1.3 THE SCHRODINGER EQUATION 7 predicted using DFT and observed experimentally are in good agreement, giving a strong indication of the accuracy of these calculations. The dissociation reaction predicted by Umemoto et al.'s calculations has important implications for creating good models of planetary formation.At the simplest level,it gives new information about what materials exist inside large planets.The calculations predict,for example,that the center of Uranus or Neptune can contain MgSiO3,but that the cores of Jupiter or Saturn will not.At a more detailed level,the thermodynamic properties of the materials can be used to model phenomena such as convection inside planets. Umemoto et al.speculated that the dissociation reaction above might severely limit convection inside "dense-Saturn,"a Saturn-like planet that has been discovered outside the solar system with a mass of ~67 Earth masses. A legitimate concern about theoretical predictions like the reaction above is that it is difficult to envision how they can be validated against experimental data.Fortunately,DFT calculations can also be used to search for similar types of reactions that occur at pressures that are accessible experimentally.By using this approach,it has been predicted that NaMgF3 goes through a series of trans- formations similar to MgSiO3;namely,a perovskite to postperovskite transition at some pressure above ambient and then dissociation in NaF and MgF2 at higher pressures.This dissociation is predicted to occur for pressures around 0.4 Mbar, far lower than the equivalent pressure for MgSiO3.These predictions suggest an avenue for direct experimental tests of the transformation mechanism that DFT calculations have suggested plays a role in planetary formation. We could fill many more pages with research vignettes showing how DFT calculations have had an impact in many areas of science.Hopefully,these three examples give some flavor of the ways in which DFT calculations can have an impact on scientific understanding.It is useful to think about the common features between these three examples.All of them involve materials in their solid state,although the first example was principally concerned with the interface between a solid and a gas.Each example generated information about a physical problem that is controlled by the properties of materials on atomic length scales that would be (at best)extraordinarily challenging to probe experimentally.In each case,the calculations were used to give infor- mation not just about some theoretically ideal state,but instead to understand phenomena at temperatures,pressures,and chemical compositions of direct relevance to physical applications. 1.3 THE SCHRODINGER EQUATION By now we have hopefully convinced you that density functional theory is a useful and interesting topic.But what is it exactly?We begin with
predicted using DFT and observed experimentally are in good agreement, giving a strong indication of the accuracy of these calculations. The dissociation reaction predicted by Umemoto et al.’s calculations has important implications for creating good models of planetary formation. At the simplest level, it gives new information about what materials exist inside large planets. The calculations predict, for example, that the center of Uranus or Neptune can contain MgSiO3, but that the cores of Jupiter or Saturn will not. At a more detailed level, the thermodynamic properties of the materials can be used to model phenomena such as convection inside planets. Umemoto et al. speculated that the dissociation reaction above might severely limit convection inside “dense-Saturn,” a Saturn-like planet that has been discovered outside the solar system with a mass of 67 Earth masses. A legitimate concern about theoretical predictions like the reaction above is that it is difficult to envision how they can be validated against experimental data. Fortunately, DFT calculations can also be used to search for similar types of reactions that occur at pressures that are accessible experimentally. By using this approach, it has been predicted that NaMgF3 goes through a series of transformations similar to MgSiO3; namely, a perovskite to postperovskite transition at some pressure above ambient and then dissociation in NaF and MgF2 at higher pressures.4 This dissociation is predicted to occur for pressures around 0.4 Mbar, far lower than the equivalent pressure for MgSiO3. These predictions suggest an avenue for direct experimental tests of the transformation mechanism that DFT calculations have suggested plays a role in planetary formation. We could fill many more pages with research vignettes showing how DFT calculations have had an impact in many areas of science. Hopefully, these three examples give some flavor of the ways in which DFT calculations can have an impact on scientific understanding. It is useful to think about the common features between these three examples. All of them involve materials in their solid state, although the first example was principally concerned with the interface between a solid and a gas. Each example generated information about a physical problem that is controlled by the properties of materials on atomic length scales that would be (at best) extraordinarily challenging to probe experimentally. In each case, the calculations were used to give information not just about some theoretically ideal state, but instead to understand phenomena at temperatures, pressures, and chemical compositions of direct relevance to physical applications. 1.3 THE SCHRO¨ DINGER EQUATION By now we have hopefully convinced you that density functional theory is a useful and interesting topic. But what is it exactly? We begin with 1.3 THE SCHRO¨ DINGER EQUATION 7
