DENSITY FUNCTIONAL THEORY A PRACTICAL INTRODUCTION David S.Sholl lanice A.Steckel WILEY
DENSITY FUNCTIONAL THEORY A Practical Introduction DAVID S.SHOLL Georgia Institute of Technology JANICE A.STECKEL National Energy Technology Laboratory WILEY A JOHN WILEY SONS,INC.,PUBLICATION
DENSITY FUNCTIONAL THEORY A Practical Introduction DAVID S. SHOLL Georgia Institute of Technology JANICE A. STECKEL National Energy Technology Laboratory
CONTENTS Preface xi 1 What Is Density Functional Theory? 1 1.1 How to Approach This Book,1 1.2 Examples of DFT in Action,2 1.2.1 Ammonia Synthesis by Heterogeneous Catalysis,2 1.2.2 Embrittlement of Metals by Trace Impurities,4 1.2.3 Materials Properties for Modeling Planetary Formation,6 1.3 The Schrodinger Equation,7 1.4 Density Functional Theory-From Wave Functions to Electron Density,10 1.5 Exchange-Correlation Functional,14 1.6 The Quantum Chemistry Tourist,16 1.6.1 Localized and Spatially Extended Functions,16 1.6.2 Wave-Function-Based Methods,18 1.6.3 Hartree-Fock Method,19 1.6.4 Beyond Hartree-Fock,23 1.7 What Can DFT Not Do?,28 1.8 Density Functional Theory in Other Fields,30 1.9 How to Approach This Book (Revisited),30 References,31 Further Reading,32
CONTENTS Preface xi 1 What Is Density Functional Theory? 1 1.1 How to Approach This Book, 1 1.2 Examples of DFT in Action, 2 1.2.1 Ammonia Synthesis by Heterogeneous Catalysis, 2 1.2.2 Embrittlement of Metals by Trace Impurities, 4 1.2.3 Materials Properties for Modeling Planetary Formation, 6 1.3 The Schro¨dinger Equation, 7 1.4 Density Functional Theory—From Wave Functions to Electron Density, 10 1.5 Exchange –Correlation Functional, 14 1.6 The Quantum Chemistry Tourist, 16 1.6.1 Localized and Spatially Extended Functions, 16 1.6.2 Wave-Function-Based Methods, 18 1.6.3 Hartree –Fock Method, 19 1.6.4 Beyond Hartree –Fock, 23 1.7 What Can DFT Not Do?, 28 1.8 Density Functional Theory in Other Fields, 30 1.9 How to Approach This Book (Revisited), 30 References, 31 Further Reading, 32 v
i CONTENTS 2 DFT Calculations for Simple Solids 35 2.1 Periodic Structures,Supercells,and Lattice Parameters,35 2.2 Face-Centered Cubic Materials,39 2.3 Hexagonal Close-Packed Materials,41 2.4 Crystal Structure Prediction,43 2.5 Phase Transformations,44 Exercises,46 Further Reading,47 Appendix Calculation Details,47 3 Nuts and Bolts of DFT Calculations 49 3.1 Reciprocal Space and k Points,50 3.1.1 Plane Waves and the Brillouin Zone,50 3.1.2 Integrals in k Space,53 3.1.3 Choosing k Points in the Brillouin Zone,55 3.1.4 Metals-Special Cases in k Space,59 3.1.5 Summary of k Space,60 3.2 Energy Cutoffs,61 3.2.1 Pseudopotentials,63 3.3 Numerical Optimization,65 3.3.1 Optimization in One Dimension,65 3.3.2 Optimization in More than One Dimension,69 3.3.3 What Do I Really Need to Know about Optimization?,73 3.4 DFT Total Energies-An Iterative Optimization Problem,73 3.5 Geometry Optimization,75 3.5.1 Internal Degrees of Freedom,75 3.5.2 Geometry Optimization with Constrained Atoms,78 3.5.3 Optimizing Supercell Volume and Shape,78 Exercises,79 References,80 Further Reading,80 Appendix Calculation Details,81 4 DFT Calculations for Surfaces of Solids 83 4.1 Importance of Surfaces,83 4.2 Periodic Boundary Conditions and Slab Models,84 4.3 Choosing k Points for Surface Calculations,87 4.4 Classification of Surfaces by Miller Indices,88 4.5 Surface Relaxation,94 4.6 Calculation of Surface Energies,96
2 DFT Calculations for Simple Solids 35 2.1 Periodic Structures, Supercells, and Lattice Parameters, 35 2.2 Face-Centered Cubic Materials, 39 2.3 Hexagonal Close-Packed Materials, 41 2.4 Crystal Structure Prediction, 43 2.5 Phase Transformations, 44 Exercises, 46 Further Reading, 47 Appendix Calculation Details, 47 3 Nuts and Bolts of DFT Calculations 49 3.1 Reciprocal Space and k Points, 50 3.1.1 Plane Waves and the Brillouin Zone, 50 3.1.2 Integrals in k Space, 53 3.1.3 Choosing k Points in the Brillouin Zone, 55 3.1.4 Metals—Special Cases in k Space, 59 3.1.5 Summary of k Space, 60 3.2 Energy Cutoffs, 61 3.2.1 Pseudopotentials, 63 3.3 Numerical Optimization, 65 3.3.1 Optimization in One Dimension, 65 3.3.2 Optimization in More than One Dimension, 69 3.3.3 What Do I Really Need to Know about Optimization?, 73 3.4 DFT Total Energies—An Iterative Optimization Problem, 73 3.5 Geometry Optimization, 75 3.5.1 Internal Degrees of Freedom, 75 3.5.2 Geometry Optimization with Constrained Atoms, 78 3.5.3 Optimizing Supercell Volume and Shape, 78 Exercises, 79 References, 80 Further Reading, 80 Appendix Calculation Details, 81 4 DFT Calculations for Surfaces of Solids 83 4.1 Importance of Surfaces, 83 4.2 Periodic Boundary Conditions and Slab Models, 84 4.3 Choosing k Points for Surface Calculations, 87 4.4 Classification of Surfaces by Miller Indices, 88 4.5 Surface Relaxation, 94 4.6 Calculation of Surface Energies, 96 vi CONTENTS
