Simulating the physical world: Hierarchical modeling from quantum mechanics to fluid dynamics

626 Pages·2007·3.655 MB·English

by  Herman J. C. Berendsen

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SIMULATING THE PHYSICAL WORLD
The simulation of physical systems requires a simplified, hierarchical approach,
whichmodelseachlevelfromtheatomistictothemacroscopicscale.Fromquan-
tum mechanics to fluid dynamics, this book systematically treats the broad scope
ofcomputermodelingandsimulations,describingthefundamentaltheorybehind
each level of approximation. Berendsen evaluates each stage in relation to their
applications giving the reader insight into the possibilities and limitations of the
models. Practical guidance for applications and sample programs in Python are
provided.Withastrongemphasisonmolecularmodelsinchemistryandbiochem-
istry, this book will be suitable for advanced undergraduate and graduate courses
onmolecularmodelingandsimulationwithinphysics,biophysics,physicalchem-
istryandmaterialsscience.Itwillalsobeausefulreferencetoallthoseworkingin
the field. Additional resources for this title including solutions for instructors and
programsareavailableonlineatwww.cambridge.org/9780521835275.
Herman J. C. Berendsen is Emeritus Professor of Physical Chemistry at
the University of Groningen. His research focuses on biomolecular modeling and
computer simulations of complex systems. He has taught hierarchical modeling
worldwideandishighlyregardedinthisfield.

