1.6 THE QUANTUM CHEMISTRY TOURIST 27 scaling if you notice from Table 1.2 that a reasonable basis set for even a tiny molecule like CH4 includes hundreds of basis functions.The computational expense of high-level wave-function-based methods means that these calcu- lations are feasible for individual organic molecules containing 10-20 atoms,but physical systems larger than this fall into either the"very challen- ging”or“computationally infeasible”categories. This brings our brief tour of quantum chemistry almost to an end.As the title of this book suggests,we are going to focus throughout the book on den- sity functional theory calculations.Moreover,we will only consider methods based on spatially periodic functions-the so-called plane-wave methods. Plane-wave methods are the method of choice in almost all situations where the physical material of interest is an extended crystalline material rather than an isolated molecule.As we stated above,it is not appropriate to view methods based on periodic functions as"right"and methods based on spatially localized functions as"wrong"(or vice versa).In the long run,it will be a great advantage to you to understand both classes of methods since having access to a wide range of tools can only improve your chances of solving significant scientific problems.Nevertheless,if you are interested in applying compu- tational methods to materials other than isolated molecules,then plane-wave DFT is an excellent place to start. It is important for us to emphasize that DFT calculations can also be per- formed using spatially localized functions-the results in Tables 1.1 and 1.2 are examples of this kind of calculation.Perhaps the main difference between DFT calculations using periodic and spatially localized functions lies in the exchange-correlation functionals that are routinely used.In Section 1.4 we defined the exchange-correlation functional by what it does not include-it is the parts of the complete energy functional that are left once we separate out the contributions that can be written in simple ways.Our discussion of the HF method,however,indicates that it is possible to treat the exchange part of the problem in an exact way,at least in principle.The most commonly used functionals in DFT calculations based on spatially localized basis func- tions are "hybrid"functionals that mix the exact results for the exchange part of the functional with approximations for the correlation part.The B3LYP func- tional is by far the most widely used of these hybrid functionals.The B stands for Becke,who worked on the exchange part of the problem,the LYP stands for Lee,Yang,and Parr,who developed the correlation part of the functional, and the 3 describes the particular way that the results are mixed together. Unfortunately,the form of the exact exchange results mean that they can be efficiently implemented for applications based on spatially localized functions but not for applications using periodic functions!Because of this fact,the func- tionals that are commonly used in plane-wave DFT calculations do not include contributions from the exact exchange results
scaling if you notice from Table 1.2 that a reasonable basis set for even a tiny molecule like CH4 includes hundreds of basis functions. The computational expense of high-level wave-function-based methods means that these calculations are feasible for individual organic molecules containing 10 –20 atoms, but physical systems larger than this fall into either the “very challenging” or “computationally infeasible” categories. This brings our brief tour of quantum chemistry almost to an end. As the title of this book suggests, we are going to focus throughout the book on density functional theory calculations. Moreover, we will only consider methods based on spatially periodic functions—the so-called plane-wave methods. Plane-wave methods are the method of choice in almost all situations where the physical material of interest is an extended crystalline material rather than an isolated molecule. As we stated above, it is not appropriate to view methods based on periodic functions as “right” and methods based on spatially localized functions as “wrong” (or vice versa). In the long run, it will be a great advantage to you to understand both classes of methods since having access to a wide range of tools can only improve your chances of solving significant scientific problems. Nevertheless, if you are interested in applying computational methods to materials other than isolated molecules, then plane-wave DFT is an excellent place to start. It is important for us to emphasize that DFT calculations can also be performed using spatially localized functions—the results in Tables 1.1 and 1.2 are examples of this kind of calculation. Perhaps the main difference between DFT calculations using periodic and spatially localized functions lies in the exchange –correlation functionals that are routinely used. In Section 1.4 we defined the exchange –correlation functional by what it does not include—it is the parts of the complete energy functional that are left once we separate out the contributions that can be written in simple ways. Our discussion of the HF method, however, indicates that it is possible to treat the exchange part of the problem in an exact way, at least in principle. The most commonly used functionals in DFT calculations based on spatially localized basis functions are “hybrid” functionals that mix the exact results for the exchange part of the functional with approximations for the correlation part. The B3LYP functional is by far the most widely used of these hybrid functionals. The B stands for Becke, who worked on the exchange part of the problem, the LYP stands for Lee, Yang, and Parr, who developed the correlation part of the functional, and the 3 describes the particular way that the results are mixed together. Unfortunately, the form of the exact exchange results mean that they can be efficiently implemented for applications based on spatially localized functions but not for applications using periodic functions! Because of this fact, the functionals that are commonly used in plane-wave DFT calculations do not include contributions from the exact exchange results. 1.6 THE QUANTUM CHEMISTRY TOURIST 27
