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

46 24 Chapter 2. Theory smaller than the number of significant overlap densities φ µ φ ν generated from the AO basis set. A speed-up by more than a factor of 10 is generally achieved and the computational expense is typically dominated by the numerical quadrature for the evaluation of the XC energy and its matrix elements [127]. The necessity to compute at most three-center ERIs is furthermore very appealing since these can be evaluated with a much simpler algorithm than for the general case [173]. The determination of the expansion coefficients requires the inversion of the matrix V whose dimension grows linearly with the system size. The computational effort for such a matrix inversion grows cubically with the dimension of the matrix. Although the prefactor is rather small, the O(N 3 ) scaling of this matrix inversion might therefore become dominant for very large systems. The actual determination of the expansion coefficients from Eq. (2.55) is done in O(N 2 ) operations in each SCF iteration. There have also been efforts to restrict the fit basis in such density fitting methods. They are based on a partitioning of the total electron density into contributions from different regions in space. The approach of Baerends et al. [153] relies on the decomposition into atom pair densities according to ρ(r) = A B(2 δ AB )ρ AB (r), (2.60) with ρ AB (r) = µ A,ν B P µa ν B φ µa (r)φ νb (r). (2.61) The additional subscript (A, B) indicates the particular atom on which an AO is located. Each of these atom pair densities is fitted individually using only fit functions located on the respective atoms A and B. Thus, the fitting procedure requires O(N 2 ) inversions of matrices V [AB] of size-independent dimension. At the same time the integrals of the inhomogeneity of the system of linear equations (2.54) for the determination of the expansion coefficients reduce to two-center integrals of the type b µ Aν B p A = (Ω pa φ µa φ νb ). The resulting formal O(N 2 ) scaling for the fitting procedure reduces to linear scaling in the asymptotic regime because the number of significant atom pair densities is O(N) [126, 174]. For the evaluation of the approximated matrix elements Ṽh,µν according to Eq. (2.59), however, three-index ERIs b µν p A of the Hartree potential = (Ω pa φ µ φ ν ) are still

47 2.3. Review of Density Fitting Methods 25 required. This yields a formal O(N 3 ) scaling behavior for the evaluation of the ERIs which reduces to O(N 2 ) in the asymptotic limit, just as for methods fitting the complete electron density. If the matrix elements of the Hartree potential are evaluated by numerical quadrature, as is actually done in the work of Baerends et al. [153], then the Hartree potential has to be first evaluated at each grid point. This is a formal O(N 2 ) process because the number of fitting functions scales linearly as in the worst case does the number of sampling points for the potential. Because the Coulomb interaction is long range, this scaling behavior cannot be reduced by simple cut-off schemes [126, 174]. The subsequent evaluation of the matrix elements is a formal O(N 3 ) process which can be reduced to O(N) in the asymptotic limit by employing cut-off techniques similar to the ones used for the XC quadrature. The density fitting procedure can also be implemented in O(N) versions by employing a divide-and-conquer (DAC) approach for the partitioning of the electron density [ ]. The formal and asymptotic scaling behavior for the evaluation of the matrix elements of the Hartree potential, however, remains in any case O(N 3 ) and O(N 2 ), respectively. Note that simple fitting expressions which work with a different expansion basis for different parts of a subdivided electron density are neither robust nor variational. It should finally be mentioned that a subdivision of the electron density enables a trivial parallelization of the density fitting process. Auxiliary basis sets for the density expansion The choice of an auxiliary basis for the expansion of the electron density is certainly a topic that requires extensive and lengthy experimentation. However, in order to reproduce SCF total energies with good accuracy, it has been shown that the reproduction of the behavior of the exact expansion near the atomic nuclei is important [157]. Therefore, the expansion basis set should include one-center products of the AO basis set, which account for by far the largest contributions to the total electron density. Consequently, the auxiliary functions should cover the range of the sums of the exponents and angular momentum quantum numbers of the original basis functions. In practice, it is sufficient to use a relatively short, even-tempered expansion ranging between the highest and lowest exponent given by this recipe [ ]. Efficient and at the same time accurate auxiliary basis

48 26 Chapter 2. Theory sets for the density expansion are available [127, 183]. The exponents of these basis sets fulfill a dependence similar to the definition of an even-tempered basis and have been optimized by exploiting the variational property of the approximated energy expression, Eq. (2.49). An increase of computational efficiency has recently been achieved by the substitution of a large part of the expansion functions by so-called Poisson functions. Although in this case the total number of expansion functions required for the density fit increases, the major part of the ERIs become simple overlap integrals without further approximation [128, 184] Fit of Overlap Densities The most general approach to the fit of individual overlap densities φ µ φ ν ρ µν was suggested by Vahtras et al. [157]. In contrast to the projection of diatomic differential overlap (PDDO) approach by Newton [152] and the limited expansion of diatomic overlap (LEDO) approach by Billingsley and Bloor [158], their method uses the same fit basis {Ω p } of atom-centered expansion functions for all overlap densities to be fitted. In analogy to Eq. (2.37) for the expansion of the complete electron density, the expansion of individual overlap densities can thus be written as ρ µν (r) ρ µν (r) = p Ω p (r) d µν p = Ω d µν. (2.62) For the sake of simplicity it will be assumed in the following that the Coulomb norm [Eq. (2.35)] is used for the definition of the scalar product. expansion coefficients d µν p are obtained from the system of linear equations Then the Vd µν = b µν (2.63) with solution d µν = V 1 b µν, (2.64) where V pq = (Ω p Ω q ) and b µν p = (Ω p ρ µν ) are two- and three-index ERIs, similar to Eqs. (2.54) and (2.55) for the fit of the complete electron density. Taking the symmetry of the matrix V into account, any four-index ERI can thus be approximated as (ρ µν ρ κλ ) ( ρ µν ρ κλ ) = d µν Vd κλ = b µν V 1 b κλ. (2.65)

49 2.3. Review of Density Fitting Methods 27 Incidentally, the same expression is obtained if a conventional resolution of the identity (RI) is introduced into the ERIs, 1 = Ω p Vpq 1 Ω q p,q p,q Ω p V 1 pq Ω q, (2.66) where the approximate equality reflects the incompleteness of the expansion basis. Therefore, density fitting is frequently denoted as RI approximation by some authors. As a matter of fact, this approximation is an inner projection similar to the method of Beebe and Linderberg which is based on the Cholesky decomposition of the ERI matrix [185, 186]. Here, however, the projection is formulated in terms of an auxiliary set of basis functions {Ω p }, rather than to generate the projection manifold explicitly from the original overlap densities. Note, that density fitting mathematically resembles a (truncated) RI only in the specific case where the weight operator W used in the metric of the expansion basis is the same as in the target integral. Furthermore, RIs do not offer a framework in which to discuss fitting criteria, constraints or robust fitting. An analysis of the error (ρ µν ρ κλ ) ( ρ µν ρ κλ ) = (ρ µν ρ µν ρ κλ ρ κλ ) + (ρ µν ρ µν ρ κλ ) + ( ρ µν ρ κλ ρ κλ ) (2.67) between the exact and the approximate ERIs shows that it contains terms linear and quadratic in the fitting error ρ µν ρ µν. However, if the Coulomb norm has been employed, the expansion coefficients are obtained from Eq. (2.64) and the following equality holds, ( ρ µν ρ κλ ) = d µν Vd κλ = b µν d κλ = (ρ µν ρ κλ ) = d µν b κλ = ( ρ µν ρ κλ ). (2.68) Thus the linear error terms in Eq. (2.67) vanish and the approximation is robust. It is important to realize that this holds only if the Coulomb norm has been employed. Insertion of the expansion for the overlap densities [Eq. (2.62)] into the expression for the electron density [Eq. (2.16)] yields ρ(r) = p Ω p (r) d p = Ω d (2.69)

50 28 Chapter 2. Theory for the approximate electron density. This expression is identical to Eq. (2.37) for the fit of the total electron density. Here, however, the expansion coefficients d p for the electron density have been obtained by contraction of the elements of the density matrix P with the expansion coefficients d µν p according to d p = µ,ν In analogy to Eq. (2.46), a vector b with components for the overlap densities d µν p P µν. (2.70) b p = (Ω p ρ) = µ,ν b µν p P µν (2.71) can be defined. Similar relations as derived for the fit of the complete electron density also hold for the fit of individual overlap densities. Specifically, the approximated Hartree energy is given as Ẽ h = 1 2 ( ρ ρ) = 1 2 d Vd, (2.72) and the approximated matrix elements of the Hartree potential are given as ( ) Ṽ h,µν = Ẽh d = Vd = d µν Vd = ( ρ µν ρ). (2.73) P µν P µν The derivatives of the expansion coefficients d p with respect to the elements P µν of the density matrix yield the expansion coefficients d µν p for the overlap densities ρ µν, as can be seen from Eq. (2.70). While these need not be explicitly determined in methods fitting the complete electron density, a comparison of Eqs. (2.58) and (2.64) shows, that also in that case they do appear implicitly in the expression for the approximated matrix elements of the Hartree potential. Using Eqs. (2.63), (2.70) and (2.71), Eqs. (2.72) and (2.73) can be rewritten as Ẽ h = 1 2 b d = 1 (ρ ρ) (2.74) 2 and Ṽ h,µν = b µν d = (ρ µν ρ). (2.75) It is very important to note that the step leading from Eqs. (2.72) and (2.73) to Eqs. (2.74) and (2.75) can only be performed if

51 2.3. Review of Density Fitting Methods 29 a) the fit basis {Ω p } is identical for all overlap densities ρ µν and b) the Coulomb norm [Eq. (2.35)] has been employed for the fitting procedure. Only in this case the simple approximation of the ERIs and thus of the Hartree energy is identical to the robust expression. Note that the approximated ERIs are not a bound to the exact ERIs, i.e. they can be smaller or larger in magnitude. The approximated Hartree energy, however, is a lower bound to the exact Hartree energy, just as in methods fitting the complete electron density. Requirements to the expansion basis Now it is possible to draw an interesting conclusion. As a fit for each individual overlap density ρ µν is performed, one might expect from Eq. (2.73) that the fit basis has to be sufficiently large to reproduce any particular member of the set of overlap densities with good accuracy. However, this is not the case as becomes evident by comparing Eqs. (2.23), (2.73) and (2.75). It can be easily seen that the matrix elements V h,µν of the Hartree potential are well approximated if a good fit for the total electron density is obtained, i.e. if ρ and ρ are sufficiently close. According to Eqs. (2.69) and (2.70), it is sufficient for the representation of the complete electron density to determine expansion coefficients d µν p corresponding to density matrix elements P µν of significant magnitude. It is obvious that the fit of the total electron density is a much easier task than the fit of each overlap density generated by the AO basis functions, thus requiring a much smaller fit basis. The most extreme example is perhaps a closed-shell atom which has a spherically symmetric electron density. for a density fit. Hence, s-type functions are sufficient It has been found that even electron densities of molecules can be fitted quite accurately by short expansions of spherical Gaussians located on the nuclei [162]. Clearly, this would be an extremely poor basis for a fit of individual AO products. However, it is sufficient for a good approximation of the matrix elements of the two-electron Coulomb part of such products. It should be noted that the demand on the expansion basis may be a lot higher in other types of calculations, where products that have nothing to do with the density expansion must also be accurately represented. This is the case, e.g., if density fitting is employed for the HF exchange (without reducing the formal

52 30 Chapter 2. Theory scaling behavior), which requires products φ µ ψ i between AO basis functions and occupied MOs to be fitted. In the case of MP2 calculations, products of the type ψ i ψ a between occupied and virtual MOs have to be fitted. Ten-no and Iwata successfully employed one-center AO products φ µ A φ µ as auxiliary functions in A HF and MCSCF calculations [132, 137] and optimized auxiliary basis sets are available both for HF [135] and MP2 [140, 141, 144] calculations. Scaling behavior Although the approach of Vahtras et al., just like methods fitting the complete electron density, is not very demanding with respect to the fit basis, it has a rather unfavorable formal scaling behavior. The number of overlap densities ρ µν grows as O(N 2 ) with the system size. For each of these overlap densities the system of linear equations (2.63) has to be solved. Since the dimension of the fit basis also grows linearly with the system size, the numerical effort for obtaining the expansion coefficients for a particular overlap density scales as O(N 2 ), once the matrix V has been inverted. Altogether, one therefore ends up with the same formal O(N 4 ) scaling just as the unapproximated method. However, the prefactor might be rather favorable and, in contrast to methods fitting the total electron density, the fit does not have to be repeated in every SCF cycle. It is clear, that the fit has to be performed only for products φ µ φ ν of AOs with significant overlap, the number of which is of O(N) in the limit of large molecules. Therefore, the asymptotic scaling for the fitting process is O(N 3 ) due to the matrix inversion like for the fit of the complete electron density Fit of Overlap Densities with a Restricted Expansion Basis It seems desirable to avoid the increase of the fit basis with system size for the fit of individual overlap densities which would lead to formal O(N 2 ) scaling. Indeed, the success of the expansion technique for diatomic systems suggests [182] that it is sufficient to use a local expansion in Eq. (2.62), i.e., to use a sparse (blocked) set of expansion coefficients {d µν p }, with coefficients only for the expansion functions Ω p on the two centers where φ µ and φ ν are centered. Such an expansion for