CONTENTS vii 4.7 Symmetric and Asymmetric Slab Models,98 4.8 Surface Reconstruction,100 4.9 Adsorbates on Surfaces,103 4.9.1 Accuracy of Adsorption Energies,106 4.10 Effects of Surface Coverage,107 Exercises,110 References,111 Further Reading,111 Appendix Calculation Details,112 5 DFT Calculations of Vibrational Frequencies 113 5.1 Isolated Molecules,114 5.2 Vibrations of a Collection of Atoms,117 5.3 Molecules on Surfaces,120 5.4 Zero-Point Energies,122 5.5 Phonons and Delocalized Modes,127 Exercises,128 Reference,128 Further Reading,128 Appendix Calculation Details,129 6 Calculating Rates of Chemical Processes Using Transition State Theory 131 6.1 One-Dimensional Example,132 6.2 Multidimensional Transition State Theory,139 6.3 Finding Transition States,142 6.3.1 Elastic Band Method,144 6.3.2 Nudged Elastic Band Method,145 6.3.3 Initializing NEB Calculations,147 6.4 Finding the Right Transition States,150 6.5 Connecting Individual Rates to Overall Dynamics,153 6.6 Quantum Effects and Other Complications,156 6.6.1 High Temperatures/Low Barriers,156 6.6.2 Quantum Tunneling,157 6.6.3 Zero-Point Energies,157 Exercises,158 Reference,159 Further Reading,159 Appendix Calculation Details,160
4.7 Symmetric and Asymmetric Slab Models, 98 4.8 Surface Reconstruction, 100 4.9 Adsorbates on Surfaces, 103 4.9.1 Accuracy of Adsorption Energies, 106 4.10 Effects of Surface Coverage, 107 Exercises, 110 References, 111 Further Reading, 111 Appendix Calculation Details, 112 5 DFT Calculations of Vibrational Frequencies 113 5.1 Isolated Molecules, 114 5.2 Vibrations of a Collection of Atoms, 117 5.3 Molecules on Surfaces, 120 5.4 Zero-Point Energies, 122 5.5 Phonons and Delocalized Modes, 127 Exercises, 128 Reference, 128 Further Reading, 128 Appendix Calculation Details, 129 6 Calculating Rates of Chemical Processes Using Transition State Theory 131 6.1 One-Dimensional Example, 132 6.2 Multidimensional Transition State Theory, 139 6.3 Finding Transition States, 142 6.3.1 Elastic Band Method, 144 6.3.2 Nudged Elastic Band Method, 145 6.3.3 Initializing NEB Calculations, 147 6.4 Finding the Right Transition States, 150 6.5 Connecting Individual Rates to Overall Dynamics, 153 6.6 Quantum Effects and Other Complications, 156 6.6.1 High Temperatures/Low Barriers, 156 6.6.2 Quantum Tunneling, 157 6.6.3 Zero-Point Energies, 157 Exercises, 158 Reference, 159 Further Reading, 159 Appendix Calculation Details, 160 CONTENTS vii
viii CONTENTS 7 Equilibrium Phase Diagrams from Ab Initio Thermodynamics 163 7.1 Stability of Bulk Metal Oxides,164 7.1.1 Examples Including Disorder-Configurational Entropy,169 7.2 Stability of Metal and Metal Oxide Surfaces,172 7.3 Multiple Chemical Potentials and Coupled Chemical Reactions,174 Exercises,175 References,176 Further Reading,176 Appendix Calculation Details,177 8 Electronic Structure and Magnetic Properties 179 8.1 Electronic Density of States,179 8.2 Local Density of States and Atomic Charges,186 8.3 Magnetism,188 Exercises,190 Further Reading,191 Appendix Calculation Details,192 9 Ab Initio Molecular Dynamics 193 9.1 Classical Molecular Dynamics,193 9.1.1 Molecular Dynamics with Constant Energy,193 9.1.2 Molecular Dynamics in the Canonical Ensemble,196 9.1.3 Practical Aspects of Classical Molecular Dynamics,197 9.2 Ab Initio Molecular Dynamics,198 9.3 Applications of Ab Initio Molecular Dynamics,201 9.3.1 Exploring Structurally Complex Materials: Liquids and Amorphous Phases,201 9.3.2 Exploring Complex Energy Surfaces,204 Exercises,207 Reference,207 Further Reading,207 Appendix Calculation Details,208
