Preface xiii
1 Introduction - Studying Systems from Two Viewpoints 1
2 Classical Mechanics and Numerical Methods 5
2.1 Mechanics -The Study of Motion 5
2.2 Classical Newtonian Mechanics 6
2.3 Analytical Solutions of Newton's Equations and Phase Space 8
2.3.1 Motion of an Object Under Constant Gravitational Force 8
2.3.2 One-Dimensional Harmonic Oscillator 10
2.3.3 Radial Force Functions in Three Dimensions 12
2.3.4 Motion Under the Influence of a Drag Force 15
2.4 Numerical Solution of Newton's Equations: The Euler Method 17
2.5 More Efficient Numerical Algorithms for Solving Newton's Equations 20
2.5.1 The Verlet Algorithm 20
2.5.2 The Leapfrog Algorithm 21
2.5.3 The Velocity Verlet Algorithm 22
2.5.4 Considerations for Numerical Solution of the Equations of Motion 23
2.6 Examples of Using Numerical Methods for Solving Newton's Equations of Motion 25
2.6.1 Motion Near the Earth's Surface Under Constant Gravitational Force 25
2.6.2 One-Dimensional Harmonic Oscillator 26
2.7 Numerical Solution of the Equations of Motion for Many-Atom Systems 28
2.8 The Lagrangian and Hamiltonian Formulations of Classical Mechanics 29
Chapter 2 Appendices 32
2.A.1 Separation of Motion in Two-Particle Systems with Radial Forces 32
2.A.2 Motion Under Spherically Symmetric Forces 33
3 Intra- and Intermolecular Potentials in Simulations 39
3.1 Introduction - Electrostatic Forces Between Atoms 39
3.2 Quantum Mechanics and Molecular Interactions 40
3.2.1 The Schroedinger Equation 40
3.2.2 The Born-Oppenheimer Approximation 42
3.3 Classical Intramolecular Potential Energy Functions from Quantum Mechanics 44
3.3.1 Intramolecular Potentials 45
3.3.2 Bond Stretch Potentials 48
3.3.3 Angle Bending Potentials 51
3.3.4 Torsional Potentials 51
3.3.5 The 1-4, 1-5, and Farther Intramolecular Interactions 53
3.4 Intermolecular Potential Energies 54
3.4.1 Electrostatic Interactions 54
3.4.1.1 The Point Charge Approximation 55
3.4.1.2 The Multipole Description of Charge Distribution 59
3.4.1.3 Polarizability 61
3.4.2 van der Waals Interactions 63
3.5 Force Fields 64
3.5.1 Water Force Fields 64
3.5.2 The AMBER Force Field 66
3.5.3 The OPLS Force Field 68
3.5.4 The CHARMM Force Field 69
3.5.5 Other Force Fields 69
Chapter 3 Appendices 71
3.A.1 The Born-Oppenheimer Approximation to Determine the Nuclear Schroedinger Equation 71
4 The Mechanics of Molecular Dynamics 73
4.1 Introduction 73
4.2 Simulation Cell Vectors 73
4.3 Simulation Cell Boundary Conditions 75
4.4 Short-Range Intermolecular Potentials 79
4.4.1 Cutoff Radius and the Minimum Image Convention 79
4.4.2 Neighbor Lists 82
4.5 Long-Range Intermolecular Potentials: Ewald Sums 84
4.6 Simulating Rigid Molecules 88
Chapter 4 Appendices 92
4.A.1 Fourier Transform of Gaussian and Error Functions 92
4.A.2 Electrostatic Force Expression from the Ewald Summation Technique 94
4.A.3 The Method of Lagrange Undetermined Multipliers 95
4.A.4 Lagrangian Multiplier for Constrained Dynamics 98
5 Probability Theory and Molecular Simulations 101
5.1 Introduction: Deterministic and Stochastic Processes 101
5.2 Single Variable Probability Distributions 103
5.2.1 Discrete Stochastic Variables 103
5.2.2 Continuous Stochastic Variables 104
5.3 Multivariable Distributions: Independent Variables and Convolution 106
5.4 The Maxwell-Boltzmann Velocity Distribution 111
5.4.1 The Concept of Temperature from the Mechanical Analysis of an Ideal Gas 112
5.4.2 The Maxwell-Boltzmann Distribution of Velocities for an Ideal Gas 115
5.4.3 Energy Distributions for Collections of Molecules in an Ideal Gas 120
5.4.4 Generating Initial Velocities in Molecular Simulations 123
5.5 Phase Space Description of an Ideal Gas 125
Chapter 5 Appendices 127
5.A.1 Normalization, Mean, and Standard Deviation of the Gaussian Function 127
5.A.2 Convolution of Gaussian Functions 128
5.A.3 The Virial Equation and the Microscopic Mechanical View of Pressure 131
5.A.4 Useful Mathematical Relations and Integral Formulas 133
5.A.4.1 Stirling's Approximation for N! 133
5.A.4.2 Exponential Integrals 134
5.A.4.3 Gaussian Integrals 134
5.A.4.4 Beta Function Integrals 134
5.A.5 Energy Distribution forThree Molecules 135
5.A.6 Deriving the Box-Muller Formula for Generating a Gaussian Distribution 136
6 Statistical Mechanics in Molecular Simulations 139
6.1 Introduction 139
6.2 Discrete States in Quantum Mechanical Systems 140
6.3 Distributions of a System Among Discrete Energy States 142
6.4 Systems with Non-interacting Molecules: The -Space Approach 145
6.5 Interacting Systems and Ensembles: The -Space Approach and the Canonical Ensemble 148
6.5.1 Thermodynamics Quantities 152
