WHAT IS DENSITY FUNCTIONAL THEORY? yourself in a research setting or so you can interact knowledgeably with collaborators who use these methods. An analogy related to cars may be useful here.Before you learned how to drive,it was presumably clear to you that you can accomplish many useful things with the aid of a car.For you to use a car,it is important to understand the basic concepts that control cars (you need to put fuel in the car regularly, you need to follow basic traffic laws,etc.)and spend time actually driving a car in a variety of road conditions.You do not,however,need to know every detail of how fuel injectors work,how to construct a radiator system that efficiently cools an engine,or any of the other myriad of details that are required if you were going to actually build a car.Many of these details may be important if you plan on undertaking some especially difficult car-related project such as,say,driving yourself across Antarctica,but you can make it across town to a friend's house and back without understanding them. With this book,we hope you can learn to"drive across town"when doing your own calculations with a DFT package or when interpreting other people's calculations as they relate to physical questions of interest to you.If you are interested in "building a better car"by advancing the cutting edge of method development in this area,then we applaud your enthusiasm.You should continue reading this chapter to find at least one surefire project that could win you a Nobel prize,then delve into the books listed in the Further Reading at the end of the chapter. At the end of most chapters we have given a series of exercises,most of which involve actually doing calculations using the ideas described in the chapter.Your knowledge and ability will grow most rapidly by doing rather than by simply reading,so we strongly recommend doing as many of the exer- cises as you can in the time available to you. 1.2 EXAMPLES OF DFT IN ACTION Before we even define what density functional theory is,it is useful to relate a few vignettes of how it has been used in several scientific fields.We have chosen three examples from three quite different areas of science from the thousands of articles that have been published using these methods.These specific examples have been selected because they show how DFT calcu- lations have been used to make important contributions to a diverse range of compelling scientific questions,generating information that would be essen- tially impossible to determine through experiments. 1.2.1 Ammonia Synthesis by Heterogeneous Catalysis Our first example involves an industrial process of immense importance:the catalytic synthesis of ammonia (NH3).Ammonia is a central component of
yourself in a research setting or so you can interact knowledgeably with collaborators who use these methods. An analogy related to cars may be useful here. Before you learned how to drive, it was presumably clear to you that you can accomplish many useful things with the aid of a car. For you to use a car, it is important to understand the basic concepts that control cars (you need to put fuel in the car regularly, you need to follow basic traffic laws, etc.) and spend time actually driving a car in a variety of road conditions. You do not, however, need to know every detail of how fuel injectors work, how to construct a radiator system that efficiently cools an engine, or any of the other myriad of details that are required if you were going to actually build a car. Many of these details may be important if you plan on undertaking some especially difficult car-related project such as, say, driving yourself across Antarctica, but you can make it across town to a friend’s house and back without understanding them. With this book, we hope you can learn to “drive across town” when doing your own calculations with a DFT package or when interpreting other people’s calculations as they relate to physical questions of interest to you. If you are interested in “building a better car” by advancing the cutting edge of method development in this area, then we applaud your enthusiasm. You should continue reading this chapter to find at least one surefire project that could win you a Nobel prize, then delve into the books listed in the Further Reading at the end of the chapter. At the end of most chapters we have given a series of exercises, most of which involve actually doing calculations using the ideas described in the chapter. Your knowledge and ability will grow most rapidly by doing rather than by simply reading, so we strongly recommend doing as many of the exercises as you can in the time available to you. 1.2 EXAMPLES OF DFT IN ACTION Before we even define what density functional theory is, it is useful to relate a few vignettes of how it has been used in several scientific fields. We have chosen three examples from three quite different areas of science from the thousands of articles that have been published using these methods. These specific examples have been selected because they show how DFT calculations have been used to make important contributions to a diverse range of compelling scientific questions, generating information that would be essentially impossible to determine through experiments. 1.2.1 Ammonia Synthesis by Heterogeneous Catalysis Our first example involves an industrial process of immense importance: the catalytic synthesis of ammonia (NH3). Ammonia is a central component of 2 WHAT IS DENSITY FUNCTIONAL THEORY?
