《材料设计与计算 Materials Design and Computation》课程教学资源（参考书籍）DENSITY FUNCTIONAL THEORY - A Practical Introduction（DAVID S. SHOLL、JANICE A. STECKEL）

12 WHAT IS DENSITY FUNCTIONAL THEORY? A useful way to write down the functional described by the Hohenberg- Kohn theorem is in terms of the single-electron wave functions,(r). Remember from Eq.(1.2)that these functions collectively define the electron density,n(r).The energy functional can be written as E[)]Eknown[)]+Exc[)], (1.3) where we have split the functional into a collection of terms we can write down in a simple analytical form,Eknown[)],and everything else,Exc.The "known"terms include four contributions: Eonw,】= ∑gpr+小emo (1.4) The terms on the right are,in order,the electron kinetic energies,the Coulomb interactions between the electrons and the nuclei,the Coulomb interactions between pairs of electrons,and the Coulomb interactions between pairs of nuclei.The other term in the complete energy functional,Exc[)],is the exchange-correlation functional,and it is defined to include all the quantum mechanical effects that are not included in the "known"terms. Let us imagine for now that we can express the as-yet-undefined exchange- correlation energy functional in some useful way.What is involved in finding minimum energy solutions of the total energy functional?Nothing we have presented so far really guarantees that this task is any easier than the formid- able task of fully solving the Schrodinger equation for the wave function. This difficulty was solved by Kohn and Sham,who showed that the task of finding the right electron density can be expressed in a way that involves sol- ving a set of equations in which each equation only involves a single electron. The Kohn-Sham equations have the form [p2+vm+we+Kc四lM=aMe (1.5) These equations are superficially similar to Eq.(1.1).The main difference is that the Kohn-Sham equations are missing the summations that appear inside the full Schrodinger equation [Eq.(1.1)].This is because the solution of the Kohn-Sham equations are single-electron wave functions that depend on only three spatial variables,(r).On the left-hand side of the Kohn-Sham equations there are three potentials,V,VH,and Vxc.The first

A useful way to write down the functional described by the Hohenberg – Kohn theorem is in terms of the single-electron wave functions, ci(r). Remember from Eq. (1.2) that these functions collectively define the electron density, n(r). The energy functional can be written as E[{ci }] ¼ Eknown[{ci}] þ EXC[{ci}], (1:3) where we have split the functional into a collection of terms we can write down in a simple analytical form, Eknown[{ci }], and everything else, EXC. The “known” terms include four contributions: Eknown[{ci}] ¼ h2 m X i ð c i r2 cid3 r þ ð V(r)n(r) d3 r þ e2 2 ð ð n(r)n(r0 ) jr r0 j d3 r d3 r0 þ Eion: (1:4) The terms on the right are, in order, the electron kinetic energies, the Coulomb interactions between the electrons and the nuclei, the Coulomb interactions between pairs of electrons, and the Coulomb interactions between pairs of nuclei. The other term in the complete energy functional, EXC[{ci}], is the exchange –correlation functional, and it is defined to include all the quantum mechanical effects that are not included in the “known” terms. Let us imagine for now that we can express the as-yet-undefined exchange – correlation energy functional in some useful way. What is involved in finding minimum energy solutions of the total energy functional? Nothing we have presented so far really guarantees that this task is any easier than the formid￾able task of fully solving the Schro¨dinger equation for the wave function. This difficulty was solved by Kohn and Sham, who showed that the task of finding the right electron density can be expressed in a way that involves sol￾ving a set of equations in which each equation only involves a single electron. The Kohn –Sham equations have the form h2 2m r2 þ V(r) þ VH(r) þ VXC(r) ci (r) ¼ 1ici(r): (1:5) These equations are superficially similar to Eq. (1.1). The main difference is that the Kohn–Sham equations are missing the summations that appear inside the full Schro¨dinger equation [Eq. (1.1)]. This is because the solution of the Kohn–Sham equations are single-electron wave functions that depend on only three spatial variables, ci (r). On the left-hand side of the Kohn–Sham equations there are three potentials, V, VH, and VXC. The first 12 WHAT IS DENSITY FUNCTIONAL THEORY?