28 WHAT IS DENSITY FUNCTIONAL THEORY? 1.7 WHAT CAN DFT NOT DO? It is very important to come to grips with the fact that practical DFT calcu- lations are not exact solutions of the full Schrodinger equation.This inexact- ness exists because the exact functional that the Hohenberg-Kohn theorem applies to is not known.So any time you (or anyone else)performs a DFT cal- culation,there is an intrinsic uncertainty that exists between the energies calculated with DFT and the true ground-state energies of the Schrodinger equation.In many situations,there is no direct way to estimate the magnitude of this uncertainty apart from careful comparisons with experimental measure- ments.As you read further through this book,we hope you will come to appreciate that there are many physical situations where the accuracy of DFT calculations is good enough to make powerful predictions about the prop- erties of complex materials.The vignettes in Section 1.2 give several examples of this idea.We discuss the complicated issue of the accuracy of DFT calcu- lations in Chapter 10. There are some important situations for which DFT cannot be expected to be physically accurate.Below,we briefly discuss some of the most common problems that fall into this category.The first situation where DFT calculations have limited accuracy is in the calculation of electronic excited states.This can be understood in a general way by looking back at the statement of the Hohenberg-Kohn theorems in Section 1.4;these theorems only apply to the ground-state energy.It is certainly possible to make predictions about excited states from DFT calculations,but it is important to remember that these predictions are not-theoretically speaking-on the same footing as similar predictions made for ground-state properties. A well-known inaccuracy in DFT is the underestimation of calculated band gaps in semiconducting and insulating materials.In isolated molecules,the energies that are accessible to individual electrons form a discrete set (usually described in terms of molecular orbitals).In crystalline materials,these ener- gies must be described by continuous functions known as energy bands.The simplest definition of metals and insulators involves what energy levels are available to the electrons in the material with the highest energy once all the low-energy bands are filled in accordance with the Pauli exclusion principle. If the next available electronic state lies only at an infinitesimal energy above the highest occupied state,then the material is said to be a metal.If the next available electronic state sits a finite energy above the highest occu- pied state,then the material is not a metal and the energy difference between these two states is called the band gap.By convention,materials with"large" band gaps(i.e.,band gaps of multiple electron volts)are called insulators while materials with "small"band gaps are called semiconductors.Standard DFT calculations with existing functionals have limited accuracy for band gaps
1.7 WHAT CAN DFT NOT DO? It is very important to come to grips with the fact that practical DFT calculations are not exact solutions of the full Schro¨dinger equation. This inexactness exists because the exact functional that the Hohenberg –Kohn theorem applies to is not known. So any time you (or anyone else) performs a DFT calculation, there is an intrinsic uncertainty that exists between the energies calculated with DFT and the true ground-state energies of the Schro¨dinger equation. In many situations, there is no direct way to estimate the magnitude of this uncertainty apart from careful comparisons with experimental measurements. As you read further through this book, we hope you will come to appreciate that there are many physical situations where the accuracy of DFT calculations is good enough to make powerful predictions about the properties of complex materials. The vignettes in Section 1.2 give several examples of this idea. We discuss the complicated issue of the accuracy of DFT calculations in Chapter 10. There are some important situations for which DFT cannot be expected to be physically accurate. Below, we briefly discuss some of the most common problems that fall into this category. The first situation where DFT calculations have limited accuracy is in the calculation of electronic excited states. This can be understood in a general way by looking back at the statement of the Hohenberg –Kohn theorems in Section 1.4; these theorems only apply to the ground-state energy. It is certainly possible to make predictions about excited states from DFT calculations, but it is important to remember that these predictions are not—theoretically speaking—on the same footing as similar predictions made for ground-state properties. A well-known inaccuracy in DFT is the underestimation of calculated band gaps in semiconducting and insulating materials. In isolated molecules, the energies that are accessible to individual electrons form a discrete set (usually described in terms of molecular orbitals). In crystalline materials, these energies must be described by continuous functions known as energy bands. The simplest definition of metals and insulators involves what energy levels are available to the electrons in the material with the highest energy once all the low-energy bands are filled in accordance with the Pauli exclusion principle. If the next available electronic state lies only at an infinitesimal energy above the highest occupied state, then the material is said to be a metal. If the next available electronic state sits a finite energy above the highest occupied state, then the material is not a metal and the energy difference between these two states is called the band gap. By convention, materials with “large” band gaps (i.e., band gaps of multiple electron volts) are called insulators while materials with “small” band gaps are called semiconductors. Standard DFT calculations with existing functionals have limited accuracy for band gaps, 28 WHAT IS DENSITY FUNCTIONAL THEORY?
