The Limited Expansion of Diatomic Overlap Density Functional Theory (LEDO-DFT):

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1 The Limited Expansion of Diatomic Overlap Density Functional Theory (LEDO-DFT): Development and Implementation of Algorithms, Optimization of Auxiliary Orbitals and Benchmark Calculations Den Naturwissenschaftlichen Fakultäten der Friedrich Alexander-Universität Erlangen Nürnberg zur Erlangung des Doktorgrades vorgelegt von Andreas Walter Götz aus Nürnberg

2 Als Dissertation genehmigt von den Naturwissenschaftlichen Fakultäten der Universität Erlangen Nürnberg Tag der mündlichen Prüfung: 09. August 2005 Vorsitzender der Promotionskommission: Erstberichterstatter: Zweitberichterstatter: Prof. Dr. D.-P. Häder Prof. Dr. A. Görling Prof. Dr. P. Otto

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5 Ricordiamoci in grazia che il cercar la costituzione del mondo è de maggiori e de piú nobil problemi che sieno in natura... (Let us remember, please, that the search for the constitution of the world is one of the greatest and noblest problems presented by nature... ) Galileo Galilei, Dialogo sopra i due massimi sistemi del mondo (Dialogue concerning the two chief world systems), 1632.

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7 In memory of Prof. Dr. Bernd A. Heß

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9 For Pat

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11 Acknowledgements It is a great pleasure to be able to thank all the people who have helped me in various ways with this dissertation, and whose support has been invaluable over the past three and a half years. In the first place I would like to thank my supervisor and academic teacher Prof. Dr. Bernd A. Heß ( ) who offered me the opportunity to work in the interesting field of density fitting methods in density functional theory. He has always been a faithful source of knowledge and inspiration. I deeply appreciate his diverse and generous support and the excellent working conditions he provided at the Chair of Theoretical Chemistry. To Prof. Dr. Andreas Görling I would like to express my sincere gratitude for having me given the possibility to continue my work after the early death of Prof. Dr. Bernd A. Heß. Without his support, it would not have been possible to finish this thesis in its present form. Special thanks are due to Dr. Christian Kollmar who developed the theoretical framework on which this work is based. Many helpful and interesting ideas originated from the frequent fruitful discussions with him. His advice and scientific guidance have certainly been decisive for the success of this work. I am furthermore very grateful to him for critical comments on the first version of this manuscript. I would also like to thank my colleagues of the groups of Prof. Dr. Bernd A. Heß and Prof. Dr. Andreas Görling for the friendly atmosphere. I am grateful to Dr. Wolfgang Hieringer for proof-reading parts of the final version of this manuscript, to Dr. Nico van Eikema Hommes for his assistance with any computer problems and to our secretary Leo Steinbauer for her friendly help with any bureaucratic problems and the constant supply with delicious cakes and cookies. During the time of this thesis I have also been working on other subjects which are not presented in this thesis. In this context I would like to thank Dr. Carsten Kind for his collaboration on the chrysene project during which I learned much about computational quantum chemistry. I furthermore would like to thank Dr. Patrícia Pinto from the group of Prof. Dr. Ulrich Zenneck and Dr. Frank

12 Lauderbach from the group of Prof. Dr. Dieter Sellmann ( ) for the stimulating cooperations which showed how fruitful the interplay between experiment and theory can be. Financial support by the Graduiertenkolleg GRK 312 Homogener und heterogener Elektronentransfer of the University of Erlangen (Doktoranden- Stipendium, November 2001 to April 2002; fellowships for the participation in the Summer School in Molecular Physics and Quantum Chemistry, Oxford/UK, 2002 and the Winter School entitled Large Molecules: Linear Scaling and Related Electronic Structure Calculations Methods, Helsinki/Finland, 2002) and by the Deutscher Akademischer Austauschdienst (fellowship for the participation in the European Summer School in Quantum Chemistry, ESQC-03, Tjörnarp/Sweden, 2003) is greatly acknowledged. This work has been prepared in the time between November 2001 and June 2005 at the Chair of Theoretical Chemistry of the Friedrich Alexander-Universität Erlangen Nürnberg under the supervision of Prof. Dr. Bernd A. Heß ( July 17th 2004) and Prof. Dr. Andreas Görling.

13 Parts of this thesis have been published previously: [1] A. W. Götz, C. Kollmar, B. A. Hess. Analytical gradients for LEDO-DFT, Molec. Phys. 103, (2005). [2] A. W. Götz, C. Kollmar, B. A. Hess. Optimization of Auxiliary Basis Sets for the LEDO Expansion and a Projection Technique for LEDO-DFT, J. Comput. Chem. 26, (2005).

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15 Contents List of Abbreviations List of Symbols v vii 1 Introduction Background of this work Outline of this work Theory Density Functional Theory Computational Aspects of DFT Calculations Review of Density Fitting Methods Fit of an Arbitrary Charge Distribution Fit of the Complete Electron Density Fit of Overlap Densities Fit of Overlap Densities with a Restricted Expansion Basis The LEDO Expansion Measuring the Quality of the LEDO Fit The LEDO Expansion Basis Near-linear Dependences LEDO-DFT Analytical Gradients for LEDO-DFT A Projection Operator Formalism for LEDO-DFT Implementation Framework and General Features i

16 ii Contents 3.2 SCF Energy Calculation Integral Evaluation and Prescreening LEDO Expansion Coefficients Hartree Contribution Exchange-Correlation Contribution Analytical Gradients Integral Derivative Evaluation Hartree Contribution LEDO Contribution Exchange-Correlation Contribution The Projection Technique Numerical Results Optimization of Auxiliary Orbitals for the SVP Basis Set Preliminary Investigations Homonuclear Diatomic Overlap Densities Distance Dependence Heteronuclear Diatomic Overlap Densities Recommended Exponents Efficiency of the A Priori Elimination Accuracy of LEDO-DFT Small Molecules Larger Molecules Linear Alkanes Critical Cases Assessment of the Projection Technique Some Real Life Examples Efficiency of LEDO-DFT Summary 99 6 Outlook Zusammenfassung 107 A Auxiliary Orbitals for the atom pairs CF, CH and CP 113