1.4 DENSITY FUNCTIONAL THEORY FROM WAVE FUNCTIONS 13 of these also appeared in the full Schrodinger equation (Eq.(1.1))and in the "known"part of the total energy functional given above (Eq.(1.4)).This potential defines the interaction between an electron and the collection of atomic nuclei.The second is called the Hartree potential and is defined by n(r) (1.6) This potential describes the Coulomb repulsion between the electron being considered in one of the Kohn-Sham equations and the total electron density defined by all electrons in the problem.The Hartree potential includes a so- called self-interaction contribution because the electron we are describing in the Kohn-Sham equation is also part of the total electron density,so part of VH involves a Coulomb interaction between the electron and itself.The self- interaction is unphysical,and the correction for it is one of several effects that are lumped together into the final potential in the Kohn-Sham equations, Vxc,which defines exchange and correlation contributions to the single- electron equations.Vxc can formally be defined as a "functional derivative" of the exchange-correlation energy: Vxc(r)= δExc(r) (1.7) δn(r) The strict mathematical definition of a functional derivative is slightly more subtle than the more familiar definition of a function's derivative,but concep- tually you can think of this just as a regular derivative.The functional deriva- tive is written using 6 rather than d to emphasize that it not quite identical to a normal derivative. If you have a vague sense that there is something circular about our discus- sion of the Kohn-Sham equations you are exactly right.To solve the Kohn- Sham equations,we need to define the Hartree potential,and to define the Hartree potential we need to know the electron density.But to find the electron density,we must know the single-electron wave functions,and to know these wave functions we must solve the Kohn-Sham equations.To break this circle, the problem is usually treated in an iterative way as outlined in the following algorithm: 1.Define an initial,trial electron density,n(r). 2.Solve the Kohn-Sham equations defined using the trial electron density to find the single-particle wave functions,(r). 3.Calculate the electron density defined by the Kohn-Sham single- particle wave functions from step 2,nks(r)=2(r(r)

of these also appeared in the full Schro¨dinger equation (Eq. (1.1)) and in the “known” part of the total energy functional given above (Eq. (1.4)). This potential defines the interaction between an electron and the collection of atomic nuclei. The second is called the Hartree potential and is defined by VH(r) ¼ e2 ð n(r0 ) jr r0 j d3 r0 : (1:6) This potential describes the Coulomb repulsion between the electron being considered in one of the Kohn –Sham equations and the total electron density defined by all electrons in the problem. The Hartree potential includes a so￾called self-interaction contribution because the electron we are describing in the Kohn –Sham equation is also part of the total electron density, so part of VH involves a Coulomb interaction between the electron and itself. The self￾interaction is unphysical, and the correction for it is one of several effects that are lumped together into the final potential in the Kohn–Sham equations, VXC, which defines exchange and correlation contributions to the single￾electron equations. VXC can formally be defined as a “functional derivative” of the exchange –correlation energy: VXC(r) ¼ dEXC(r) dn(r) : (1:7) The strict mathematical definition of a functional derivative is slightly more subtle than the more familiar definition of a function’s derivative, but concep￾tually you can think of this just as a regular derivative. The functional deriva￾tive is written using d rather than d to emphasize that it not quite identical to a normal derivative. If you have a vague sense that there is something circular about our discus￾sion of the Kohn –Sham equations you are exactly right. To solve the Kohn – Sham equations, we need to define the Hartree potential, and to define the Hartree potential we need to know the electron density. But to find the electron density, we must know the single-electron wave functions, and to know these wave functions we must solve the Kohn –Sham equations. To break this circle, the problem is usually treated in an iterative way as outlined in the following algorithm: 1. Define an initial, trial electron density, n(r). 2. Solve the Kohn –Sham equations defined using the trial electron density to find the single-particle wave functions, ci(r). 3. Calculate the electron density defined by the Kohn –Sham single￾particle wave functions from step 2, nKS(r) ¼ 2 P i c i (r)ci (r). 1.4 DENSITY FUNCTIONAL THEORY FROM WAVE FUNCTIONS 13

