22 WHAT IS DENSITY FUNCTIONAL THEORY? single-electron equation are found,X;(x)for j 1,...,N,and the total wave function is formed from the Slater determinant of these spin orbitals. To actually solve the single-electron equation in a practical calculation, we have to define the spin orbitals using a finite amount of information since we cannot describe an arbitrary continuous function on a computer. To do this,we define a finite set of functions that can be added together to approximate the exact spin orbitals.If our finite set of functions is written as φ1(x),φ2(x),.,Φx(x),then we can approximate the spin orbitals as X(x) (1.16) When using this expression,we only need to find the expansion coefficients, oji,for i 1,...,K and j 1,...,N to fully define all the spin orbitals that are used in the HF method.The set of functions (x),2(x),...,(x)is called the basis set for the calculation.Intuitively,you can guess that using a larger basis set (i.e.,increasing K)will increase the accuracy of the calcu- lation but also increase the amount of effort needed to find a solution. Similarly,choosing basis functions that are very similar to the types of spin orbitals that actually appear in real materials will improve the accuracy of an HF calculation.As we hinted at in Section 1.6.1,the characteristics of these functions can differ depending on the type of material that is being considered. We now have all the pieces in place to perform an HF calculation-a basis set in which the individual spin orbitals are expanded,the equations that the spin orbitals must satisfy,and a prescription for forming the final wave func- tion once the spin orbitals are known.But there is one crucial complication left to deal with;one that also appeared when we discussed the Kohn-Sham equations in Section 1.4.To find the spin orbitals we must solve the single- electron equations.To define the Hartree potential in the single-electron equations,we must know the electron density.But to know the electron den- sity,we must define the electron wave function,which is found using the indi- vidual spin orbitals!To break this circle,an HF calculation is an iterative procedure that can be outlined as follows: L.Make an initial estimate of the spin orbitalsX(x)=∑f,ajnφ，(x)by specifying the expansion coefficients,aji 2.From the current estimate of the spin orbitals,define the electron density,n(r). 3.Using the electron density from step 2,solve the single-electron equations for the spin orbitals
single-electron equation are found, xj (x) for j 1, ... , N, and the total wave function is formed from the Slater determinant of these spin orbitals. To actually solve the single-electron equation in a practical calculation, we have to define the spin orbitals using a finite amount of information since we cannot describe an arbitrary continuous function on a computer. To do this, we define a finite set of functions that can be added together to approximate the exact spin orbitals. If our finite set of functions is written as f1(x), f2(x), ... , fK(x), then we can approximate the spin orbitals as xj (x) ¼ X K i 1 aj,ifi(x): (1:16) When using this expression, we only need to find the expansion coefficients, aj,i, for i 1, ... , K and j 1, ... , N to fully define all the spin orbitals that are used in the HF method. The set of functions f1(x), f2(x), ... , fK(x) is called the basis set for the calculation. Intuitively, you can guess that using a larger basis set (i.e., increasing K) will increase the accuracy of the calculation but also increase the amount of effort needed to find a solution. Similarly, choosing basis functions that are very similar to the types of spin orbitals that actually appear in real materials will improve the accuracy of an HF calculation. As we hinted at in Section 1.6.1, the characteristics of these functions can differ depending on the type of material that is being considered. We now have all the pieces in place to perform an HF calculation—a basis set in which the individual spin orbitals are expanded, the equations that the spin orbitals must satisfy, and a prescription for forming the final wave function once the spin orbitals are known. But there is one crucial complication left to deal with; one that also appeared when we discussed the Kohn –Sham equations in Section 1.4. To find the spin orbitals we must solve the singleelectron equations. To define the Hartree potential in the single-electron equations, we must know the electron density. But to know the electron density, we must define the electron wave function, which is found using the individual spin orbitals! To break this circle, an HF calculation is an iterative procedure that can be outlined as follows: 1. Make an initial estimate of the spin orbitals xj (x) ¼ PK i 1 aj,ifi(x) by specifying the expansion coefficients, aj,i: 2. From the current estimate of the spin orbitals, define the electron density, n(r0 ): 3. Using the electron density from step 2, solve the single-electron equations for the spin orbitals. 22 WHAT IS DENSITY FUNCTIONAL THEORY?