7 Equilibrium Phase Diagrams from Ab Initio Thermodynamics 163 7.1 Stability of Bulk Metal Oxides, 164 7.1.1 Examples Including Disorder—Configurational Entropy, 169 7.2 Stability of Metal and Metal Oxide Surfaces, 172 7.3 Multiple Chemical Potentials and Coupled Chemical Reactions, 174 Exercises, 175 References, 176 Further Reading, 176 Appendix Calculation Details, 177 8 Electronic Structure and Magnetic Properties 179 8.1 Electronic Density of States, 179 8.2 Local Density of States and Atomic Charges, 186 8.3 Magnetism, 188 Exercises, 190 Further Reading, 191 Appendix Calculation Details, 192 9 Ab Initio Molecular Dynamics 193 9.1 Classical Molecular Dynamics, 193 9.1.1 Molecular Dynamics with Constant Energy, 193 9.1.2 Molecular Dynamics in the Canonical Ensemble, 196 9.1.3 Practical Aspects of Classical Molecular Dynamics, 197 9.2 Ab Initio Molecular Dynamics, 198 9.3 Applications of Ab Initio Molecular Dynamics, 201 9.3.1 Exploring Structurally Complex Materials: Liquids and Amorphous Phases, 201 9.3.2 Exploring Complex Energy Surfaces, 204 Exercises, 207 Reference, 207 Further Reading, 207 Appendix Calculation Details, 208 viii CONTENTS
CONTENTS 议 10 Accuracy and Methods beyond "Standard"Calculations 209 10.1 How Accurate Are DFT Calculations?,209 10.2 Choosing a Functional,215 10.3 Examples of Physical Accuracy,220 10.3.1 Benchmark Calculations for Molecular Systems-Energy and Geometry,220 10.3.2 Benchmark Calculations for Molecular Systems-Vibrational Frequencies,221 10.3.3 Crystal Structures and Cohesive Energies,222 10.3.4 Adsorption Energies and Bond Strengths,223 10.4 DFT+X Methods for Improved Treatment of Electron Correlation,224 10.4.1 Dispersion Interactions and DFT-D,225 10.4.2 Self-Interaction Error,Strongly Correlated Electron Systems,and DFT+U,227 10.5 Larger System Sizes with Linear Scaling Methods and Classical Force Fields,229 10.6 Conclusion,230 References,231 Further Reading,232 Index 235
10 Accuracy and Methods beyond “Standard” Calculations 209 10.1 How Accurate Are DFT Calculations?, 209 10.2 Choosing a Functional, 215 10.3 Examples of Physical Accuracy, 220 10.3.1 Benchmark Calculations for Molecular Systems—Energy and Geometry, 220 10.3.2 Benchmark Calculations for Molecular Systems—Vibrational Frequencies, 221 10.3.3 Crystal Structures and Cohesive Energies, 222 10.3.4 Adsorption Energies and Bond Strengths, 223 10.4 DFTþX Methods for Improved Treatment of Electron Correlation, 224 10.4.1 Dispersion Interactions and DFT-D, 225 10.4.2 Self-Interaction Error, Strongly Correlated Electron Systems, and DFTþU, 227 10.5 Larger System Sizes with Linear Scaling Methods and Classical Force Fields, 229 10.6 Conclusion, 230 References, 231 Further Reading, 232 Index 235 CONTENTS ix
PREFACE The application of density functional theory (DFT)calculations is rapidly becoming a "standard tool"for diverse materials modeling problems in physics,chemistry,materials science,and multiple branches of engineering. Although a number of highly detailed books and articles on the theoretical foundations of DFT are available,it remains difficult for a newcomer to these methods to rapidly learn the tools that allow him or her to actually perform calculations that are now routine in the fields listed above.This book aims to fill this gap by guiding the reader through the applications of DFT that might be considered the core of continually growing scientific litera- ture based on these methods.Each chapter includes a series of exercises to give readers experience with calculations of their own. We have aimed to find a balance between brevity and detail that makes it possible for readers to realistically plan to read the entire text.This balance inevitably means certain technical details are explored in a limited way.Our choices have been strongly influenced by our interactions over multiple years with graduate students and postdocs in chemical engineering,physics, chemistry,materials science,and mechanical engineering at Carnegie Mellon University and the Georgia Institute of Technology.A list of Further Reading is provided in each chapter to define appropriate entry points to more detailed treatments of the area.These reading lists should be viewed as identifying highlights in the literature,not as an effort to rigorously cite all relevant work from the thousands of studies that exist on these topics. 扩