WHAT IS DENSITY FUNCTIONAL THEORY? the observation that one of the most profound scientific advances of the twentieth century was the development of quantum mechanics and the repeated experimental observations that confirmed that this theory of matter describes,with astonishing accuracy,the universe in which we live. In this section,we begin a review of some key ideas from quantum mech- anics that underlie DFT (and other forms of computational chemistry).Our goal here is not to present a complete derivation of the techniques used in DFT.Instead,our goal is to give a clear,brief,introductory presentation of the most basic equations important for DFT.For the full story,there are a number of excellent texts devoted to quantum mechanics listed in the Further Reading section at the end of the chapter. Let us imagine a situation where we would like to describe the properties of some well-defined collection of atoms-you could think of an isolated molecule or the atoms defining the crystal of an interesting mineral.One of the fundamental things we would like to know about these atoms is their energy and,more importantly,how their energy changes if we move the atoms around.To define where an atom is,we need to define both where its nucleus is and where the atom's electrons are.A key observation in applying quantum mechanics to atoms is that atomic nuclei are much heavier than indi- vidual electrons;each proton or neutron in a nucleus has more than 1800 times the mass of an electron.This means,roughly speaking,that electrons respond much more rapidly to changes in their surroundings than nuclei can.As a result,we can split our physical question into two pieces.First,we solve, for fixed positions of the atomic nuclei,the equations that describe the electron motion.For a given set of electrons moving in the field of a set of nuclei,we find the lowest energy configuration,or state,of the electrons.The lowest energy state is known as the ground state of the electrons,and the separation of the nuclei and electrons into separate mathematical problems is the Born- Oppenheimer approximation.If we have M nuclei at positions R1,...,RM, then we can express the ground-state energy,E,as a function of the positions of these nuclei,E(R1,...,RM).This function is known as the adiabatic potential energy surface of the atoms.Once we are able to calculate this potential energy surface we can tackle the original problem posed above-how does the energy of the material change as we move its atoms around? One simple form of the Schrodinger equation-more precisely,the time- independent,nonrelativistic Schrodinger equation-you may be familiar with is H=Ev.This equation is in a nice form for putting on a T-shirt or a coffee mug,but to understand it better we need to define the quantities that appear in it.In this equation,H is the Hamiltonian operator and is a set of solutions,or eigenstates,of the Hamiltonian.Each of these solutions
the observation that one of the most profound scientific advances of the twentieth century was the development of quantum mechanics and the repeated experimental observations that confirmed that this theory of matter describes, with astonishing accuracy, the universe in which we live. In this section, we begin a review of some key ideas from quantum mechanics that underlie DFT (and other forms of computational chemistry). Our goal here is not to present a complete derivation of the techniques used in DFT. Instead, our goal is to give a clear, brief, introductory presentation of the most basic equations important for DFT. For the full story, there are a number of excellent texts devoted to quantum mechanics listed in the Further Reading section at the end of the chapter. Let us imagine a situation where we would like to describe the properties of some well-defined collection of atoms—you could think of an isolated molecule or the atoms defining the crystal of an interesting mineral. One of the fundamental things we would like to know about these atoms is their energy and, more importantly, how their energy changes if we move the atoms around. To define where an atom is, we need to define both where its nucleus is and where the atom’s electrons are. A key observation in applying quantum mechanics to atoms is that atomic nuclei are much heavier than individual electrons; each proton or neutron in a nucleus has more than 1800 times the mass of an electron. This means, roughly speaking, that electrons respond much more rapidly to changes in their surroundings than nuclei can. As a result, we can split our physical question into two pieces. First, we solve, for fixed positions of the atomic nuclei, the equations that describe the electron motion. For a given set of electrons moving in the field of a set of nuclei, we find the lowest energy configuration, or state, of the electrons. The lowest energy state is known as the ground state of the electrons, and the separation of the nuclei and electrons into separate mathematical problems is the Born – Oppenheimer approximation. If we have M nuclei at positions R1, ... , RM, then we can express the ground-state energy, E, as a function of the positions of these nuclei, E(R1, ... , RM). This function is known as the adiabatic potential energy surface of the atoms. Once we are able to calculate this potential energy surface we can tackle the original problem posed above—how does the energy of the material change as we move its atoms around? One simple form of the Schro¨dinger equation—more precisely, the timeindependent, nonrelativistic Schro¨dinger equation—you may be familiar with is Hc ¼ Ec. This equation is in a nice form for putting on a T-shirt or a coffee mug, but to understand it better we need to define the quantities that appear in it. In this equation, H is the Hamiltonian operator and c is a set of solutions, or eigenstates, of the Hamiltonian. Each of these solutions, 8 WHAT IS DENSITY FUNCTIONAL THEORY?
