Variational Methods in Molecular Modeling (eBook)

Dr. Jianzhong Wu is a professor of Chemical Engineering and a cooperating faculty member of Mathematics Department at the University of California, Riverside. His research is focused on the development and application of statistical-mechanical methods, in particular density functional theory, for predicting the microscopic structure and physiochemical properties of confined fluids, soft materials and biological systems.

Series Editor’s Preface 6
Preface 7
Contents 9
Editor and Contributors 10
Variational Methods in Statistical Thermodynamics---A Pedagogical Introduction 12
1 Introduction 12
2 The Variational Nature of Thermodynamics 13
3 The Variational Origin of Statistical Thermodynamics 16
4 The Method of Steepest Descent 18
5 The Gibbs-Bogoliubov-Feynman Variational Principle 21
6 A Toy Example 22
7 Mean-Field Solution for the Interacting Ising Model 26
8 The Poisson-Boltzmann Equation 30
9 Fluctuations 35
10 Summary and Perspective 39
References 40
Square-Gradient Model for Inhomogeneous Systems: From Simple Fluids to Microemulsions, Polymer Blends and Electronic Structure 41
1 Introduction 41
2 Statistical Mechanics 42
3 Density Functional Theory (DFT) 44
4 Square-Gradient Approximation (SGA) 47
5 Simple Fluids 49
6 Microemulsions 54
7 Polymer Blends 57
8 Electronic Systems 60
9 Summary 65
References 73
Classical Density Functional Theory for Molecular Systems 75
1 Molecular Models and Force Fields 75
2 Statistical Mechanics for Polyatomic Systems 79
2.1 Molecular Configuration and Interaction Sites 79
2.2 Grand Partition Function 81
2.3 Molecular Density and Molecular Correlation Functions 82
2.4 Site Density and Site Correlation Functions 84
2.5 Classical Models for Polyatomic Ideal-gas Systems 85
3 Density Functional Theory 87
3.1 Hohenberg-Kohn-Mermin Theorem 88
3.2 DFT for Ideal-Gas Systems 89
3.3 Excess Helmholtz Energy 90
3.4 Direct Correlation Function 91
3.5 Exact Functionals and Approximations 92
4 Interaction Site Formulism 95
4.1 Variational Principle 95
4.2 Reference Systems in Site Formalism 95
4.3 Direct Site Correlation Functions 98
4.4 Reference Interaction Site Model 99
4.5 Site Density Profiles 101
4.6 Thermodynamic Potentials 102
5 The Bridge Functional and Universality Ansatz 103
6 Perspectives 105
References 106
Classical Density Functional Theory of Polymer Fluids 110
1 Introduction 110
2 Classical Density Functional for Polymers 111
2.1 End Segment Distributions 115
2.2 Estimating the Excess Free Energy: Accounting for Steric Interactions 115
2.3 The Lennard-Jones Chain Model 118
2.4 Solving the Density Functional Equations 119
3 Density Functional Theory for Polydisperse Semi-flexible Polymers 119
3.1 Application to Semi-flexible Polymer Solution Films 124
3.2 Interaction Free Energy Between Non-adsorbing Surfaces 125
3.3 Interaction Free Energy Between Adsorbing Surfaces 127
3.4 Approaching the Rod-Like Limit 130
4 Other Polymeric Architectures 133
4.1 Using the PDFT for Room Temperature Ionic Liquids 133
5 A Classical Density Functional Theory for RTILs 135
5.1 Dealing with Electrostatic Correlations 136
5.2 Numerical Solution via End Segment Distribution 138
5.3 RTILs at Planar Electrodes: Comparison of the DFT with Simulations 141
6 Conclusion 143
References 143
Variational Perturbation Theory for Electrolyte Solutions 146
1 Introduction 146
2 Development of the Field Theory 147
2.1 Electrostatics 147
2.2 Partition Function 149
2.3 Dispersion Interactions 150
2.4 Mean-Field Approximation 151
3 Variational Perturbation Approximation 153
4 Point Charge Model 155
4.1 Mean-Field Approximation 156
4.2 Debye-Hückel Theory 156
4.3 Stability of the Point Charge Model 158
4.4 Dielectric Interfaces 160
5 Conclusions 162
References 163
Self-consistent Field Theory of Inhomogeneous Polymeric Systems 164