17 Contents iii B Accuracy of LEDO-DFT for a test set of 142 small molecules 117 List of Tables 125 List of Figures 127 Bibliography 129

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19 List of Abbreviations 6-31G Pople s split-valence basis set au Hartree atomic units AO atomic orbital BP86 Becke Perdew exchange-correlation functional CC2 approximate second order coupled cluster model COO canonically orthogonalized orbital DFT density functional theory ERI electron repulsion integral GGA generalized gradient approximation GTF Gaussian type function HF Hartree Fock HOMO highest occupied molecular orbital KS Kohn Sham LCAO linear combination of atomic orbitals LDA local density approximation LEDO limited expansion of diatomic overlap LUMO lowest unoccupied molecular orbital MCSCF multi-configurational self-consistent field MO molecular orbital MP2 second order Møller Plesset perturbation theory PVM parallel virtual machine RI resolution of the identity RMS root-mean-square SCF self-consistent field STF Slater type function v

20 vi List of Abbreviations SVD SVP XC singular value decomposition Ahlrichs split-valence plus polarization basis set exchange-correlation

21 List of Symbols ψ(r) φ(r) χ(r) Ω(r) Λ(r) η(r) molecular orbital atomic orbital canonically orthogonalized orbital density fitting expansion function density fitting auxiliary function density fitting auxiliary orbital α, β perturbational parameters, e.g. a nuclear coordinate ζ Ξ(r) ρ(r) and ρ µν (r) d p and d µν p l N exponent of a basis function arbitrary charge distribution electron density and overlap density φ µ (r) φ µ (r) expansion coefficients for electron density and overlap densities angular momentum quantum number number of AO basis functions r position vector (x, y, z) r 12 r 1 r 2 W, W weight operator H KS, H KS Kohn Sham operator Q, Q projection operator P S f g f O g (f g) first order reduced density matrix overlap matrix dr f(r) g(r) dr1 dr 2 f(r 1 ) O(r 1, r 2 ) g(r 2 ) dr1 dr 2 f(r 1 ) r12 1 g(r 2 ) Laplace operator, norm of a difference vector W pq and V pq Ω p W Ω q and (Ω p Ω q ) vii

22 viii List of Symbols a µν p and b µν p Ω p W φ µ φ ν and (Ω p φ µ φ ν ) E h, v h (r), V h Hartree energy and potential E xc, v xc (r), V xc E ext, v ext (r), V ext exchange-correlation energy and potential external energy and potential i, j indices denoting MOs or basis functions of a shell µ, ν, κ, λ indices denoting AO basis functions or auxiliary orbitals k, l, m, n indices denoting shells of basis functions p, q indices denoting density fitting expansion functions A, B, C, D, K indices denoting atoms

23 Chapter 1 Introduction This thesis describes the development, implementation, and benchmarking of efficient algorithms for electronic structure calculations with the limited expansion of diatomic overlap density functional theory (LEDO-DFT) [1], a novel formalism in the framework of Kohn Sham DFT (KS-DFT) [2, 3] which exhibits a favorable scaling behavior. The general importance of quantum chemical methods, in particular of KS-DFT, and the relevance of the associated scaling behavior of the computational effort with system size are briefly sketched in this introduction followed by an outline of the contents of this work. 1.1 Background of this work Today there is little doubt that the foundation for all of low-energy physics, chemistry and biology lies in the quantum theory of electrons and atomic nuclei. Therefore, the equations of quantum mechanics should be attempted to be solved, if such complex processes as occurring in real materials shall be described with high precision. Unfortunately, these equations are far too complicated to be solved analytically for all but the simplest (and hence most trivial) systems. The only chance to bring the power of quantum mechanics to investigations in the field of chemistry and related sciences is to solve the equations in a numerical fashion by a computational modeling of the systems of interest. Electronic structure theory, i.e., the theoretical prediction of material properties from quantum chemical methods without recourse to empirical parameters, 1

24 2 Chapter 1. Introduction has advanced significantly in the recent years, serving basic science as well as applied research in numerous industries. In many fields, computer simulations have aided, stimulated, and sometimes even replaced experimental investigations. Nevertheless, many fundamental problems still continue to challenge scientists. The same complexity which precludes the exact analytical solution also results in a highly unfavorable scaling of computational effort and required resources. The computational demands of exact calculations, e.g., with the full configuration interaction (full CI) method, grow exponentially with the size of the system under investigation. Evidently, approximations that represent a reasonable compromise between efficiency and accuracy are needed if one desires to carry out electronic structure calculations on sizable molecules. A number of well-controlled approximations which do not sacrifice the predictive power of the parameter-free nature of quantum mechanical calculations but exhibit polynomial rather than exponential scaling are routinely available nowadays. Among these are popular wave function based quantum chemical methods which include the effects of electron correlation in an approximate fashion such as the second order Møller Plesset perturbation theory (MP2) and coupled cluster theory including all single, double and perturbative triple excitations (CCSD(T)). However, these correlated ab initio methods still exhibit a scaling behavior with system size of at least O(N 5 ). Here, N denotes the number of basis functions employed in the calculation which in general is proportional to the number of atoms or electrons. Doubling the size of the investigated system therefore results in an increase of the required computing resources by a factor of at least 32, thus severely limiting the applicability of these accurate methods to large molecules. This is where density functional theory (DFT) comes into play. The promise of DFT lies in its ability to fold into a local exchange-correlation operator the most difficult aspects of electronic structure theory. In the Kohn Sham formulation of DFT, the complicated many-electron Schrödinger equation is replaced by an equivalent set of self-consistent one-electron equations [3]. It is precisely this feature, of treating complex many-body systems in principle exactly with only the computational expense of a self-consistent-field (SCF) calculation, which has been so appealing in KS-DFT. In the last decade electronic structure methods based on KS-DFT have evolved to a powerful quantum chemical tool and are

