1.6 THE QUANTUM CHEMISTRY TOURIST 17 2.0 1.8 1.6 f(x) 1.4 1.2 41.0 0.8 0.6 0.4 0.2 0.0 -10-8-6 0 6810 Figure 1.1 Example of spatially localized functions defined in the text. Spatially localized functions are an extremely useful framework for thinking about the quantum chemistry of isolated molecules because the wave functions of isolated molecules really do decay to zero far away from the molecule. But what if we are interested in a bulk material such as the atoms in solid silicon or the atoms beneath the surface of a metal catalyst?We could still use spatially localized functions to describe each atom and add up these func- tions to describe the overall material,but this is certainly not the only way for- ward.A useful alternative is to use periodic functions to describe the wave functions or electron densities.Figure 1.2 shows a simple example of this idea by plotting f(x)=fi(x)+(x)+f(x), where =sm2(孕）. =3o() f(x)=sin2(mx). The resulting function is periodic;that is f(x+4n)=f(x)
Spatially localized functions are an extremely useful framework for thinking about the quantum chemistry of isolated molecules because the wave functions of isolated molecules really do decay to zero far away from the molecule. But what if we are interested in a bulk material such as the atoms in solid silicon or the atoms beneath the surface of a metal catalyst? We could still use spatially localized functions to describe each atom and add up these functions to describe the overall material, but this is certainly not the only way forward. A useful alternative is to use periodic functions to describe the wave functions or electron densities. Figure 1.2 shows a simple example of this idea by plotting f(x) ¼ f1(x) þ f2(x) þ f3(x), where f1(x) ¼ sin2 px 4 , f2(x) ¼ 1 3 cos2 px 2 , f3(x) ¼ 1 10 sin2 (px): The resulting function is periodic; that is f(x þ 4n) ¼ f(x), Figure 1.1 Example of spatially localized functions defined in the text. 1.6 THE QUANTUM CHEMISTRY TOURIST 17
18 WHAT IS DENSITY FUNCTIONAL THEORY? 1.4 f(x) 1.2 1.0 0.8- 0.6 0.4 0.2 0.0 -8 6 -2 0 6 Figure 1.2 Example of spatially periodic functions defined in the text. for any integer n.This type of function is useful for describing bulk materials since at least for defect-free materials the electron density and wave function really are spatially periodic functions. Because spatially localized functions are the natural choice for isolated molecules,the quantum chemistry methods developed within the chemistry community are dominated by methods based on these functions.Conversely, because physicists have historically been more interested in bulk materials than in individual molecules,numerical methods for solving the Schrodinger equation developed in the physics community are dominated by spatially periodic functions.You should not view one of these approaches as "right" and the other as"wrong"as they both have advantages and disadvantages. 1.6.2 Wave-Function-Based Methods A second fundamental classification of quantum chemistry calculations can be made according to the quantity that is being calculated.Our introduction to DFT in the previous sections has emphasized that in DFT the aim is to com- pute the electron density,not the electron wave function.There are many methods,however,where the object of the calculation is to compute the full electron wave function.These wave-function-based methods hold a crucial advantage over DFT calculations in that there is a well-defined hierarchy of methods that,given infinite computer time,can converge to the exact solution of the Schrodinger equation.We cannot do justice to the breadth of this field in just a few paragraphs,but several excellent introductory texts are available
for any integer n. This type of function is useful for describing bulk materials since at least for defect-free materials the electron density and wave function really are spatially periodic functions. Because spatially localized functions are the natural choice for isolated molecules, the quantum chemistry methods developed within the chemistry community are dominated by methods based on these functions. Conversely, because physicists have historically been more interested in bulk materials than in individual molecules, numerical methods for solving the Schro¨dinger equation developed in the physics community are dominated by spatially periodic functions. You should not view one of these approaches as “right” and the other as “wrong” as they both have advantages and disadvantages. 1.6.2 Wave-Function-Based Methods A second fundamental classification of quantum chemistry calculations can be made according to the quantity that is being calculated. Our introduction to DFT in the previous sections has emphasized that in DFT the aim is to compute the electron density, not the electron wave function. There are many methods, however, where the object of the calculation is to compute the full electron wave function. These wave-function-based methods hold a crucial advantage over DFT calculations in that there is a well-defined hierarchy of methods that, given infinite computer time, can converge to the exact solution of the Schro¨dinger equation. We cannot do justice to the breadth of this field in just a few paragraphs, but several excellent introductory texts are available Figure 1.2 Example of spatially periodic functions defined in the text. 18 WHAT IS DENSITY FUNCTIONAL THEORY?
