, and Euler-Lagrange's
Equations ............................................. 417
A.2 Suggestions for Coding and Visualization .............. 418
A.3 Parallelization by MPI ................................ 421
A.4 Maxwell-Boltzmann Distribution ........................ 425
A.5 Parameters ............................................ 428
References ................................................. 431
Index ......................................................... 467
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