1.2 EXAMPLES OF DFT IN ACTION 3 fertilizers for agriculture,and more than 100 million tons of ammonia are produced commercially each year.By some estimates,more than 1%of all energy used in the world is consumed in the production of ammonia.The core reaction in ammonia production is very simple: N2+3H2→2NH3. To get this reaction to proceed,the reaction is performed at high tempera- tures (>400C)and high pressures (>100 atm)in the presence of metals such as iron (Fe)or ruthenium (Ru)that act as catalysts.Although these metal catalysts were identified by Haber and others almost 100 years ago, much is still not known about the mechanisms of the reactions that occur on the surfaces of these catalysts.This incomplete understanding is partly because of the structural complexity of practical catalysts.To make metal catalysts with high surface areas,tiny particles of the active metal are dispersed throughout highly porous materials.This was a widespread application of nanotechno- logy long before that name was applied to materials to make them sound scientifically exciting!To understand the reactivity of a metal nanoparticle, it is useful to characterize the surface atoms in terms of their local coordination since differences in this coordination can create differences in chemical reactivity;surface atoms can be classified into "types"based on their local coordination.The surfaces of nanoparticles typically include atoms of various types (based on coordination),so the overall surface reactivity is a compli- cated function of the shape of the nanoparticle and the reactivity of each type of atom. The discussion above raises a fundamental question:Can a direct connec- tion be made between the shape and size of a metal nanoparticle and its activity as a catalyst for ammonia synthesis?If detailed answers to this question can be found,then they can potentially lead to the synthesis of improved catalysts. One of the most detailed answers to this question to date has come from the DFT calculations of Honkala and co-workers,who studied nanoparticles of Ru.Using DFT calculations,they showed that the net chemical reaction above proceeds via at least 12 distinct steps on a metal catalyst and that the rates of these steps depend strongly on the local coordination of the metal atoms that are involved.One of the most important reactions is the breaking of the N2 bond on the catalyst surface.On regions of the catalyst surface that were similar to the surfaces of bulk Ru (more specifically,atomically flat regions),a great deal of energy is required for this bond-breaking reaction, implying that the reaction rate is extremely slow.Near Ru atoms that form a common kind of surface step edge on the catalyst,however,a much smaller amount of energy is needed for this reaction.Honkala and co-workers used additional DFT calculations to predict the relative stability of many different local coordinations of surface atoms in Ru nanoparticles in a way that allowed
fertilizers for agriculture, and more than 100 million tons of ammonia are produced commercially each year. By some estimates, more than 1% of all energy used in the world is consumed in the production of ammonia. The core reaction in ammonia production is very simple: N2 þ 3H2 ! 2NH3: To get this reaction to proceed, the reaction is performed at high temperatures (.4008C) and high pressures (.100 atm) in the presence of metals such as iron (Fe) or ruthenium (Ru) that act as catalysts. Although these metal catalysts were identified by Haber and others almost 100 years ago, much is still not known about the mechanisms of the reactions that occur on the surfaces of these catalysts. This incomplete understanding is partly because of the structural complexity of practical catalysts. To make metal catalysts with high surface areas, tiny particles of the active metal are dispersed throughout highly porous materials. This was a widespread application of nanotechnology long before that name was applied to materials to make them sound scientifically exciting! To understand the reactivity of a metal nanoparticle, it is useful to characterize the surface atoms in terms of their local coordination since differences in this coordination can create differences in chemical reactivity; surface atoms can be classified into “types” based on their local coordination. The surfaces of nanoparticles typically include atoms of various types (based on coordination), so the overall surface reactivity is a complicated function of the shape of the nanoparticle and the reactivity of each type of atom. The discussion above raises a fundamental question: Can a direct connection be made between the shape and size of a metal nanoparticle and its activity as a catalyst for ammonia synthesis? If detailed answers to this question can be found, then they can potentially lead to the synthesis of improved catalysts. One of the most detailed answers to this question to date has come from the DFT calculations of Honkala and co-workers,1 who studied nanoparticles of Ru. Using DFT calculations, they showed that the net chemical reaction above proceeds via at least 12 distinct steps on a metal catalyst and that the rates of these steps depend strongly on the local coordination of the metal atoms that are involved. One of the most important reactions is the breaking of the N2 bond on the catalyst surface. On regions of the catalyst surface that were similar to the surfaces of bulk Ru (more specifically, atomically flat regions), a great deal of energy is required for this bond-breaking reaction, implying that the reaction rate is extremely slow. Near Ru atoms that form a common kind of surface step edge on the catalyst, however, a much smaller amount of energy is needed for this reaction. Honkala and co-workers used additional DFT calculations to predict the relative stability of many different local coordinations of surface atoms in Ru nanoparticles in a way that allowed 1.2 EXAMPLES OF DFT IN ACTION 3