53 2.3. Review of Density Fitting Methods 31 diatomic overlap densities φ µa φ νb ρ µa ν B given by a product of AOs centered on atoms A and B can be written as φ µa (r)φ νb (r) p A Ω pa (r) d µ Aν B p A + q B Ω qb (r) d µ Aν B q B, (2.76) or, in a shorthand notation, as ρ µa ν B ρ µa ν B = Ω A dµ Aν B A + Ω B dµ Aν B B = Ω [AB] dµ Aν B [AB]. (2.77) Note that the case A = B is assumed to be handled in analogy to Eq. (2.76) as φ µ A (r)φ µ (r) Ω A pa (r) d µ A µ A p A. (2.78) p A The expansion functions {Ω p } are restricted to be located on atoms A and B. Indeed it is not obvious that an expansion function located on an atom C far away from atoms A and B should improve the fit significantly. Furthermore, the possible benefit in fitting diatomic overlap densities ρ µa ν B obtained by including expansion functions from other atoms C A, B is heavily system dependent. Since the presence of atoms C close to ρ µa ν B is accidental, the quality of the fit could vary from one system to another depending on the accidental surrounding of a pair AB by other atoms. A reliable expansion basis will have to be able to fit an overlap density ρ µa ν B accurately, with approximately the same precision, under all circumstances, also in cases where other atoms are remote or not present at all, for instance in the diatomic molecule AB. A restriction of the expansion basis like in Eq. (2.76) makes the quality of the fit independent of the molecular environment. In analogy to Eq. (2.54) for the fit of the complete electron density ρ and Eq. (2.63) for the fit of individual overlap densities ρ µν, the expansion coefficients d µ Aν B p A and d µ Aν B q B are obtained from the system of linear equations V AA d µ Aν B A V BA d µ Aν B A + V AB d µ Aν B B + V BB d µ Aν B B = b µ Aν B A = b µ Aν B B (2.79) if the Coulomb norm [Eq. (2.35)] is employed, i.e. for W = r12 1. The definition of the matrix V and the vector b µ Aν B is the same as for the fit of individual overlap densities ρ µν, i.e. V pa q B = (Ω pa Ω qb ) and b µ Aν B p A = (Ω pa ρ µa ν B ). Comparison of

54 32 Chapter 2. Theory Eqs. (2.63) and (2.79) shows that the latter involves certain blocks of the complete matrix V depending on the location of the AOs forming the overlap densities to be fitted. To put it in more mathematical terms, the overlap densities are now projected onto subspaces of the total space spanned by the expansion basis {Ω p }. These subspaces can differ for different overlap densities. Of course it is also possible to use distinct sets {Ω AB p A } of expansion functions on an atom A for different atom pairs AB containing that atom depending on e.g. the type of the partner atom B or the interatomic distance. The system of linear equations (2.79) can be rewritten as with solution where and Introducing the notation V [AB] d µ Aν B [AB] d µ Aν B [AB] V [AB] = V AA = b µ Aν B [AB] (2.80) = V 1 [AB] bµ Aν B [AB], (2.81) d µ Aν B [AB] = b µ Aν B [AB] = V BA V AB V BB dµ Aν B A d µ Aν B B bµ Aν B A V [ABCD] = V AC b µ Aν B B V BC any four-center ERI can be approximated as, (2.82) (2.83) (2.84) V AD V BD (ρ µa ν B ρ κc λ D ) ( ρ µa ν B ρ κc λ D ) = d µ Aν B [AB] The error between the exact and the approximated ERIs,, (2.85) V [ABCD] d κ Cλ D [CD]. (2.86) (ρ µa ν B ρ κc λ D ) ( ρ µa ν B ρ κc λ D ) = (ρ µa ν B ρ µa ν B ρ κc λ D ρ κc λ D ) + (ρ µa ν B ρ µa ν B ρ κc λ D ) + ( ρ µa ν B ρ κc λ D ρ κc λ D ), (2.87)

55 2.3. Review of Density Fitting Methods 33 contains terms that are linear and quadratic in the fitting error ρ µa ν B ρ µa ν B. It is important to realize that even if the Coulomb norm has been employed, the linear error terms do not vanish for the general case of four-index ERIs because the expansion basis differs for each atom pair. In order to get rid of the linear error terms, the robust approximation ( ρ µa ν B ρ κc λ D ) rob = (ρ µa ν B ρ κc λ D ) + ( ρ µa ν B ρ κc λ D ) ( ρ µa ν B ρ κc λ D ) = b µ Aν B [CD] d κ Cλ D [CD] + d µ Aν B [AB] b κ Cλ D [AB] d µ Aν B [AB] V [ABCD] d κ Cλ D [CD] (2.88) has to be invoked. However, this requires the evaluation of three-center ERIs of the type b µ Aν B p C = (Ω pc ρ µa ν B ) which vitiates the advantage of the restriction of the expansion basis. For approximated two-center ERIs of the type ( ρ µa ν B ρ κa λ B ) the following equality holds, ( ρ µa ν B ρ κa λ B ) = d µ Aν B [AB] = b µ Aν B [AB] = d µ Aν B [AB] V [ABAB] d κ Aλ B [AB] d κ Aλ B [AB] = (ρ µa ν B ρ κa λ B ) b κ Aλ B [AB] = ( ρ µa ν B ρ κa λ B ), (2.89) since V [ABAB] = V [AB]. Thus, for this kind of two-center ERIs the linear error terms in Eq. (2.87) vanish and the simple approximation of Eq. (2.86) becomes identical to the robust approximation of Eq. (2.88). Just like in the approach of Baerends et al. [153], the total electron density can be decomposed into atom pair densities according to ρ(r) = A B ρ AB (r), (2.90) with ρ AB (r) = (2 δ AB )P µa ν B ρ µa ν B (r). (2.91) µ A,ν B Using expansion (2.77) for the diatomic overlap densities in expression (2.91) for the atom pair densities leads to ρ(r) = ρ AB (r) = Ω [AB] d [AB] (2.92) A B A B for the approximated electron density. Here, the expansion coefficients d AB p A for an atom pair density ρ AB have been obtained by contraction of the elements of the

56 34 Chapter 2. Theory density matrix P with the expansion coefficients d µ Aν B p A densities ρ µa ν B according to for the diatomic overlap d [AB] = (2 δ AB )d µaνb [AB] P µ A ν B. (2.93) µ A,ν B Eqs. (2.90) to (2.93) are valid for the most general case of distinct sets {Ω AB p A } and {Ω AB q B } of expansion functions for each different atom pair AB. The approximated Hartree energy is then given as Ẽ h = 1 2 ( ρ ρ) = 1 2 A B C D d [AB] V [ABCD]d [CD] (2.94) and the approximated matrix elements of the Hartree potential are given as Ṽ h,µa ν B = Ẽh = ( ) d [CD] V [CDEF ] d [EF ] P µa ν B P µa ν C D E F B (2.95) = C D d µ Aν B [AB] V [ABCD]d [CD] = ( ρ µa ν B ρ). Taking into account that the expansion coefficients have been obtained according to Eq. (2.81) and sorting out the atom pairs AB from the sum over the atom pairs CD, Eqs. (2.94) and (2.95) can be rewritten as and Ẽ h = 1 b [AB] 2 d [AB] A B A B = 1 (ρ AB ρ AB ) A B A B C D C A D B C D C A D B d [AB] V [ABCD]d [CD] ( ρ AB ρ CD ) (2.96) Ṽ h,µa ν B = b µ Aν B [AB] d [AB] + C D C A D B = (ρ µa ν B ρ AB ) + C D C A D B d µ Aν B [AB] V [ABCD] d [CD] ( ρ µa ν B ρ CD ). (2.97) The expression for the approximated electron density [Eq. (2.92)] can be simplified for the specific case in which the same sets {Ω pa } and {Ω qb } of expansion functions are used for all atom pairs AB, i.e. for the case in which the set {Ω pa }

57 2.3. Review of Density Fitting Methods 35 for an atom pair AB does not vary for differing partner atoms B or interatomic distances. Then Eqs. (2.69) to (2.73) for the general fit of individual overlap densities ρ µν are still valid if it is taken into account that the expansion coefficients d µ Aν B p C vanish for C A, B. Eqs. (2.74) and (2.75), however, do not hold any longer. In analogy to Eqs. (2.37) and (2.69) the approximate density can be written in this case as ρ(r) = p Ω p (r) d p = Ω d, (2.98) while the expansion coefficients d p for the electron density are now given by the restricted summation d pa = ( d µaνb p A B,µ A,ν B + d ν Bµ A p A ) PµA ν B. (2.99) This result is obtained by some straightforward manipulation after inserting the restricted expansion for the overlap densities [Eq. (2.76)] into the expression for the electron density [Eq. (2.16)]. It should be mentioned that the two terms in Eq. (2.99) appear because unlike the elements of the density matrix P, the expansion coefficients d µ Aν B p A exchange, i.e. d µ Aν B p A for the overlap densities are not symmetric with respect to index d ν Bµ A p A. Eqs. (2.72) and (2.73) then read as Ẽ h = 1 2 ( ρ ρ) = 1 2 d Vd = 1 d A 2 V AB d B (2.100) A,B and Ṽ h,µa ν B = Ẽh P µa ν B ( d C ) = V P CD d D µa ν C,D B = { } d µ Aν B A V AD + d µ Aν B B V BD d D = ( ρ µa ν B ρ). (2.101) D Sorting out the atoms A and B from the sum over D, Eq. (2.79) can still be used to reformulate the corresponding one- and two-center terms. With the definition of the approximated one-center electron densities ρ A (r) = Ω A d A, (2.102)

58 36 Chapter 2. Theory this leads to Ṽ h,µa ν B = b µ Aν B A d A + b µ Aν B B + { D A,B d B d µ Aν B A } V AD + d µ Aν B B V BD d D = (ρ µa ν B ρ A ) + (ρ µa ν B ρ B ) + D A,B for the approximated matrix elements of the Hartree potential. ( ρ µa ν B ρ D ) (2.103) Requirements to the expansion basis Comparing Eq. (2.23) and Eqs. (2.97) or (2.103) shows that now an expansion basis capable of fitting each individual overlap density ρ µa ν B with sufficient accuracy is required. This is the price to pay for the restriction of the basis. In Eqs. (2.97) and (2.103) the error arising from insufficient fits of individual overlap densities ρ µa ν B shows up in the last term on the right hand side. This term and thus the corresponding error vanishes for diatomic molecules and becomes larger and larger with increasing size of the molecule. Due to this term the approximated Hartree energy is not variational, i.e. it is not a bound to the exact Hartree energy. Robust fitting expressions could be invoked, however, this would require the evaluation of three-center ERIs as discussed above. Scaling behavior It has already been noted that a restriction of the expansion basis as described above leads to a formal O(N 2 ) scaling behavior because only one- and two-center ERIs of the type V pa q B and b µ Aν B p A have to be evaluated. The fact that at maximum two-center ERIs are required is especially appealing since highly optimized algorithms exist which allow for a several times more efficient evaluation of twocenter ERIs as compared to their four-center analogues [187]. The density fitting procedure itself also exhibits a formal O(N 2 ) scaling behavior since it requires the inversion of O(N 2 ) matrices V [AB] of size-independent dimension. A parallelization of the density fitting procedure is trivial since the expansion coefficients are determined independently for each atom pair AB. Since in the limit of large

59 2.4. The LEDO Expansion 37 molecules the number of significant diatomic overlap densities ρ µa ν B grows linearly with increasing system size, the asymptotic scaling for the density fitting procedure is O(N). The asymptotic scaling behavior for the evaluation of the ERIs or equivalently the Hartree energy, however, remains O(N 2 ) due to the long-range nature of the Coulomb interaction. 2.4 The LEDO Expansion Although one-center overlap densities can be projected onto an expansion basis, the error from such an approximation is unjustified, in so far as same-center densities describe the majority of the electron density, and because the cost for the evaluation of integrals involving two-center overlap densities will asymptotically become dominant. The idea of the LEDO expansion [158] therefore is, to approximate only the remainder of the density by projecting two-center overlap densities φ µa φ νb ρ µa ν B onto a set {Ω p } of auxiliary basis functions centered on the two atoms giving rise to the overlap densities according to (cf. Sec ) φ µa (r)φ νb (r) p A Ω pa (r) d µ Aν B p A + q B Ω qb (r) d µ Aν B q B, A > B. (2.104) This can be written in a shorthand notation as ρ k ρ k = p Ω p (r) d pk = Ω d k (2.105) and is just an expansion as described by Eqs. (2.76) and (2.77) with the restriction to unique atom pairs A > B. Again, d k is the coefficient vector that minimizes the norm of the difference vector ρ k ρ k between the exact and the approximate overlap density. It has been shown that the optimal choice for the weight operator in this fitting process is W = r 1 12 [1], i.e. to employ the Coulomb norm k (W = r 1 12 ) = (ρ k ρ k ρ k ρ k ). (2.106) The minimization of this norm with respect to the coefficients d k dk k = 0 results in the system of linear equations by setting (Ω p Ω q )d pk = (Ω p ρ k ) (2.107) q

60 38 Chapter 2. Theory for the determination of the expansion coefficients d pk. Switching back to a notation explicitly stating the atom pair of the diatomic overlap densities to be fitted and collecting the coefficient vectors d µ Aν B [AB] a matrix D [AB] this can be rewritten as [Eq. (2.83)] into the columns of V [AB] D [AB] = B [AB], A > B, (2.108) where the coefficient matrix V [AB] contains ERIs of the type (Ω p Ω q ) as defined in Eq. (2.82) and the inhomogeneity matrix B [AB] has the form of the matrix D [AB] but contains ERIs of the type (Ω p ρ k ). Further constraints may be imposed but it was shown that they do not improve the quality of the fit [1] Measuring the Quality of the LEDO Fit From Eqs. (2.105) and (2.107) follows the equality ( ρ k ρ k ) = ( ρ k ρ k ) (2.109) which allows to express the norm k of the difference vector ρ k ρ k as k = (ρ k ρ k ) ( ρ k ρ k ) = (ρ k ρ k ) p d pk (Ω p ρ k ). (2.110) Since the norm of a vector is always positive, the approximate integrals of Eq. (2.109) are always smaller in magnitude than the exact integrals (ρ k ρ k ). Note that k = 0 implies ρ k (r) = ρ k (r), i.e. an exact fit of the overlap density ρ k, because only the zero vector has a zero norm. Eq. (2.110) therefore represents an appropriate method to measure the quality of the LEDO expansion for a given overlap density ρ k [188]. It is convenient to define the norm la m B according to la m B = ila j mb, (2.111) i la,j mb where φ ila and φ jmb are AO basis functions φ µa and φ νb belonging to the shells l A and m B on atoms A and B, respectively. A shell is here defined as a set of basis functions sharing the same exponents, contraction pattern and angular momentum quantum number. Choosing spherical harmonics as a basis for the angular