25 1.1. Background of this work 3 playing an increasing role in the ongoing effort for the determination of accurate properties of large molecular systems. Numerous electrical, magnetic, and structural properties of molecules and materials have been calculated using DFT, and the extent to which DFT has contributed to the science of molecules is reflected by the 1998 Nobel Prize in Chemistry, which was awarded to Walter Kohn [4], the founding father of DFT, and John Pople [5], who was instrumental in implementing quantum chemical methods in computational chemistry. Nowadays, sophisticated density functionals are available in widely accessible commercial and non-commercial computer programs. Scientists applying DFT can rely on a wealth of information that has been gathered in the past and which documents the applicability and the reliability of DFT methods for molecular calculations. Especially for transition metal complexes, accuracies have been achieved that are comparable or superior to correlated wave function based theories [6 8]. A straightforward implementation of KS-DFT has an O(N 4 ) scaling behavior. Reduction to O(N 3 ) is possible if density fitting is applied, i.e., if the electron density is expanded into an auxiliary basis set. At this point it is important to distinguish between formal and asymptotic (or effective) scaling, because the actual computational cost as a function of system size is not necessarily related to the formal scaling properties. The formal scaling behavior results from a direct implementation of the equations appearing in a given formalism. However, in many cases technical tricks can be employed which exploit physical properties of the system under investigation like, e.g., that certain interactions between regions separated far enough are negligible. The prevention of the calculation of such negligible quantities can lead to a lower effective scaling behavior in the limit of large molecules which therefore is referred to as asymptotic scaling. Furthermore, a prefactor is always associated with the scaling behavior. It is clear that this prefactor will vary for different formalisms even if they share the same scaling behavior and that it will also depend on the efficiency of the algorithms used for the implementation. For sufficiently large molecular systems it is always the asymptotic scaling behavior together with the associated prefactor that determines the real cost of a calculation. The asymptotic scaling behavior for the setup of the secular matrix is O(N 2 ) for KS-DFT, with and without density fitting approximations. Thus, for KS-DFT it is always the prefactor

26 4 Chapter 1. Introduction that determines the computational cost for extended systems. In the case of conventional density fitting the lower formal scaling leads to a smaller prefactor and speedups by more than a factor of 10 as compared to unapproximated KS- DFT calculations. Recently, a novel formalism denoted as LEDO-DFT with the nice property of a formal scaling behavior as low as O(N 2 ) for the setup of the secular matrix has been presented by Kollmar and Hess [1]. This method therefore has the potential to further reduce the prefactor associated with the asymptotic O(N 2 ) scaling of KS-DFT calculations. An implementation into the MOLPRO [9] program package demonstrates the accuracy of LEDO-DFT with respect to energetics and structural parameters for a little test set of small molecules containing H, C, N and O atoms. The authors employed a preliminary auxiliary basis set for all calculations and made use of numerical gradients for the structure optimizations. Later, a rudimentary implementation of LEDO-DFT for single point calculations with limited features has been realized in the program package TURBOMOLE [10, 11] by the author during the course of his Diplomarbeit [12] (Master s Thesis). 1.2 Outline of this work This thesis builds upon the work of Kollmar and Hess [1] and extends the TUR- BOMOLE implementation mentioned above. Its main objectives have been: 1. to find a way for the systematic optimization of auxiliary basis sets 2. the optimization of auxiliary basis sets for a larger set of atom types 3. the implementation of analytical gradients for LEDO-DFT 4. the tuning of the implementation with respect to efficiency 5. the objective assessment of LEDO-DFT with respect to its (a) accuracy and (b) computational cost

27 1.2. Outline of this work 5 The text consists of three main parts. In chapter 2 the complete theory as necessary for a deeper understanding of all aspects of LEDO-DFT is presented. Because the process of creating a program often entails as much research as developing the theory [13], chapter 3 is completely devoted to a description of the algorithms for the implementation of the formalism. In chapter 4, numerical results obtained with the implementation presented in chapter 3 are presented. First, the optimization of auxiliary orbitals for the LEDO expansion is described. Then, the accuracy and the efficiency of LEDO-DFT are critically assessed. A short overview is given at the beginning of each of these chapters to better guide the reader through the text. In chapter 5, the results obtained so far are summarized and some general conclusions are drawn. Finally, an outline of the direction for future work building upon this thesis is given in chapter 6.

28 6 Chapter 1. Introduction

29 Chapter 2 Theory This chapter deals with the theoretical aspects underlying the LEDO-DFT formalism, which is intended to facilitate first-principles calculations in the framework of KS-DFT. Being based on the LEDO expansion, LEDO-DFT can be interpreted as a special kind of density fitting method. Therefore, after a concise introduction to DFT (Sec. 2.1), the main bottlenecks of DFT calculations are briefly summarized alongside with a description of established techniques to reduce their computational cost (Sec. 2.2). This is followed by a detailed review of density fitting methods (Sec. 2.3), actually being in wide use in DFT, which finally leads to a description of the LEDO expansion (Sec. 2.4). Next, the LEDO-DFT formalism is worked out (Sec. 2.5), followed by a presentation of the analytical gradients for the energy expression (Sec. 2.6). Finally, a projection technique to improve the SCF convergence behavior of LEDO-DFT calculations is presented in the last section of this chapter. After having presented all necessary ingredients, the description of the actual implementation will be given in chapter Density Functional Theory The roots of modern density functional theory (DFT) [7, 14 22] can be traced back to the early days of quantum theory when Thomas and Fermi used models for the electronic structure of atoms which depend only on the electron density [23, 24]. Later the Hartree Fock Slater or Xα method was introduced to be 7

