Wu Variational Methods in Molecular Modeling

1. Auflage 2017
ISBN: 978-981-10-2502-0
Verlag: Springer Nature Singapore
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 331 Seiten

Reihe: Molecular Modeling and Simulation

This book presents tutorial overviews for many applications of variational methods to molecular modeling. Topics discussed include the Gibbs-Bogoliubov-Feynman variational principle, square-gradient models, classical density functional theories, self-consistent-field theories, phase-field methods, Ginzburg-Landau and Helfrich-type phenomenological models, dynamical density functional theory, and variational Monte Carlo methods. Illustrative examples are given to facilitate understanding of the basic concepts and quantitative prediction of the properties and rich behavior of diverse many-body systems ranging from inhomogeneous fluids, electrolytes and ionic liquids in micropores, colloidal dispersions, liquid crystals, polymer blends, lipid membranes, microemulsions, magnetic materials and high-temperature superconductors. 
All chapters are written by leading experts in the field and illustrated with tutorial examples for their practical applications to specific subjects. With emphasis placed on physical understanding rather than on rigorous mathematical derivations, the content is accessible to graduate students and researchers in the broad areas of materials science and engineering, chemistry, chemical and biomolecular engineering, applied mathematics, condensed-matter physics, without specific training in theoretical physics or calculus of variations.

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.

Autoren/Hrsg.