61 2.4. The LEDO Expansion 39 part of the AOs, the index i la designates magnetic quantum numbers corresponding to the angular momentum quantum number of shell l A. The advantage of la m B is that it is invariant under unitary transformations of the basis functions and thus invariant under rotations of the coordinate system. Furthermore, the norm l Am B shall be defined by summing over all shells up to l A and m B, l Am B = l A l A =1 m B m B =1 l A m B, (2.112) and the norm AB for an atom pair A > B by summing over all shells l A and m B, AB = la max m max B l A =1 m B =1 la m B. (2.113) The rotational invariance mentioned above also holds for l Am B and AB. The norm AB has all properties required for an optimization procedure. This leads to the expressions and AB ζ p i AB c p i = 0 (2.114) = 0 (2.115) for the determination of the exponents ζ p i and contraction coefficients c p i of the auxiliary functions Ω p [188]. The derivatives can be obtained numerically and the optimization of the auxiliary functions can be carried out by a relaxation step based on these derivatives. Note that a good LEDO expansion basis should minimize the norm la m B for all possible combinations of shell pairs for a given AO basis set that can appear in molecular calculations. A systematic approach for this complex task will be described in chapter The LEDO Expansion Basis In principle the LEDO expansion basis {Ω pa } can be chosen arbitrarily, bearing in mind that rotational invariance should be guaranteed. In the original context [158], the LEDO expansion basis consists of all one-center overlap densities

62 40 Chapter 2. Theory {φ µ A φ µ }, with µ A A µ A. This is a very good start, but in general the fit of the diatomic overlap densities ρ µa ν B has to be improved by supplementing this expansion basis with an additional set of auxiliary functions {Λ ra } [1]. These can either be chosen as orbitals products {Λ ra } = {η µ A φ µ, η A µ η A µ } arising from A a set of auxiliary orbitals {η µa } or as a set of independent functions. Up to now only the first choice has been explored [1, 158, 188, 189]. A major drawback of this form of the expansion basis is the rapid increase of the number of expansion functions since each auxiliary orbital η µa has to be multiplied with the complete AO basis set {φ µa }. Moreover, near-linear dependences are introduced in a rather uncontrollable fashion and some of the non-linear dependent new AO products might not contribute to a significant improvement of the density fit at all. However, an implementation based on one-center AO products for the LEDO expansion basis and using auxiliary orbitals is straightforward since the required integrals are readily available in common quantum chemistry program packages. The optimization of such auxiliary orbitals for the Ahlrichs SVP basis set [190] will be presented in chapter 4.1. It might be possible to dispense with the one-center overlap densities in the expansion basis altogether and to base the fit exclusively on a properly chosen auxiliary basis {Λ ra }, as is done in methods fitting the complete electron density [127, 128, 153, 155] (cf. Sec ). However, no use of such auxiliary functions was made in this work Near-linear Dependences A serious technical difficulty is encountered in solving the system of linear equations (2.108) for the determination of the LEDO expansion coefficients D [AB] as a result of rounding errors and near-linear dependences in the LEDO expansion basis {Ω p }. This problem is not unique to the LEDO expansion and in principle all density fitting methods are affected. It requires special attention and there are several ways to deal with it. A standard procedure is to perform an eigenvector decomposition and to project out small eigenvalues. This approach is essentially a singular value decomposition (SVD) [191, 192] of a square matrix and has already been employed by Billingsley and Bloor for the LEDO expansion [158]. The drawback of the SVD is that it is computationally intensive. A very clever

63 2.5. LEDO-DFT 41 way of dealing with this problem is the perturbative approximation proposed by Eichkorn et al. [127] in the framework of conventional density fitting in which a small shift δ = 10 9 is applied to the diagonal elements of the coefficient matrix V [AB] of the system of linear equations (2.108). In the case of the LEDO expansion a one-step procedure which is accurate up to δ is satisfactory [189] instead of the proposed two-step procedure which is accurate up to δ 2 [127]. Another solution to the problem is the elimination of the near-linear dependences from the expansion basis via a modified Cholesky factorization of the coefficient matrix V [AB] [1, 185]. Like the SVD, this procedure is computationally rather expensive. However, it turns out that the near-linear dependent expansion functions have to be determined only once for each atom type by a modified Cholesky factorization of the one-center blocks of V [AB]. In practice, no additional linear dependences are encountered for atom pairs, i.e. for non-zero two-center blocks of V [AB] [188]. A list of the basis set specific linear dependent expansion functions can thus be generated and employed in all subsequent calculations with this basis set. As for the perturbative approximation one can then proceed with a standard Cholesky factorization which is the method of choice for the solution of a system of linear equations with a symmetric and positive definite coefficient matrix like V [AB] [191, 193]. The major advantage of this a priori elimination, however, is the reduction of the actual size of the LEDO expansion basis thus greatly reducing the computational demand for the determination of the LEDO expansion coefficients D [AB] [188]. This is very important since the number of expansion functions grows quickly upon inclusion of auxiliary orbitals, especially if these have a high angular momentum quantum number. The efficiency and robustness of the a priori elimination will be demonstrated in Sec LEDO-DFT There have been several attempts to formulate non-empirical MO theories of both HF [194, 195] and DFT types [196] on the basis of the Rüdenberg approximation [197], but none of these seems to have resulted in a successful implementation widely used by the chemical community. Obviously the Rüdenberg approximation is not accurate enough to serve as a basis for a non-empirical MO method [1].

64 42 Chapter 2. Theory However, this failure does not discredit the principal idea to expand diatomic overlap densities in terms of one-center auxiliary functions, like it is done in the LEDO expansion. At the time the LEDO approach was published, Slater orbitals were still widely used in molecular electronic structure calculations. The LEDO method has been developed to avoid the cumbersome evaluation of three- and four-center ERIs in a Slater basis. With the advent of Gaussian basis sets, such integrals could be evaluated rapidly and the LEDO approach seems to have been largely forgotten since. Recently, however, the LEDO approximation has been revived [1] in the framework of KS-DFT where it reveals its full potential. Starting point of the LEDO-DFT formalism is a subdivision of the complete electron density [Eq. (2.16)] into its one- and two-center terms according to ρ(r) = P µ A µ ρ A µ (r) + 2 P A µ A µa ν B ρ µa ν B (r). (2.116) A>B,µ A,ν B A,µ A,µ A Using the LEDO expansion (2.104) in Eq. (2.116), the two-center terms can be reexpressed by one-center terms. After some straightforward manipulation, the approximated electron density can be written as ρ(r) = P µ A µ ρ A µ (r) + Ω A µ A pa (r) d pa, (2.117) A,p A A,µ A,µ A where the expansion coefficients d pa for the approximated two-center part of the electron density are given by contraction of the appropriate two-center elements of the density matrix P with the LEDO expansion coefficients according to d pa = 2 B,µ A,ν B B<A d µ Aν B p A P µa ν B + 2 B,µ A,ν B B>A d ν Bµ A p A P νb µ A. (2.118) Eqs. (2.117) and (2.118) are valid for an arbitrary set {Ω pa } of LEDO expansion functions. If the LEDO expansion basis {Ω pa } contains all one-center overlap densities {φ µ A φ µ A }, i.e. for {φ µ A φ µ A } {Ω p A }, then Eq. (2.117) simplifies to ρ(r) = A,pA Ω pa (r) d pa = A Ω A d A = Ω d. (2.119) In Eq. (2.119) the one-center elements of the density matrix P are included in

65 2.5. LEDO-DFT 43 the respective elements of the expansion coefficients. These are now defined as d pa = 2 B,µ A,ν B B<A d µ Aν B p A P µa ν B + 2 d µ A µ A = g µ A µ A P µ A µ A + 2 B,µ A,ν B B>A B,µ A,ν B B<A + 2 B,µ A,ν B B>A d ν Bµ A p A P νb µ A, p A µ Aµ A, (2.120a) d µ Aν B µ A µ A P µa ν B d ν Bµ A µ A µ A P νb µ A, µ A µ A, (2.120b) g µ A µ = 1 µ A = µ A. (2.120c) A 2 µ A µ A The factor g has been introduced because the AO products φ µ A φ µ are restricted A to µ A µ A due to the symmetry with respect to exchange of the two indices while the summation for the electron density in Eq. (2.116) or Eq. (2.117) runs over all index pairs. In all following derivations it will be assumed that the approximated electron density is given by Eq. (2.119). For other cases, i.e., if the LEDO expansion basis does not contain the one-center AO products, the expressions have to be adjusted accordingly. The approximated electron density ρ can now be used in all terms of the KS energy expression (2.1) which are explicit functionals of the electron density, i.e. all terms except the kinetic energy. The variation of the resulting approximated electronic energy Ẽ is δẽ = δt s[ρ(r)] + dr δ ρ(r) {v ext (r) + ṽ h (r) + ṽ xc (r)}, (2.121) where ṽ h (r) = dr 2 r 1 12 ρ(r 2 ) (2.122) and ṽ xc (r) = δe xc[ ρ(r)] δ ρ(r) (2.123) are the approximated Hartree and XC potential, respectively. According to Eq. (2.119) the variation of the approximated density is given as δ ρ(r) = A,p A Ω pa (r) δd pa (2.124)

66 44 Chapter 2. Theory by a variation of the expansion coefficients d. Eq. (2.121) can thus be written as with δẽ = δp µa ν B φ µa 1 2 φ ν B + δd pa Ṽp KS A, (2.125) A,B,µ A,ν B A,p A Ṽ KS p A = Ṽext,p A + Ṽh,p A + Ṽxc,p A = dr Ω pa (r) {v ext (r) + ṽ h (r) + ṽ xc (r)}. (2.126) If the two-center matrix elements of the potential terms are approximated as Ṽ KS µ A ν B = p A Ṽ KS p A Ṽ KS ν B µ A = Ṽ KS µ A ν B, d µ Aν B p A + q B Ṽ KS q B d µ Aν B q B, A > B, (2.127) then it can be easily shown using Eq. (2.120) that d pa Ṽp KS A = P µa ν B Ṽµ KS A ν B. (2.128) A,p A µ A,ν B Using Eq. (2.128) in Eq. (2.125), the energy variation δẽ terms of a variation of the elements of the density matrix P, δẽ = can be expressed in A,B,µ A ν B δp µa ν B HKS µ A ν B, (2.129) with H KS µ A ν B = φ µa 1 2 φ ν B + Ṽ KS µ A ν B. (2.130) Adding the orthonormality constraints, the variational condition results in the matrix eigenvalue equation H KS C = SCε, (2.131) which is just the approximated equivalent of the conventional KS matrix eigenvalue equation (2.20). The essence of the present formalism lies in the exceptionally simple scheme for the construction of the secular matrix H KS. Apart from the matrix elements of the kinetic energy operator, only the one-center terms Ṽ KS p A, the number of which scales linearly with the number of basis functions, have to be evaluated

67 2.5. LEDO-DFT 45 Table 2.1: Scaling behavior of the various parts required for the setup of the approximated potential contribution ṼKS to the KS matrix for LEDO-DFT. Scaling behavior formal asymptotic Ṽ ext,pa = dr Ω pa (r)v ext (r) O(N 2 ) O(N 2 ) Ṽ h,pa = dr Ω pa (r)ṽ h (r) = d qb (Ω pa Ω qb ) B,q B O(N 2 ) O(N 2 ) Ṽ xc,pa = dr Ω pa (r)ṽ xc (r) O(N 2 ) O(N) Ṽ KS µ A ν B = A,p A Ṽ KS p A d µ Aν B p A + B,q B Ṽ KS q B d µ Aν B q B O(N 2 ) O(N) explicitly, because the two-center elements of the matrix ṼKS are obtained from Eq. (2.127). Using Eq. (2.126) and inserting the approximated electron density [Eq. (2.119)] into the approximated Hartree potential [Eq. (2.122)], the expression for the approximated Hartree contribution to the one-center elements of the KS matrix reads as Ṽ h,pa = d qb (Ω pa Ω qb ). (2.132) B,q B Now it can be easily seen that the formation of the matrix H KS is formally an O(N 2 ) process since only one- and two-center ERIs of the type (Ω pa Ω qb ) are required and the numerical integration of the XC contribution is limited to onecenter terms. The asymptotic scaling for this step will remain O(N 2 ) due to the long-range nature of the Coulomb interaction. The number of two-center terms Ṽ KS µ A ν B grows quadratically with the number of basis functions, but since they are obtained from Eq. (2.127), the cost for the evaluation of each is independent of the system size, resulting in formal O(N 2 ) scaling. For large interatomic distances, the overlap densities ρ µa ν B and thus the corresponding expansion coefficients will all be zero and this step will eventually scale linearly for sufficiently d µ Aν B p A extended systems. The scaling behavior of the various steps is illustrated in Table 2.1. In practice the LEDO approximation will not be applied to the matrix elements of the external potential v ext since the evaluation of these is not costly

68 46 Chapter 2. Theory and already scales asymptotically as O(N 2 ). Recalling the facts discussed in Sec about the fit of individual overlap densities with a restricted expansion basis, it is clear that an approximation based on the LEDO expansion cannot be robust. This can as well be seen if one considers that each diatomic overlap density is fitted with an error independent of the system size. Since the number of interacting overlap densities increases as O(N 4 ) for small molecules and asymptotically as O(N 2 ), the error introduced by the LEDO approximation increases in a non-linear fashion with system size. Therefore, a failure of LEDO-DFT might be observed for very large molecular systems. However, it is also clear that this is not a general problem because the errors will be small if a sufficiently accurate fit is obtained and the point of breakdown can be shifted by improving the expansion basis. It is furthermore evident that it is sufficient to obtain an expansion basis which guarantees numerical stability up to system sizes for which linear scaling fast multipole techniques can be efficiently applied to treat the interaction of distant overlap densities. It should be noted that while the LEDO-DFT energy expression is variational, the LEDO expansion coefficients D are not determined in a variational fashion. Therefore the approximated energy is not a bound to the exact energy and the derivatives D α of the LEDO expansion coefficients will appear in the expression of the analytical gradient. Finally, it should be mentioned that LEDO-DFT is not appropriate for the use with hybrid density functionals [1]. This is due to the fact that the density fit can be applied efficiently only to energy terms involving the electron density and not, however, to terms requiring the density matrix like the kinetic energy or the HFlike (orbital) exchange. This also precludes the use of the LEDO approximation in HF theory. One could still use the LEDO expansion for an approximation of the ERIs in the exchange part, thus still ending up with two-center integrals only, but the elegance of the formalism and the favorable O(N 2 ) scaling is lost. The reason why the LEDO expansion has been forgotten may just be due to the fact that it had originally been intended as an integral approximation in HF theory [158], whereas its full power can be realized only in the context of KS-DFT.