1 Introduction 164
2 Formulation of SCFT 165
2.1 Molecular Model of A/B Homopolymers 166
2.2 Field-Theoretical Formulation of the Partition Function 167
2.3 The Single-Chain Partition Function and the Propagators 170
2.4 Functional Derivatives of the Propagators 171
3 Canonical Ensemble and Helmholtz Free Energy Functional 173
3.1 SCFT in Canonical Ensemble 173
3.2 Homogeneous Phase 176
4 Grand-Canonical Ensemble and Grand Potential Functional 177
4.1 SCFT in Grand-Canonical Ensemble 177
4.2 Homogeneous Phase 179
5 Summary of SCFT 180
6 Ginzburg-Landau Free Energy Functional of Polymer Blends 182
7 Conclusion and Discussions 186
References 188
Variational Methods for Biomolecular Modeling 190
1 Introduction 190
2 Variational Multiscale Methods for Biomolecular Electrostatics and Solvation 193
2.1 Polar Solvation Free Energy 195
2.2 Nonpolar Solvation Free Energy 196
2.3 Governing Equations 198
2.4 Computational Simulations and Summary 199
3 Variational Methods for Pattern Formation in Bilayer Membranes 203
3.1 Classical Phase Field Models 204
3.2 Geodesic Curvature Based Membrane Models 205
3.3 Computational Simulations and Summary 211
4 Variational Methods for Curvature Induced Protein Localization in Bilayer Membranes 215
4.1 Lagrangian Formulation 216
4.2 Eulerian Formulation 218
4.3 Computational Simulations and Summary 220
5 Conclusions 223
References 224
A Theoretician's Approach to Nematic Liquid Crystals and Their Applications 231
1 Introduction 231
2 Continuum Theories for Nematic Liquid Crystals 234
2.1 The Landau-de Gennes Theory 236
2.2 The Ericksen Theory 241
2.3 The Oseen-Frank Theory 241
3 The Planar Bistable Device 243
3.1 A Two-Dimensional Oseen-Frank Model 244
3.2 A Landau-de Gennes Approach 252
4 Conclusions 259
References 260
Dynamical Density Functional Theory for Brownian Dynamics of Colloidal Particles 263
1 Introduction 263
2 Density Functional Theory (DFT) in Equilibrium 264
2.1 Basics 264
2.2 DFT of Freezing 265
2.3 Approximations for the Density Functional 266
3 Brownian Dynamics (BD) 269
3.1 Noninteracting Brownian Particles 269
3.2 Interacting Brownian Particles 271
4 Dynamical Density Functional Theory (DDFT) 272
5 An Example: Crystal Growth on Patterned Substrates 276
6 Hydrodynamic Interactions 277
7 Rod-Like Particles 280
7.1 Statistical Mechanics of Rod-Like Particles 280
7.2 Density Functional Theory 282
7.3 Brownian Dynamics of Rod-Like Particles and DDFT 283
8 Recent Developments 285
8.1 Derivation of the Phase Field Crystal (PFC) Model from DDFT 285
8.2 More Recent Applications of DDFT 285
8.3 DDFT for Active Brownian Particles 286
8.4 Modern Derivation of DDFT Using Projection Operator Techniques and Extended DDFT (EDDFT) 287
9 Conclusions 289
References 289
Introduction to the Variational Monte Carlo Method in Quantum Chemistry and Physics 293
1 Overview of Quantum Monte Carlo Methods 293
2 Variational Monte Carlo: A Basic Introduction 295
2.1 The Variational Principle 295
2.2 The Metropolis Monte Carlo Method 296
2.3 Putting the Two Together: Variational Monte Carlo 300
3 VMC Wave Functions 302
4 Wave Function Optimization 303
4.1 Choosing the Cost Function 304
4.2 Minimizing the Variational Energy 304
5 A Selection of State-of-the-Art VMC Algorithms 306
5.1 The Linear Method 306
5.2 Stochastic Reconfiguration 310
6 Applications of Variational Monte Carlo Methods in Physics and Chemistry 312
6.1 Quantum Chemistry 312
6.2 Excited States 313
6.3 The Hubbard Model 315
7 Summary and Outlook 316
References 316
Appendix: Calculus of Variations 322
A.1 Functional 322
A.2 Variational Problem 323
A.3 Functional Derivative 325
A.4 Chain Rules of Functional Derivative 326
A.5 Higher-Order Functional Derivatives and Functional Taylor Expansion 327
A.6 Functional Integration 328
A.7 Functional of a Multidimensional Function 328
A.8 An Illustrative Example 329