Wu, Jianzhong

Inhaltsverzeichnis

1;Series Editor’s Preface;6
2;Preface;7
3;Contents;9
4;Editor and Contributors;10
5;Variational Methods in Statistical Thermodynamics---A Pedagogical Introduction;12
5.1;1 Introduction;12
5.2;2 The Variational Nature of Thermodynamics;13
5.3;3 The Variational Origin of Statistical Thermodynamics;16
5.4;4 The Method of Steepest Descent;18
5.5;5 The Gibbs-Bogoliubov-Feynman Variational Principle;21
5.6;6 A Toy Example;22
5.7;7 Mean-Field Solution for the Interacting Ising Model;26
5.8;8 The Poisson-Boltzmann Equation;30
5.9;9 Fluctuations;35
5.10;10 Summary and Perspective;39
5.11;References;40
6;Square-Gradient Model for Inhomogeneous Systems: From Simple Fluids to Microemulsions, Polymer Blends and Electronic Structure;41
6.1;1 Introduction;41
6.2;2 Statistical Mechanics;42
6.3;3 Density Functional Theory (DFT);44
6.4;4 Square-Gradient Approximation (SGA);47
6.5;5 Simple Fluids;49
6.6;6 Microemulsions;54
6.7;7 Polymer Blends;57
6.8;8 Electronic Systems;60
6.9;9 Summary;65
6.10;References;73
7;Classical Density Functional Theory for Molecular Systems;75
7.1;1 Molecular Models and Force Fields;75
7.2;2 Statistical Mechanics for Polyatomic Systems;79
7.2.1;2.1 Molecular Configuration and Interaction Sites;79
7.2.2;2.2 Grand Partition Function;81
7.2.3;2.3 Molecular Density and Molecular Correlation Functions;82
7.2.4;2.4 Site Density and Site Correlation Functions;84
7.2.5;2.5 Classical Models for Polyatomic Ideal-gas Systems;85
7.3;3 Density Functional Theory;87
7.3.1;3.1 Hohenberg-Kohn-Mermin Theorem;88
7.3.2;3.2 DFT for Ideal-Gas Systems;89
7.3.3;3.3 Excess Helmholtz Energy;90
7.3.4;3.4 Direct Correlation Function;91
7.3.5;3.5 Exact Functionals and Approximations;92
7.4;4 Interaction Site Formulism;95
7.4.1;4.1 Variational Principle;95
7.4.2;4.2 Reference Systems in Site Formalism;95
7.4.3;4.3 Direct Site Correlation Functions;98
7.4.4;4.4 Reference Interaction Site Model;99
7.4.5;4.5 Site Density Profiles;101
7.4.6;4.6 Thermodynamic Potentials;102
7.5;5 The Bridge Functional and Universality Ansatz;103
7.6;6 Perspectives;105
7.7;References;106
8;Classical Density Functional Theory of Polymer Fluids;110
8.1;1 Introduction;110
8.2;2 Classical Density Functional for Polymers;111
8.2.1;2.1 End Segment Distributions;115
8.2.2;2.2 Estimating the Excess Free Energy: Accounting for Steric Interactions;115
8.2.3;2.3 The Lennard-Jones Chain Model;118
8.2.4;2.4 Solving the Density Functional Equations;119
8.3;3 Density Functional Theory for Polydisperse Semi-flexible Polymers;119
8.3.1;3.1 Application to Semi-flexible Polymer Solution Films;124
8.3.2;3.2 Interaction Free Energy Between Non-adsorbing Surfaces;125
8.3.3;3.3 Interaction Free Energy Between Adsorbing Surfaces;127
8.3.4;3.4 Approaching the Rod-Like Limit;130
8.4;4 Other Polymeric Architectures;133
8.4.1;4.1 Using the PDFT for Room Temperature Ionic Liquids;133
8.5;5 A Classical Density Functional Theory for RTILs;135
8.5.1;5.1 Dealing with Electrostatic Correlations;136
8.5.2;5.2 Numerical Solution via End Segment Distribution;138
8.5.3;5.3 RTILs at Planar Electrodes: Comparison of the DFT with Simulations;141
8.6;6 Conclusion;143
8.7;References;143
9;Variational Perturbation Theory for Electrolyte Solutions;146
9.1;1 Introduction;146
9.2;2 Development of the Field Theory;147
9.2.1;2.1 Electrostatics;147
9.2.2;2.2 Partition Function;149
9.2.3;2.3 Dispersion Interactions;150
9.2.4;2.4 Mean-Field Approximation;151
9.3;3 Variational Perturbation Approximation;153
9.4;4 Point Charge Model;155
9.4.1;4.1 Mean-Field Approximation;156
9.4.2;4.2 Debye-Hückel Theory;156
9.4.3;4.3 Stability of the Point Charge Model;158
9.4.4;4.4 Dielectric Interfaces;160
9.5;5 Conclusions;162
9.6;References;163
10;Self-consistent Field Theory of Inhomogeneous Polymeric Systems;164
10.1;1 Introduction;164
10.2;2 Formulation of SCFT;165
10.2.1;2.1 Molecular Model of A/B Homopolymers;166
10.2.2;2.2 Field-Theoretical Formulation of the Partition Function;167
10.2.3;2.3 The Single-Chain Partition Function and the Propagators;170
10.2.4;2.4 Functional Derivatives of the Propagators;171