1.7 WHAT CAN DFT NOT DO?29 with errors larger than 1 eV being common when comparing with experimen- tal data.A subtle feature of this issue is that it has been shown that even the formally exact Kohn-Sham exchange-correlation functional would suffer from the same underlying problem. Another situation where DFT calculations give inaccurate results is associ- ated with the weak van der Waals (vdW)attractions that exist between atoms and molecules.To see that interactions like this exist,you only have to think about a simple molecule like CH4(methane).Methane becomes a liquid at suf- ficiently low temperatures and high enough pressures.The transportation of methane over long distances is far more economical in this liquid form than as a gas;this is the basis of the worldwide liquefied natural gas (LNG)industry. But to become a liquid,some attractive interactions between pairs of CH4 mol- ecules must exist.The attractive interactions are the van der Waals interactions, which,at the most fundamental level,occur because of correlations that exist between temporary fluctuations in the electron density of one molecule and the energy of the electrons in another molecule responding to these fluctuations. This description already hints at the reason that describing these interactions with DFT is challenging;van der Waals interactions are a direct result of long range electron correlation.To accurately calculate the strength of these interactions from quantum mechanics,it is necessary to use high-level wave-function-based methods that treat electron correlation in a systematic way.This has been done,for example,to calculate the very weak interactions that exist between pairs of H2 molecules,where it is known experimentally that energy of two H2 molecules in their most favored geometry is ~0.003 eV lower than the energy of the same molecules separated by a long distance. There is one more fundamental limitation of DFT that is crucial to appreci- ate,and it stems from the computational expense associated with solving the mathematical problem posed by DFT.It is reasonable to say that calculations that involve tens of atoms are now routine,calculations involving hundreds of atoms are feasible but are considered challenging research-level problems,and calculations involving a thousand or more atoms are possible but restricted to a small group of people developing state-of-the-art codes and using some of the world's largest computers.To keep this in a physical perspective,a droplet of water 1 um in radius contains on the order of 10 atoms.No con- ceivable increase in computing technology or code efficiency will allow DFT Development of methods related to DFT that can treat this situation accurately is an active area of research where considerable progress is being made.Two representative examples of this kind of work are P.Rinke,A.Qteish,J.Neugebauer,and M.Scheffler,Exciting Prospects for Solids: Exact Exchange Based Functional Meet Quasiparticle Energy Calculations,Phrys.Stat.Sol.245 (2008),929,and J.Uddin,J.E.Peralta,and G.E.Scuseria,Density Functional Theory Study of Bulk Platinum Monoxide,Phrys.Rev.B,71 (2005),155112
with errors larger than 1 eV being common when comparing with experimental data. A subtle feature of this issue is that it has been shown that even the formally exact Kohn –Sham exchange –correlation functional would suffer from the same underlying problem.k Another situation where DFT calculations give inaccurate results is associated with the weak van der Waals (vdW) attractions that exist between atoms and molecules. To see that interactions like this exist, you only have to think about a simple molecule like CH4 (methane). Methane becomes a liquid at suf- ficiently low temperatures and high enough pressures. The transportation of methane over long distances is far more economical in this liquid form than as a gas; this is the basis of the worldwide liquefied natural gas (LNG) industry. But to become a liquid, some attractive interactions between pairs of CH4 molecules must exist. The attractive interactions are the van der Waals interactions, which, at the most fundamental level, occur because of correlations that exist between temporary fluctuations in the electron density of one molecule and the energy of the electrons in another molecule responding to these fluctuations. This description already hints at the reason that describing these interactions with DFT is challenging; van der Waals interactions are a direct result of long range electron correlation. To accurately calculate the strength of these interactions from quantum mechanics, it is necessary to use high-level wave-function-based methods that treat electron correlation in a systematic way. This has been done, for example, to calculate the very weak interactions that exist between pairs of H2 molecules, where it is known experimentally that energy of two H2 molecules in their most favored geometry is 0.003 eV lower than the energy of the same molecules separated by a long distance.8 There is one more fundamental limitation of DFT that is crucial to appreciate, and it stems from the computational expense associated with solving the mathematical problem posed by DFT. It is reasonable to say that calculations that involve tens of atoms are now routine, calculations involving hundreds of atoms are feasible but are considered challenging research-level problems, and calculations involving a thousand or more atoms are possible but restricted to a small group of people developing state-of-the-art codes and using some of the world’s largest computers. To keep this in a physical perspective, a droplet of water 1 mm in radius contains on the order of 1011 atoms. No conceivable increase in computing technology or code efficiency will allow DFT k Development of methods related to DFT that can treat this situation accurately is an active area of research where considerable progress is being made. Two representative examples of this kind of work are P. Rinke, A. Qteish, J. Neugebauer, and M. Scheffler, Exciting Prospects for Solids: Exact Exchange Based Functional Meet Quasiparticle Energy Calculations, Phys. Stat. Sol. 245 (2008), 929, and J. Uddin, J. E. Peralta, and G. E. Scuseria, Density Functional Theory Study of Bulk Platinum Monoxide, Phys. Rev. B, 71 (2005), 155112. 1.7 WHAT CAN DFT NOT DO? 29