30 8 Chapter 2. Theory used in studies on systems with more than one atom. From todays perspective it represents one of the first density functional methods. It emerged from the work of Slater [25] who proposed to replace the complicated, non-local exchange term of the Hartree Fock (HF) method by the approximate local exchange potential of Dirac [26] which is simply given by ρ 1/3. However, it was not until 1964 when DFT was put on a firm basis by the celebrated theorems of Hohenberg and Kohn [2], later generalized by Levy [27, 28], which state that all properties of an electronic system are functionals of the ground state electron density. In particular, the ground state energy of an electron system in an external potential, i.e., the potential due to a set of nuclei in a given arrangement, can be found by minimizing the functional of the total electronic energy with respect to variations in the density. Parr and Yang [14] have reviewed how major chemical concepts follow from the existence of such a functional. Unfortunately, it is very difficult to develop sufficiently accurate density functionals, in particular, for the kinetic energy. This is a problem that already plagued the Thomas Fermi approach. Although the Hohenberg Kohn theorems give a theoretically sound basis for the Thomas Fermi approach and DFT in general, the practical importance of DFT might not have risen above that of the Thomas Fermi model if it had not been for the ingenious idea of Kohn and Sham [3] to obtain the real, interacting electronic density from an auxiliary system of hypothetical, non-interacting electrons as described below. The crucial point is that the electron density of this Kohn Sham (KS) model system is identical to the electron density of the system of real, interacting electrons. In KS theory [3], the total electronic energy is given as E = T s [ρ(r)] + dr ρ(r)v ext (r) + 1 dr 1 dr 2 ρ(r 1 )r12 1 ρ(r 2 ) + E xc [ρ(r)], (2.1) 2 with r 12 = r 1 r 2. The various terms represent, in order, the kinetic energy of the KS system (the fictitious system of non-interacting electrons), the interaction between electrons and the external potential v ext generated by the nuclei and external fields, the Hartree energy arising from the Coulomb interactions of the electrons, and the remainder of the total energy which is referred to as the exchange-correlation (XC) energy E xc. Note that E xc, which is a crucial quantity in DFT, is defined with the help of the KS system and not with respect

31 2.1. Density Functional Theory 9 to the HF wave function as in traditional ab initio methods. The electron density is obtained from the KS wave function, i.e., from a reference wave function of non-interacting electrons which is simply a Slater determinant built from the eigenfunctions ψ i of a single-particle Hamiltonian. For a closed-shell system one obtains occ. ρ(r) = 2 ψ i (r) 2, (2.2) the sum extending over all occupied KS or molecular orbitals (MOs) ψ i. i extension to the spin-polarized case is straightforward and shall not be considered here. The functional T s representing the kinetic energy of the non-interacting electrons is then given as An occ. T s [ρ(r)] = 2 ψ i 1 2 ψ i. (2.3) i It must not be identified with the exact kinetic energy of the real system of interacting electrons. The important point is that it is a convenient and fairly good approximation to the exact kinetic energy based on the KS orbitals. The remainder of the exact kinetic energy is taken into account as part of the XC energy E xc. According to the second Hohenberg Kohn theorem [2] the ground state electron density minimizes E[ρ] and hence must satisfy the Euler-Lagrange equation Here, and δe[ρ(r)] δρ(r) = δt s[ρ(r)] δρ(r) v h (r) = + v ext (r) + v h (r) + v xc (r) = µ. (2.4) dr 2 r 1 12 ρ(r 2 ) (2.5) v xc (r) = δe xc[ρ(r)], (2.6) δρ(r) are the Hartree and the XC potential, respectively. The Lagrange multiplier µ takes the constraint dr ρ(r) = n (2.7) into account, where n is the number of electrons, thus guaranteeing charge conservation. Since for T s as explicit density functional no sufficiently accurate approximations are available it is written as an orbital functional [Eq. (2.3)] and

32 10 Chapter 2. Theory therefore Eq. (2.4) cannot be solved directly, i.e., one cannot directly minimize Eq. (2.1) with respect to ρ in order to obtain the ground state energy and density. Instead, this minimization is performed indirectly. For the non-interacting electrons moving in a potential v s (r), the corresponding Euler-Lagrange equation is δe s [ρ(r)] δρ(r) = δt s[ρ(r)] δρ(r) + v s (r) = µ. (2.8) The density solving this equation is ρ s. Comparing Eq. (2.8) with Eq. (2.4) shows that both have the same solution ρ = ρ s, if v s is chosen to be v s (r) = v ext (r) + v h (r) + v xc (r). (2.9) Consequently, the density of the interacting (many-body) system in the external potential v ext can be obtained by solving the one-electron Schrödinger equations with the KS single-particle Hamiltonian H KS ψ i (r) = ε i ψ i (r) (2.10) H KS = v s(r) = v ext(r) + v h (r) + v xc (r) (2.11) and the canonical MO energies ε i. Once the KS orbitals have been determined from Eq. (2.10), the total electronic energy of the real, interacting system can be obtained from Eq. (2.1). Eq. (2.10) is somewhat deceptive, in that it looks like a simple single-particle Schrödinger equation. However, two features bring out the full many-body character of the problem. One is that Eq. (2.10) has to be solved self-consistently since both v h and v xc depend on the electron density ρ which is a function of the KS orbitals ψ i. The other is the incomplete knowledge of the XC energy density functional E xc. Exchange-correlation functionals Classical electronic structure theories have always started with the exact Hamiltonian operator, and used approximations for the wave function, usually with a single Slater determinant as a starting point. The foundation of KS-DFT begins with an approximate energy expression, the refinements placed only on the XC