1.6 THE QUANTUM CHEMISTRY TOURIST 19 and are listed in the Further Reading section at the end of this chapter.The strong connections between DFT and wave-function-based methods and their importance together within science was recognized in 1998 when the Nobel prize in chemistry was awarded jointly to Walter Kohn for his work developing the foundations of DFT and John Pople for his groundbreaking work on developing a quantum chemistry computer code for calculating the electronic structure of atoms and molecules.It is interesting to note that this was the first time that a Nobel prize in chemistry or physics was awarded for the development of a numerical method (or more precisely,a class of numerical methods)rather than a distinct scientific discovery.Kohn's Nobel lecture gives a very readable description of the advantages and disadvantages of wave-function-based and DFT calculations.3 Before giving a brief discussion of wave-function-based methods,we must first describe the common ways in which the wave function is described. We mentioned earlier that the wave function of an N-particle system is an N-dimensional function.But what,exactly,is a wave function?Because we want our wave functions to provide a quantum mechanical description of a system of N electrons,these wave functions must satisfy several mathematical properties exhibited by real electrons.For example,the Pauli exclusion principle prohibits two electrons with the same spin from existing at the same physical location simultaneously.*We would,of course,like these properties to also exist in any approximate form of the wave function that we construct. 1.6.3 Hartree-Fock Method Suppose we would like to approximate the wave function of N electrons.Let us assume for the moment that the electrons have no effect on each other.If this is true,the Hamiltonian for the electrons may be written as hi (1.10) where h;describes the kinetic and potential energy of electron i.The full elec- tronic Hamiltonian we wrote down in Eq.(1.1)takes this form if we simply neglect electron-electron interactions.If we write down the Schrodinger Spin is a quantum mechanical property that does not appear in classical mechanics.An electron can have one of two distinct spins,spin up or spin down.The full specification of an electron's state must include both its location and its spin.The Pauli exclusion principle only applies to electrons with the same spin state
and are listed in the Further Reading section at the end of this chapter. The strong connections between DFT and wave-function-based methods and their importance together within science was recognized in 1998 when the Nobel prize in chemistry was awarded jointly to Walter Kohn for his work developing the foundations of DFT and John Pople for his groundbreaking work on developing a quantum chemistry computer code for calculating the electronic structure of atoms and molecules. It is interesting to note that this was the first time that a Nobel prize in chemistry or physics was awarded for the development of a numerical method (or more precisely, a class of numerical methods) rather than a distinct scientific discovery. Kohn’s Nobel lecture gives a very readable description of the advantages and disadvantages of wave-function-based and DFT calculations.5 Before giving a brief discussion of wave-function-based methods, we must first describe the common ways in which the wave function is described. We mentioned earlier that the wave function of an N-particle system is an N-dimensional function. But what, exactly, is a wave function? Because we want our wave functions to provide a quantum mechanical description of a system of N electrons, these wave functions must satisfy several mathematical properties exhibited by real electrons. For example, the Pauli exclusion principle prohibits two electrons with the same spin from existing at the same physical location simultaneously.‡ We would, of course, like these properties to also exist in any approximate form of the wave function that we construct. 1.6.3 Hartree –Fock Method Suppose we would like to approximate the wave function of N electrons. Let us assume for the moment that the electrons have no effect on each other. If this is true, the Hamiltonian for the electrons may be written as H ¼ X N i 1 hi, (1:10) where hi describes the kinetic and potential energy of electron i. The full electronic Hamiltonian we wrote down in Eq. (1.1) takes this form if we simply neglect electron–electron interactions. If we write down the Schro¨dinger ‡ Spin is a quantum mechanical property that does not appear in classical mechanics. An electron can have one of two distinct spins, spin up or spin down. The full specification of an electron’s state must include both its location and its spin. The Pauli exclusion principle only applies to electrons with the same spin state. 1.6 THE QUANTUM CHEMISTRY TOURIST 19