10.3;3 Canonical Ensemble and Helmholtz Free Energy Functional;173
10.3.1;3.1 SCFT in Canonical Ensemble;173
10.3.2;3.2 Homogeneous Phase;176
10.4;4 Grand-Canonical Ensemble and Grand Potential Functional;177
10.4.1;4.1 SCFT in Grand-Canonical Ensemble;177
10.4.2;4.2 Homogeneous Phase;179
10.5;5 Summary of SCFT;180
10.6;6 Ginzburg-Landau Free Energy Functional of Polymer Blends;182
10.7;7 Conclusion and Discussions;186
10.8;References;188
11;Variational Methods for Biomolecular Modeling;190
11.1;1 Introduction;190
11.2;2 Variational Multiscale Methods for Biomolecular Electrostatics and Solvation;193
11.2.1;2.1 Polar Solvation Free Energy;195
11.2.2;2.2 Nonpolar Solvation Free Energy;196
11.2.3;2.3 Governing Equations;198
11.2.4;2.4 Computational Simulations and Summary;199
11.3;3 Variational Methods for Pattern Formation in Bilayer Membranes;203
11.3.1;3.1 Classical Phase Field Models;204
11.3.2;3.2 Geodesic Curvature Based Membrane Models;205
11.3.3;3.3 Computational Simulations and Summary;211
11.4;4 Variational Methods for Curvature Induced Protein Localization in Bilayer Membranes;215
11.4.1;4.1 Lagrangian Formulation;216
11.4.2;4.2 Eulerian Formulation;218
11.4.3;4.3 Computational Simulations and Summary;220
11.5;5 Conclusions;223
11.6;References;224
12;A Theoretician's Approach to Nematic Liquid Crystals and Their Applications;231
12.1;1 Introduction;231
12.2;2 Continuum Theories for Nematic Liquid Crystals;234
12.2.1;2.1 The Landau-de Gennes Theory;236
12.2.2;2.2 The Ericksen Theory;241
12.2.3;2.3 The Oseen-Frank Theory;241
12.3;3 The Planar Bistable Device;243
12.3.1;3.1 A Two-Dimensional Oseen-Frank Model;244
12.3.2;3.2 A Landau-de Gennes Approach;252
12.4;4 Conclusions;259
12.5;References;260
13;Dynamical Density Functional Theory for Brownian Dynamics of Colloidal Particles;263
13.1;1 Introduction;263
13.2;2 Density Functional Theory (DFT) in Equilibrium;264
13.2.1;2.1 Basics;264
13.2.2;2.2 DFT of Freezing;265
13.2.3;2.3 Approximations for the Density Functional;266
13.3;3 Brownian Dynamics (BD);269
13.3.1;3.1 Noninteracting Brownian Particles;269
13.3.2;3.2 Interacting Brownian Particles;271
13.4;4 Dynamical Density Functional Theory (DDFT);272
13.5;5 An Example: Crystal Growth on Patterned Substrates;276
13.6;6 Hydrodynamic Interactions;277
13.7;7 Rod-Like Particles;280
13.7.1;7.1 Statistical Mechanics of Rod-Like Particles;280
13.7.2;7.2 Density Functional Theory;282
13.7.3;7.3 Brownian Dynamics of Rod-Like Particles and DDFT;283
13.8;8 Recent Developments;285
13.8.1;8.1 Derivation of the Phase Field Crystal (PFC) Model from DDFT;285
13.8.2;8.2 More Recent Applications of DDFT;285
13.8.3;8.3 DDFT for Active Brownian Particles;286
13.8.4;8.4 Modern Derivation of DDFT Using Projection Operator Techniques and Extended DDFT (EDDFT);287
13.9;9 Conclusions;289
13.10;References;289
14;Introduction to the Variational Monte Carlo Method in Quantum Chemistry and Physics;293
14.1;1 Overview of Quantum Monte Carlo Methods;293
14.2;2 Variational Monte Carlo: A Basic Introduction;295
14.2.1;2.1 The Variational Principle;295
14.2.2;2.2 The Metropolis Monte Carlo Method;296
14.2.3;2.3 Putting the Two Together: Variational Monte Carlo;300
14.3;3 VMC Wave Functions;302
14.4;4 Wave Function Optimization;303
14.4.1;4.1 Choosing the Cost Function;304
14.4.2;4.2 Minimizing the Variational Energy;304
14.5;5 A Selection of State-of-the-Art VMC Algorithms;306
14.5.1;5.1 The Linear Method;306
14.5.2;5.2 Stochastic Reconfiguration;310
14.6;6 Applications of Variational Monte Carlo Methods in Physics and Chemistry;312
14.6.1;6.1 Quantum Chemistry;312
14.6.2;6.2 Excited States;313
14.6.3;6.3 The Hubbard Model;315
14.7;7 Summary and Outlook;316
14.8;References;316
15;Appendix: Calculus of Variations;322
15.1;A.1 Functional;322
15.2;A.2 Variational Problem;323
15.3;A.3 Functional Derivative;325
15.4;A.4 Chain Rules of Functional Derivative;326
15.5;A.5 Higher-Order Functional Derivatives and Functional Taylor Expansion;327
15.6;A.6 Functional Integration;328
15.7;A.7 Functional of a Multidimensional Function;328
15.8;A.8 An Illustrative Example;329