33 2.1. Density Functional Theory 11 term. The success of the KS method therefore hinges on the availability of good approximations to E xc. For many years the most widely used scheme has been the so-called local density approximation (LDA), E LDA xc [ρ(r)] = dr ε xc (ρ(r)), (2.12) where ε xc is the XC energy density 1 in a homogeneous electron gas, known with great accuracy from quantum Monte-Carlo calculations [29]. This approximation is obviously valid in the limit of slowly varying densities, but has proven its accuracy for a wide range of systems. The functional Exc LDA exchange and correlation contributions, can be divided into E LDA xc [ρ(r)] = Ex LDA [ρ(r)] + Ec LDA [ρ(r)], (2.13) where the exchange contribution is given by the Dirac exchange energy functional [26]. A popular analytical form of the correlation contribution has been given by Vosko, Wilk and Nusair (VWN) [30], others are due to Perdew and Zunger (PZ81) [31] and Perdew and Wang (PW92) [32]. For practical purposes all LDA functionals are next to equivalent. More recently, however, these approximations to E xc have been much improved by the introduction of the generalized gradient approximation (GGA), which supplements the LDA term with one that depends explicitly on the gradients of the density, Exc GGA [ρ(r)] = dr F xc (ρ(r), ρ(r)). (2.14) It was the advent of GGAs that made DFT popular among chemists since these provided for the first time a level of accuracy which allows for the quantitative discussion of chemical bonds. Popular gradient corrections are the ones by Becke (B88) [33] and Perdew and Wang (PW86) [34] for the exchange contribution and the ones by Perdew (P86) [35], Becke (B88C) [36], and Lee, Yang and Parr (LYP) [37] for the correlation contribution. Perdew et al. (PW91) [38], Perdew, 1 Sometimes the XC energy per particle is used instead of the XC energy density. In that case an additional ρ(r) appears under the integral on the right hand side of Eq. (2.12).

34 12 Chapter 2. Theory Burke and Wang (PBW96) [39] and Perdew, Burke and Ernzerhof (PBE) [40] have given widely used expressions for the complete XC energy. Many other GGA functionals are available, and new ones continue to appear. Some of these approximate functionals were constructed in a semiempirical fashion with many parameters fitted to chemical data, others were constructed to incorporate key properties of the exact XC energy. The latter functionals usually perform well for both small and extended systems. While the semiempirical functionals can achieve remarkably high accuracy for atoms and molecules, they are typically less accurate for surfaces and solids [41]. The most widespread functional among chemists is probably the one called B3LYP [42]. It is a so-called hybrid functional meaning that some portion of exact exchange, i.e. non-local HF-like exchange expressed with KS orbitals, is mixed into the expression of the XC energy as introduced by Becke [43]. Another beyond-gga development is the emergence of so-called meta-ggas, which depend, in addition to the density and its derivatives, also on the Laplacian of the density or the KS kinetic energy density [44, 45]. These functionals have given favorable results even when compared to the best GGAs [41, 46, 47], but the full potential of this type of approximation has not yet been explored systematically. Finally, it should be mentioned that in the framework of orbital dependent density functionals, an exact treatment of the exchange contribution is feasible [48]. Future developments will be aimed at a description of correlation effects compatible to exact exchange (hyper-ggas) [47, 49]. Basis set methods Although in principle the MOs can be represented on a grid and the KS equations solved numerically, in molecular calculations the MOs are in general expanded into a set {φ µ } of atom-centered basis functions or atomic orbitals (AOs) according to ψ i (r) = µ C i µφ µ (r). (2.15)

35 2.1. Density Functional Theory 13 Within this linear combination of atomic orbitals (LCAO) expansion the electron density [Eq. (2.2)] is then given as ρ(r) = µ,ν P µν φ µ (r)φ ν (r), (2.16) where P is the first order reduced density matrix with elements occ. P µν = 2 CµC i ν. i (2.17) The expression for the total electronic energy [Eq. (2.1)] now reads as E = µ,ν P µν φ µ 1 2 +v ext φ ν i µ,ν,κ,λ P µν P κλ (φ µ φ ν φ κ φ λ )+E xc [ρ(r)]. (2.18) In Eq. (2.18) the charge density notation for the electron repulsion integrals (ERIs), has been employed. (φ µ φ ν φ κ φ λ ) = dr 1 dr 2 φ µ (r 1 )φ ν (r 1 )r 1 12 φ κ (r 2 )φ λ (r 2 ), (2.19) The variational parameters in this energy expression are the MO coefficients or the elements of the density matrix and application of the variational principle leads to the matrix eigenvalue equation H KS C = SCε (2.20) instead of the operator equation (2.10). In Eq. (2.20) H KS is the matrix representation of the KS operator (2.11) in the AO basis, H KS µν = φ µ H KS φ ν, (2.21) S is the overlap matrix with elements S µν = φ µ φ ν, and the matrix C contains as elements C µi the MO coefficients C i µ which are obtained by solving the KS equation (2.20). The matrix ε of the Lagrange multipliers for the orthonormality constraints is a diagonal matrix which can be identified with the KS matrix in the canonical MO basis and its diagonal elements represent the corresponding MO energies ε i of the KS orbitals ψ i. Due to decades of experience with HF calculations much is known about the construction of basis functions φ µ that are suitable for molecules. Almost all of