1.3 THE SCHRODINGER EQUATION 9 has an associated eigenvalue,E.a real number*that satisfies the eigenvalue equation.The detailed definition of the Hamiltonian depends on the physical system being described by the Schrodinger equation.There are several well-known examples like the particle in a box or a harmonic oscillator where the Hamiltonian has a simple form and the Schrodinger equation can be solved exactly.The situation we are interested in where multiple electrons are interacting with multiple nuclei is more complicated.In this case,a more complete description of the Schrodinger is 业=E (1.1) Here,m is the electron mass.The three terms in brackets in this equation define,in order,the kinetic energy of each electron,the interaction energy between each electron and the collection of atomic nuclei,and the interaction energy between different electrons.For the Hamiltonian we have chosen,is the electronic wave function,which is a function of each of the spatial coordi- nates of each of the N electrons,so=(r1,...,r),and E is the ground- state energy of the electrons."*The ground-state energy is independent of time,so this is the time-independent Schrodinger equation. Although the electron wave function is a function of each of the coordinates of all N electrons,it is possible to approximate as a product of individual electron wave functions,=(r(r),...,(r).This expression for the wave function is known as a Hartree product,and there are good motivations for approximating the full wave function into a product of individual one- electron wave functions in this fashion.Notice that N,the number of electrons, is considerably larger than M,the number of nuclei,simply because each atom has one nucleus and lots of electrons.If we were interested in a single molecule of CO2,the full wave function is a 66-dimensional function(3 dimensions for each of the 22 electrons).If we were interested in a nanocluster of 100 Pt atoms, the full wave function requires more the 23,000 dimensions!These numbers should begin to give you an idea about why solving the Schrodinger equation for practical materials has occupied many brilliant minds for a good fraction of a century. "The value of the functions are complex numbers,but the eigenvalues of the Schrodinger equation are real numbers. For clarity of presentation,we have neglected electron spin in our description.In a complete presentation,each electron is defined by three spatial variables and its spin. "The dynamics of electrons are defined by the time dependent Schrodinger equation, ih(/or)=H.The appearance of i=v 1 in this equation makes it clear that the wave func tion is a complex valued function,not a real valued function
cn, has an associated eigenvalue, En, a real number that satisfies the eigenvalue equation. The detailed definition of the Hamiltonian depends on the physical system being described by the Schro¨dinger equation. There are several well-known examples like the particle in a box or a harmonic oscillator where the Hamiltonian has a simple form and the Schro¨dinger equation can be solved exactly. The situation we are interested in where multiple electrons are interacting with multiple nuclei is more complicated. In this case, a more complete description of the Schro¨dinger is h2 2m X N i 1 r2 i þX N i 1 V(ri) þX N i 1 X j,i U(ri, rj) " #c ¼ Ec: (1:1) Here, m is the electron mass. The three terms in brackets in this equation define, in order, the kinetic energy of each electron, the interaction energy between each electron and the collection of atomic nuclei, and the interaction energy between different electrons. For the Hamiltonian we have chosen, c is the electronic wave function, which is a function of each of the spatial coordinates of each of the N electrons, so c ¼ c(r1, ... , rN), and E is the groundstate energy of the electrons. The ground-state energy is independent of time, so this is the time-independent Schro¨dinger equation.† Although the electron wave function is a function of each of the coordinates of all N electrons, it is possible to approximate c as a product of individual electron wave functions, c ¼ c1(r)c2(r), ... , cN(r). This expression for the wave function is known as a Hartree product, and there are good motivations for approximating the full wave function into a product of individual oneelectron wave functions in this fashion. Notice that N, the number of electrons, is considerably larger than M, the number of nuclei, simply because each atom has one nucleus and lots of electrons. If we were interested in a single molecule of CO2, the full wave function is a 66-dimensional function (3 dimensions for each of the 22 electrons). If we were interested in a nanocluster of 100 Pt atoms, the full wave function requires more the 23,000 dimensions! These numbers should begin to give you an idea about why solving the Schro¨dinger equation for practical materials has occupied many brilliant minds for a good fraction of a century. The value of the functions cn are complex numbers, but the eigenvalues of the Schro¨dinger equation are real numbers. For clarity of presentation, we have neglected electron spin in our description. In a complete presentation, each electron is defined by three spatial variables and its spin. † The dynamics of electrons are defined by the time dependent Schro¨dinger equation, ih(@c=@t) ¼ Hc. The appearance of i ¼ 1 p in this equation makes it clear that the wave func tion is a complex valued function, not a real valued function. 1.3 THE SCHRO¨ DINGER EQUATION 9
