Understanding Molecular Simulation,

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 Nosé-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