36 14 Chapter 2. Theory this continues to hold in DFT, a fact that has greatly contributed to the popularity of DFT in chemistry. The most popular types of basis functions φ µ for molecular calculations can be classified with respect to their behavior as a function of the radial coordinate into Slater type functions (STFs) [50], which decay exponentially from their origin, and Gaussian type functions (GTFs) [51], which have a Gaussian behavior. While STFs more closely resemble the true behavior of atomic wave functions, GTFs are much easier to handle numerically. In particular an analytical evaluation of the ERIs is possible in an efficient manner. This is the reason for the predominance of GTFs not only in conventional ab initio methods but also in programs for molecular DFT calculations [52]. 2.2 Computational Aspects of DFT Calculations In general, there are three important bottlenecks in Gaussian-based KS-DFT calculations. These are the evaluation of the Hartree energy, the XC quadrature and the diagonalization of the KS Hamiltonian. In this section, a short description of these bottlenecks along with a summary of existing techniques to speed up the respective part of the calculations shall be given, before density fitting methods are discussed thoroughly in the subsequent section. Evaluation of the Hartree energy In Gaussian-based KS theory the Hartree energy E h = 1 2 (ρ ρ) = 1 2 µ,ν,κ,λ P µν P κλ (φ µ φ ν φ κ φ λ ) (2.22) and the corresponding contributions to the KS matrix, i.e. the matrix elements of the Hartree potential (frequently termed Coulomb matrix) V h,µν = E h P µν = κ,λ P κλ (φ µ φ ν φ κ φ λ ) = (φ µ φ ν ρ) (2.23) have to be computed from four-index ERIs. Formally, this is an O(N 4 ) computational process, but the number of integrals to be evaluated in the asymptotic

37 2.2. Computational Aspects of DFT Calculations 15 limit of extended systems scales as O(N 2 ) if appropriate prescreening of the ERIs is employed [53 56]. However, despite huge advances in integral evaluation techniques [57 72], the prefactor for this quadratic scaling is large enough to make the evaluation of the ERIs the rate-limiting step even for moderate-sized systems. As a consequence a wide range of refined technical tricks and elegant approximations have been devised to reduce the effective scaling and enable the rapid evaluation of the Hartree energy and the matrix elements of the Hartree potential. Ahmadi and Almlöf have developed an algorithm for the evaluation of the Coulomb matrix in which the density matrix is summed into the underlying Gaussian integral formulas with substantial benefits in the floating point operation (FLOP) cost [73]. The J matrix engine [74 76] from Head-Gordon et al. and the Quantum Chemical Tree Code (QCTC) [77] work in a similar fashion. The latter is a modified version of the McMurchie/Davidson [61] algorithm for the ERI evaluation which is based upon a representation of the density in a Hermite Gaussian type function (HGTF) basis and has been reported to lead to sub-o(n 2 ) scalings [77]. All of these algorithms do not require any approximations. For a review about fast methods for computing the Coulomb matrix in a basis of GTFs up to the year 1996 the reader is referred to reference [78]. An evaluation of the Coulomb potential on an integration grid by explicit use of the Poisson equation has also been explored, but it does not lead to competitive methods [79]. Other research efforts have focused on methods that provide approximations to the ERIs. The Fourier transform Coulomb (FTC) method [80] is based on an intermediate discrete Fourier transformation of the electron density. It is in some sense similar to the Gaussian and augmented plane-wave (GAPW) method [81] and was shown to exhibit near-linear asymptotic scaling while retaining high numerical accuracy. Important progress with respect to the Coulomb problem has been achieved by generalization of the fast multipole method (FMM) [82, 83] to Gaussian charge distributions, which leads to linear scaling for the asymptotic limit of very large systems [84 89]. In this method the computation of the Coulomb interaction is divided into a near-field and a far-field part, the latter of which is treated by accurate multipole approximations and the former by exact analytical integration. The recursive bisection method (RBM) [90] works in a

38 16 Chapter 2. Theory similar fashion and exhibits the same asymptotic scaling behavior. The linear scaling KWIK/CASE algorithm [91 96] relies on a similar partitioning, but the long-range part is computed by Fourier summations. In practice, however, much of the computational effort for molecules of chemical interest does not fall into the asymptotic regime, and the underlying formal scaling behavior which determines the prefactor has a significant quantitative effect on performance. The analytical integration of the four-index ERIs is still an expensive part of the calculation, and efficient algorithms are required to achieve high computational throughput. Fitting procedures for the electron density are widely used in DFT calculations to avoid the unfavorable formal O(N 4 ) scaling behavior in the processing of the ERIs and shall be reviewed in detail in Sec The pseudospectral approximation [97 101] is a different approach in a similar spirit. This method uses a grid as an auxiliary basis to expand the electron density (or the potential from it), which leads to the same formal O(N 3 ) scaling behavior as conventional density fitting procedures. It should be mentioned that density fitting has already been successfully employed for the evaluation of the nearfield interactions in order to reduce the prefactor of the fast multipole method [102]. If hybrid DFT functionals are used, the HF-like (orbital) exchange energy has to be evaluated from four-index ERIs like the Hartree energy. However, for non-metallic systems with a large HOMO-LUMO gap the density matrix is decaying exponentially [103]. This has lead to the development of O(N) methods like ONX [ ], LinK [107] and others [108] which exploit the fast decaying nature of the exchange interaction. Exchange-correlation quadrature The XC energy E xc in DFT is given by Eq. (2.14). As indicated in this formula, the XC functional F xc can depend both on the electron density ρ and its gradient ρ. In general this dependence is very complicated and the XC integrals cannot be solved analytically, even if the density is expressed in a basis of Gaussian charge distributions [Eq. (2.16)]. Instead, they are evaluated using an integration grid.

