SIMULATING THE PHYSICAL WORLD

The simulation of physical systems requires a simpli¬ed, hierarchical approach,

which models each level from the atomistic to the macroscopic scale. From quan-

tum mechanics to ¬‚uid dynamics, this book systematically treats the broad scope

of computer modeling and simulations, describing the fundamental theory behind

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. With a strong emphasis on molecular models in chemistry and biochem-

istry, this book will be suitable for advanced undergraduate and graduate courses

on molecular modeling and simulation within physics, biophysics, physical chem-

istry and materials science. It will also be a useful reference to all those working in

the ¬eld. Additional resources for this title including solutions for instructors and

programs are available online at www.cambridge.org/9780521835275.

H e r m a n J . C . B e r e n d s e n 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

worldwide and is highly regarded in this ¬eld.

SIMULATING THE PHYSICAL WORLD

Hierarchical Modeling from Quantum

Mechanics to Fluid Dynamics

HERMAN J. C. BERENDSEN

Emeritus Professor of Physical Chemistry,

University of Groningen, the Netherlands

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University Press

The Edinburgh Building, Cambridge CB2 8RU, UK

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

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ISBN-10 0-521-83527-5

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ISBN-13 978-0-521-54294-4

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Contents

Preface page xi

Symbols, units and constants xv

1

Part I A Modeling Hierarchy for Simulations

3

1 Introduction

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

19

2 Quantum mechanics: principles and relativistic e¬ects

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

39

3 From quantum to classical mechanics: when and how

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

77

dinger equation

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

109

5 Dynamics of mixed quantum/classical systems

5.1 Introduction 109

5.2 Quantum dynamics in a non-stationary potential 114

5.3 Embedding in a classical environment 129

139

6 Molecular dynamics

6.1 Introduction 139

6.2 Boundary conditions of the system 140

6.3 Force ¬eld 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

211

7 Free energy, entropy and potential of mean force

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

249

8 Stochastic dynamics: reducing degrees of freedom

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 ¬‚uctuation“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