PREFACE The application of density functional theory (DFT) calculations is rapidly becoming a “standard tool” for diverse materials modeling problems in physics, chemistry, materials science, and multiple branches of engineering. Although a number of highly detailed books and articles on the theoretical foundations of DFT are available, it remains difficult for a newcomer to these methods to rapidly learn the tools that allow him or her to actually perform calculations that are now routine in the fields listed above. This book aims to fill this gap by guiding the reader through the applications of DFT that might be considered the core of continually growing scientific literature based on these methods. Each chapter includes a series of exercises to give readers experience with calculations of their own. We have aimed to find a balance between brevity and detail that makes it possible for readers to realistically plan to read the entire text. This balance inevitably means certain technical details are explored in a limited way. Our choices have been strongly influenced by our interactions over multiple years with graduate students and postdocs in chemical engineering, physics, chemistry, materials science, and mechanical engineering at Carnegie Mellon University and the Georgia Institute of Technology. A list of Further Reading is provided in each chapter to define appropriate entry points to more detailed treatments of the area. These reading lists should be viewed as identifying highlights in the literature, not as an effort to rigorously cite all relevant work from the thousands of studies that exist on these topics. xi
xii PREFACE One important choice we made to limit the scope of the book was to focus solely on one DFT method suitable for solids and spatially extended materials, namely plane-wave DFT.Although many of the foundations of plane-wave DFT are also relevant to complementary approaches used in the chemistry community for isolated molecules,there are enough differences in the appli- cations of these two groups of methods that including both approaches would only have been possible by significantly expanding the scope of the book.Moreover,several resources already exist that give a practical "hands- on"introduction to computational chemistry calculations for molecules. Our use of DFT calculations in our own research and our writing of this book has benefited greatly from interactions with numerous colleagues over an extended period.We especially want to acknowledge J.Karl Johnson (University of Pittsburgh),Aravind Asthagiri (University of Florida),Dan Sorescu (National Energy Technology Laboratory),Cathy Stampfl(University of Sydney),John Kitchin (Carnegie Mellon University), and Duane Johnson (University of Illinois).We thank Jeong-Woo Han for his help with a number of the figures.Bill Schneider(University of Notre Dame),Ken Jordan (University of Pittsburgh),and Taku Watanabe (Georgia Institute of Technology)gave detailed and helpful feedback on draft versions.Any errors or inaccuracies in the text are,of course,our responsibility alone. DSS dedicates this book to his father and father-in-law,whose love of science and curiosity about the world are an inspiration.JAS dedicates this book to her husband,son,and daughter. DAVID SHOLL Georgia Institute of Technology. Atlanta,GA,USA JAN STECKEL National Energy Technology Laboratory, Pittsburgh,PA,USA
One important choice we made to limit the scope of the book was to focus solely on one DFT method suitable for solids and spatially extended materials, namely plane-wave DFT. Although many of the foundations of plane-wave DFT are also relevant to complementary approaches used in the chemistry community for isolated molecules, there are enough differences in the applications of these two groups of methods that including both approaches would only have been possible by significantly expanding the scope of the book. Moreover, several resources already exist that give a practical “handson” introduction to computational chemistry calculations for molecules. Our use of DFT calculations in our own research and our writing of this book has benefited greatly from