1.6 THE QUANTUM CHEMISTRY TOURIST 23 4.If the spin orbitals found in step 3 are consistent with orbitals used in step 2,then these are the solutions to the HF problem we set out to calculate.If not,then a new estimate for the spin orbitals must be made and we then return to step 2. This procedure is extremely similar to the iterative method we outlined in Section 1.4 for solving the Kohn-Sham equations within a DFT calculation. Just as in our discussion in Section 1.4,we have glossed over many details that are of great importance for actually doing an HF calculation.To identify just a few of these details:How do we decide if two sets of spin orbitals are similar enough to be called consistent?How can we update the spin orbitals in step 4 so that the overall calculation will actually converge to a solution?How large should a basis set be?How can we form a useful initial estimate of the spin orbitals?How do we efficiently find the expansion coefficients that define the solutions to the single-electron equations?Delving into the details of these issues would take us well beyond our aim in this section of giving an overview of quantum chemistry methods,but we hope that you can appreciate that reasonable answers to each of these questions can be found that allow HF calculations to be performed for physically interesting materials. 1.6.4 Beyond Hartree-Fock The Hartree-Fock method provides an exact description of electron exchange. This means that wave functions from HF calculations have exactly the same properties when the coordinates of two or more electrons are exchanged as the true solutions of the full Schrodinger equation.If HF calculations were possible using an infinitely large basis set,the energy of N electrons that would be calculated is known as the Hartree-Fock limit.This energy is not the same as the energy for the true electron wave function because the HF method does not correctly describe how electrons influence other electrons.More succinctly,the HF method does not deal with electron correlations. As we hinted at in the previous sections,writing down the physical laws that govern electron correlation is straightforward,but finding an exact description of electron correlation is intractable for any but the simplest systems.For the purposes of quantum chemistry,the energy due to electron correlation is defined in a specific way:the electron correlation energy is the difference between the Hartree-Fock limit and the true (non-relativistic)ground-state energy.Quantum chemistry approaches that are more sophisticated than the HF method for approximately solving the Schrodinger equation capture some part of the electron correlation energy by improving in some way upon one of the assumptions that were adopted in the Hartree-Fock approach
4. If the spin orbitals found in step 3 are consistent with orbitals used in step 2, then these are the solutions to the HF problem we set out to calculate. If not, then a new estimate for the spin orbitals must be made and we then return to step 2. This procedure is extremely similar to the iterative method we outlined in Section 1.4 for solving the Kohn –Sham equations within a DFT calculation. Just as in our discussion in Section 1.4, we have glossed over many details that are of great importance for actually doing an HF calculation. To identify just a few of these details: How do we decide if two sets of spin orbitals are similar enough to be called consistent? How can we update the spin orbitals in step 4 so that the overall calculation will actually converge to a solution? How large should a basis set be? How can we form a useful initial estimate of the spin orbitals? How do we efficiently find the expansion coefficients that define the solutions to the single-electron equations? Delving into the details of these issues would take us well beyond our aim in this section of giving an overview of quantum chemistry methods, but we hope that you can appreciate that reasonable answers to each of these questions can be found that allow HF calculations to be performed for physically interesting materials. 1.6.4 Beyond Hartree –Fock The Hartree –Fock method provides an exact description of electron exchange. This means that wave functions from HF calculations have exactly the same properties when the coordinates of two or more electrons are exchanged as the true solutions of the full Schro¨dinger equation. If HF calculations were possible using an infinitely large basis set, the energy of N electrons that would be calculated is known as the Hartree –Fock limit. This energy is not the same as the energy for the true electron wave function because the HF method does not correctly describe how electrons influence other electrons. More succinctly, the HF method does not deal with electron correlations. As we hinted at in the previous sections, writing down the physical laws that govern electron correlation is straightforward, but finding an exact description of electron correlation is intractable for any but the simplest systems. For the purposes of quantum chemistry, the energy due to electron correlation is defined in a specific way: the electron correlation energy is the difference between the Hartree –Fock limit and the true (non-relativistic) ground-state energy. Quantum chemistry approaches that are more sophisticated than the HF method for approximately solving the Schro¨dinger equation capture some part of the electron correlation energy by improving in some way upon one of the assumptions that were adopted in the Hartree –Fock approach. 1.6 THE QUANTUM CHEMISTRY TOURIST 23