20 WHAT IS DENSITY FUNCTIONAL THEORY? equation for just one electron based on this Hamiltonian,the solutions would satisfy hx =EX. (1.11) The eigenfunctions defined by this equation are called spin orbitals.For each single-electron equation there are multiple eigenfunctions,so this defines a set of spin orbitals X (xi)(j 1,2,...)where xi is a vector of coordinates that defines the position of electron i and its spin state (up or down).We will denote the energy of spin orbital x(x)by E.It is useful to label the spin orbitals so that the orbital with j 1 has the lowest energy,the orbital with j 2 has the next highest energy,and so on.When the total Hamiltonian is simply a sum of one-electron operators,hi,it follows that the eigenfunctions of H are products of the one-electron spin orbitals: 4(x1,,Xw)=X(X1)X2X2)…Xw(Kw). (1.12) The energy of this wave function is the sum of the spin orbital energies,E= Ej++Eiv.We have already seen a brief glimpse of this approximation to the N-electron wave function,the Hartree product,in Section 1.3. Unfortunately,the Hartree product does not satisfy all the important criteria for wave functions.Because electrons are fermions,the wave function must change sign if two electrons change places with each other.This is known as the antisymmetry principle.Exchanging two electrons does not change the sign of the Hartree product,which is a serious drawback.We can obtain a better approximation to the wave function by using a Slater determinant. In a Slater determinant,the N-electron wave function is formed by combining one-electron wave functions in a way that satisfies the antisymmetry principle. This is done by expressing the overall wave function as the determinant of a matrix of single-electron wave functions.It is best to see how this works for the case of two electrons.For two electrons,the Slater determinant is 2)=Idet)x) 2 Xk(X1)Xx(X1) 2X:) X(x2X() (1.13) The coefficient of (1/2)is simply a normalization factor.This expression builds in a physical description of electron exchange implicitly;it changes sign if two electrons are exchanged.This expression has other advantages. For example,it does not distinguish between electrons and it disappears if two electrons have the same coordinates or if two of the one-electron wave functions are the same.This means that the Slater determinant satisfies
equation for just one electron based on this Hamiltonian, the solutions would satisfy hx ¼ Ex: (1:11) The eigenfunctions defined by this equation are called spin orbitals. For each single-electron equation there are multiple eigenfunctions, so this defines a set of spin orbitals xj (xi) ( j 1, 2, ...) where xi is a vector of coordinates that defines the position of electron i and its spin state (up or down). We will denote the energy of spin orbital xj (xi) by Ej. It is useful to label the spin orbitals so that the orbital with j 1 has the lowest energy, the orbital with j 2 has the next highest energy, and so on. When the total Hamiltonian is simply a sum of one-electron operators, hi, it follows that the eigenfunctions of H are products of the one-electron spin orbitals: c(x1, ... , xN) ¼ xj1 (x1)xj2 (x2) xjN (xN): (1:12) The energy of this wave function is the sum of the spin orbital energies, E ¼ Ej1 þþ EjN . We have already seen a brief glimpse of this approximation to the N-electron wave function, the Hartree product, in Section 1.3. Unfortunately, the Hartree product does not satisfy all the important criteria for wave functions. Because electrons are fermions, the wave function must change sign if two electrons change places with each other. This is known as the antisymmetry principle. Exchanging two electrons does not change the sign of the Hartree product, which is a serious drawback. We can obtain a better approximation to the wave function by using a Slater determinant. In a Slater determinant, the N-electron wave function is formed by combining one-electron wave functions in a way that satisfies the antisymmetry principle. This is done by expressing the overall wave function as the determinant of a matrix of single-electron wave functions. It is best to see how this works for the case of two electrons. For two electrons, the Slater determinant is c(x1, x2) ¼ 1 2 p det xj (x1) xj (x2) xk(x1) xk(x1) ¼ 1 2 p xj (x1)xk(x1) xj (x2)xk(x1) h i: (1:13) The coefficient of (1/ p2) is simply a normalization factor. This expression builds in a physical description of electron exchange implicitly; it changes sign if two electrons are exchanged. This expression has other advantages. For example, it does not distinguish between electrons and it disappears if two electrons have the same coordinates or if two of the one-electron wave functions are the same. This means that the Slater determinant satisfies 20 WHAT IS DENSITY FUNCTIONAL THEORY?