69 2.6. Analytical Gradients for LEDO-DFT Analytical Gradients for LEDO-DFT The calculation of energy derivatives of a molecule with respect to nuclear displacements [ ] is one of the most routinely used features of modern quantum chemical software. Gradient methods allow for the calculation of forces acting on the nuclei, which in turn render it possible to locate the equilibrium structure of a molecule and transition states between minima, and thus to map out the course of chemical reactions [ ]. Analytical gradient-based optimization methods are almost an order of magnitude faster than optimization algorithms that do not use gradients. The second derivatives of the electronic energy with respect to nuclear displacements can efficiently be obtained as numerical derivatives of the analytically determined gradients [208], thus enabling the calculation of harmonic force fields which in turn can be used to predict vibrational spectra. Therefore, the expression for the analytical gradient for LEDO-DFT is of high interest and shall be presented in the following. Various analytical expressions for the derivatives of the electronic energy in KS-DFT have been elaborated in great detail by Komornicki and Fitzgerald [209] and can be adapted to the LEDO-DFT formalism. The derivation of the analytical gradient of the LEDO-DFT energy is actually identical to one which has been developed earlier for a similar formalism based on the Rüdenberg instead of the LEDO expansion [196] and the final expression has already been presented in the original LEDO-DFT contribution [1]. Considering the gradient of a KS reference wave function with respect to a perturbational parameter α such as, e.g., a nuclear coordinate, one may write for the MO coefficients [209] C = C 0 UG, (2.133) where C 0 is the MO coefficient matrix at the reference geometry. The matrix U allows for unitary transformations of the MOs while the matrix G explicitly depends only on the nuclear coordinates and thus ensures orthogonality of the MOs for all values of the perturbational parameters. Both U and G are of course identical to the unit matrix at the reference point. As a general expression for the derivative of the MOs at the reference geometry one thus obtains C α = C 0 (U α + G α ). (2.134)

70 48 Chapter 2. Theory The derivative U α vanishes in case of a variationally optimized wave function. There are many ways to define the matrix G but it is most obvious to choose it symmetric, which results in [210] G = (C 0 SC 0 ) 1/2. (2.135) The first derivative at the reference geometry is now given by [210] G α = 1 2 C0 S α C 0. (2.136) Formal differentiation of the KS energy expression of a closed shell system in MO basis yields [209] occ. E α = 2 ψ i v occ. ext ψ i (α) + 2 (ψ i ψ i ψ j ψ j ) (α) i + 2 i occ. ψ i v xc ψ i (α) + 4 Hit KS G α ti. i,t ij (2.137) Putting the upper index α in parentheses indicates that only the basis functions have to be differentiated. Taking into account that the matrix representation of the KS operator in the canonical MO basis is diagonal, with the diagonal elements just being the MO energies ε i, one obtains occ. occ. Hit KS G α ti = ε i G α ii (2.138) i,t for the last term on the right hand side of Eq. (2.137). As can be seen from Eq. (2.136), this term accounts for the geometry dependence of the metric of the AO basis set. Eq. (2.137) can now be used to derive the expression for the analytical gradient of the LEDO-DFT energy. i In contrast to conventional KS-DFT, the electron density is not given by Eq. (2.2). However, this expression may be used as a formal notation for the electron density if it is taken into account that the electron density is not obtained by simply plugging in the LCAO expansion (2.15) for the MOs, which leads to Eq. (2.16), but rather by subsequent application of the LEDO approximation which is given by Eq. (2.104). In that case, expressions (2.2) and (2.119) for the electron density without and with LEDO approximation,

71 2.6. Analytical Gradients for LEDO-DFT 49 respectively, are completely equivalent and Eq. (2.137) can be employed. Thus, inserting the LCAO ansatz Eq. (2.15) in the matrix elements of Eq. (2.137) and applying the LEDO approximation [Eq. (2.104)] to all terms except the kinetic energy results in the expression Ẽ α = P νb µ A φ µa 1 2 φ ν B α A,B,µ A,ν B + [ ( α ( d pa dr Ω pa (r)v ext (r)) + A,p A + 1 d pa d qb (Ω pa Ω qb ) α + d α p 2 A Ṽp KS A + 4 A,B,p A,q B A,p A i = T α s + Ẽα ext + Ẽα xc + Ẽα h + E α LEDO + G α ) ] (α) dr Ω pa (r)ṽ xc (r) ε i G α ii (2.139) for the analytical gradient of the LEDO-DFT energy. Here, the expansion coefficients d pa of the approximated electron density are defined as in Eq. (2.120). The same result is of course obtained by insertion of Eq. (2.119) for the approximated electron density into all terms of Eq. (2.137) which explicitly depend on the electron density, i.e., all terms except the kinetic energy. The first four terms of this expression are analogous to the energy expression, the integrals just being replaced by their derivatives. The sixth term G α accounts for the geometry dependence of the metric of the AO basis, as discussed above. All of these terms have similar or identical counterparts in conventional KS-DFT implementations. The only new term is ELEDO α containing the derivative of the vector d. Although LEDO-DFT is variational, so that the derivative of the density matrix P does not appear, the vector d contains the LEDO expansion coefficients [Eq. (2.120)] which do depend on the structure of the molecule. The derivatives of the LEDO expansion coefficients are obtained by differentiating Eq. (2.107) resulting in an analogous system of linear equations, (Ω p Ω q )d α qk = (Ω p ρ k ) α q which can be written as q (Ω p Ω q ) α d qk, (2.140) V [AB] D α [AB] = B α [AB] V α [AB]D [AB], A > B. (2.141) Note that the coefficient matrix is identical to the one needed for the determination of the LEDO expansion coefficients D [AB], which means that in the course

72 50 Chapter 2. Theory of a structure optimization only one factorization has to be performed per optimization step. Furthermore, derivatives of at most two-center ERIs appear in the expression of the analytical gradient of the LEDO-DFT energy, resulting in a formal O(N 2 ) computational cost for its evaluation. Essentially, any statement about the scaling behavior of LEDO-DFT calculations also holds for the evaluation of the analytical energy gradient. 2.7 A Projection Operator Formalism for LEDO-DFT Due to the reasons outlined in Sec. 2.5, for large molecular systems the numerical errors introduced by the LEDO approximation can lead to an uncontrollable behavior of the SCF process. It has already been stated that these errors can always be reduced by an improvement of the LEDO expansion basis or by using robust expressions for the approximated ERIs. In the following, however, a projection technique for the elimination of near-linear dependent AOs shall be introduced, which can be applied in order to guarantee for SCF convergence of LEDO-DFT calculations without resorting to the aforementioned remedies. Although AO basis sets commonly used in MO calculations are linearly independent in the strict sense of the term it may happen that they are approximately linearly dependent from a numerical point of view [151]. As a measure for the degree of near-linear dependence it is convenient to choose the condition number of the overlap matrix S which represents the metric of the AO basis set {φ µ } [211]. The condition number of the overlap matrix given by the ratio of the highest to the lowest eigenvalue is not invariant with respect to local orthogonalization, i.e., orthogonalization of the AOs located on a particular atom. However, the changes of the condition number due to local orthogonalization are comparatively small and do not affect the present discussion. Löwdin [151] approached the problem of near-linear dependence by considering the canonically orthogonalized orbitals (COOs). Arranging the AO basis functions φ µ and the COOs χ k as row vectors Φ = {φ 1, φ 2,... } and χ = {χ 1, χ 2,... },

73 2.7. A Projection Operator Formalism for LEDO-DFT 51 respectively, the latter can be written as χ = ΦUe 1/2. (2.142) U is a unitary matrix diagonalizing the overlap matrix S with the eigenvalues being arranged in a diagonal matrix e, U SU = e. (2.143) The squares of the coefficients of the COOs do not add up to one as it would be the case for an orthogonal basis, instead one has 2 U µk = 1. (2.144) e k µ e 1/2 k Near-linear dependence occurs if there are eigenvalues e k close to zero. According to Eq. (2.144), the AO coefficients of the corresponding eigenvectors can become very large. These are just the eigenvectors which need to be eliminated from the basis if numerical problems are to be avoided. The projection scheme will be outlined in the following. A threshold c max for the condition number is defined and, starting with the eigenvector belonging to the lowest eigenvalue, as many eigenvectors are removed as is necessary to fulfill the condition e N e M+1 < c max (2.145) if N is the dimension of the AO basis set and the eigenvalues e k are arranged in increasing order. Thus eliminating M eigenvectors, the remaining ones are used to form a projection operator Q = N k=m+1 χ k χ k (2.146) which acts on the KS operator H KS from the left and from the right, H KS,proj = Q H KS Q = N k,l=m+1 χ k χ k H KS χ l χ l. (2.147) Performing the projection in the AO basis, the projected KS matrix reads as H KS,proj = Q H KS Q = SPH KS PS (2.148)

74 52 Chapter 2. Theory with the matrix elements of P being given as P µν = N k=m+1 U µk U νk e k. (2.149) Suhai et al. [212] made plausible why this projection scheme should dampen the effect of numerical inaccuracies in the evaluation of the KS matrix. First, it should be noted that the projection reduces the dimension of the basis from N to N M as can be seen from Eq. (2.147) thus raising the electronic energy. However, this effect will be small because the eliminated basis vectors contribute mainly to the space of virtual orbitals. Following Suhai et al. [212], the KS matrix in the AO basis can be decomposed in an exact part and a contribution arising from numerical errors, H KS = H KS,exact + H KS. (2.150) To clarify the effects of numerical errors it is necessary to transform the KS matrix from the non-orthogonal AO basis to an orthogonal basis. Choosing the basis of COOs one obtains H KS,COO = e 1/2 U H KS Ue 1/2. (2.151) Thus, the error part of the KS matrix is given by H KS,COO kl = 1 e 1/2 k e1/2 l µ,ν U µku νl H KS µν. (2.152) It is easily recognized from Eq. (2.152) that the presence of small eigenvalues of the overlap matrix leads to an amplification of the inaccuracies in the KS matrix. Cycle after cycle the errors increase and the SCF procedure can become unstable. Thus, it is desirable to eliminate the eigenvectors belonging to such small eigenvalues. The use of this technique is inexpensive and requires only minor modifications to the program.

75 Chapter 3 Implementation For the practical application the LEDO-DFT formalism including analytical gradients and the projection technique, which have been presented in the last chapter in Secs. 2.4 to 2.7, have been turned into a working implementation. In this chapter a brief overview of this implementation is given and some important details are highlighted. Possible and desirable extensions will be discussed in chapter Framework and General Features All implementations presented in this work 1 [188, 189] have been carried out in the framework of the program package TURBOMOLE [10, 11] in version 5.1. The programs RIDFT and RDGRAD have been chosen for the implementation of the energy and the gradient calculations, respectively. These are the programs for KS-DFT calculations within the RI approximation (conventional density fitting) and therefore appear as a natural choice for the implementation of LEDO- DFT, although the programs DSCF and GRAD could have been used as well. These latter programs allow to perform HF and unapproximated KS-DFT calcu- 1 The SCF part is based on a preliminary implementation of LEDO-DFT which had been realized in the framework of a Diplomarbeit (Master s Thesis) [12]. It was restricted to Cartesian Gaussian functions as AO basis and the LEDO approximation was limited to the Hartree term. In this version the perturbative approach of Ahlrichs and the a priori elimination for the determination of the LEDO expansion coefficients (cf. Sec ), auxiliary orbitals to augment the LEDO expansion basis and integral prescreening have not been available. 53

76 54 Chapter 3. Implementation lations. The projection technique has been implemented both into the program RIDFT and DSCF. The programming language FORTRAN 77 [213] has been used throughout, apart from the dynamical memory management which is handled by FORTRAN 90 [214] statements. All required quantities are completely kept in the core memory during the calculations. While this might not be the best choice for future applications, it is completely sufficient for the purposes of this work. Communication between the programs RIDFT and RDGRAD, which is necessary for efficient LEDO-DFT calculations, has been realized via files. Extensive use of BLAS [215] and LAPACK [216] linear algebra routines is made whenever possible and the translational invariance of the ERIs [ ] is fully exploited in the gradient program. Using a prescreening of the ERIs based on the Schwarz inequality [55, 220], it is possible to determine the LEDO expansion coefficients and derivatives thereof only for non-negligible overlap densities. Point group symmetry is not exploited in the present implementation. However, this is not considered as a major drawback since most large molecular systems of interest actually do not possess any symmetry elements other than C 1. The LEDO approximation has been implemented both for the Hartree and the XC contributions for energy and gradient calculations, with the option to restrict the LEDO approximation to the Hartree term. All other contributions to the KS matrix, the energy and the gradient, are treated conventionally as implemented in TURBOMOLE. The LEDO expansion basis {Ω p } can be augmented by a set of auxiliary orbitals {η µ } and the full Cartesian components (i.e. 6 d-type, 10 f- type etc.) of the Gaussians are always used in the LEDO expansion. Apart from the modified Cholesky factorization [1], the perturbative approximation [127] or the a priori elimination [188] can be employed in combination with a standard Cholesky factorization for the determination of the LEDO expansion coefficients and their derivatives. Finally, the menu-driven input program DEFINE has been adapted in such a way that input files appropriate for LEDO-DFT calculations with TURBOMOLE can be created with ease.