24 WHAT IS DENSITY FUNCTIONAL THEORY? How do more advanced quantum chemical approaches improve on the HF method?The approaches vary,but the common goal is to include a description of electron correlation.Electron correlation is often described by "mixing"into the wave function some configurations in which electrons have been excited or promoted from lower energy to higher energy orbitals.One group of methods that does this are the single-determinant methods in which a single Slater determinant is used as the reference wave function and excitations are made from that wave function.Methods based on a single reference determinant are formally known as "post-Hartree-Fock"methods.These methods include configuration interaction(CD,coupled cluster (CC),Moller-Plesset perturbation theory (MP),and the quadratic configuration interaction (QCD) approach.Each of these methods has multiple variants with names that describe salient details of the methods.For example,CCSD calculations are coupled-cluster calculations involving excitations of single electrons (S), and pairs of electrons (double-D),while CCSDT calculations further include excitations of three electrons(triples-T).Moller-Plesset perturbation theory is based on adding a small perturbation (the correlation potential)to a zero- order Hamiltonian (the HF Hamiltonian,usually).In the Moller-Plesset perturbation theory approach,a number is used to indicate the order of the perturbation theory,so MP2 is the second-order theory and so on. Another class of methods uses more than one Slater determinant as the reference wave function.The methods used to describe electron correlation within these calculations are similar in some ways to the methods listed above These methods include multiconfigurational self-consistent field (MCSCF), multireference single and double configuration interaction (MRDCD,and N-electron valence state perturbation theory (NEVPT)methods.s The classification of wave-function-based methods has two distinct com- ponents:the level of theory and the basis set.The level of theory defines the approximations that are introduced to describe electron-electron interactions This is described by the array of acronyms introduced in the preceding para- graphs that describe various levels of theory.It has been suggested,only half-jokingly,that a useful rule for assessing the accuracy of a quantum chem- istry calculation is that"the longer the acronym,the better the level of theory." The second,and equally important,component in classifying wave-function- based methods is the basis set.In the simple example we gave in Section 1.6.1 of a spatially localized function,we formed an overall function by adding together three individual functions.If we were aiming to approximate a par- ticular function in this way,for example,the solution of the Schrodinger sThis may be a good time to remind yourself that this overview of quantum chemistry is meant to act something like a phrase book or travel guide for a foreign city.Details of the methods listed here may be found in the Further Reading section at the end of this chapter
How do more advanced quantum chemical approaches improve on the HF method? The approaches vary, but the common goal is to include a description of electron correlation. Electron correlation is often described by “mixing” into the wave function some configurations in which electrons have been excited or promoted from lower energy to higher energy orbitals. One group of methods that does this are the single-determinant methods in which a single Slater determinant is used as the reference wave function and excitations are made from that wave function. Methods based on a single reference determinant are formally known as “post –Hartree –Fock” methods. These methods include configuration interaction (CI), coupled cluster (CC), Møller –Plesset perturbation theory (MP), and the quadratic configuration interaction (QCI) approach. Each of these methods has multiple variants with names that describe salient details of the methods. For example, CCSD calculations are coupled-cluster calculations involving excitations of single electrons (S), and pairs of electrons (double—D), while CCSDT calculations further include excitations of three electrons (triples—T). Møller –Plesset perturbation theory is based on adding a small perturbation (the correlation potential) to a zeroorder Hamiltonian (the HF Hamiltonian, usually). In the Møller –Plesset perturbation theory approach, a number is used to indicate the order of the perturbation theory, so MP2 is the second-order theory and so on. Another class of methods uses more than one Slater determinant as the reference wave function. The methods used to describe electron correlation within these calculations are similar in some ways to the methods listed above. These methods include multiconfigurational self-consistent field (MCSCF), multireference single and double configuration interaction (MRDCI), and N-electron valence state perturbation theory (NEVPT) methods.§ The classification of wave-function-based methods has two distinct components: the level of theory and the basis set. The level of theory defines the approximations that are introduced to describe electron –electron interactions. This is described by the array of acronyms introduced in the preceding paragraphs that describe various levels of theory. It has been suggested, only half-jokingly, that a useful rule for assessing the accuracy of a quantum chemistry calculation is that “the longer the acronym, the better the level of theory.”6 The second, and equally important, component in classifying wave-functionbased methods is the basis set. In the simple example we gave in Section 1.6.1 of a spatially localized function, we formed an overall function by adding together three individual functions. If we were aiming to approximate a particular function in this way, for example, the solution of the Schro¨dinger § This may be a good time to remind yourself that this overview of quantum chemistry is meant to act something like a phrase book or travel guide for a foreign city. Details of the methods listed here may be found in the Further Reading section at the end of this chapter. 24 WHAT IS DENSITY FUNCTIONAL THEORY?