30 WHAT IS DENSITY FUNCTIONAL THEORY? calculations to directly examine collections of atoms of this size.As a result, anyone using DFT calculations must clearly understand how information from calculations with extremely small numbers of atoms can be connected with information that is physically relevant to real materials. 1.8 DENSITY FUNCTIONAL THEORY IN OTHER FIELDS For completeness,we need to point out that the name density functional theory is not solely applied to the type of quantum mechanics calculations we have described in this chapter.The idea of casting problems using functionals of density has also been used in the classical theory of fluid thermodynamics. In this case,the density of interest is the fluid density not the electron density, and the basic equation of interest is not the Schrodinger equation.Realizing that these two distinct scientific communities use the same name for their methods may save you some confusion if you find yourself in a seminar by a researcher from the other community. 1.9 HOW TO APPROACH THIS BOOK (REVISITED) We began this chapter with an analogy about learning to drive to describe our aims for this book.Now that we have introduced much of the terminology associated with DFT and quantum chemistry calculations,we can state the subject matter and approach of the book more precisely.The remaining chap- ters focus on using plane-wave DFT calculations with commonly applied functionals to physical questions involving bulk materials,surfaces,nanopar- ticles,and molecules.Because codes to perform these plane-wave calculations are now widely available,we aim to introduce many of the issues associated with applying these methods to interesting scientific questions in a computa- tionally efficient way. The book has been written with two audiences in mind.The primary audi- ence is readers who are entering a field of research where they will perform DFT calculations (and perhaps other kinds of computational chemistry or materials modeling)on a daily basis.If this describes you,it is important that you perform as many of the exercises at the end of the chapters as possible. These exercises have been chosen to require relatively modest computational resources while exploring most of the key ideas introduced in each chapter. Simply put,if your aim is to enter a field where you will perform calculations, then you must actually do calculations of your own,not just read about other people's work.As in almost every endeavor,there are many details that are best learned by experience.For readers in this group,we recommend reading through every chapter sequentially
calculations to directly examine collections of atoms of this size. As a result, anyone using DFT calculations must clearly understand how information from calculations with extremely small numbers of atoms can be connected with information that is physically relevant to real materials. 1.8 DENSITY FUNCTIONAL THEORY IN OTHER FIELDS For completeness, we need to point out that the name density functional theory is not solely applied to the type of quantum mechanics calculations we have described in this chapter. The idea of casting problems using functionals of density has also been used in the classical theory of fluid thermodynamics. In this case, the density of interest is the fluid density not the electron density, and the basic equation of interest is not the Schro¨dinger equation. Realizing that these two distinct scientific communities use the same name for their methods may save you some confusion if you find yourself in a seminar by a researcher from the other community. 1.9 HOW TO APPROACH THIS BOOK (REVISITED) We began this chapter with an analogy about learning to drive to describe our aims for this book. Now that we have introduced much of the terminology associated with DFT and quantum chemistry calculations, we can state the subject matter and approach of the book more precisely. The remaining chapters focus on using plane-wave DFT calculations with commonly applied functionals to physical questions involving bulk materials, surfaces, nanoparticles, and molecules. Because codes to perform these plane-wave calculations are now widely available, we aim to introduce many of the issues associated with applying these methods to interesting scientific questions in a computationally efficient way. The book has been written with two audiences in mind. The primary audience is readers who are entering a field of research where they will perform DFT calculations (and perhaps other kinds of computational chemistry or materials modeling) on a daily basis. If this describes you, it is important that you perform as many of the exercises at the end of the chapters as possible. These exercises have been chosen to require relatively modest computational resources while exploring most of the key ideas introduced in each chapter. Simply put, if your aim is to enter a field where you will perform calculations, then you must actually do calculations of your own, not just read about other people’s work. As in almost every endeavor, there are many details that are best learned by experience. For readers in this group, we recommend reading through every chapter sequentially. 30 WHAT IS DENSITY FUNCTIONAL THEORY?