77 3.2. SCF Energy Calculation SCF Energy Calculation For the energy calculation two major parts can be distinguished, namely the determination of the LEDO expansion coefficients D before the SCF cycle (cf. Sec. 2.4) and the evaluation of the Hartree and XC contributions within the LEDO approximation during the SCF cycle (cf. Sec. 2.5). First, all ERIs V and B are precalculated and stored in two arrays in the main memory. Eq. (2.108) is then solved for each atom pair A > B subsequently and the factorized coefficient matrices V 1 [AB] and the LEDO expansion coefficients D [AB] are written to disk for later use in the gradient program. The expansion coefficients D are stored in the array for the ERIs B which are no longer needed. The ERIs V are needed in the SCF cycle in order to evaluate the approximated one-center elements of the Hartree contribution to the KS matrix according to Eq. (2.132). At this point, there is the option to calculate ERIs of the type (ρ k ρ k ) as required for the evaluation of the norm k [Eq. (2.110)] and la m B [Eq. (2.111)], if the quality of the LEDO fit is to be assessed. Otherwise the program enters the SCF cycle. During each SCF cycle the vector d is calculated from the LEDO expansion coefficients D and the density matrix P [Eq. (2.120)]. Next, the approximated one-center matrix elements of the Hartree potential Ṽ h are obtained from the ERIs V and the vector d [Eq. (2.132)]. The approximated one-center elements of the XC potential Ṽxc are calculated by numerical quadrature of the approximated electron density given by Eq. (2.119), and are added to the one-center elements of Ṽ h. Note that one-center elements have to be evaluated for one-center AO products φ µ A φ µ and auxiliary functions Λ A r A, i.e. one-center products with auxiliary orbitals η µa of the type η µ A φ µ and η A µ η A µ. The two-center elements of A the approximated potential contributions Ṽh,xc = Ṽh + Ṽxc to the KS matrix are finally obtained from its one-center elements and the LEDO expansion coefficients D [Eq. (2.127)]. Once the SCF is converged, the one-center elements of the Hartree and XC contribution Ṽh,xc to the KS matrix and the vector d are written to disk for use in the gradient program. A scheme of the program flow is depicted in Fig A transition to an integral direct algorithm is straightforward, but has not

78 56 Chapter 3. Implementation calculate T s and V ext calculate ERIs V and B loop over atom pairs A > B solve V [AB] D [AB] = B [AB] write V 1 [AB] and D [AB] to disk guess for initial P form vector d from D and P get new P form Ṽh A = B V ABd B calculate Ṽxc A from ρ = Ω d form Ṽh,xc AB = Ṽh,xc [AB] D [AB] convergence? solve secular equations write vector d and Ṽh,xc A to disk Figure 3.1: Flow chart of the LEDO-DFT implementation into the SCF program RIDFT. Dashed boxes indicate parts of the program that have been left unchanged. Here, ṼA h and Ṽxc A represent the one-center elements corresponding to atom A and Ṽh,xc [AB] the one-center elements corresponding to atoms A and B. been realized within this work. In a direct algorithm, the ERIs V can be treated directly, while the LEDO expansion coefficients D should be kept in main memory or written to disk because of the high cost for their determination. The magnitude of the elements of the density vector d the ERIs V are contracted with to obtain the one-center elements of the approximated Hartree potential should then be accounted for during the SCF, and only those ERIs actually be recalculated which give rise to a significant contribution to the KS matrix. Further savings within such a scheme, especially towards the end of the SCF procedure, can be expected if only the difference in the vector d between two SCF cycles is considered. In what follows, some important details of the various steps described above

79 3.2. SCF Energy Calculation 57 shall be discussed Integral Evaluation and Prescreening The Schwarz inequality (ρ µa ν B ρ κc λ D ) (ρ µa ν B ρ µa ν B ) (ρ κc λ D ρ κc λ D ) (3.1) provides a strict upper bound for the magnitude of ERIs and has proven successful and reliable for integral prescreening in electronic structure calculations [55, 56, 220]. This bound works properly because the ERIs define a scalar product in a vector space with metric r 1 12 [cf. Eq. (2.35)] and the ERIs (ρ µa ν B ρ µa ν B ) thus represent the norm of the vectors (cf. also the discussions in Sec. 2.3). From Eq. (3.1) it is clear that an ERI (ρ µa ν B ρ κc λ D ) will always be small in magnitude if one of the associated overlap densities ρ µa ν B or ρ κc λ D is negligible. Therefore, the magnitude of the ERIs of the type (ρ µa ν B ρ µa ν B ) is an appropriate quantity to decide if a diatomic overlap density ρ µa ν B is readily implemented. can be neglected. Such a prescreening In LEDO-DFT, ERIs V of the type (Ω pa Ω qb ) and ERIs B of the type (Ω pa ρ µa ν B ) are required. The ERIs V will never be negligible due to the long range nature of the Coulomb interaction and because the norm of the LEDO expansion functions Ω p (or the one-center overlap densities they consist of) is non-negligible. The case is different, of course, for the ERIs B. These can be neglected for all diatomic overlap densities ρ µa ν B the norm of which is smaller than a given threshold. In fact, all computations related to negligible diatomic overlap densities ρ µa ν B are superfluous. Thus, prescreening for such diatomic overlap densities has tremendous consequences for various parts of a LEDO-DFT calculation. The AO basis functions in TURBOMOLE are organized into shells l A, each shell being defined as a set of basis functions φ ila sharing the same exponents, contraction pattern and angular momentum quantum number (cf. also Sec ). The integral routines actually evaluate all ERIs (k A l B m C n D ) belonging to a shell quadruple at the same time. The advantage of this procedure is that certain auxiliary functions employed during the integral evaluation are common to all

80 58 Chapter 3. Implementation members of a shell and computational efficiency is thereby enhanced [59]. Prescreening for the ERIs B therefore has to take place on the level of shell pairs l A m B which can safely be neglected if the largest overlap density ρ ila j mb belonging to that shell pair is smaller than a given threshold. Eliminating complete shell pairs by such a prescreening has furthermore the advantage that rotational invariance remains guaranteed. If all shell pairs l A m B of an atom pair A > B are small in magnitude, then no density fit has to be carried out for that atom pair and all LEDO-related operations of that atom pair can be completely skipped. This will eventually lead to linear scaling of the fitting procedure for extended systems. While, due to the ERIs V, this prescreening does not affect the O(N 2 ) scaling behavior of LEDO-DFT, it greatly reduces the associated prefactor. In the present implementation, first all ERIs (ρ µa ν B ρ µa ν B ) required for the prescreening are evaluated in order to set up index vectors for negligible overlap densities, shell pairs and atom pairs. Next, a special loop structure is executed, in which the ERIs V and B are evaluated for each atom pair A > B subsequently and stored in two arrays in the main memory. Integral prescreening on the basis of shell pairs and atom pairs as described above is applied for the evaluation of the ERIs B, reducing the computational effort and memory requirement of LEDO-DFT calculations. Organizing the evaluation of the ERIs according to atom pairs A > B is not only convenient for incore calculations, but also enables an easy transition to an integral direct algorithm in which the determination of the LEDO expansion coefficients D [Eq. (2.108)] will have to be embedded into the loop which is executed for the evaluation of the ERIs. The index vectors for the prescreening are kept in main memory and are written to disk for later use in the SCF cycle and the gradient program as described in the subsequent sections LEDO Expansion Coefficients Once the ERIs V and B have been evaluated, the LEDO expansion coefficients D are determined, again for each atom pair A > B subsequently. Due to the prescreening of negligible overlap densities ρ µa ν B, the inhomogeneity B [AB] of the system of linear equations (2.108) to be solved will contain the less vectors b µ Aν B [AB] the more overlap densities ρ µa ν B can be neglected. If an atom pair can be skipped

81 3.2. SCF Energy Calculation 59 completely, the reduction in the computational effort is even more pronounced since the coefficient matrix V [AB] does not have to be factorized at all. Among the methods mentioned in Sec to prevent problems with numerical instabilities during the determination of the LEDO expansion coefficients, the a priori elimination deserves some comments because it affects various steps of a LEDO-DFT calculation. In principle, all ERIs over near-linear dependent expansion functions are dispensable and do not have to be evaluated at all. However, the a priori elimination does not guarantee that complete shell pairs l A l A are eliminated from the expansion basis {Ω pa } = {φ µ A φ µ, η A µ φ A µ, η A µ η A µ }. Therefore, A an integral prescreening for the ERIs V and B due to the near-linear dependences cannot be employed in practice. Furthermore, for the evaluation of the Hartree term during the SCF [Eq. (2.132)], all ERIs V over one-center AO products φ µ A φ µ are always needed, because these contribute directly to the one-center A elements of the KS matrix and the one-center elements of the density matrix P entering the corresponding elements of the density expansion vector d [Eq. (2.120b)] are in general not equal to zero. This would require to distinguish between near-linear dependences within the space of one-center AO products and products containing auxiliary orbitals η µa for the exploitation of the a priori elimination during the evaluation of the ERIs V, which would strongly reduce the clarity of the implementation and therefore has not been pursued. Thus, all integrals are evaluated and the a priori elimination is only employed in the subsequent step of the determination of the LEDO expansion coefficients where it strongly reduces the cost of the linear algebra operations. Again, the case is different for the ERIs B. Once these have been evaluated, they do not have to be stored for near-linear dependent expansion functions Ω p. The vectors b µ Aν B [AB] therefore get shorter which results in a considerable reduction of the memory resources needed for the storage of the integrals B and the LEDO expansion coefficients D, respectively. Thus, in the present implementation the ERIs V are stored for the complete (near-linear dependent) LEDO expansion basis while the ERIs B and the LEDO expansion coefficients D are stored only for linear independent LEDO expansion functions. For the determination of the LEDO expansion coefficients, only the ERIs V [AB] belonging to linear independent expansion functions Ω p are copied

82 60 Chapter 3. Implementation to an array and factorized by a standard Cholesky procedure. As mentioned in Sec. (2.4.3) this results in great computational savings for the factorization of the coefficient matrix and the solution of the system of linear equations (2.108) by back-substitution Hartree Contribution For the evaluation of the one-center matrix elements of the Hartree contribution Ṽ h to the KS matrix, the density expansion vector d is required [Eq. (2.132)]. For the setup of the vector d [Eq. (2.120)] during the SCF cycle, both nearlinear dependences and negligible overlap densities ρ µa ν B (or complete atom pairs A > B) are taken into consideration for the diatomic contributions. As discussed above the contributions of the one-center overlap densities ρ µ A µ of AO basis A functions [Eq. (2.120b)] have to be included even if the corresponding LEDO expansion functions are near-linear dependent. In the implementation, elements of d belonging to near-linear dependent auxiliary functions Λ p are simply set to zero so that the vector d contains elements for the complete LEDO expansion basis as needed for the evaluation of the one-center elements of the approximated Hartree contribution. Next, these matrix elements are obtained as [Eq. (2.132)] Ṽ h A = B V AB d B (3.2) and stored in an auxiliary vector v. The approximated Hartree energy is evaluated as Ẽ h = 1 2 ( ρ ρ) = 1 d 2 AṼh A = 1 2 d v. (3.3) Unless the LEDO approximation is requested to be restricted to the Hartree contribution, all following operations are treated together with the XC contribution as described in the following section. A Exchange-Correlation Contribution In order to understand the implementation of the LEDO approximation for the XC contribution, it is instructive to give an outline of how this part is handled exactly. An overview of the algorithm used in TURBOMOLE has been given by

83 3.2. SCF Energy Calculation 61 van Wüllen [114]. The XC integrals that need to be evaluated for the setup of the KS matrix within the GGA approximation are given as [114] ( δfxc V xc,µν = dr δρ φ µφ ν + 1 ) δf xc ρ δ ρ ρ (φ µφ ν ). (3.4) As described in Sec. 2.2, these integrals are solved by numerical quadrature on a grid. A special loop structure is used in the TURBOMOLE code to calculate the electron density [Eq. (2.16)] at the grid points r j ρ(r j ) = µ,ν P µν φ µ (r j )φ ν (r j ) (3.5) which takes advantage of the sparseness of the density matrix P as well as of the almost vanishing values of basis functions φ µ (r j ) in the time determining steps [113, 114]. There are O(N) grid points r j and O(N 2 ) overlap densities φ µ φ ν entering the expression for the electron density. Thus the computation times formally scale as O(N 3 ). However, there are only O(N) significant overlap densities in the asymptotic limit of very large molecules. Furthermore the (average) number M of basis functions φ µ which have a non-negligible contribution to a given grid point r j is essentially independent of the size of the molecule making the computational effort per grid point a constant. The numerical quadrature thus effectively scales only linearly with the molecular size. The LEDO approximation in a certain way implicitly exploits the sparseness of the density matrix as the electron density is expressed using only O(N) atomcentered expansion functions [Eq. (2.119)], ρ(r j ) p d p Ω p (r j ), (3.6) reducing the formal scaling to O(N 2 ). The (average) number M LEDO of expansion functions Ω p which have a non-negligible contribution to a given grid point is of course independent of the molecular size, as well. The asymptotical scaling behavior therefore is also linear. If, however, for each grid point M LEDO < M 2, then the prefactor will be smaller for LEDO-DFT. The elements of the density matrix P belonging to non-negligible overlap densities φ µ φ ν are stored in a linear array in TURBOMOLE and are addressed via an index vector, as are the elements of the KS matrix. The algorithm of

84 62 Chapter 3. Implementation the numerical quadrature routines [114] is outlined below (ω j are the quadrature weights): loop over all grid points j calculate values φ k and gradients φ k of those AO basis functions φ µ which contribute to the grid point j (1 k M) M calculate auxiliary vector s k = P kl φ l ρ = end loop k=1 l=1 M M s k φ k, ρ = 2 s k φ k k=1 δf xc f 1 = ω j δρ, f 1 δf xc 2 = ω j ρ δ ρ calculate auxiliary vectors t k = f 2 ρ φ k and s k = f 1 φ k + t k update KS matrix: H KS kl = H KS kl + s k φ l + t l φ k Note that only a small fraction of time is spent on the evaluation of the density functional itself and its (functional) derivatives. The XC energy and the integral of the electron density can also be evaluated at no extra cost [114]. The LEDO approximation requires only slight modifications of the algorithm sketched above. Since the set {Ω pa } of LEDO expansion functions on an atom A is given by all one-center products of AO basis functions φ µa and auxiliary orbitals η µa, one merely has to assure that the auxiliary orbitals η µ are known in the subroutines for the numerical quadrature so that their values and gradients can be calculated. It should be mentioned that the use of auxiliary orbitals creates a computational overhead for this step, since the number M of nonnegligible auxiliary orbitals at a given grid point simply adds to the number M of non-negligible AO basis functions. Instead of the density matrix P, the density vector d has to be handed over to the XC subroutines while the restriction of the summations to the one-center terms is simply achieved by an appropriate usage of the index vector mentioned above. Thus, in the present implementation all onecenter elements of the XC part of the KS matrix as well as the respective auxiliary elements necessary to evaluate the two-center elements are obtained. In order to