1.6 THE QUANTUM CHEMISTRY TOURIST 25 equation,we could always achieve this task more accurately by using more functions in our sum.Using a basis set with more functions allows a more accurate representation of the true solution but also requires more compu- tational effort since the numerical coefficients defining the magnitude of each function's contribution to the net function must be calculated.Just as there are multiple levels of theory that can be used,there are many possible ways to form basis sets. To illustrate the role of the level of theory and the basis set,we will look at two properties of a molecule of CH4,the C-H bond length and the ionization energy.Experimentally,the C-H bond length is 1.094A7 and the ionization energy for methane is 12.61 eV.First,we list these quantities calculated with four different levels of theory using the same basis set in Table 1.1.Three of the levels of theory shown in this table are wave-function-based,namely HF, MP2,and CCSD.We also list results from a DFT calculation using the most popular DFT functional for isolated molecules,that is,the B3LYP functional. (We return at the end of this section to the characteristics of this functional.) The table also shows the computational time needed for each calculation nor- malized by the time for the HF calculation.An important observation from this column is that the computational time for the HF and DFT calculations are approximately the same-this is a quite general result.The higher levels of theory,particularly the CCSD calculation,take considerably more compu- tational time than the HF (or DFT)calculations. All of the levels of theory listed in Table 1.1 predict the C-H bond length with accuracy within 1%.One piece of cheering information from Table 1.1 is that the DFT method predicts this bond length as accurately as the much more computationally expensive CCSD approach.The error in the ionization energy predicted by HF is substantial,but all three of the other methods give better predictions.The higher levels of theory (MP2 and CCSD)give considerably more accurate results for this quantity than DFT. Now we look at the properties of CH4 predicted by a set of calculations in which the level of theory is fixed and the size of the basis set is varied. TABLE 1.1 Computed Properties of CH Molecule for Four Levels of Theory Using pVTZ Basis Set Level of CH Percent lonization Percent Relative Theory (A) Error (ev) Error Time HF 1.085 0.8 11.49 8.9 1 DFT (B3LYP) 1.088 0.5 12.46 1.2 1 MP2 1.085 0.8 12.58 0.2 2 CCSD 1.088 0.5 12.54 0.5 18 Erors are defined relative to the experimental value
equation, we could always achieve this task more accurately by using more functions in our sum. Using a basis set with more functions allows a more accurate representation of the true solution but also requires more computational effort since the numerical coefficients defining the magnitude of each function’s contribution to the net function must be calculated. Just as there are multiple levels of theory that can be used, there are many possible ways to form basis sets. To illustrate the role of the level of theory and the basis set, we will look at two properties of a molecule of CH4, the C –H bond length and the ionization energy. Experimentally, the C –H bond length is 1.094 A˚ 7 and the ionization energy for methane is 12.61 eV. First, we list these quantities calculated with four different levels of theory using the same basis set in Table 1.1. Three of the levels of theory shown in this table are wave-function-based, namely HF, MP2, and CCSD. We also list results from a DFT calculation using the most popular DFT functional for isolated molecules, that is, the B3LYP functional. (We return at the end of this section to the characteristics of this functional.) The table also shows the computational time needed for each calculation normalized by the time for the HF calculation. An important observation from this column is that the computational time for the HF and DFT calculations are approximately the same—this is a quite general result. The higher levels of theory, particularly the CCSD calculation, take considerably more computational time than the HF (or DFT) calculations. All of the levels of theory listed in Table 1.1 predict the C –H bond length with accuracy within 1%. One piece of cheering information from Table 1.1 is that the DFT method predicts this bond length as accurately as the much more computationally expensive CCSD approach. The error in the ionization energy predicted by HF is substantial, but all three of the other methods give better predictions. The higher levels of theory (MP2 and CCSD) give considerably more accurate results for this quantity than DFT. Now we look at the properties of CH4 predicted by a set of calculations in which the level of theory is fixed and the size of the basis set is varied. TABLE 1.1 Computed Properties of CH4 Molecule for Four Levels of Theory Using pVTZ Basis Seta Level of Theory C H (A˚ ) Percent Error Ionization (eV) Percent Error Relative Time HF 1.085 0.8 11.49 8.9 1 DFT (B3LYP) 1.088 0.5 12.46 1.2 1 MP2 1.085 0.8 12.58 0.2 2 CCSD 1.088 0.5 12.54 0.5 18 a Errors are defined relative to the experimental value. 1.6 THE QUANTUM CHEMISTRY TOURIST 25