39 2.2. Computational Aspects of DFT Calculations 17 This is a set of points r j = (x j, y j, z j ) and non-negative weights ω j such that dr f(r) ω j f(r j ). (2.24) j The XC energy which is actually computed is thus given by E GGA xc = j ω j F xc (ρ(r j ), ρ(r j )). (2.25) The integrand is usually partitioned over atomic points using a weight scheme [109, 110], and a further decomposition into radial and angular components of each atomic contribution is introduced [ ]. The formal scaling for the setup of an integration grid with such a weight scheme is O(N 3 ). Other steps in the numerical quadrature also yield O(N 3 ) scaling simply because the number of atomic-based grid points grows linearly with molecular size and their contributions need to be evaluated over every pair φ µ φ ν of AO basis functions. Advantages of such a numerical integration are the possibility to control and systematically enhance the numerical accuracy and the insensitivity of the integration itself to the type of density functional in use. The numerical XC quadrature has long been recognized as being intrinsically linear scaling due to the fast decaying nature of the basis functions used [114, 115]. Much attention has also been focused on the construction of the grid with respect to the efficiency of the resulting code and linear scaling quadrature grid construction is feasible [116]. Diagonalization The matrix eigenvalue equation (2.20) is usually solved by diagonalization of the KS matrix. Today, it turns out that the diagonalization steps which scale as O(N 3 ) are becoming more and more important, although the prefactor is very small. This is partly because it is almost trivial to make use of parallel processing when constructing the Kohn Sham matrix, while this is more difficult when diagonalizing. As a result, there are methods available to replace the diagonalization step by procedures with a more favorable scaling behavior, like conjugate gradient [ ] or quasi Newton approaches [121, 122].

40 18 Chapter 2. Theory 2.3 Review of Density Fitting Methods In this section, density fitting methods shall be reviewed. It has already been mentioned that these are widely used in DFT. Implementations are documented for the programs DGauss [123], DeFT [124], ParaGauss [125], ADF [126], Turbomole [127], Molpro [128], Orca [129] and Magic [130] and are most likely available in virtually any other KS-DFT program, even if not explicitly documented in the scientific literature. Furthermore, density fitting procedures have been successfully applied as integral approximations in the framework of HF calculations [ ] for both the Coulomb problem and the HF exchange. Density fitting has also proven to offer significant advantages for multi-configurational SCF (MC- SCF) [132, 137], second order Møller Plesset perturbation theory (MP2) [ ], CC2 [143, 144], coupled cluster [145, 146] and, more recently, explicitly correlated MP2-R12 [ ] calculations Fit of an Arbitrary Charge Distribution Consider an arbitrary charge distribution Ξ(r). Suppose that a model distribution Ξ(r) = p Ω p (r) d p = Ω d (2.26) shall be formed by expanding this charge distribution into a set {Ω p } of auxiliary functions. d is the coefficient vector that minimizes the norm of the difference vector Ξ Ξ between the exact and the approximated charge distribution. Several choices are possible for defining this norm, generally having the form (W) = dr 1 dr 2 [Ξ(r 1 ) Ξ(r ] [ 1 ) W(r 1, r 2 ) Ξ(r 2 ) Ξ(r ] 2 ) = Ξ W Ξ 2 Ξ W Ξ + Ξ W Ξ. (2.27) W is the weight operator whose Fourier transform has to be positive [150]. This norm is minimized with respect to the coefficients d p by setting d (W) = 0 resulting in the system of linear equations Ω p W Ω q d q = Ω p W Ξ (2.28) q

41 2.3. Review of Density Fitting Methods 19 for the determination of the expansion coefficients d p. This can be rewritten in matrix notation as Wd = a (2.29) with the solution d = W 1 a, (2.30) where W pq = Ω p W Ω q and a p = Ω p W Ξ. The matrix W is strictly positive definite if the expansion functions Ω p are linearly independent which guarantees the mathematical existence of the solution (2.30). Any prescribed accuracy in fitting the charge distribution Ξ(r) can thus be achieved if the expansion basis is sufficiently flexible and approaches completeness. Additional constraints, e.g. such that the total charge of Ξ(r) is exactly reproduced, can be easily imposed by the use of Lagrangian multipliers. In the following, however, this option shall not be considered. The approximated charge distribution is then given as Ξ(r) = Ω d = Ω W 1 a. (2.31) The generalized least squares fitting procedure described here can also be interpreted as a projection of the charge distribution Ξ(r) onto a basis {Ω p} orthonormal in the linear vector space with metric W which is obtained by symmetric orthonormalization [151] as Ω = W 1/2 Ω. (2.32) The projection becomes Ξ(r) = p Ω p Ω p W Ξ = Ω Ω W Ξ = Ω W 1 a, (2.33) in analogy with Eq. (2.31). In the context of DFT, the charge distribution that shall be approximated is the electron density ρ(r) in order to facilitate the evaluation of the Hartree energy [Eq. (2.22)]. For doing so, the choice of the weight operator W is very important. Using W = δ(r 12 ) [ ] amounts to minimizing the least-squares error in the charge distribution 2, (W = δ(r 12 )) = dr Ξ(r) Ξ(r) 2. (2.34) 2 δ is the Dirac delta function