SIMULATING THE PHYSICAL WORLD
Hierarchical Modeling from Quantum
Mechanics to Fluid Dynamics
HERMAN J. C. BERENDSEN
EmeritusProfessorofPhysicalChemistry,
UniversityofGroningen,theNetherlands
CAMBRIDGEUNIVERSITYPRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
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Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521835275
© H. J. C. Berendsen 2007
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2007
ISBN-13 978-0-511-29491-4 eBook (EBL)
ISBN-10 0-511-29491-3 eBook (EBL)
ISBN-13 978-0-521-83527-5 hardback
ISBN-10 0-521-83527-5 hardback
ISBN-13 978-0-521-54294-4 paperback
ISBN-10 0-521-54294-4 paperback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
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Contents
Preface page xi
Symbols, units and constants xv
Part I A Modeling Hierarchy for Simulations 1
1 Introduction 3
1.1 What is this book about? 3
1.2 A modeling hierarchy 9
1.3 Trajectories and distributions 13
1.4 Further reading 14
2 Quantum mechanics: principles and relativistic effects 19
2.1 The wave character of particles 19
2.2 Non-relativistic single free particle 23
2.3 Relativistic energy relations for a free particle 25
2.4 Electrodynamic interactions 31
2.5 Fermions, bosons and the parity rule 36
3 From quantum to classical mechanics: when and how 39
3.1 Introduction 39
3.2 From quantum to classical dynamics 42
3.3 Path integral quantum mechanics 44
3.4 Quantum hydrodynamics 64
3.5 Quantum corrections to classical behavior 70
4 Quantum chemistry: solving the time-independent Schr¨o-
dinger equation 77
4.1 Introduction 77
4.2 Stationary solutions of the TDSE 78
4.3 The few-particle problem 79
4.4 The Born–Oppenheimer approximation 97
v
vi Contents
4.5 The many-electron problem of quantum chemistry 98
4.6 Hartree–Fock methods 99
4.7 Density functional theory 102
4.8 Excited-state quantum mechanics 105
4.9 Approximate quantum methods 106
4.10 Nuclear quantum states 107
5 Dynamics of mixed quantum/classical systems 109
5.1 Introduction 109
5.2 Quantum dynamics in a non-stationary potential 114
5.3 Embedding in a classical environment 129
6 Molecular dynamics 139
6.1 Introduction 139
6.2 Boundary conditions of the system 140
6.3 Force field descriptions 149
6.4 Solving the equations of motion 189
6.5 Controlling the system 194
6.6 Replica exchange method 204
6.7 Applications of molecular dynamics 207
7 Free energy, entropy and potential of mean force 211
7.1 Introduction 211
7.2 Free energy determination by spatial integration 213
7.3 Thermodynamic potentials and particle insertion 218
7.4 Free energy by perturbation and integration 221
7.5 Free energy and potentials of mean force 227
7.6 Reconstruction of free energy from PMF 231
7.7 Methods to derive the potential of mean force 234
7.8 Free energy from non-equilibrium processes 239
8 Stochastic dynamics: reducing degrees of freedom 249
8.1 Distinguishing relevant degrees of freedom 249
8.2 The generalized Langevin equation 251
8.3 The potential of mean force 255
8.4 Superatom approach 256
8.5 The fluctuation–dissipation theorem 257
8.6 Langevin dynamics 263
8.7 Brownian dynamics 268
8.8 Probability distributions and Fokker–Planck equations 269
8.9 Smart Monte Carlo methods 272
8.10 How to obtain the friction tensor 274
Contents vii
9 Coarse graining from particles to fluid dynamics 279
9.1 Introduction 279
9.2 The macroscopic equations of fluid dynamics 281
9.3 Coarse graining in space 288
9.4 Conclusion 295
10 Mesoscopic continuum dynamics 297
10.1 Introduction 297
10.2 Connection to irreversible thermodynamics 298
10.3 The mean field approach to the chemical potential 301
11 Dissipative particle dynamics 305
11.1 Representing continuum equations by particles 307
11.2 Prescribing fluid parameters 308
11.3 Numerical solutions 309
11.4 Applications 309
Part II Physical and Theoretical Concepts 313
12 Fourier transforms 315
12.1 Definitions and properties 315
12.2 Convolution and autocorrelation 316
12.3 Operators 317
12.4 Uncertainty relations 318
12.5 Examples of functions and transforms 320
12.6 Discrete Fourier transforms 323
12.7 Fast Fourier transforms 324
12.8 Autocorrelation and spectral density from FFT 325
12.9 Multidimensional Fourier transforms 331
13 Electromagnetism 335
13.1 Maxwell’s equation for vacuum 335
13.2 Maxwell’s equation for polarizable matter 336
13.3 Integrated form of Maxwell’s equations 337
13.4 Potentials 337
13.5 Waves 338
13.6 Energies 339
13.7 Quasi-stationary electrostatics 340
13.8 Multipole expansion 353
13.9 Potentials and fields in non-periodic systems 362
13.10 Potentials and fields in periodic systems of charges 362
viii Contents
14 Vectors, operators and vector spaces 379
14.1 Introduction 379
14.2 Definitions 380
14.3 Hilbert spaces of wave functions 381
14.4 Operators in Hilbert space 382
14.5 Transformations of the basis set 384
14.6 Exponential operators and matrices 385
14.7 Equations of motion 390
14.8 The density matrix 392
15 Lagrangian and Hamiltonian mechanics 397
15.1 Introduction 397
15.2 Lagrangian mechanics 398
15.3 Hamiltonian mechanics 399
15.4 Cyclic coordinates 400
15.5 Coordinate transformations 401
15.6 Translation and rotation 403
15.7 Rigid body motion 405
15.8 Holonomic constraints 417
16 Review of thermodynamics 423
16.1 Introduction and history 423
16.2 Definitions 425
16.3 Thermodynamic equilibrium relations 429
16.4 The second law 432
16.5 Phase behavior 433
16.6 Activities and standard states 435
16.7 Reaction equilibria 437
16.8 Colligative properties 441
16.9 Tabulated thermodynamic quantities 443
16.10 Thermodynamics of irreversible processes 444
17 Review of statistical mechanics 453
17.1 Introduction 453
17.2 Ensembles and the postulates of statistical mechanics 454
17.3 Identification of thermodynamical variables 457
17.4 Other ensembles 459
17.5 Fermi–Dirac, Bose–Einstein and Boltzmann statistics 463
17.6 The classical approximation 472
17.7 Pressure and virial 479
17.8 Liouville equations in phase space 492
17.9 Canonical distribution functions 497