# Understanding Molecular Simulation

**From Algorithms to Applications**

**By**

- Daan Frenkel, FOM Institute for Atomic and Molecular Physics, The Netherlands
- Berend Smit, University of Amsterdam, Amsterdam, The Netherlands

**Understanding Molecular Simulation: From Algorithms to Applications** explains the physics behind the "recipes" of molecular simulation for materials science. Computer simulators are continuously confronted with questions concerning the choice of a particular technique for a given application. A wide variety of tools exist, so the choice of technique requires a good understanding of the basic principles. More importantly, such understanding may greatly improve the efficiency of a simulation program. The implementation of simulation methods is illustrated in pseudocodes and their practical use in the case studies used in the text.Since the first edition only five years ago, the simulation world has changed significantly -- current techniques have matured and new ones have appeared. This new edition deals with these new developments; in particular, there are sections on:脗路 Transition path sampling and diffusive barrier crossing to simulaterare events脗路 Dissipative particle dynamic as a course-grained simulation technique脗路 Novel schemes to compute the long-ranged forces脗路 Hamiltonian and non-Hamiltonian dynamics in the context constant-temperature and constant-pressure molecular dynamics simulations脗路 Multiple-time step algorithms as an alternative for constraints脗路 Defects in solids脗路 The pruned-enriched Rosenbluth sampling, recoil-growth, and concerted rotations for complex molecules脗路 Parallel tempering for glassy HamiltoniansExamples are included that highlight current applications and the codes of case studies are available on the World Wide Web. Several new examples have been added since the first edition to illustrate recent applications. Questions are included in this new edition. No prior knowledge of computer simulation is assumed.

View full description### Audience

Graduate students in physics and materials science departments studying molecular simulation techniques; scientists in the fields of polymers, materials science, and applied physics.

### Book information

- Published: October 2001
- Imprint: ACADEMIC PRESS
- ISBN: 978-0-12-267351-1

### Reviews

"芒聙娄brilliantly maintains a balance between explaining the physical phenomena and performing computations. Its marvelous writing style invites scientists and students to deepen their knowledge of MD simulations."--**ComputingReviews.com, January 11, 2013**

"... this book brilliantly lays down the scientific foundations of the simulational approach ..."--**Prof. Kurt Binder in Physics World, 1997**

"... a treasure. The book is a marvellous mix of just enough formalism with an informal and readable style, sufficient detail to understand methodological advances, appropriate mathematics ..."--**Prof. Mark A. Ratner in Physics Today, 1997**

### Table of Contents

Preface to the Second Edition

Preface

List of Symbols

1 Introduction

Part I Basics

2 Statistical Mechanics

2.1 Entropy and Temperature

2.2 Classical Statistical Mechanics

2.3 Questions and Exercises

3 Monte Carlo Simulations

3.1 The Monte Carlo Method

3.2 A Basic Monte Carlo Algorithm

3.3 Trial Moves

3.4 Applications

3.5 Questions and Exercises

4 Molecular Dynamics Simulations

4.1 Molecular Dynamics: the Idea

4.2 Molecular Dynamics: a Program

4.3 Equations of Motion

4.4 Computer Experiments

4.5 Some Applications

4.6 Questions and Exercises

Part II Ensembles

5 Monte Carlo Simulations in Various Ensembles

5.1 General Approach

5.2 Canonical Ensemble

5.3 Microcanonical Monte Carlo

5.4 Isobaric-Isothermal Ensemble

5.5 Isotension-Isothermal Ensemble

5.6 Grand-Canonical Ensemble

5.7 Questions and Exercises

6 Molecular Dynamics in Various Ensembles

6.1 Molecular Dynamics at Constant Temperature

6.2 Molecular Dynamics at Constant Pressure

6.3 Questions and Exercises

Part III Free Energies and Phase Equilibria

7 Free Energy Calculations

7.1 Thermodynamic Integration

7.2 Chemical Potentials

7.3 Other Free Energy Methods

7.4 Umbrella Sampling

7.5 Questions and Exercises

8 The Gibbs Ensemble

8.1 The Gibbs Ensemble Technique

8.2 The Partition Function

8.3 Monte Carlo Simulations

8.4 Applications

8.5 Questions and Exercises

9 Other Methods to Study Coexistence

9.1 Semigrand Ensemble

9.2 Tracing Coexistence Curves

10 Free Energies of Solids

10.1 Thermodynamic Integration

10.2 Free Energies of Solids

10.3 Free Energies of Molecular Solids

10.4 Vacancies and Interstitials

11 Free Energy of Chain Molecules

11.1 Chemical Potential as Reversible Work

11.2 Rosenbluth Sampling

Part IV Advanced Techniques

12 Long-Range Interactions

12.1 Ewald Sums

12.2 Fast Multipole Method

12.3 Particle Mesh Approaches

12.4 Ewald Summation in a Slab Geometry

13 Biased Monte Carlo Schemes

13.1 Biased Sampling Techniques

13.2 Chain Molecules

13.3 Generation of Trial Orientations

13.4 Fixed Endpoints

13.5 Beyond Polymers

13.6 Other Ensembles

13.7 Recoil Growth

13.8 Questions and Exercises

14 Accelerating Monte Carlo Sampling

14.1 Parallel Tempering

14.2 Hybrid Monte Carlo

14.3 Cluster Moves

15 Tackling Time-Scale Problems

15.1 Constraints

15.2 On-the-Fly Optimization: Car-Parrinello Approach

15.3 Multiple Time Steps

16 Rare Events

16.1 Theoretical Background

16.2 Bennett-Chandler Approach

16.3 Diffusive Barrier Crossing

16.4 Transition Path Ensemble

16.5 Searching for the Saddle Point

17 Dissipative Particle Dynamics

17.1 Description of the Technique

17.2 Other Coarse-Grained Techniques

Part V Appendices

A Lagrangian and Hamiltonian

A.1 Lagrangian

A.2 Hamiltonian

A.3 Hamilton Dynamics and Statistical Mechanics

B Non-Hamiltonian Dynamics

B.1 Theoretical Background

B.2 Non-Hamiltonian Simulation of the N, V, T Ensemble

B.3 The N, P, T Ensemble

C Linear Response Theory

C.1 Static Response

C.2 Dynamic Response

C.3 Dissipation

C.4 Elastic Constants

D Statistical Errors

D.1 Static Properties: System Size

D.2 Correlation Functions

D.3 Block Averages

E Integration Schemes

E.1 Higher-Order Schemes

E.2 Nos脙漏-Hoover Algorithms

F Saving CPU Time

F.1 Verlet List

F.2 Cell Lists

F.3 Combining the Verlet and Cell Lists

F.4 Efficiency

G Reference States

G.1 Grand-Canonical Ensemble Simulation

H Statistical Mechanics of the Gibbs Ensemble

H.1 Free Energy of the Gibbs Ensemble

H.2 Chemical Potential in the Gibbs Ensemble

I Overlapping Distribution for Polymers

J Some General Purpose Algorithms

K Small Research Projects

K.1 Adsorption in Porous Media

K.2 Transport Properties in Liquids

K.3 Diffusion in a Porous Media

K.4 Multiple-Time-Step Integrators

K.5 Thermodynamic Integration

L Hints for Programming

Bibliography

Author Index

Index