85 3.3. Analytical Gradients 63 save computation time, the one-center elements of the XC contribution are added to the one-center elements of the Hartree contribution which have been evaluated before and stored on the auxiliary vector v, so that all further operations can be executed for both contributions together. The one-center elements Ṽh,xc A(AO) belonging to products of AO basis functions are then copied to the KS matrix and the near-linear dependent elements are removed from the auxiliary vector v. Next, the two-center matrix elements Ṽh,xc AB XC potential are obtained from the one-center elements Ṽh,xc A (2.127)] Ṽ h,xc AB of the approximated Hartree and according to [Eq. = Ṽh,xc [AB] D[AB] = v [AB] D [AB] (3.7) by just one call to the respective subroutine. In Eq. (3.7), Ṽ h,xc [AB] or v [AB] denote the vector of the one-center terms of atoms A and B. At this step both prescreening for small overlap densities ρ µa ν B (or atom pairs A > B) and the near-linear dependences are taken into account. Once the SCF is converged, the linear independent one-center elements of Ṽh,xc contained in the auxiliary vector v are written to disk for later use in the gradient program. Larger modifications to the numerical quadrature routines are necessary if auxiliary functions are to be employed instead of auxiliary orbitals. Nevertheless the changes are obvious and shall not be discussed in further detail here. 3.3 Analytical Gradients The calculation of the analytical energy gradients for LEDO-DFT according to Eq. (2.139) is organized as follows. In the gradient program, first the density vector d is read from disk. Then, the following steps are performed for each atom pair A > B subsequently: The integral derivatives VAB α are precomputed. Contraction with the appropriate parts of the density vector d yields the Hartree contribution Ẽ α h for this atom pair to the gradient [Eq. (2.139)]. Then the LEDO expansion coefficients D [AB] are read from disk, the integral derivatives B α [AB] computed, and the inhomogeneity of the system of linear equations (2.141) is formed. After reading the factorized coefficient matrix V 1 [AB] from disk, Eq. (2.141) is solved for the derivatives D α [AB] of the LEDO expansion coefficients. These are used together with the density matrix P to obtain the contribution of the atom pair

86 64 Chapter 3. Implementation calculate T α, E α ext, Gα read vector d from disk loop over atom pairs A > B calculate ERI derivatives V α AB form contribution to gradient: Ẽ α h,ab = d A Vα AB d B read LEDO expansion coefficients D [AB] from disk calculate B α [AB] and form inhomogen.: B α [AB] Vα [AB] D [AB] read factorized coefficient matrix from disk V 1 [AB] solve for derivatives of LEDO expansion ( coefficients: ) B α [AB] V α D α [AB] =V 1 [AB] [AB] D [AB] form contribution of atom pair A > B to d α form contribution to inhomogenity: V α [AB] D [AB] read Ṽh,xc A E α LEDO = A dα A and form Ṽ h,xc A calculate Ṽ α xc from ρ = Ω d Figure 3.2: Flow chart of the LEDO-DFT implementation into the gradient program RDGRAD. Dashed boxes indicate parts of the program that have been left unchanged. A > B to the derivative vector d α which is formed in analogy to Eq. (2.118). Once the contributions of all atom pairs to d α are known, the one-center elements of Ṽh,xc are read from disk to obtain the contribution ELEDO α to the gradient [Eq. (2.139)]. It should be noted that, since the LEDO approximation is not applied to the external potential, ELEDO α contains only contributions arising from the Hartree and the XC terms. Finally, the XC contribution Ẽα xc to the gradient is calculated by numerical quadrature of the approximated electron density ρ. A scheme of the program flow is depicted in Fig. 3.2 and some important details of the various steps described above shall be discussed in the following sections Integral Derivative Evaluation For a general four-center ERI (φ µa φ νb φ κc φ λd ) there are 12 distinct derivatives with respect to the Cartesian components of the coordinates of the centers K of

87 3.3. Analytical Gradients 65 the basis functions φ µk. However, the integrals and integral derivatives remain invariant under translation and rotation of all of the centers K appearing in the integral. Therefore, the sum of these derivative terms for any Cartesian component must equal zero [ ], K i K (φ µa φ νb φ κc φ λd ) = 0, i = x, y, z. (3.8) The evaluation of the derivatives of the ERIs can thus be simplified since the derivatives of an n-center ERI with respect to one of the centers can be calculated from the derivatives with respect to the remaining n 1 centers. This reduces the maximum number of derivative ERIs from 12 to 9 times the number of ordinary ERIs. Furthermore, the derivatives of one-center ERIs are always zero. For LEDO-DFT, derivatives of the ERIs V and B are required [Eqs. (2.139) and (2.140)]. The ERIs V are one- and two-center ERIs. The derivatives of the one-center ERIs V AA are all zero. For the two-center ERIs V AB, only the derivatives with respect to one of the centers α = {x A, y A, z A } have to be evaluated. The derivatives with respect to the other center β = {x B, y B, z B } are then given as All ERIs B are two-center ERIs and therefore V β AB = Vα AB. (3.9) B β [AB] = Bα [AB]. (3.10) In the implementation, the translational invariance of the ERIs as described above is fully exploited, i.e. only derivatives with respect to the Cartesian components of the coordinates of one of the nuclei of the two-center ERIs are computed. During the calculation of the derivative integrals B α [AB], prescreening for negligible diatomic overlap densities ρ µa ν B as described for the evaluation of the respective ERIs is applied. Furthermore, the a priori elimination is employed for B α [AB], i.e., only derivative integrals over linear independent LEDO expansion functions are stored in memory, while the ERI derivatives VAB α are stored for all LEDO expansion functions as needed for an easy evaluation of the Hartree contribution Ẽ α h to the gradient.

88 66 Chapter 3. Implementation Hartree Contribution Once the integral derivatives VAB α have been calculated, the computation of the approximated Hartree contribution Ẽh α = d pa d qb (Ω pa Ω qb ) α = 1 d A 2 Vα ABd B (3.11) A,B,p A,q B to the gradient [Eq. (2.139)] is straightforward. Since the one-center ERI derivatives are zero, the summation can be restricted to all unique atom pairs, i.e. A,B Ẽh α = d A Vα ABd B. (3.12) A>B Due to the translational invariance of the ERIs, only the gradient contributions of one atom appearing in the two-center ERIs have to be evaluated explicitly according to Eq. (3.12). The gradient contributions of the other atom are obtained as Ẽ β h = Ẽα h. (3.13) In the implementation, the Hartree contribution is evaluated for each atom pair A > B subsequently as described and added directly to the gradient vector LEDO Contribution For the computation of the LEDO contribution Ẽ α LEDO = A,p A d α p A Ṽ h,xc p A = A d α A Ṽ h,xc A (3.14) to the analytical energy gradient [Eq. (2.139)], the derivatives d α of the density expansion vector are required. These have to be computed from the density matrix P and the derivatives D α of the LEDO expansion coefficients. The latter are obtained from the system of linear equations (2.141). Thus, for each atom pair A > B, first the inhomogeneity of this system of linear equations has to be calculated. The fact that the one-center blocks VAA α and Vα BB of Vα [AB] are zero can be exploited during the matrix multiplication V[AB] α D [AB] which is required for this step. Due to the translational invariance of the ERI derivatives appearing in the inhomogeneity, only the derivatives of the LEDO expansion coefficients with

89 3.3. Analytical Gradients 67 respect to the Cartesian components α of the coordinates of one of the atoms of each atom pair A > B have to be determined. The derivatives with respect to the Cartesian components β of the coordinates of the other atom are obtained as D β [AB] = Dα [AB]. (3.15) Thus, only three instead of six systems of linear equations have to be solved for each atom pair A > B and therefore the translational invariance is equally important for the determination of the LEDO expansion coefficients as for the evaluation of the ERI derivatives. Of course, just as the LEDO expansion coefficients D [AB] (and the ERIs B [AB] and their derivatives B α [AB]), their derivatives D α [AB] have to be determined only for non-negligible diatomic overlap densities ρ µa ν B. Once the derivatives D α [AB] of the LEDO expansion coefficients for an atom pair A > B are known, the contribution of this atom pair to the gradient of the density vector can be calculated as [cf. Eq. (2.118)] (d α p A ) AB = 2 (d p µaνb A µ A,ν B ) α P µa ν B + 2 µ A,ν B (d νbµa p A ) α P νb µ A. (3.16) As for the setup of the density vector d, both near-linear dependences and negligible overlap densities ρ µa ν B (or complete atom pairs A > B) can be taken into account during this step. It should be noted that d α always needs to be computed only for linear independent LEDO expansion functions since, unlike for the density vector d [Eq. (2.120)], there are no contributions from density matrix elements P µ A µ of one-center overlap densities ρ A µ. From Eq. (3.15) follows for A ν A the contribution of atom pair A > B to d α, that (d β A ) AB = (d α A) AB. (3.17) Thus, for an atom pair A > B, only the contributions of atom A to the derivatives of the density vector have to be evaluated explicitly according to Eq. (3.16), while the contributions of the atom B are obtained from Eq. (3.17). Each element of the vector d α has contributions from all atom pairs A > B, unless the perturbed nucleus and the nucleus of the LEDO expansion function are so far apart, that all diatomic overlap densities are negligible. Thus, in principle a complete derivative vector d α has to be kept in memory for each perturbational parameter α. Once

90 68 Chapter 3. Implementation the contributions of all atom pairs A > B to the derivatives d α of the density vector have been calculated, the LEDO contribution ELEDO α to the gradient [Eq. (2.139)] can be evaluated by contraction with the one-center elements of the Hartree and XC contributions Ṽh,xc to the KS matrix according to Eq. (3.14). It should be noted that this step can only be performed after the contributions of all atoms pair A > B to the derivative vectors d α are known. The implementation has been realized as sketched above Exchange-Correlation Contribution The final step of a gradient run is the evaluation of the XC contribution to the analytical gradient [Eq. (2.139)]. The XC integrals that need to be evaluated within the GGA scheme without LEDO approximation are given as [114] E α = 2 µ,ν P µν ( δfxc dr δρ φ µφ (α) ν + 1 ρ δf xc δ ρ ρ ( ) ) φ µ φ (α) ν. (3.18) The numerical quadrature routines of the gradient code of TURBOMOLE work very similarly to the ones that handle the XC contributions to the KS matrix and the total energy [114] (cf. Sec ): loop over all grid points j calculate φ k, φ k and 2 φ k, α β α, β = x, y, z (1 k M) M calculate auxiliary vector s k = P kl φ l ρ = k=1 l=1 M M s k φ k, ρ = 2 s k φ k k=1 δf xc f 1 = ω j δρ, f 1 δf xc 2 = ω j ρ δ ρ M calculate auxiliary vector t k = P kl ( ρ φ l ) loop k = 1 to M E = E + (f 1 s k + f 2 t k )φ (x k) k + f 2 s k [ ρ φ (x k) k ] x k x k l=1

91 3.4. The Projection Technique 69 E = E + (f 1 s k + f 2 t k )φ (y k) k + f 2 s k [ ρ φ (y k) y k y k E = E + (f 1 s k + f 2 t k )φ (z k) k + f 2 s k [ ρ φ (z k) z k z k end loop end loop Just as for the SCF part,the LEDO approximation requires only slight modifications of the algorithm sketched above. These modifications have to ensure that in the routines for the numerical quadrature the auxiliary orbitals are known so that their values and gradients at the grid points can be calculated. Furthermore, instead of the density matrix P, the density vector d has to be handed over to the XC subroutines. Finally, the restriction of the summations to the correct one-center terms (i.e. to the LEDO expansion functions) is achieved by an appropriate usage of the index vector which is used for addressing the basis functions and the respective density matrix elements as mentioned in Sec k ] k ] 3.4 The Projection Technique The implementation of the projection technique outlined in Sec. 2.7 is straightforward and does not need extensive comments. It requires in the first step a diagonalization of the overlap matrix S before the start of the SCF cycle. After having determined the number M of COOs to be projected out according to Eq. (2.145), the eigenvectors u k of the overlap matrix belonging to the M lowest eigenvalues e k have to be renormalized according to ũ k = u k e 1/2 k. (3.19) Collecting the renormalized eigenvectors ũ k into the columns of a rectangular matrix Ũ, the projection operator is then given as [Eqs. (2.148) and (2.149)] ) Q = 1 (ŨŨ S (3.20) where 1 is the unit matrix. The application of this projection operator to the KS matrix H KS during the SCF cycle is achieved by two simple matrix multiplications.

92 70 Chapter 3. Implementation

93 Chapter 4 Numerical Results In this chapter numerical results obtained with the implementation of LEDO- DFT as described in chapter 3 shall be presented. Because auxiliary orbitals or auxiliary fit functions are mandatory to obtain reasonable results with LEDO- DFT, the first section of this chapter deals with the optimization of the former [188]. After describing some general observations made during preliminary investigations (Sec ), a concise illustration of the strategy for the optimization of auxiliary orbitals for the Ahlrichs split valence plus polarization (SVP) [190] basis set is given (Secs to 4.1.5). This is followed by a presentation of the efficiency of the a priori elimination for the SVP basis set with the optimized auxiliary orbitals (Sec ). Next, the accuracy of LEDO-DFT for the SVP basis in combination with these optimized auxiliary orbitals is verified [188, 189] (Sec. 4.2). A large test set of small molecules (Sec ) and linear alkanes as representatives for larger molecules (Sec ) are chosen as benchmark systems. Furthermore, the accuracy of LEDO-DFT in conjunction with the projection technique is assessed for some critical test molecules (Sec ). The accuracy of LEDO-DFT calculations with the SVP basis set and the optimized auxiliary orbitals together with the projection technique for some interesting test molecules which represent typical real life examples is also investigated (Sec ). Finally, the efficiency of LEDO-DFT is demonstrated [188] by comparing timings with conventional DFT calculations and calculations employing the RI-J technique (conventional density fitting) (Sec. 4.3). 71

94 72 Chapter 4. Numerical Results 4.1 Optimization of Auxiliary Orbitals for the SVP Basis Set It has already been pointed out several times in this work, that the LEDO expansion basis consisting of all one-center AO products {φ µ A φ µ } has to be augmented A by a set {Λ ra } of auxiliary functions in order to obtain accurate results with LEDO-DFT (cf. Sec ). Of course, the auxiliary functions {Λ ra } needed for the density fit have to be determined for each AO basis set and each element separately. As discussed in Sec , in this work a set {η µa } of auxiliary orbitals leading to an additional set {Λ ra } = {η µ A φ µ, η A µ η A µ } of one-center A overlap densities in the LEDO expansion basis has been employed. The LEDO expansion basis is therefore given by the set {Ω pa } = {φ µ A φ µ A, η µ A φ µ A, η µ A η µ A } of one-center product functions. Auxiliary orbitals consisting of uncontracted Gaussians have already been determined in an empirical manner by Kollmar and Hess [1] for the 6-31G split valence basis set [221] for the elements H, C, N and O. In this work, a systematic procedure for the optimization of auxiliary orbitals based on the norms presented in Sec has been devised. Although applicable to any basis set, the focus is on the SVP [190] basis set, which in general gives more accurate results than the smaller 6-31G basis set. In fact, the SVP basis set can be considered as one of the best basis sets available for DFT calculations on very large molecular systems. The procedure and the results of the optimizations are presented in what follows Preliminary Investigations Preliminary investigations employing a numerical integration technique for the determination of the quality of the LEDO fit had shown that diatomic overlap densities ρ µa ν B involving only core functions seem to be in general uncritical with respect to the accuracy of the fit. This can be attributed to the fact that core functions are rapidly decaying with increasing distance. Similarly uncritical are valence functions with an angular momentum quantum number l if the basis set contains valence functions or polarization functions with angular momentum quantum number l > l but similar exponent, i.e. similar radial decay properties.