interactions with numerous colleagues over an extended period. We especially want to acknowledge J. Karl Johnson (University of Pittsburgh), Aravind Asthagiri (University of Florida), Dan Sorescu (National Energy Technology Laboratory), Cathy Stampfl (University of Sydney), John Kitchin (Carnegie Mellon University), and Duane Johnson (University of Illinois). We thank Jeong-Woo Han for his help with a number of the figures. Bill Schneider (University of Notre Dame), Ken Jordan (University of Pittsburgh), and Taku Watanabe (Georgia Institute of Technology) gave detailed and helpful feedback on draft versions. Any errors or inaccuracies in the text are, of course, our responsibility alone. DSS dedicates this book to his father and father-in-law, whose love of science and curiosity about the world are an inspiration. JAS dedicates this book to her husband, son, and daughter. DAVID SHOLL Georgia Institute of Technology, Atlanta, GA, USA JAN STECKEL National Energy Technology Laboratory, Pittsburgh, PA, USA xii PREFACE
WHAT IS DENSITY FUNCTIONAL THEORY? 1.1 HOW TO APPROACH THIS BOOK There are many fields within the physical sciences and engineering where the key to scientific and technological progress is understanding and controlling the properties of matter at the level of individual atoms and molecules. Density functional theory is a phenomenally successful approach to finding solutions to the fundamental equation that describes the quantum behavior of atoms and molecules,the Schrodinger equation,in settings of practical value.This approach has rapidly grown from being a specialized art practiced by a small number of physicists and chemists at the cutting edge of quantum mechanical theory to a tool that is used regularly by large numbers of research- ers in chemistry,physics,materials science,chemical engineering,geology, and other disciplines.A search of the Science Citation Index for articles pub- lished in 1986 with the words"density functional theory"in the title or abstract yields less than 50 entries.Repeating this search for 1996 and 2006 gives more than 1100 and 5600 entries,respectively. Our aim with this book is to provide just what the title says:an introduction to using density functional theory (DFT)calculations in a practical context. We do not assume that you have done these calculations before or that you even understand what they are.We do assume that you want to find out what is possible with these methods,either so you can perform calculations Density Functional Theory:A Practical Introduction.By David S.Sholl and Janice A.Steckel Copyright C 2009 John Wiley Sons,Inc
1 WHAT IS DENSITY FUNCTIONAL THEORY? 1.1 HOW TO APPROACH THIS BOOK There are many fields within the physical sciences and engineering where the key to scientific and technological progress is understanding and controlling the properties of matter at the level of individual atoms and molecules. Density functional theory is a phenomenally successful approach to finding solutions to the fundamental equation that describes the quantum behavior of atoms and molecules, the Schro¨dinger equation, in settings of practical value. This approach has rapidly grown from being a specialized art practiced by a small number of physicists and chemists at the cutting edge of quantum mechanical theory to a tool that is used regularly by large numbers of researchers in chemistry, physics, materials science, chemical engineering, geology, and other disciplines. A search of the Science Citation Index for articles published in 1986 with the words “density functional theory” in the title or abstract yields less than 50 entries. Repeating this search for 1996 and 2006 gives more than 1100 and 5600 entries, respectively. Our aim with this book is to provide just what the title says: an introduction to using density functional theory (DFT) calculations in a practical context. We do not assume that you have done these calculations before or that you even understand what they are. We do assume that you want to find out what is possible with these methods, either so you can perform calculations Density Functional Theory: A Practical Introduction. By David S. Sholl and Janice A. Steckel Copyright # 2009 John Wiley & Sons, Inc. 1