1.6 THE QUANTUM CHEMISTRY TOURIST 21 the conditions of the Pauli exclusion principle.The Slater determinant may be generalized to a system of N electrons easily;it is the determinant of an Nx N matrix of single-electron spin orbitals.By using a Slater deter- minant,we are ensuring that our method for solving the Schrodinger equation will include exchange.Unfortunately,this is not the only kind of electron correlation that we need to describe in order to arrive at good compu- tational accuracy. The description above may seem a little unhelpful since we know that in any interesting system the electrons interact with one another.The many different wave-function-based approaches to solving the Schrodinger equation differ in how these interactions are approximated.To understand the types of approxi- mations that can be used,it is worth looking at the simplest approach,the Hartree-Fock method,in some detail.There are also many similarities between Hartree-Fock calculations and the DFT calculations we have described in the previous sections,so understanding this method is a useful way to view these ideas from a slightly different perspective. In a Hartree-Fock (HF)calculation,we fix the positions of the atomic nuclei and aim to determine the wave function of N-interacting electrons. The first part of describing an HF calculation is to define what equations are solved.The Schrodinger equation for each electron is written as e+Wm+=6风 (1.14) The third term on the left-hand side is the same Hartree potential we saw in Eq.(1.5: VH(r)=e2 (r) d Jr rl (1.15) In plain language,this means that a single electron "feels"the effect of other electrons only as an average,rather than feeling the instantaneous repulsive forces generated as electrons become close in space.If you compare Eg.(1.14)with the Kohn-Sham equations,Eq.(1.5),you will notice that the only difference between the two sets of equations is the additional exchange-correlation potential that appears in the Kohn-Sham equations. To complete our description of the HF method,we have to define how the solutions of the single-electron equations above are expressed and how these solutions are combined to give the N-electron wave function.The HF approach assumes that the complete wave function can be approximated using a single Slater determinant.This means that the N lowest energy spin orbitals of the
the conditions of the Pauli exclusion principle. The Slater determinant may be generalized to a system of N electrons easily; it is the determinant of an N N matrix of single-electron spin orbitals. By using a Slater determinant, we are ensuring that our method for solving the Schro¨dinger equation will include exchange. Unfortunately, this is not the only kind of electron correlation that we need to describe in order to arrive at good computational accuracy. The description above may seem a little unhelpful since we know that in any interesting system the electrons interact with one another. The many different wave-function-based approaches to solving the Schro¨dinger equation differ in how these interactions are approximated. To understand the types of approximations that can be used, it is worth looking at the simplest approach, the Hartree –Fock method, in some detail. There are also many similarities between Hartree –Fock calculations and the DFT calculations we have described in the previous sections, so understanding this method is a useful way to view these ideas from a slightly different perspective. In a Hartree –Fock (HF) calculation, we fix the positions of the atomic nuclei and aim to determine the wave function of N-interacting electrons. The first part of describing an HF calculation is to define what equations are solved. The Schro¨dinger equation for each electron is written as h2 2m r2 þ V(r) þ VH(r) xj (x) ¼ Ejxj (x): (1:14) The third term on the left-hand side is the same Hartree potential we saw in Eq. (1.5): VH(r) ¼ e2 ð n(r0 ) jr r0 j d3 r0 : (1:15) In plain language, this means that a single electron “feels” the effect of other electrons only as an average, rather than feeling the instantaneous repulsive forces generated as electrons become close in space. If you compare Eq. (1.14) with the Kohn–Sham equations, Eq. (1.5), you will notice that the only difference between the two sets of equations is the additional exchange –correlation potential that appears in the Kohn –Sham equations. To complete our description of the HF method, we have to define how the solutions of the single-electron equations above are expressed and how these solutions are combined to give the N-electron wave function. The HF approach assumes that the complete wave function can be approximated using a single Slater determinant. This means that the N lowest energy spin orbitals of the 1.6 THE QUANTUM CHEMISTRY TOURIST 21