4 WHAT IS DENSITY FUNCTIONAL THEORY? them to predict the detailed shape of the nanoparticles as a function of particle size.This prediction makes a precise connection between the diameter of a Ru nanoparticle and the number of highly desirable reactive sites for breaking N2 bonds on the nanoparticle.Finally,all of these calculations were used to develop an overall model that describes how the individual reaction rates for the many different kinds of metal atoms on the nanoparticle's surfaces couple together to define the overall reaction rate under realistic reaction con- ditions.At no stage in this process was any experimental data used to fit or adjust the model,so the final result was a truly predictive description of the reaction rate of a complex catalyst.After all this work was done,Honkala et al.compared their predictions to experimental measurements made with Ru nanoparticle catalysts under reaction conditions similar to industrial con- ditions.Their predictions were in stunning quantitative agreement with the experimental outcome. 1.2.2 Embrittlement of Metals by Trace Impurities It is highly likely that as you read these words you are within I m of a large number of copper wires since copper is the dominant metal used for carrying electricity between components of electronic devices of all kinds.Aside from its low cost,one of the attractions of copper in practical applications is that it is a soft,ductile metal.Common pieces of copper (and other metals)are almost invariably polycrystalline,meaning that they are made up of many tiny domains called grains that are each well-oriented single crystals.Two neigh- boring grains have the same crystal structure and symmetry,but their orien- tation in space is not identical.As a result,the places where grains touch have a considerably more complicated structure than the crystal structure of the pure metal.These regions,which are present in all polycrystalline materials, are called grain boundaries. It has been known for over 100 years that adding tiny amounts of certain impurities to copper can change the metal from being ductile to a material that will fracture in a brittle way (i.e.,without plastic deformation before the fracture).This occurs,for example,when bismuth (Bi)is present in copper (Cu)at levels below 100 ppm.Similar effects have been observed with lead (Pb)or mercury (Hg)impurities.But how does this happen?Qualitatively, when the impurities cause brittle fracture,the fracture tends to occur at grain boundaries,so something about the impurities changes the properties of grain boundaries in a dramatic way.That this can happen at very low concen- trations of Bi is not completely implausible because Bi is almost completely insoluble in bulk Cu.This means that it is very favorable for Bi atoms to seg- regate to grain boundaries rather than to exist inside grains,meaning that the
them to predict the detailed shape of the nanoparticles as a function of particle size. This prediction makes a precise connection between the diameter of a Ru nanoparticle and the number of highly desirable reactive sites for breaking N2 bonds on the nanoparticle. Finally, all of these calculations were used to develop an overall model that describes how the individual reaction rates for the many different kinds of metal atoms on the nanoparticle’s surfaces couple together to define the overall reaction rate under realistic reaction conditions. At no stage in this process was any experimental data used to fit or adjust the model, so the final result was a truly predictive description of the reaction rate of a complex catalyst. After all this work was done, Honkala et al. compared their predictions to experimental measurements made with Ru nanoparticle catalysts under reaction conditions similar to industrial conditions. Their predictions were in stunning quantitative agreement with the experimental outcome. 1.2.2 Embrittlement of Metals by Trace Impurities It is highly likely that as you read these words you are within 1 m of a large number of copper wires since copper is the dominant metal used for carrying electricity between components of electronic devices of all kinds. Aside from its low cost, one of the attractions of copper in practical applications is that it is a soft, ductile metal. Common pieces of copper (and other metals) are almost invariably polycrystalline, meaning that they are made up of many tiny domains called grains that are each well-oriented single crystals. Two neighboring grains have the same crystal structure and symmetry, but their orientation in space is not identical. As a result, the places where grains touch have a considerably more complicated structure than the crystal structure of the pure metal. These regions, which are present in all polycrystalline materials, are called grain boundaries. It has been known for over 100 years that adding tiny amounts of certain impurities to copper can change the metal from being ductile to a material that will fracture in a brittle way (i.e., without plastic deformation before the fracture). This occurs, for example, when bismuth (Bi) is present in copper (Cu) at levels below 100 ppm. Similar effects have been observed with lead (Pb) or mercury (Hg) impurities. But how does this happen? Qualitatively, when the impurities cause brittle fracture, the fracture tends to occur at grain boundaries, so something about the impurities changes the properties of grain boundaries in a dramatic way. That this can happen at very low concentrations of Bi is not completely implausible because Bi is almost completely insoluble in bulk Cu. This means that it is very favorable for Bi atoms to segregate to grain boundaries rather than to exist inside grains, meaning that the 4 WHAT IS DENSITY FUNCTIONAL THEORY?