26 WHAT IS DENSITY FUNCTIONAL THEORY? TABLE 1.2 Properties of CH Calculated Using DFT(B3LYP)with Four Different Basis Sets" Number of Basis C H Percent lonization Percent Relative Basis Set Functions (A) Error (ev) Error Time STO 3G 27 1.097 0.3 12.08 4.2 cc pVDZ 61 1.100 0.6 12.34 2.2 1 cc pVTZ 121 1.088 0.5 12.46 1.2 2 cc pVQZ 240 1.088 0.5 12.46 1.2 13 "Errors are defined relative to the experimental value.Time is defined relative to the STO 3G calculation. Table 1.2 contains results of this kind using DFT calculations with the B3LYP functional in each case.There is a complicated series of names associated with different basis sets.Without going into the details,let us just say that STO-3G is a very common "minimal"basis set while cc-pVDZ,cc-pVTZ,and cc- pVQZ(D stands for double,T for triple,etc.)is a popular series of basis sets that have been carefully developed to be numerically efficient for molecu- lar calculations.The table lists the number of basis functions used in each cal- culation and also the computational time relative to the most rapid calculation. All of the basis sets listed in Table 1.2 give C-H bond lengths that are within 1%of the experimental value.The ionization energy,however,becomes sig- nificantly more accurate as the size of the basis set becomes larger. One other interesting observation from Table 1.2 is that the results for the two largest basis sets,pVTZ and pVQZ,are identical(at least to the numerical precision we listed in the table).This occurs when the basis sets include enough functions to accurately describe the solution of the Schrodinger equation,and when it occurs the results are said to be"converged with respect to basis set."When it happens,this is a good thing!An unfortunate fact of nature is that a basis set that is large enough for one level of theory,say DFT,is not necessarily large enough for higher levels of theory.So the results in Table 1.2 do not imply that the pVTZ basis set used for the CCSD calcu- lations in Table 1.1 were converged with respect to basis set. In order to use wave-function-based methods to converge to the true sol- ution of the Schrodinger equation,it is necessary to simultaneously use a high level of theory and a large basis set.Unfortunately,this approach is only feasible for calculations involving relatively small numbers of atoms because the computational expense associated with these calculations increases rapidly with the level of theory and the number of basis functions. For a basis set with N functions,for example,the computational expense of a conventional HF calculation typically requires ~Noperations,while a con- ventional coupled-cluster calculation requires~Noperations.Advances have been made that improve the scaling of both HF and post-HF calculations.Even with these improvements,however you can appreciate the problem with
Table 1.2 contains results of this kind using DFT calculations with the B3LYP functional in each case. There is a complicated series of names associated with different basis sets. Without going into the details, let us just say that STO-3G is a very common “minimal” basis set while cc-pVDZ, cc-pVTZ, and ccpVQZ (D stands for double, T for triple, etc.) is a popular series of basis sets that have been carefully developed to be numerically efficient for molecular calculations. The table lists the number of basis functions used in each calculation and also the computational time relative to the most rapid calculation. All of the basis sets listed in Table 1.2 give C –H bond lengths that are within 1% of the experimental value. The ionization energy, however, becomes significantly more accurate as the size of the basis set becomes larger. One other interesting observation from Table 1.2 is that the results for the two largest basis sets, pVTZ and pVQZ, are identical (at least to the numerical precision we listed in the table). This occurs when the basis sets include enough functions to accurately describe the solution of the Schro¨dinger equation, and when it occurs the results are said to be “converged with respect to basis set.” When it happens, this is a good thing! An unfortunate fact of nature is that a basis set that is large enough for one level of theory, say DFT, is not necessarily large enough for higher levels of theory. So the results in Table 1.2 do not imply that the pVTZ basis set used for the CCSD calculations in Table 1.1 were converged with respect to basis set. In order to use wave-function-based methods to converge to the true solution of the Schro¨dinger equation, it is necessary to simultaneously use a high level of theory and a large basis set. Unfortunately, this approach is only feasible for calculations involving relatively small numbers of atoms because the computational expense associated with these calculations increases rapidly with the level of theory and the number of basis functions. For a basis set with N functions, for example, the computational expense of a conventional HF calculation typically requires N4 operations, while a conventional coupled-cluster calculation requires N7 operations. Advances have been made that improve the scaling of both HF and post-HF calculations. Even with these improvements, however you can appreciate the problem with TABLE 1.2 Properties of CH4 Calculated Using DFT (B3LYP) with Four Different Basis Setsa Basis Set Number of Basis Functions C H (A˚ ) Percent Error Ionization (eV) Percent Error Relative Time STO 3G 27 1.097 0.3 12.08 4.2 1 cc pVDZ 61 1.100 0.6 12.34 2.2 1 cc pVTZ 121 1.088 0.5 12.46 1.2 2 cc pVQZ 240 1.088 0.5 12.46 1.2 13 a Errors are defined relative to the experimental value. Time is defined relative to the STO 3G calculation. 26 WHAT IS DENSITY FUNCTIONAL THEORY?