Density Functional Theory A Practical Introduction

A concise and rigorous introduction to the applications of DFT calculations

In the newly revised second edition of Density Functional Theory: A Practical Introduction, the authors deliver a concise and easy-to-follow introduction to the key concepts and practical applications of density functional theory (DFT) with an emphasis on plane-wave DFT. The authors draw on decades of experience in the field, offering students from a variety of backgrounds a balanced approach between accessibility and rigor, creating a text that is highly digestible in its entirety.

This new edition:

• Discusses in more detail the accuracy of DFT calculations and the choice of functionals
• Adds an overview of the wide range of available DFT codes
• Contains more examples on the use of DFT for high throughput materials calculations
• Puts more emphasis on computing phase diagrams and on open ensemble methods widely used in electrochemistry
• Is significantly extended to cover calculation beyond standard DFT, e.g., dispersion-corrected DFT, DFT+U, time-dependent DFT

Perfect for graduate students and postdoctoral candidates in physics and engineering, Density Functional Theory: A Practical Introduction will also earn a place in the libraries of researchers and practitioners in chemistry, materials science, and mechanical engineering.

David S. Sholl is a Professor of Chemical and & Biomolecular Engineering at the Georgia Institute of Technology, where holds the Michael Tennenbaum Family Chair and remains a GRA Eminent Scholar in Energy Sustainability.

Janice A. Steckel is a Physical Scientist at the United States Department of Energy, National Energy Technology Laboratory in Pittsburgh, Pennsylvania.

1 What Is Density Functional Theory?

1.1 How to Approach This Book

1.2 Examples of DFT in Action

1.2.1 Ammonia Synthesis by Heterogeneous Catalysis

1.2.2 Embrittlement of Metals by Trace Impurities

1.2.3 Materials Properties for Modeling Planetary Formation

1.2.4 High Throughput/Big Data Case Study

1.3 The Schrödinger Equation

1.4 Density Functional Theory—From Wave Functions to Electron Density

1.5 Exchange– Correlation Functional

1.6 The Quantum Chemistry Tourist

1.6.1 Localized and Spatially Extended Functions

1.6.2 Wave-Function-Based Methods

1.6.3 Hartree– Fock Method

1.6.4 Beyond Hartree–Fock

1.7 What Can DFT Not Do?

1.8 Which DFT Code Should I Use?

1.9 Density Functional Theory in Other Fields

1.10 How to Approach This Book

2 DFT Calculations for Simple Solids

2.1 Periodic Structures, Supercells, and Lattice Parameters

2.2 Face-Centered Cubic Materials

2.3 Hexagonal Close-Packed Materials

2.4 Crystal Structure Prediction

2.5 Phase Transformations

Exercises

3 Nuts and Bolts of DFT Calculations

3.1 Reciprocal Space and k Points

3.1.1 Plane Waves and the Brillouin Zone

3.1.2 Integrals in k Space

3.1.3 Choosing k Points in the Brillouin Zone

3.1.4 Metals—Special Cases in k Space; DFT+U

3.1.5 Summary of k Space

3.2 Energy Cutoffs

3.2.1 Pseudopotentials

3.3 Numerical Optimization

3.3.1 Optimization in One Dimension

3.3.2 Optimization in More than One Dimension

3.3.3 What Do I Really Need to Know about Optimization?

3.4 DFT Total Energies—An Iterative Optimization Problem

3.5 Geometry Optimization

3.5.1 Internal Degrees of Freedom

3.5.2 Geometry Optimization with Constrained Atoms

3.5.3 Optimizing Supercell Volume and Shape

Appendix: Calculation Details

4 Thinking About Accuracy and Choosing Functionals for DFT Calculations

4.1 How Accurate Are DFT Calculations?

4.2 Choosing a Functional

4.3 Examples of Physical Accuracy

4.3.1 Benchmark Calculations for Molecular Systems—Energy and Geometry

4.3.2 Benchmark Calculations for Molecular Systems—Vibrational Frequencies

4.3.3 Crystal Structures and Cohesive Energies

4.3.4 Adsorption Energies and Bond Strengths

4.4 How to Use the Rest of this Book

5 DFT Calculations for Surfaces of Solids and Interfaces in Crystals

5.1 Importance of Surfaces

5.2 Periodic Boundary Conditions and Slab Models

5.3 Choosing k Points for Surface Calculations

5.4 Classification of Surfaces by Miller Indices

5.5 Surface Relaxation

5.6 Calculation of Surface Energies

5.7 Symmetric and Asymmetric Slab Models

5.8 Surface Reconstruction

5.9 Adsorbates on Surfaces

5.9.1 Accuracy of Adsorption Energies

5.10 Effects of Surface Coverage

5.11 Grain Boundaries in Solids

Exercises

Appendix: Calculation Details

6 DFT Calculations of Vibrational Frequencies

6.1 Isolated Molecules

6.2 Vibrations of a Collection of Atoms

6.3 Molecules on Surfaces

6.4 Zero-Point Energies

6.5 Phonons and Delocalized Modes

Exercises

7 Calculating Rates of Chemical Processes Using Transition State Theory

7.1 One-Dimensional Example

7.2 Multidimensional Transition State Theory

7.3 Finding Transition States

7.3.1 Elastic Band Method

7.3.2 Nudged Elastic Band Method and the Dimer Method

7.3.3 Initializing NEB Calculations

7.4 Finding the Right Transition States

7.5 Connecting Individual Rates to Overall Dynamics

7.6 Quantum Effects and Other Complications

7.6.1 High Temperatures/Low Barriers

7.6.2 Quantum Tunneling

7.6.3 Zero-Point Energies

Exercises

Appendix: Calculation Details

8 Equilibrium Phase Diagrams and Electrochemistry with Open Ensemble Methods

8.1 Stability of Bulk Metal Oxides

8.1.1 Examples Including Disorder—Configurational Entropy

8.2 Stability of Metal and Metal Oxide Surfaces

8.3 Multiple Chemical Potentials and Coupled Chemical Reactions

8.4 DFT for Electrochemistry

Exercises

Appendix: Calculation Details

9 Electronic Structure and Magnetic Properties

9.1 Electronic Density of States

9.2 Local Density of States and Atomic Charges

9.3 Magnetism

Exercises

10 Ab Initio Molecular Dynamics

10.1 Classical Molecular Dynamics

10.1.1 Molecular Dynamics with Constant Energy

10.1.2 Molecular Dynamics in the Canonical Ensemble

10.1.3 Practical Aspects of Classical Molecular Dynamics

10.2 Ab Initio Molecular Dynamics: Gaussian Basis Sets in Non-Plane Wave Codes

10.3 Applications of Ab Initio Molecular Dynamics

10.3.1 Exploring Structurally Complex Materials: Liquids and Amorphous Phases

10.3.2 Exploring Complex Energy Surfaces

10.4 Time-Dependent Density Functional Theory

Exercises

Appendix: Calculation Details

11 Methods beyond “Standard” Calculations

11.1 Choosing a Functional (Revisited)

11.2 Estimating Uncertainties in DFT Results Using the BEEF Approach

11.3 DFT+X Methods for Improved Treatment of Electron Correlation

11.3.1 Dispersion Interactions and DFT-D and D2, D3, TS methods

11.4 Self-Interaction Error, Strongly Correlated Electron Systems, and DFT+U

11.5 RPA

11.6 Larger System Sizes with Linear Scaling Methods and Classical Force Fields

11.7 Conclusion

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