1.2 EXAMPLES OF DFT IN ACTION 5 local concentration of Bi at grain boundaries can be much higher than the net concentration in the material as a whole. Can the changes in copper caused by Bi be explained in a detailed way?As you might expect for an interesting phenomena that has been observed over many years,several alternative explanations have been suggested.One class of explanations assigns the behavior to electronic effects.For example,a Bi atom might cause bonds between nearby Cu atoms to be stiffer than they are in pure Cu,reducing the ability of the Cu lattice to deform smoothly.A second type of electronic effect is that having an impurity atom next to a grain boundary could weaken the bonds that exist across a boundary by chan- ging the electronic structure of the atoms,which would make fracture at the boundary more likely.A third explanation assigns the blame to size effects, noting that Bi atoms are much larger than Cu atoms.If a Bi atom is present at a grain boundary,then it might physically separate Cu atoms on the other side of the boundary from their natural spacing.This stretching of bond dis- tances would weaken the bonds between atoms and make fracture of the grain boundary more likely.Both the second and third explanations involve weakening of bonds near grain boundaries,but they propose different root causes for this behavior.Distinguishing between these proposed mechanisms would be very difficult using direct experiments. Recently,Schweinfest,Paxton,and Finnis used DFT calculations to offer a definitive description of how Bi embrittles copper;the title of their study gives away the conclusion.They first used DFT to predict stress-strain relationships for pure Cu and Cu containing Bi atoms as impurities.If the bond stiffness argu- ment outlined above was correct,the elastic moduli of the metal should be increased by adding Bi.In fact,the calculations give the opposite result,immedi- ately showing the bond-stiffening explanation to be incorrect.In a separate and much more challenging series of calculations,they explicitly calculated the cohe- sion energy of a particular grain boundary that is known experimentally to be embrittled by Bi.In qualitative consistency with experimental observations, the calculations predicted that the cohesive energy of the grain boundary is greatly reduced by the presence of Bi.Crucially,the DFT results allow the elec- tronic structure of the grain boundary atoms to be examined directly.The result is that the grain boundary electronic effect outlined above was found to not be the cause of embrittlement.Instead,the large change in the properties of the grain boundary could be understood almost entirely in terms of the excess volume introduced by the Bi atoms,that is,by a size effect.This reasoning suggests that Cu should be embrittled by any impurity that has a much larger atomic size than Cu and that strongly segregates to grain boundaries.This description in fact correctly describes the properties of both Pb and Hg as impurities in Cu,and,as mentioned above,these impurities are known to embrittle Cu
local concentration of Bi at grain boundaries can be much higher than the net concentration in the material as a whole. Can the changes in copper caused by Bi be explained in a detailed way? As you might expect for an interesting phenomena that has been observed over many years, several alternative explanations have been suggested. One class of explanations assigns the behavior to electronic effects. For example, a Bi atom might cause bonds between nearby Cu atoms to be stiffer than they are in pure Cu, reducing the ability of the Cu lattice to deform smoothly. A second type of electronic effect is that having an impurity atom next to a grain boundary could weaken the bonds that exist across a boundary by changing the electronic structure of the atoms, which would make fracture at the boundary more likely. A third explanation assigns the blame to size effects, noting that Bi atoms are much larger than Cu atoms. If a Bi atom is present at a grain boundary, then it might physically separate Cu atoms on the other side of the boundary from their natural spacing. This stretching of bond distances would weaken the bonds between atoms and make fracture of the grain boundary more likely. Both the second and third explanations involve weakening of bonds near grain boundaries, but they propose different root causes for this behavior. Distinguishing between these proposed mechanisms would be very difficult using direct experiments. Recently, Schweinfest, Paxton, and Finnis used DFT calculations to offer a definitive description of how Bi embrittles copper; the title of their study gives away the conclusion.2 They first used DFT to predict stress–strain relationships for pure Cu and Cu containing Bi atoms as impurities. If the bond stiffness argument outlined above was correct, the elastic moduli of the metal should be increased by adding Bi. In fact, the calculations give the opposite result, immediately showing the bond-stiffening explanation to be incorrect. In a separate and much more challenging series of calculations, they explicitly calculated the cohesion energy of a particular grain boundary that is known experimentally to be embrittled by Bi. In qualitative consistency with experimental observations, the calculations predicted that the cohesive energy of the grain boundary is greatly reduced by the presence of Bi. Crucially, the DFT results allow the electronic structure of the grain boundary atoms to be examined directly. The result is that the grain boundary electronic effect outlined above was found to not be the cause of embrittlement. Instead, the large change in the properties of the grain boundary could be understood almost entirely in terms of the excess volume introduced by the Bi atoms, that is, by a size effect. This reasoning suggests that Cu should be embrittled by any impurity that has a much larger atomic size than Cu and that strongly segregates to grain boundaries. This description in fact correctly describes the properties of both Pb and Hg as impurities in Cu, and, as mentioned above, these impurities are known to embrittle Cu. 1.2 EXAMPLES OF DFT IN ACTION 5