10 WHAT IS DENSITY FUNCTIONAL THEORY? The situation looks even worse when we look again at the Hamiltonian,H. The term in the Hamiltonian defining electron-electron interactions is the most critical one from the point of view of solving the equation.The form of this contribution means that the individual electron wave function we defined above,(r),cannot be found without simultaneously considering the individual electron wave functions associated with all the other electrons. In other words,the Schrodinger equation is a many-body problem. Although solving the Schrodinger equation can be viewed as the fundamen- tal problem of quantum mechanics,it is worth realizing that the wave function for any particular set of coordinates cannot be directly observed.The quantity that can (in principle)be measured is the probability that the N electrons are at a particular set of coordinates,r1,...,rN.This probability is equal to (r1,...,r)(r1,...,rN),where the asterisk indicates a complex conju- gate.A further point to notice is that in experiments we typically do not care which electron in the material is labeled electron 1,electron 2,and so on.Moreover,even if we did care,we cannot easily assign these labels. This means that the quantity of physical interest is really the probability that a set of Nelectrons in any order have coordinates r,...,rN.A closely related quantity is the density of electrons at a particular position in space,n(r).This can be written in terms of the individual electron wave functions as nr)=2r4c). (1.2) Here,the summation goes over all the individual electron wave functions that are occupied by electrons,so the term inside the summation is the probability that an electron in individual wave function(r)is located at position r.The factor of 2 appears because electrons have spin and the Pauli exclusion prin- ciple states that each individual electron wave function can be occupied by two separate electrons provided they have different spins.This is a purely quantum mechanical effect that has no counterpart in classical physics.The point of this discussion is that the electron density,n(r),which is a function of only three coordinates,contains a great amount of the information that is actually physically observable from the full wave function solution to the Schrodinger equation,which is a function of 3N coordinates. 1.4 DENSITY FUNCTIONAL THEORY-FROM WAVE FUNCTIONS TO ELECTRON DENSITY The entire field of density functional theory rests on two fundamental math- ematical theorems proved by Kohn and Hohenberg and the derivation of a
The situation looks even worse when we look again at the Hamiltonian, H. The term in the Hamiltonian defining electron –electron interactions is the most critical one from the point of view of solving the equation. The form of this contribution means that the individual electron wave function we defined above, ci(r), cannot be found without simultaneously considering the individual electron wave functions associated with all the other electrons. In other words, the Schro¨dinger equation is a many-body problem. Although solving the Schro¨dinger equation can be viewed as the fundamental problem of quantum mechanics, it is worth realizing that the wave function for any particular set of coordinates cannot be directly observed. The quantity that can (in principle) be measured is the probability that the N electrons are at a particular set of coordinates, r1, ... , rN. This probability is equal to c (r1, ... , rN)c(r1, ... , rN), where the asterisk indicates a complex conjugate. A further point to notice is that in experiments we typically do not care which electron in the material is labeled electron 1, electron 2, and so on. Moreover, even if we did care, we cannot easily assign these labels. This means that the quantity of physical interest is really the probability that a set of N electrons in any order have coordinates r1, ... , rN. A closely related quantity is the density of electrons at a particular position in space, n(r). This can be written in terms of the individual electron wave functions as n(r) ¼ 2 X i c i (r)ci(r): (1:2) Here, the summation goes over all the individual electron wave functions that are occupied by electrons, so the term inside the summation is the probability that an electron in individual wave function ci(r) is located at position r. The factor of 2 appears because electrons have spin and the Pauli exclusion principle states that each individual electron wave function can be occupied by two separate electrons provided they have different spins. This is a purely quantum mechanical effect that has no counterpart in classical physics. The point of this discussion is that the electron density, n(r), which is a function of only three coordinates, contains a great amount of the information that is actually physically observable from the full wave function solution to the Schro¨dinger equation, which is a function of 3N coordinates. 1.4 DENSITY FUNCTIONAL THEORY—FROM WAVE FUNCTIONS TO ELECTRON DENSITY The entire field of density functional theory rests on two fundamental mathematical theorems proved by Kohn and Hohenberg and the derivation of a 10 WHAT IS DENSITY FUNCTIONAL THEORY?