42 20 Chapter 2. Theory At first sight, this seems very attractive since the integrals W pq and a p involved in the fitting process reduce to simple overlap integrals. However, it was found that approaches using W = r 1 12 are at least an order of magnitude more accurate [ ]. These approaches minimize the least squares error of the electric field because [155] (W = r12 1 ) = (Ξ Ξ Ξ Ξ) = 1 4π dr E(r) Ẽ(r) 2, (2.35) where E and Ẽ are the electric fields arising from the exact and the fitted charge distribution. Expression (2.35) is frequently denoted as Coulomb norm. This form of fitting has also been suggested by Billingsley and Bloor [158], Whitten [159] and Fortunelli and Salvetti [160] for the approximation of overlap densities, by Hall for point charge models [161] and molecular electron densities [162, 163], and is in wide use nowadays. It has not yet been established whether the idea of minimizing the least squares error in the Hartree potential v h of the charge distribution, (W = r 12 ) = dr v h (r) ṽ h (r) 2, (2.36) by using the weight operator W = r 12 [164], offers any advantage Fit of the Complete Electron Density All current DFT implementations exploiting density fits rely on the expansion of the complete electron density, Eq. (2.16), according to ρ(r) = p Ω p (r) d p = Ω d (2.37) into a set {Ω p } of atom-centered auxiliary functions. This is desirable because the AO product basis {φ µ φ ν } in which the exact electron density is expanded is nearly linear dependent and grows as O(N 2 ) whereas the dimension of the basis {Ω p } increases only as O(N). As described above, the expansion coefficients are obtained from the system of linear equations Wd = a (2.38)

43 2.3. Review of Density Fitting Methods 21 with the solution d = W 1 a. (2.39) The elements a p of the inhomogeneity vector are now defined as a p = Ω p W ρ = µ,ν a µν p P µν, (2.40) with the three-index integrals a µν p = Ω p W φ µ φ ν. (2.41) The Hartree energy, given by Eq. (2.22), is then approximated as Ẽ h = 1 2 ( ρ ρ) = 1 2 d Vd, (2.42) where V pq = (Ω p Ω q ) (2.43) is a two-index ERI. An analysis of the error E h Ẽh = 1 2 {(ρ ρ) ( ρ ρ)} = 1 {(ρ ρ ρ ρ) + 2(ρ ρ ρ)} (2.44) 2 between the exact and the fitted Hartree energy shows that it contains terms linear and quadratic in the fitting error ρ ρ in the electron density. The form of Eq. (2.44) therefore suggests the alternative approximation Ẽ rob h = (ρ ρ) 1 2 ( ρ ρ) = b d 1 2 d Vd (2.45) which goes back to Dunlap et al. [155] In Eq. (2.45) the elements of the vector b are defined as b p = (Ω p ρ) = µ,ν b µν p P µν (2.46) with the three-index ERIs b µν p = (Ω p φ µ φ ν ). (2.47) Fitting expressions of this kind are called robust since the error E h Ẽrob h = 1 2 {(ρ ρ) 2(ρ ρ) + ( ρ ρ)} = 1 (ρ ρ ρ ρ) (2.48) 2

44 22 Chapter 2. Theory is always quadratic in the fitting error ρ ρ. The robust energy expression (2.45) is also variational because it is a lower bound to the exact Hartree energy, Ẽ rob h E h, (2.49) which follows directly from Eq. (2.48). In other words, the expansion coefficients d p have been determined in such a way as to maximize the robust approximation to the Hartree energy. Therefore, derivatives of the expansion coefficients will not appear in expressions for the gradient of the energy if robust fitting is performed. The concepts of robust and variational fitting in various contexts have been thoroughly discussed by Dunlap [155, ]. For a generally fitted case one obtains Ṽ h,µν = Ẽh P µν = 1 P µν 2 d Vd = ( d P µν ) Vd (2.50) for the matrix elements of the Hartree potential and the derivatives of the fitting coefficients are required. These are given as d = W 1 a = W 1 a µν, (2.51) P µν P µν which follows from Eqs. (2.39) and (2.40). The approximated matrix elements of the Hartree potential thus become Ṽ h,µν = a µν W 1 Vd. (2.52) For robust fitting the approximated matrix elements are given as Ṽh,µν rob = Ẽrob h P µν = b µν d + a µν W 1 {b Vd}. (2.53) It is not possible to say much about the quality of the fitted matrix elements Ṽh,µν given by Eq. (2.52). For the robust approximation Ṽ h,µν rob [Eq. (2.53)], however, the first term on the right hand side is (φ µ φ ν ρ) and the term in the brackets is (Ω ρ ρ). Thus, the approximation of the matrix elements will be good, if the fit for the electron density is sufficiently accurate. If the Coulomb norm Eq. (2.35) is employed in the fitting procedure, then W = V and a = b and the expansion coefficients are obtained from the system of linear equations Vd = b (2.54)

45 2.3. Review of Density Fitting Methods 23 with the solution d = V 1 b. (2.55) The elements of V are two-index ERIs as defined in Eq. (2.43) and b contains three-index ERIs [Eqs. (2.46) and (2.47)]. Now the following equality holds [cf. Eq. (2.54)], ( ρ ρ) = d Vd = b d = (ρ ρ). (2.56) Thus the linear term in the error of the fitted Hartree energy [Eq. (2.44)] vanishes and the robust and the simple approximation to the Hartree energy become identical, Ẽ rob h = Ẽh = 1 2 d Vd. (2.57) This explains the observation that the Coulomb norm is the best. The derivatives of the fitting coefficients with respect to the elements of the density matrix are in this case given as d P µν = V 1 b µν (2.58) and the approximated matrix elements of the Hartree potential become ( ) d Ṽ h,µν = Vd = b µν d = (φ µ φ ν ρ). (2.59) P µν This expression is of course identical to the robust approximation and the error in the matrix elements will be small if the fit of the electron density is good. Scaling behavior Before discussing methods fitting individual overlap densities, some conclusions about the scaling behavior of methods fitting the complete electron density shall be drawn. For simplicity, it will be assumed that the Coulomb norm has been employed. The formal scaling behavior is O(N 3 ) for the evaluation of the ERIs since only two- and three-index quantities V pq and b µν p have to be evaluated. Due to the long-range nature of the Coulomb interaction the asymptotic scaling behavior remains O(N 2 ) like for the four-index ERIs, however with a much smaller prefactor. The number of ERIs that need to be explicitly evaluated is much smaller because the number of required fitting functions Ω p is considerably