95 4.1. Optimization of Auxiliary Orbitals for the SVP Basis Set 73 Most critical are overlap densities involving high-l valence functions and polarization functions. It turned out that an inclusion of one uncontracted Gaussian for each set of these critical functions improves the fit significantly. The auxiliary orbitals η µa should have an angular momentum quantum number l which is increased by one with respect to the functions involved in the overlap densities to be fitted while the radial decay properties should be as similar as possible. If a split valence AO basis set contains polarization functions, a sufficiently accurate fit of the overlap densities involving the more compact valence functions is in general achieved. Auxiliary orbitals have to be provided only for the diffuse valence functions with the highest angular momentum quantum number l and the polarization function. This general recipe can be easily employed for any basis set. Here, the focus is on the Ahlrichs SVP [190] basis set. As already pointed out, it suffices to provide two uncontracted Gaussians as auxiliary orbitals. First we should like to present results for homonuclear diatomic overlap densities and investigate the distance dependence of the quality of the LEDO fit. The observations made shall then be extended to heteronuclear diatomic overlap densities Homonuclear Diatomic Overlap Densities One might think that it is best to optimize the exponents of both auxiliary orbitals simultaneously by minimizing the norm AA according to Eq. (2.114). However, we found a two-step procedure more appropriate in which we first optimize the exponent of the auxiliary orbital for the valence shell. Keeping this exponent fixed, we subsequently optimize the auxiliary orbital for the polarization functions. For the purpose of optimizing the exponent of the auxiliary orbital for the shell l A we consider the norm l Al A [Eq. (2.112)]. Let us explain the procedure taking the carbon atom as an example. The SVP basis set for the carbon atom is of the [3s2p1d] type, i.e., it consists of three s, two p and one d shell. We therefore will need one d type and one f type auxiliary orbital to improve the fit of overlap densities involving, respectively, the more diffuse p shell and the d shell. Let us enumerate the shells according to increasing l quantum number and according to increasing diffuseness. If we add

96 74 Chapter 4. Numerical Results p5p / CC / ζ(d ) ζ( f ) Figure 4.1: CC at an interatomic distance of 120 pm: Norm 5p5p with varying exponent ζ(d) of the d type auxiliary orbital (left). Norm CC with varying exponent ζ(f) of the f type auxiliary orbital employing a d type auxiliary orbital with ζ(d) = 0.16 (right). the type of the shell to the index we end up denoting the shells of the carbon atom as 1s, 2s, 3s, 4p, 5p and 6d. The shell 5p is the diffuse valence shell of p type and the shell 6d constitutes the polarization functions of d type for which we wish to optimize auxiliary orbitals. In order to optimize the exponent ζ(d) of the d type auxiliary orbital we minimize the norm 5p5p. This norm excludes overlap densities involving basis functions of the 6d shell for which no auxiliary orbital has been optimized yet. Once the exponent ζ(d) has been determined, it is kept fixed and an f type auxiliary orbital is added whose exponent ζ(f) is optimized by minimizing the norm 6d6d = CC. This procedure will not yield the minimum of the norm AA, however, it guarantees for a better fit of the core and valence functions which we consider more important since integrals of these overlap densities will be contracted with elements of the first-order reduced density matrix that are in general larger in magnitude than matrix elements involving polarization functions. The quality of the fit of the overlap densities in which polarization functions participate is less critical with respect to the SCF convergence and the accuracy of the results. We started our investigations at an interatomic distance of 120 pm which approximately corresponds to a CC triple bond [222]. A plot of the norm 5p5p with varying ζ(d) can be found in Figure 4.1. The minimum for ζ(d) = 0.16 leads to an excellent fit with a value of for 5p5p as compared to

97 4.1. Optimization of Auxiliary Orbitals for the SVP Basis Set ζ( f ) 0.9 ζ( f ) ζ(d ) ζ(d ) Figure 4.2: CC at an interatomic distance of 120 pm: Contour plots of the norm 5p5p (left, isovalues between and ) and CC (right, isovalues between and ) under variation of the exponents ζ(d) and ζ(f) of the d and f type auxiliary orbitals. The minimum for the independent, consecutive optimization of the exponents is indicated by a cross. without auxiliary orbital. Note that this exponent is very similar to the exponent of the 5p shell which is approximately An exponent of 0.8 does not enhance the LEDO expansion basis since the d type polarization function has the same exponent and the peak at 0.8 represents the value of 5p5p without any auxiliary orbital. Figure 4.1 also contains a plot of the norm CC under variation of ζ(f) employing a d type auxiliary orbital with the previously determined exponent ζ(d) = 0.16 being kept fixed. A good fit of the overlap densities involving the polarization functions is more difficult to achieve. Nevertheless, the minimum at ζ(f) = 0.85 leads to an acceptable value of for CC, which is two orders of magnitude better than the value of without f type auxiliary orbital. Note that the 6d shell has a very similar exponent of 0.8. Contour plots of the norm 5p5p and CC under variation of both exponents near the minimum are shown in Figure 4.2. The surface is very flat in the proximity of the minimum. Therefore, a variation of the exponents within a small range will not affect the quality of the LEDO fit. While the value of CC is reduced from without auxiliary orbitals to at the minimum, the value of for the independently optimized exponents is only slightly higher. We furthermore note that this value is mainly determined by ζ(f) while ζ(d) has less influence. The contour plot of 5p5p, on the other

98 76 Chapter 4. Numerical Results hand, shows that the quality of the fit of all overlap densities not involving the 6d polarization functions is almost independent from ζ(f) at least in the vicinity of ζ(f) = Since the exponents of the auxiliary orbitals are not strongly coupled, an independent, consecutive optimization is justified Distance Dependence Crucial for the success of LEDO-DFT is a sufficient accuracy of the fit of the diatomic overlap densities ρ µa ν B up to fairly large interatomic distances. We therefore have to consider interatomic distances ranging from bond distances up to distances at which all overlap densities are essentially zero all over space. Figure 4.3 shows the distance dependence of the best exponents ζ(d) and ζ(f) obtained by minimizing 5p5p and CC, respectively. The calculations for ζ(f) have been performed employing the previously determined best ζ(d) at each distance. We can clearly see that the best value for the exponents is indeed distance dependent. However, the distance dependence of ζ(d) is not very strong and there is no significant deterioration of the quality of the fit for a fixed exponent ζ(d) if it is chosen properly (Fig. 4.3, bottom left). The distance dependence of the best exponent ζ(f) is much more pronounced. The quality of the fit does not change much for a fixed exponent ζ(f) in the region between 120 pm and around 250 pm (Fig. 4.3, bottom right). This is the region in which overlap densities involving the 6d shell are still significant. For larger distances these overlap densities become insignificant and the norm CC becomes almost identical to the norm 5p5p. A reduction of the exponent ζ(f) leads to an improvement of the fit of overlap densities involving the 5p shell which is more diffuse than the 6d shell and thus to a smaller value of CC for interatomic distances larger than 250 pm. Based upon these observations the value of a fixed ζ(f) should be reduced as compared to the best value for short interatomic distances and we recommend exponents of ζ(d) = 0.16 and ζ(f) = 0.78 for the auxiliary orbitals. These values should be compared to the exponents that conforming with the preliminary investigations (Sec ) would be chosen to be ζ(d) = and ζ(f) = 0.8 according to the exponents of the 5p and 6d shells, respectively. Thus, our detailed investigation confirms the empirical procedure for the choice of auxiliary orbitals which has

99 4.1. Optimization of Auxiliary Orbitals for the SVP Basis Set best ζ(d ) best ζ( f ) interatomic distance / pm interatomic distance / pm best ζ(d ), ζ( f ) ζ(d ) = 0.16, ζ( f ) = p5p / best ζ(d ) ζ(d ) = 0.16 CC / interatomic distance / pm interatomic distance / pm 500 Figure 4.3: Distance dependence of ζ(d) (left) and ζ(f) (right) for CC. The best exponent is shown on top while the corresponding values of 5p5p and CC for the best and fixed exponents are shown below. If not stated otherwise, the calculations for ζ(f) have been performed employing the previously determined best ζ(d) at each distance. been presented in Sec Note that these fixed exponents lead to a reduction of the norm CC by two orders of magnitude at all interatomic distances. It is clear that an improvement of the fit at large interatomic distances can be easily achieved by inclusion of an additional f type auxiliary orbital with an exponent ζ (f) = 0.16 that is identical to the exponent ζ(d) of the d type auxiliary orbital Heteronuclear Diatomic Overlap Densities In the following we are going to analyze whether the auxiliary orbitals which have been optimized for homonuclear atom pairs are able to yield reasonable results for heteronuclear atom pairs, as well. For this purpose we investigate a pair of carbon and nitrogen atoms at a typical CN double bond distance of 122 pm.

100 78 Chapter 4. Numerical Results ζ(d ) of N ζ( f ) of N ζ(d ) of C ζ( f ) of C Figure 4.4: CN at an interatomic distance of 122 pm: Contour plots of the norm 5p5p under variation of the exponents ζ(d) (left, isovalues between and ) and CN under variation of the exponents ζ(f) (right, isovalues between and ) of carbon and nitrogen. The minima for the exponents optimized for the homonuclear atom pairs CC and NN are indicated by a cross. Figure 4.4 contains contour plots of the norms 5p5p and CN under variation of the exponents ζ(d) and ζ(f) of carbon and nitrogen. The calculations for CN have been performed employing d type auxiliary orbitals optimized for the homonuclear atom pairs of carbon and nitrogen. The value of the norm 5p5p is reduced from without auxiliary orbitals to at the minimum. Since the minimum is very shallow, the exponents ζ(d) obtained for the homonuclear atom pairs result in almost the same value for 5p5p. The norm CN is reduced from without auxiliary orbitals to at the minimum as compared to the only slightly larger value of for the auxiliary orbitals optimized for the homonuclear atom pairs. Similar results have been obtained for the atom pair CF as a more polar example and CH and CP as examples for pairs of atoms from different periods (cf. Appendix A). We conclude that the auxiliary orbitals which have been optimized for homonuclear atom pairs are very well suitable for heteronuclear atom pairs, as well. Molecular test calculations corroborate this statement (cf. Sec. 4.2).

101 4.1. Optimization of Auxiliary Orbitals for the SVP Basis Set 79 Table 4.1: Recommended exponents of the two sets of auxiliary orbitals used in the LEDO expansion for the SVP basis set as optimized by the procedure described in Secs and l H Li Be B C N O F Na Mg Al Si P S Cl The angular momentum quantum number is denoted as l Recommended Exponents Auxiliary orbitals for the elements H, Li F and Na Cl have been determined along the lines described above and are presented in Table 4.1. A similar behavior of the distance dependence of the exponents and the norms employed for the optimization of the exponents has been observed for all elements under consideration with changes that are regular across the periodic table. The procedure presented is therefore generally applicable. The addition of two sets of auxiliary orbitals of angular momentum quantum number and exponents given in Table

102 80 Chapter 4. Numerical Results CC / interatomic distance / pm 500 Figure 4.5: Quality of the LEDO fit with the a priori elimination. The graph shows the distance dependence of the norm CC of the homonuclear atom pair CC. Various thresholds in between 10 3 and for the truncation of the modified Cholesky factorization have been investigated. 4.1 results in the reduction of the norm AA for homonuclear atom pairs of at least two orders of magnitude Efficiency of the A Priori Elimination The LEDO expansion basis consisting of the one-center products of the AO basis functions and the auxiliary orbitals presented in Table 4.1 is not linear independent from a numerical point of view. The most efficient way to deal with this problem is the a priori elimination of near-linear dependent LEDO expansion functions (cf. Sec ). Careful numerical analyses have led us to the conclusion that a threshold of 10 9 for the truncation of the modified Cholesky factorization [1, 185] employed for the determination of these near-linear dependences yields numerically stable results without reducing the quality of the LEDO fit. The threshold should not be chosen smaller, because otherwise numerical imprecision due to rounding errors cannot be excluded. Larger thresholds, on the other hand, can lead to the elimination of important LEDO expansion functions and thus a deterioration of the numerical accuracy in the fitting procedure. Fig. 4.5 shows the quality of the LEDO fit for the homonuclear atom pair CC at various interatomic distances with several different thresholds. It can be clearly seen that the accuracy of the fit deteriorates significantly for thresholds of 10 6 or larger,

103 4.1. Optimization of Auxiliary Orbitals for the SVP Basis Set 81 Table 4.2: Number of functions in the full LEDO expansion basis and with a priori elimination for the SVP basis set and the auxiliary orbitals of Table 4.1. full a priori full a priori full a priori H N Al Li O Si Be F P B Na S C Mg Cl especially at small interatomic distances. A threshold of 10 9, however, yields a fit quality in agreement with the numerical precision that can be attained. Table 4.2 shows the number of functions in the full LEDO expansion basis and with a priori elimination for the elements and auxiliary orbitals presented in the previous section. The full Cartesian components (i.e. 6 d-type, 10 f-type, etc.) of the Gaussians have been used in the LEDO expansion. The reduction of the dimension of the expansion basis by a priori elimination is lowest for H and highest for Al. This reduction to 76.2 % (H) and 42.4 % (Al) of its original size corresponds to a theoretical reduction of the computational cost for the determination of the LEDO expansion coefficients to 44.2 % and 7.6 %, respectively, because the computational expense for the factorization of the coefficient matrix grows cubically with its dimension. Timings for homonuclear systems of 25 atoms are shown in Fig The atoms have been arranged in five parallel rows of five atoms each and the interatomic distances have been chosen to be in the range of typical bond lengths in such a way that no atom pairs are neglected due to prescreening. However, a different number of diatomic shell pairs are neglected depending on the atom type. This explains the slight decrease of CPU times from B to F and from Al to Cl, although the number of AO basis functions and LEDO expansion functions (including linear dependences) is identical for these elements. The CPU times of the calculations with the a priori elimination include the additional overhead due to the extra sorting steps now required after