REFERENCES 31 The second audience is people who are unlikely to routinely perform their own calculations,but who work in a field where DFT calculations have become a "standard"approach.For this group,it is important to understand the language used to describe DFT calculations and the strengths and limit- ations of DFT.This situation is no different from"standard"experimental techniques such as X-ray diffraction or scanning electron microscopy,where a working knowledge of the basic methods is indispensable to a huge commu- nity of researchers,regardless of whether they personally apply these methods. If you are in this audience,we hope that this book can help you become a soph- isticated consumer of DFT results in a relatively efficient way.If you have a limited amount of time (a long plane flight,for example),we recommend that you read Chapter 3,Chapter 10,and then read whichever of Chapters 4-9 appears most relevant to you.If (when?)your flight is delayed,read one of the chapters that doesn't appear directly relevant to your specific research interests-we hope that you will learn something interesting. We have consciously limited the length of the book in the belief that the pro- spect of reading and understanding an entire book of this length is more appealing than the alternative of facing (and carrying)something the size of a large city's phone book.Inevitably,this means that our coverage of various topics is limited in scope.In particular,we do not examine the details of DFT calculations using localized basis sets beyond the cursory treatment already presented in this chapter.We also do not delve deeply into the theory of DFT and the construction of functionals.In this context,the word"introduc- tion"appears in the title of the book deliberately.You should view this book as an entry point into the vibrant world of DFT,computational chemistry,and materials modeling.By following the resources that are listed at the end of each chapter in the Further Reading section,we hope that you will continue to expand your horizons far beyond the introduction that this book gives. We have opted to defer the crucial issue of the accuracy of DFT calculations until chapter 10,after introducing the application of DFT to a wide variety of physical properties in the preceding chapters.The discussion in that chapter emphasizes that this topic cannot be described in a simplistic way.Chapter 10 also points to some of the areas in which rapid developments are currently being made in the application of DFT to challenging physical problems. REFERENCES 1.K.Honkala,A.Hellman,I.N.Remediakis,A.Logadottir,A.Carlsson,S.Dahl, C.H.Christensen,and J.K.Norskov,Ammonia Synthesis from First-Principles Calculations,Science 307 (2005),555. 2.R.Schweinfest,A.T.Paxton,and M.W.Finnis,Bismuth Embrittlement of Copper is an Atomic Size Effect,Nature 432 (2004),1008
The second audience is people who are unlikely to routinely perform their own calculations, but who work in a field where DFT calculations have become a “standard” approach. For this group, it is important to understand the language used to describe DFT calculations and the strengths and limitations of DFT. This situation is no different from “standard” experimental techniques such as X-ray diffraction or scanning electron microscopy, where a working knowledge of the basic methods is indispensable to a huge community of researchers, regardless of whether they personally apply these methods. If you are in this audience, we hope that this book can help you become a sophisticated consumer of DFT results in a relatively efficient way. If you have a limited amount of time (a long plane flight, for example), we recommend that you read Chapter 3, Chapter 10, and then read whichever of Chapters 4–9 appears most relevant to you. If (when?) your flight is delayed, read one of the chapters that doesn’t appear directly relevant to your specific research interests—we hope that you will learn something interesting. We have consciously limited the length of the book in the belief that the prospect of reading and understanding an entire book of this length is more appealing than the alternative of facing (and carrying) something the size of a large city’s phone book. Inevitably, this means that our coverage of various topics is limited in scope. In particular, we do not examine the details of DFT calculations using localized basis sets beyond the cursory treatment already presented in this chapter. We also do not delve deeply into the theory of DFT and the construction of functionals. In this context, the word “introduction” appears in the title of the book deliberately. You should view this book as an entry point into the vibrant world of DFT, computational chemistry, and materials modeling. By following the resources that are listed at the end of each chapter in the Further Reading section, we hope that you will continue to expand your horizons far beyond the introduction that this book gives. We have opted to defer the crucial issue of the accuracy of DFT calculations until chapter 10, after introducing the application of DFT to a wide variety of physical properties in the preceding chapters. The discussion in that chapter emphasizes that this topic cannot be described in a simplistic way. Chapter 10 also points to some of the areas in which rapid developments are currently being made in the application of DFT to challenging physical problems. REFERENCES 1. K. Honkala, A. Hellman, I. N. Remediakis, A. Logadottir, A. Carlsson, S. Dahl, C. H. Christensen, and J. K. Nørskov, Ammonia Synthesis from First-Principles Calculations, Science 307 (2005), 555. 2. R. Schweinfest, A. T. Paxton, and M. W. Finnis, Bismuth Embrittlement of Copper is an Atomic Size Effect, Nature 432 (2004), 1008. REFERENCES 31