14 WHAT IS DENSITY FUNCTIONAL THEORY? 4.Compare the calculated electron density,nks(r),with the electron density used in solving the Kohn-Sham equations,n(r).If the two densities are the same,then this is the ground-state electron density, and it can be used to compute the total energy.If the two densities are different,then the trial electron density must be updated in some way. Once this is done,the process begins again from step 2. We have skipped over a whole series of important details in this process (How close do the two electron densities have to be before we consider them to be the same?What is a good way to update the trial electron density? How should we define the initial density?),but you should be able to see how this iterative method can lead to a solution of the Kohn-Sham equations that is self-consistent. 1.5 EXCHANGE-CORRELATION FUNCTIONAL Let us briefly review what we have seen so far.We would like to find the ground-state energy of the Schrodinger equation,but this is extremely diffi- cult because this is a many-body problem.The beautiful results of Kohn, Hohenberg,and Sham showed us that the ground state we seek can be found by minimizing the energy of an energy functional,and that this can be achieved by finding a self-consistent solution to a set of single-particle equations.There is just one critical complication in this otherwise beautiful formulation:to solve the Kohn-Sham equations we must specify the exchange-correlation func- tion,Exc[()].As you might gather from Eqs.(1.3)and (1.4),defining Exc[f)]is very difficult.After all,the whole point of Eq.(1.4)is that we have already explicitly written down all the "easy"parts. In fact,the true form of the exchange-correlation functional whose exist- ence is guaranteed by the Hohenberg-Kohn theorem is simply not known. Fortunately,there is one case where this functional can be derived exactly: the uniform electron gas.In this situation,the electron density is constant at all points in space;that is,n(r)=constant.This situation may appear to be of limited value in any real material since it is variations in electron density that define chemical bonds and generally make materials interesting.But the uniform electron gas provides a practical way to actually use the Kohn- Sham equations.To do this,we set the exchange-correlation potential at each position to be the known exchange-correlation potential from the uni- form electron gas at the electron density observed at that position: Vxc(r)=Vecctron gas[nr小. (1.8)

4. Compare the calculated electron density, nKS(r), with the electron density used in solving the Kohn –Sham equations, n(r). If the two densities are the same, then this is the ground-state electron density, and it can be used to compute the total energy. If the two densities are different, then the trial electron density must be updated in some way. Once this is done, the process begins again from step 2. We have skipped over a whole series of important details in this process (How close do the two electron densities have to be before we consider them to be the same? What is a good way to update the trial electron density? How should we define the initial density?), but you should be able to see how this iterative method can lead to a solution of the Kohn –Sham equations that is self-consistent. 1.5 EXCHANGE –CORRELATION FUNCTIONAL Let us briefly review what we have seen so far. We would like to find the ground-state energy of the Schro¨dinger equation, but this is extremely diffi- cult because this is a many-body problem. The beautiful results of Kohn, Hohenberg, and Sham showed us that the ground state we seek can be found by minimizing the energy of an energy functional, and that this can be achieved by finding a self-consistent solution to a set of single-particle equations. There is just one critical complication in this otherwise beautiful formulation: to solve the Kohn –Sham equations we must specify the exchange –correlation func￾tion, EXC[{ci }]. As you might gather from Eqs. (1.3) and (1.4), defining EXC[{ci }] is very difficult. After all, the whole point of Eq. (1.4) is that we have already explicitly written down all the “easy” parts. In fact, the true form of the exchange –correlation functional whose exist￾ence is guaranteed by the Hohenberg –Kohn theorem is simply not known. Fortunately, there is one case where this functional can be derived exactly: the uniform electron gas. In this situation, the electron density is constant at all points in space; that is, n(r) ¼ constant. This situation may appear to be of limited value in any real material since it is variations in electron density that define chemical bonds and generally make materials interesting. But the uniform electron gas provides a practical way to actually use the Kohn – Sham equations. To do this, we set the exchange –correlation potential at each position to be the known exchange –correlation potential from the uni￾form electron gas at the electron density observed at that position: VXC(r) ¼ V electron gas XC [n(r)]: (1:8) 14 WHAT IS DENSITY FUNCTIONAL THEORY?