6 WHAT IS DENSITY FUNCTIONAL THEORY? 1.2.3 Materials Properties for Modeling Planetary Formation To develop detailed models of how planets of various sizes have formed,it is necessary to know (among many other things)what minerals exist inside planets and how effective these minerals are at conducting heat.The extreme conditions that exist inside planets pose some obvious challenges to probing these topics in laboratory experiments.For example,the center of Jupiter has pressures exceeding 40 Mbar and temperatures well above 15,000 K. DFT calculations can play a useful role in probing material properties at these extreme conditions,as shown in the work of Umemoto,Wentzcovitch, and Allen.This work centered on the properties of bulk MgSiO3,a silicate mineral that is important in planet formation.At ambient conditions, MgSiO3 forms a relatively common crystal structure known as a perovskite. Prior to Umemoto et al.'s calculations,it was known that if MgSiO3 was placed under conditions similar to those in the core-mantle boundary of Earth,it transforms into a different crystal structure known as the CalrO3 struc- ture.(It is conventional to name crystal structures after the first compound dis- covered with that particular structure,and the naming of this structure is an example of this convention.) Umemoto et al.wanted to understand what happens to the structure of MgSiO3 at conditions much more extreme than those found in Earth's core-mantle boundary.They used DFT calculations to construct a phase diagram that compared the stability of multiple possible crystal structures of solid MgSiO3.All of these calculations dealt with bulk materials.They also considered the possibility that MgSiO3 might dissociate into other compounds.These calculations predicted that at pressures of ~11 Mbar, MgSiO3 dissociates in the following way: MgSiO3 [CalrO3 structure]Mgo [CsCl structure] SiO2 [cotunnite structure]. In this reaction,the crystal structure of each compound has been noted in the square brackets.An interesting feature of the compounds on the right-hand side is that neither of them is in the crystal structure that is the stable structure at ambient conditions.MgO,for example,prefers the NaCl structure at ambi- ent conditions (i.e.,the same crystal structure as everyday table salt).The beha- vior of SiO2 is similar but more complicated;this compound goes through several intermediate structures between ambient conditions and the conditions relevant for MgSiO3 dissociation.These transformations in the structures of MgO and SiO2 allow an important connection to be made between DFT cal- culations and experiments since these transformations occur at conditions that can be directly probed in laboratory experiments.The transition pressures
1.2.3 Materials Properties for Modeling Planetary Formation To develop detailed models of how planets of various sizes have formed, it is necessary to know (among many other things) what minerals exist inside planets and how effective these minerals are at conducting heat. The extreme conditions that exist inside planets pose some obvious challenges to probing these topics in laboratory experiments. For example, the center of Jupiter has pressures exceeding 40 Mbar and temperatures well above 15,000 K. DFT calculations can play a useful role in probing material properties at these extreme conditions, as shown in the work of Umemoto, Wentzcovitch, and Allen.3 This work centered on the properties of bulk MgSiO3, a silicate mineral that is important in planet formation. At ambient conditions, MgSiO3 forms a relatively common crystal structure known as a perovskite. Prior to Umemoto et al.’s calculations, it was known that if MgSiO3 was placed under conditions similar to those in the core–mantle boundary of Earth, it transforms into a different crystal structure known as the CaIrO3 structure. (It is conventional to name crystal structures after the first compound discovered with that particular structure, and the naming of this structure is an example of this convention.) Umemoto et al. wanted to understand what happens to the structure of MgSiO3 at conditions much more extreme than those found in Earth’s core–mantle boundary. They used DFT calculations to construct a phase diagram that compared the stability of multiple possible crystal structures of solid MgSiO3. All of these calculations dealt with bulk materials. They also considered the possibility that MgSiO3 might dissociate into other compounds. These calculations predicted that at pressures of 11 Mbar, MgSiO3 dissociates in the following way: MgSiO3 [CaIrO3 structure] ! MgO [CsCl structure] þ SiO2 [cotunnite structure]: In this reaction, the crystal structure of each compound has been noted in the square brackets. An interesting feature of the compounds on the right-hand side is that neither of them is in the crystal structure that is the stable structure at ambient conditions. MgO, for example, prefers the NaCl structure at ambient conditions (i.e., the same crystal structure as everyday table salt). The behavior of SiO2 is similar but more complicated; this compound goes through several intermediate structures between ambient conditions and the conditions relevant for MgSiO3 dissociation. These transformations in the structures of MgO and SiO2 allow an important connection to be made between DFT calculations and experiments since these transformations occur at conditions that can be directly probed in laboratory experiments. The transition pressures 6 WHAT IS DENSITY FUNCTIONAL THEORY?