This page intentionally left blank SIMULATING THE PHYSICAL WORLD The simulation of physical systems requires a simplified, hierarchical approach, which models each level from the atomistic to the macroscopic scale. From quan- tum mechanics to fluid 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 field. Additional resources for this title including solutions for instructors and programs are available online at www.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 worldwide and is highly regarded in this field. 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 First published in print format ISBN-13 978-0-521-83527-5 ISBN-13 978-0-521-54294-4 ISBN-13 978-0-511-29491-4 © H. J. C. Berendsen 2007 2007 Information on this title: www.cambridge.org/9780521835275 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 p ermission of Cambrid g e University Press. ISBN-10 0-511-29491-3 ISBN-10 0-521-83527-5 ISBN-10 0-521-54294-4 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 g uarantee that any content on such websites is, or will remain, accurate or a pp ro p riate. Published in the United States of America by Cambridge University Press, New York www.cambridge.org hardback paperback paperback eBook (EBL) eBook (EBL) hardback 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 Freeenergy,entropyandpotentialofmeanforce 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 [...]... hierarchy of models for simulation from relativistic quantum mechanics to macroscopic fluid dynamics; Part II reviews the necessary mathematical, physical and chemical concepts, which are meant to provide a common background of knowledge and notation Some of these topics may be superfluous xi xii Preface to physicists or mathematicians, others to chemists The chapters of Part II could be useful in courses... a number of questions; whether a question is answerable by simulation depends on: • the state of theoretical development (models and methods of solution); • the computational capabilities; • the possibilities to implement the methods of solution in algorithms; • the possibilities to validate the model Validation means the assessment of the accuracy of the model (compared to physical reality) by critical... certain topics in their education; for this purpose exercises are included Answers and further information are available on the book’s website The subjects treated in this book, and the depth to which they are explored, necessarily reflect the personal preference and experience of the author Within this subjective selection the literature sources are restricted to the period before January 1, 2006 The overall... all the specialisms involved, many practical simulators specialize in their niche of interest, adopt – often unquestioned – the methods that are commonplace in their niche, read the literature selectively, and too often turn a blind eye on the limitations of their approaches This book tries to connect the various disciplines and expand the horizon for each field of application The basic approach is a physical. .. 1/(4π) and μ0 = 4πα2 The latter is equivalent to μ0 = 1/(ε0 c2 ), with both quantities expressed in a.u Table 7 lists the values of the basic atomic units in terms of SI units These units employ physical constants, which are not so constant as the name suggests; they depend on the definition of basic units and on the improving precision of measurements The numbers given here refer to constants published... in practice to systems that are made up of interacting atomic nuclei, which are specified by their mass, charge and spin, electrons, and photons that carry the electromagnetic interactions between the nuclei and electrons Occasionally we may wish to add gravitational interactions to the electromagnetic ones The internal structure of atomic nuclei is of no consequence for the behavior of atoms and molecules... solid angle are the dimensionless radian and steradian See Table 5 for the defined SI units All other units are derived from these basic units (Table 6) While the Syst`me International also defines the mole (with unit mol ), e being a number of entities (such as molecules) large enough to bring its total mass into the range of grams, one may express quantities of molecular size also per mole rather than per... of modeling 3 4 Introduction 1.1.2 System limitation We limit ourselves to models of the real world around us This is the realm of chemistry, biology and material sciences, and includes all industrial and practical applications We do not include the formation of stars and galaxies (stellar dynamics) or the physical processes in hot plasma on the sun’s surface (astrophysics); neither do we include the. .. system, and if the response to external influences is required, a specification ofthe external influences Both the model and the method of solution depend on the purpose of the simulation: they should be accurate and efficient The model should be chosen accordingly For example, an accurate quantum- mechanical description of the behavior of a many-particle system is not efficient for studying the flow of air... In the normal range of temperatures this limitation implies a practical division between electrons on the one hand and nuclei on the other: while all particles obey the rules of quantum mechanics, the quantum character of electrons is essential but the behavior of nuclei approaches the classical limit This distinction has far-reaching consequences, but it is rough and inaccurate For example, protons . blank SIMULATING THE PHYSICAL WORLD The simulation of physical systems requires a simplified, hierarchical approach, which models each level from the atomistic to the macroscopic scale. From quan- tum. 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. modeling and computer simulations of complex systems. He has taught hierarchical modeling worldwide and is highly regarded in this field. SIMULATING THE PHYSICAL WORLD Hierarchical Modeling from