1.5 EXCHANGE CORRELATION FUNCTIONAL 15 This approximation uses only the local density to define the approximate exchange-correlation functional,so it is called the local density approxi- mation (LDA).The LDA gives us a way to completely define the Kohn- Sham equations,but it is crucial to remember that the results from these equations do not exactly solve the true Schrodinger equation because we are not using the true exchange-correlation functional. It should not surprise you that the LDA is not the only functional that has been tried within DFT calculations.The development of functionals that more faithfully represent nature remains one of the most important areas of active research in the quantum chemistry community.We promised at the beginning of the chapter to pose a problem that could win you the Nobel prize.Here it is:Develop a functional that accurately represents nature's exact functional and implement it in a mathematical form that can be effi- ciently solved for large numbers of atoms.(This advice is a little like the Hohenberg-Kohn theorem-it tells you that something exists without provid- ing any clues how to find it.) Even though you could become a household name (at least in scientific cir- cles)by solving this problem rigorously,there are a number of approximate functionals that have been found to give good results in a large variety of phys- ical problems and that have been widely adopted.The primary aim of this book is to help you understand how to do calculations with these existing functionals.The best known class of functional after the LDA uses infor- mation about the local electron density and the local gradient in the electron density;this approach defines a generalized gradient approximation(GGA). It is tempting to think that because the GGA includes more physical information than the LDA it must be more accurate.Unfortunately,this is not always correct. Because there are many ways in which information from the gradient of the electron density can be included in a GGA functional,there are a large number of distinct GGA functionals.Two of the most widely used functionals in cal- culations involving solids are the Perdew-Wang functional (PW91)and the Perdew-Burke-Ernzerhof functional (PBE).Each of these functionals are GGA functionals,and dozens of other GGA functionals have been developed and used,particularly for calculations with isolated molecules.Because differ- ent functionals will give somewhat different results for any particular configur- ation of atoms,it is necessary to specify what functional was used in any particular calculation rather than simple referring to"a DFT calculation." Our description of GGA functionals as including information from the elec- tron density and the gradient of this density suggests that more sophisticated functionals can be constructed that use other pieces of physical information. In fact,a hierarchy of functionals can be constructed that gradually include

This approximation uses only the local density to define the approximate exchange –correlation functional, so it is called the local density approxi￾mation (LDA). The LDA gives us a way to completely define the Kohn– Sham equations, but it is crucial to remember that the results from these equations do not exactly solve the true Schro¨dinger equation because we are not using the true exchange –correlation functional. It should not surprise you that the LDA is not the only functional that has been tried within DFT calculations. The development of functionals that more faithfully represent nature remains one of the most important areas of active research in the quantum chemistry community. We promised at the beginning of the chapter to pose a problem that could win you the Nobel prize. Here it is: Develop a functional that accurately represents nature’s exact functional and implement it in a mathematical form that can be effi- ciently solved for large numbers of atoms. (This advice is a little like the Hohenberg –Kohn theorem—it tells you that something exists without provid￾ing any clues how to find it.) Even though you could become a household name (at least in scientific cir￾cles) by solving this problem rigorously, there are a number of approximate functionals that have been found to give good results in a large variety of phys￾ical problems and that have been widely adopted. The primary aim of this book is to help you understand how to do calculations with these existing functionals. The best known class of functional after the LDA uses infor￾mation about the local electron density and the local gradient in the electron density; this approach defines a generalized gradient approximation (GGA). It is tempting to think that because the GGA includes more physical information than the LDA it must be more accurate. Unfortunately, this is not always correct. Because there are many ways in which information from the gradient of the electron density can be included in a GGA functional, there are a large number of distinct GGA functionals. Two of the most widely used functionals in cal￾culations involving solids are the Perdew –Wang functional (PW91) and the Perdew –Burke–Ernzerhof functional (PBE). Each of these functionals are GGA functionals, and dozens of other GGA functionals have been developed and used, particularly for calculations with isolated molecules. Because differ￾ent functionals will give somewhat different results for any particular configur￾ation of atoms, it is necessary to specify what functional was used in any particular calculation rather than simple referring to “a DFT calculation.” Our description of GGA functionals as including information from the elec￾tron density and the gradient of this density suggests that more sophisticated functionals can be constructed that use other pieces of physical information. In fact, a hierarchy of functionals can be constructed that gradually include 1.5 EXCHANGE CORRELATION FUNCTIONAL 15