1.4 DENSITY FUNCTIONAL THEORY FROM WAVE FUNCTIONS 11 set of equations by Kohn and Sham in the mid-1960s.The first theorem,proved by Hohenberg and Kohn,is:The ground-state energy from Schrodinger's equation is a unique functional of the electron density. This theorem states that there exists a one-to-one mapping between the ground-state wave function and the ground-state electron density.To appreci- ate the importance of this result,you first need to know what a"functional"is. As you might guess from the name,a functional is closely related to the more familiar concept of a function.A function takes a value of a variable or vari- ables and defines a single number from those variables.A simple example of a function dependent on a single variable is f(x)=x2+1.A functional is similar,but it takes a function and defines a single number from the function. For example, f(x)dx, is a functional of the function f(x).If we evaluate this functional using f(x)=x2+1,we get F[f]=.So we can restate Hohenberg and Kohn's result by saying that the ground-state energy E can be expressed as E[n(r)], where n(r)is the electron density.This is why this field is known as density functional theory. Another way to restate Hohenberg and Kohn's result is that the ground-state electron density uniquely determines all properties,including the energy and wave function,of the ground state.Why is this result important?It means that we can think about solving the Schrodinger equation by finding a function of three spatial variables,the electron density,rather than a function of 3N vari- ables,the wave function.Here,by "solving the Schrodinger equation"we mean,to say it more precisely,finding the ground-state energy.So for a nanocluster of 100 Pd atoms the theorem reduces the problem from something with more than 23,000 dimensions to a problem with just 3 dimensions. Unfortunately,although the first Hohenberg-Kohn theorem rigorously proves that a functional of the electron density exists that can be used to solve the Schrodinger equation,the theorem says nothing about what the func- tional actually is.The second Hohenberg-Kohn theorem defines an important property of the functional:The electron density that minimizes the energy of the overall functional is the true electron density corresponding to the full sol- ution of the Schrodinger equation.If the "true"functional form were known, then we could vary the electron density until the energy from the functional is minimized,giving us a prescription for finding the relevant electron density. This variational principle is used in practice with approximate forms of the functional
set of equations by Kohn and Sham in the mid-1960s. The first theorem, proved by Hohenberg and Kohn, is: The ground-state energy from Schro¨dinger’s equation is a unique functional of the electron density. This theorem states that there exists a one-to-one mapping between the ground-state wave function and the ground-state electron density. To appreciate the importance of this result, you first need to know what a “functional” is. As you might guess from the name, a functional is closely related to the more familiar concept of a function. A function takes a value of a variable or variables and defines a single number from those variables. A simple example of a function dependent on a single variable is f(x) ¼ x2 þ 1. A functional is similar, but it takes a function and defines a single number from the function. For example, F[ f] ¼ ð 1 1 f(x) dx, is a functional of the function f(x). If we evaluate this functional using f(x) ¼ x2 þ 1, we get F[ f] ¼ 8 3. So we can restate Hohenberg and Kohn’s result by saying that the ground-state energy E can be expressed as E[n(r)], where n(r) is the electron density. This is why this field is known as density functional theory. Another way to restate Hohenberg and Kohn’s result is that the ground-state electron density uniquely determines all properties, including the energy and wave function, of the ground state. Why is this result important? It means that we can think about solving the Schro¨dinger equation by finding a function of three spatial variables, the electron density, rather than a function of 3N variables, the wave function. Here, by “solving the Schro¨dinger equation” we mean, to say it more precisely, finding the ground-state energy. So for a nanocluster of 100 Pd atoms the theorem reduces the problem from something with more than 23,000 dimensions to a problem with just 3 dimensions. Unfortunately, although the first Hohenberg –Kohn theorem rigorously proves that a functional of the electron density exists that can be used to solve the Schro¨dinger equation, the theorem says nothing about what the functional actually is. The second Hohenberg –Kohn theorem defines an important property of the functional: The electron density that minimizes the energy of the overall functional is the true electron density corresponding to the full solution of the Schro¨dinger equation. If the “true” functional form were known, then we could vary the electron density until the energy from the functional is minimized, giving us a prescription for finding the relevant electron density. This variational principle is used in practice with approximate forms of the functional. 1.4 DENSITY FUNCTIONAL THEORY FROM WAVE FUNCTIONS 11