6.5.2 Fluctuations in Thermodynamic Quantities in the Canonical Ensemble 154
6.5.3 Canonical Ensemble for Systems with Non-interacting Molecules 156
6.5.4 A Physical Interpretation of the Canonical Partition Function 157
6.6 Other Constraints Coupling the System to the Environment 158
6.6.1 Isothermal-Isobaric Ensemble (Fixed N, P, and T) 158
6.6.2 Grand Canonical Ensemble (Fixed , V, and T) 163
6.6.3 Microcanonical Ensemble (Fixed N, V, and E) 166
6.6.4 Isenthalpic-Isobaric Ensemble (Fixed N, P, and H) 167
6.7 Classical Statistical Mechanics 167
6.7.1 The Canonical Ensemble 167
6.7.2 The Isothermal-Isobaric Ensemble 169
6.7.3 The Grand Canonical Ensemble 169
6.7.4 The Microcanonical Ensemble 170
6.7.5 Isenthalpic-Isobaric Ensemble 170
6.8 Statistical Mechanics and Molecular Simulations 171
Chapter 6 Appendices 172
6.A.1 Quantum Mechanical Description and Determination of the Lagrange Multiplier 𝛽and Pressure for an Ideal Gas 172
6.A.2 Determination of the Lagrange Multiplier 𝛽in Systems with Interacting Molecules 174
6.A.3 Summary of Statistical Mechanical Formulas 175
7 Thermostats and Barostats 177
7.1 Introduction 177
7.2 Constant Pressure Molecular Dynamics (the Isobaric Ensembles) 178
7.2.1 Non-isotropic Volume Variation: The Parrinello-Rahman Method 184
7.3 Constant Temperature Molecular Dynamics 185
7.3.1 Extended System Method: The Nose-Hoover Thermostat 185
7.3.2 The Berendsen Thermostat 190
7.4 Combined Constant Temperature-Constant Pressure Molecular Dynamics 192
7.5 Scope of Molecular Simulations with Thermostats and Barostats 195
Chapter 7 Appendices 196
7.A.1 Andersen Barostat and the Isobaric-Isenthalpic Ensemble 196
7.A.2 The Lagrangian for a Constant Pressure System with Non-isotropic Volume Change: The Parrinello-Rahman Method 196
7.A.3 Nose Thermostat System and the Canonical Ensemble Distribution Function 197
8 Simulations of Structural and Thermodynamic Properties 199
8.1 Introduction 199
8.2 Simulations of Solids, Liquids, and Gases 200
8.2.1 Setting Up Initial Structures for Molecular Simulations of Solids, Liquids, and Gases 202
8.3 The Radial Distribution Function 205
8.4 Simulations of Solutions 211
8.5 Simulations of Biological Molecules 214
8.6 Simulation of Surface Tension 219
8.7 Structural Order Parameters 224
8.8 Statistical Mechanics and the Radial Distribution Function 227
8.9 Long-Range (Tail) Corrections to the Potential 232
Chapter 8 Appendices 233
8.A.1 Force Fields for Simulations in the Figures of Chapter 8 233
8.A.1.1 Nitrogen Force Field 233
8.A.1.2 NaCl Simulation Force Field 233
8.A.2 The PDB File Format 234
9 Simulations of Dynamic Properties 237
9.1 Introduction 237
9.2 Molecular Motions and the Mean Square Displacement 237
9.2.1 Motion in Bulk Phases 237
9.2.2 Motion in Confined Spaces and on Surfaces 244
9.3 Molecular Velocities and Time Correlation Functions 247
9.3.1 Collisions and the Velocity Autocorrelation Function 247
9.3.2 Time Correlation Functions for Stationary Systems 251
9.4 Orientation Autocorrelation Functions 251
9.5 Hydrogen Bonding Dynamics 253
9.6 Molecular Motions on Nanoparticles: The Lindemann Index 254
9.7 Microscopic Determination of Transport Coefficients 256
9.7.1 The Transport Coefficients 256
9.7.2 Nonequilibrium Molecular Dynamics Simulations of Transport Coefficients 261
9.7.3 The Green-Kubo Relations and Simulation of Transport Coefficients 261
Chapter 9 Appendices 263
9.A.1 Brownian Motion and the Langevin Equation 263
9.A.2 The Discrete Random Walk Model of Diffusion 265
9.A.3 The Solution of the Diffusion Equation 267
9.A.4 Relation Between Mean Square Displacement and Diffusion Coefficient 268
10 Monte Carlo Simulations 269
10.1 Introduction 269
10.2 The Canonical Monte Carlo Procedure 270
10.2.1.1 Determining Which Molecule to Move 273
10.2.1.2 Determining Whether a Translation or Rotation is Performed 273
10.2.1.3 Translation Moves 274
10.2.1.4 Rotational Moves 274
10.3 The Condition of Microscopic Reversibility and Importance Sampling 277
10.4 Monte Carlo Simulations in Other Ensembles 279
10.4.1 Grand Canonical Monte Carlo Simulations 279
10.4.2 Isothermal-Isobaric Monte Carlo Simulations 283
10.4.3 Biased Monte Carlo Sampling Methods 284
10.5 Gibbs Ensemble Monte Carlo Simulations 285
10.5.1 Simulations of Liquid-Gas Phase Equilibrium 285
10.6 Simulations of Gas Adsorption in Porous Solids 288
10.6.1 Simulations of the Gas Adsorption Isotherm and Heat of Adsorption 288
10.6.2 Force Fields for Gas Adsorption Simulations 291
10.6.3 Block Averaging of Data from Monte Carlo and Molecular Dynamics Simulations 291
Chapter 10 Appendices 295
10.A.1 Thermodynamic Relation for the Heat of Adsorption 295
References 297
Index 317