16 WHAT IS DENSITY FUNCTIONAL THEORY? more and more detailed physical information.More information about this hierarchy of functionals is given in Section 10.2. 1.6 THE QUANTUM CHEMISTRY TOURIST As you read about the approaches aside from DFT that exist for finding numeri- cal solutions of the Schrodinger equation,it is likely that you will rapidly encounter a bewildering array of acronyms.This experience could be a little bit like visiting a sophisticated city in an unfamiliar country.You may recog- nize that this new city is beautiful,and you definitely wish to appreciate its merits,but you are not planning to live there permanently.You could spend years in advance of your trip studying the language,history,culture,and geography of the country before your visit,but most likely for a brief visit you are more interested in talking with some friends who have already visited there,reading a few travel guides,browsing a phrase book,and perhaps trying to identify a few good local restaurants.This section aims to present an overview of quantum chemical methods on the level of a phrase book or travel guide. 1.6.1 Localized and Spatially Extended Functions One useful way to classify quantum chemistry calculations is according to the types of functions they use to represent their solutions.Broadly speaking, these methods use either spatially localized functions or spatially extended functions.As an example of a spatially localized function,Fig.1.1 shows the function f(x)=fi(x)+f(x)+f3(x), (1.9) where fi(x)=exp(x2), f2(x)=x2exp(x2/2), f=02(1x)2exp(x2/4), Figure 1.1 also shows fi,f2,and f.All of these functions rapidly approach zero for large values ofx.Functions like this are entirely appropriate for representing the wave function or electron density of an isolated atom.This example incorporates the idea that we can combine multiple individual func- tions with different spatial extents,symmetries,and so on to define an overall function.We could include more information in this final function by includ- ing more individual functions within its definition.Also,we could build up functions that describe multiple atoms simply by using an appropriate set of localized functions for each individual atom

more and more detailed physical information. More information about this hierarchy of functionals is given in Section 10.2. 1.6 THE QUANTUM CHEMISTRY TOURIST As you read about the approaches aside from DFT that exist for finding numeri￾cal solutions of the Schro¨dinger equation, it is likely that you will rapidly encounter a bewildering array of acronyms. This experience could be a little bit like visiting a sophisticated city in an unfamiliar country. You may recog￾nize that this new city is beautiful, and you definitely wish to appreciate its merits, but you are not planning to live there permanently. You could spend years in advance of your trip studying the language, history, culture, and geography of the country before your visit, but most likely for a brief visit you are more interested in talking with some friends who have already visited there, reading a few travel guides, browsing a phrase book, and perhaps trying to identify a few good local restaurants. This section aims to present an overview of quantum chemical methods on the level of a phrase book or travel guide. 1.6.1 Localized and Spatially Extended Functions One useful way to classify quantum chemistry calculations is according to the types of functions they use to represent their solutions. Broadly speaking, these methods use either spatially localized functions or spatially extended functions. As an example of a spatially localized function, Fig. 1.1 shows the function f(x) ¼ f1(x) þ f2(x) þ f3(x), (1:9) where f1(x) ¼ exp( x2), f2(x) ¼ x2 exp( x2=2), f3(x) ¼ 1 10 x2(1 x) 2 exp( x2=4). Figure 1.1 also shows f1, f2, and f3. All of these functions rapidly approach zero for large values of jxj. Functions like this are entirely appropriate for representing the wave function or electron density of an isolated atom. This example incorporates the idea that we can combine multiple individual func￾tions with different spatial extents, symmetries, and so on to define an overall function. We could include more information in this final function by includ￾ing more individual functions within its definition. Also, we could build up functions that describe multiple atoms simply by using an appropriate set of localized functions for each individual atom. 16 WHAT IS DENSITY FUNCTIONAL THEORY?