104 82 Chapter 4. Numerical Results t / sec ERIs a priori elimination (cf. Sec ) perturbative approximation (cf. Sec ) 0 H Li Be B C N O F Na Mg Al Si P S Cl Figure 4.6: CPU times (in seconds on a 2-GHz Intel Xeon processor) for the determination of the LEDO expansion coefficients for homonuclear systems of 25 atoms (300 atom pairs) with the SVP basis set and the auxiliary orbitals of Table 4.1. the evaluation of the ERIs (cf. Sec ). The observed speedup therefore should be lower than the theoretical value. In practice, the a priori elimination therefore leads to a reduction of the computational cost for the linear algebra up to 13.6 % (Al) of its original cost with the perturbative approximation (cf. Sec ). The important point, however, is that the CPU time required for the determination of the LEDO expansion coefficients is comparable to the CPU time spent during the evaluation of the ERIs. This reasonable cost balance is not given if near-linear dependences in the LEDO expansion basis are not exploited. 4.2 Accuracy of LEDO-DFT In this section, results of benchmark calculations on a variety of test systems are reported in order to demonstrate the accuracy that can be obtained with LEDO-DFT and the auxiliary orbitals presented in Table 4.1 of this work. All calculations have been performed using the gradient-corrected Becke Perdew BP86 [30, 33, 35] XC functional with the DFT programs of the TURBOMOLE program package [10, 11] in version 5.1. The LEDO-DFT calculations have been carried out with our implementation into the programs RIDFT and RDGRAD (cf. Chapter 3). The force constants have been obtained as numerical first deriva-

105 4.2. Accuracy of LEDO-DFT 83 tives of the analytical energy gradients as implemented in the parallel PVM code SNF [208]. No symmetry constraints have been imposed during the structure optimizations and the frequency analyses. The full Cartesian components of the Gaussians have been used in the LEDO expansion and the a priori elimination has been employed. The LEDO approximation has been applied to the Hartree and the XC contribution. Using a prescreening of the ERIs based on the Schwartz inequality, the LEDO expansion coefficients have been determined only for nonnegligible overlap densities. If not otherwise stated, a convergence threshold of 10 8 au for the total energy has been employed for the SCF while the structure optimizations have been considered as converged if the change in the total energy between two steps was below 10 7 au and the maximum norm of the gradient smaller than 10 4 au. These accurately converged SCF results guarantee a satisfactory accuracy in the frequency analyses of the optimized structures Small Molecules First, results of benchmark calculations on a large test set of small molecules shall be presented. There are 142 compounds in this test set and we have taken care that they represent a comprehensive collection of commonly encountered bond types for all elements for which auxiliary orbitals are available. Although the structure optimizations have been carried out in C 1 symmetry, some molecules actually do possess symmetry elements. For the analysis of the structure constants we therefore considered the mean values of bond distances and bond angles which are symmetry-equivalent and excluded structure constants which are fixed due to the symmetry, resulting in 315 and 282 values, respectively. Only molecules with non-zero dipole moment have been considered for the analysis of the accuracy of the dipole moments, leaving a set of 85 values. Of course, zero results of dipole moments for non-polar molecules should also be reproduced by LEDO-DFT. However, it is clear that non-zero results are artefacts of the scatter introduced by the symmetry-unrestricted structure optimizations and not a consequence of the LEDO approximation 1. An inclusion of non-polar molecules into the analysis of the accuracy would therefore bias the results in favor of LEDO- 1 In fact, the deviations from zero are of the order of 10 4 au both for DFT and LEDO-DFT.

106 84 Chapter 4. Numerical Results DFT. Finally, 1446 harmonic frequencies have been analyzed. The comparison of the harmonic frequencies will give the most reliable evidence of the accuracy of LEDO-DFT, since it includes the information about the curvature of the potential energy surface. Table 4.3 contains errors of computed total electronic energies and dipole moments and maximum errors of molecular structure constants and harmonic frequencies for some molecules of our test set comparing the results of LEDO- DFT with those of conventional DFT calculations with the same functional but no further approximations. The molecules which showed the largest errors for the examined quantities are included in this table. A complete list of the errors for all molecules including root-mean-square (RMS) deviations of the molecular structure constants and the harmonic frequencies is given in Appendix B. In Table 4.4 we report maximum and RMS errors for the complete test set of small molecules. The energy differences for most of the small molecules are below 10 3 au, and tend to increase with the size of the system. The largest deviation of au was found for P 4 O 10. The error in the energy per atom never exceeds au and is, in general, much lower, which is reflected by the RMS value of au. Errors in single point energies are in general of the same order as the errors in the total energies of the optimized structures and therefore have not been included in the tabulated data. Although the structural parameters deviate up to 1.37 pm and 1.9 degrees, in general the accuracy of LEDO-DFT is very good, yielding errors of the order of 0.1 pm and 0.1 degrees for the majority of the compounds. Note that the error in bond distances involving hydrogen never exceeds 0.1 pm. Although it seems that some combinations of elements are more problematic with respect to the structure constants than others, the larger deviations cannot be definitively traced back to certain atom pairs because, e.g., the PC bond is in error by 0.13 pm in P(CH 3 ) 3 and by 0.50 pm in PO(CH 3 ) 3. However, in general the larger deviations appear for weak bonds and soft angles which can be varied without large changes of the total energy of the system. Some examples are Si 2 H 6 ( d SiSi = 1.37 pm), BH 3 PH 3 ( d BP = 0.71 pm), AlH 3 NH 3 ( d AlN = 0.81 pm) and Li 2 O ( γ LiOLi = 1.9 degrees).

107 4.2. Accuracy of LEDO-DFT 85 Table 4.3: Accuracy of LEDO-DFT calculations: Errors of computed total energies ( E = E LEDO E DFT in 10 3 au) and dipole moments ( µ = µ LEDO µ DFT in 10 3 au) and maximum absolute errors in bond distances ( d in pm), bond angles ( γ in degrees) and harmonic frequencies ( ν in cm 1 ) for some small molecules. Molecule E d γ µ ν H 2 O N 2 H C 3 H CH 3 N 2 CH Glycine Cysteine Li 2 O Si 2 H Si(CH 3 ) 3 Cl P(CH 3 ) PO(CH 3 ) P 4 O Note, that 10 3 au corresponds to 219 cm 1. Given the errors in the total energies and structure constants, both the dipole moments and harmonic frequencies are in very good agreement. In Fig. 4.7 the infrared spectrum of the amino acid cysteine as calculated with LEDO-DFT and conventional DFT is presented. The agreement of the spectra is indeed remarkable. We conclude that the overall performance of LEDO-DFT with the auxiliary orbitals presented in this work (Table 4.1) is satisfactory for the molecules under consideration. The performance for the subset of classical organic compounds is very good. To proof that, in principle, the errors can be reduced to negligible values we

108 86 Chapter 4. Numerical Results Table 4.4: Accuracy of LEDO-DFT calculations: maximum absolute and rootmean-square (RMS) deviations of computed total energies ( E in 10 3 au), bond distances ( d in pm), bond angles ( γ in degrees), dipole moments ( µ in 10 3 au) and harmonic frequencies ( ν in cm 1 ) for a test set of 142 small molecules. E E/atom d γ µ ν max RMS Note, that 10 3 au corresponds to 219 cm 1. Absorption coeff. / km mol ν / cm Absorption coeff. / km mol ν / cm Figure 4.7: IR spectrum of the amino acid cysteine as calculated with LEDO- DFT (left) and conventional DFT (right). Lorentz profiles with a half-width of 10 cm 1 were used for the intensities. have repeated the structure optimizations for some critical test cases with an extended LEDO expansion basis. We have included for each atom an additional auxiliary orbital set with high angular momentum quantum number l + 1 and exponent ζ (l+1) = ζ(l) identical to the exponent ζ(l) of the auxiliary orbital with low angular momentum quantum number l, as already explained in Sec The results are summarized in Table 4.5 together with errors of calculations with the well-established RI-J approximation [127].

109 4.2. Accuracy of LEDO-DFT 87 Table 4.5: Accuracy of LEDO-DFT calculations with an extended expansion basis and RI-J calculations: Errors of computed total energies ( E = E LEDO E DFT or E = E RI J E DFT in 10 3 au) and dipole moments ( µ = µ LEDO µ DFT or µ = µ RI J µ DFT in 10 3 au) and maximum absolute errors in bond distances ( d in pm) and bond angles ( γ in degrees) for some critical test molecules. LEDO-DFT RI-J Molecule E d γ µ E d γ µ Li 2 O Si 2 H P(CH 3 ) PO(CH 3 ) P 4 O BH 3 PH AlH 3 NH Although the RI-J calculations are in all instances slightly more accurate, it is evident that with this simple scheme for the enhancement of the LEDO expansion basis practically all errors become insignificant. The only exception is Li 2 O with a large deviation of 1.2 degrees in the bond angle. However, one should consider that the RI-J approximation also shows a relatively large deviation of 0.4 degrees for this bond angle Larger Molecules Linear Alkanes In the following, the behavior of LEDO-DFT with increasing system size shall be discussed. Linear alkanes have been chosen for this purpose. The chain length has been limited to a maximum of 30 carbon atoms due to restrictions in our incore implementation of LEDO-DFT, however, general trends can be very well observed even for these moderately sized systems. Structures pre-optimized by the TINKER program [223] with the MM3 force field [224] have been used for

110 88 Chapter 4. Numerical Results Table 4.6: Accuracy of LEDO-DFT calculations: Errors of computed total energies ( E = E LEDO E DFT in 10 3 au) at the MM3 [224] and at the optimized structure and maximum absolute and root-mean-square (RMS) deviations of bond distances ( d in pm) and bond angles ( γ in degrees) for a test set of linear alkanes. d γ Molecule E MM3 E opt max RMS max RMS n-c 5 H n-c 10 H n-c 15 H n-c 20 H n-c 25 H n-c 30 H all single-point calculations and as starting point for all structure optimizations. The DFT calculations have been performed as described at the beginning of Sec. 4.2, however, the structure optimizations have been considered as converged if the change in the total energy between two steps was below 10 6 au. In Table 4.6 we report errors introduced in computed energies and molecular structure constants. While the energy difference for n-c 5 H 12 is negligible, the error in the computed total energy increases for the higher homologues. Note that the increase of this error with system size is slightly stronger than linear. In all cases the magnitude of the deviation is of the same order for the singlepoint energy at the MM3-optimized structures ( E MM3 ) and the energy at the optimized structures ( E opt ). This has already been mentioned for the test set of small molecules. The error in the computed structure parameters increases only very slowly with increasing system size and the maximum absolute deviation d of the bond distances remains very small. While the maximum absolute deviation of the bond angles γ is relatively large, the RMS values indicate that the structures

111 4.2. Accuracy of LEDO-DFT 89 are still in very good agreement. We conclude that, apart from the computed total energies, no significant deterioration of the results obtained with LEDO- DFT can be observed with increasing system size Critical Cases Assessment of the Projection Technique The results obtained with the test set of small molecules (Sec ) and the linear alkanes (Sec ) are very encouraging, however, due to the reasons discussed in Secs. 2.5 and 2.7, for large molecular systems the numerical errors introduced by the LEDO approximation can lead to problems with the SCF convergence. Here, the use of the projection technique (Sec. 2.7) for such critical systems shall be assessed. To analyze the effect of the projection technique we have performed a series of test calculations on naphthalene, anthracene, polyynes and three isomers of C 20. These compounds have been chosen since the SCF for LEDO-DFT with the present auxiliary orbitals (Table 4.1) fails to converge. Note that the polyynes have already been reported [225] to be critical with respect to the SCF convergence. The isomers of C 20 enable an analysis of the SCF convergence behavior for systems of fixed size but differing numbers of significant overlap densities ρ µa ν B. These are increasing from the ring over the bowl to the cage isomer. We therefore expect the most critical behavior for the cage isomer. The ring isomer of C 20, however, is a special case since it can be regarded as a cyclic polyyne. The starting structures for the optimizations of the ring, bowl and cage isomer have been taken from the work of Grossman et al. 2 [226] The calculations have been performed as described at the beginning of Sec For the structure optimizations of the C 20 isomers, however, the change of the total energy between two steps was only required to be smaller than 10 6 au. A threshold for the maximum allowed condition number of c max = 4000 was employed in all calculations since LEDO-DFT calculations do not converge for polyynes with chain lengths from six carbon atoms onwards and the ring isomer of C 20 for values of c max Restricting the maximum condition number to smaller values, on the other hand, leads to increasing errors. Of course, the condition number depends on the posi- 2 structures have been obtained from

112 90 Chapter 4. Numerical Results tion of the atoms, and it consequently changes during the course of a structure optimization. However, in general, these changes can be expected to be small, which means that no additional basis set superposition error (BSSE)-like effect should be observed. Indeed, we have observed that the number of omitted COOs at the beginning and at the end of a structure optimization is identical in most of the cases. In the following we should like to report how electronic energies and structural parameters are affected by the projection technique. Table 4.7 summarizes the errors which are introduced by the projection technique and by the LEDO approximation. Reference for the LEDO-DFT calculations are conventional DFT calculations employing the projection technique. Let us first comment on the errors introduced by the projection technique. As expected, for similar systems the error in the total energy grows with increasing number M of eliminated COOs. The maximum deviation of the bond distances remains very small in all cases. At first sight, the errors in bond distances for the cage isomer of C 20 seem large. However, the maximum deviation of bond distances of 0.18 pm for a calculation with conventional density fitting [127] (RI- J) is of the same order of magnitude. The errors in the total energies due to the LEDO approximation are of the order of 10 3 au. The errors in the bond distances are smaller than 0.1 pm for most molecules and only little larger for the bowl and cage isomers of C 20 with deviations of 0.23 pm and 0.19 pm, respectively. The errors introduced by the LEDO approximation are of the same order as the errors introduced by the projection technique and we therefore conclude that the application of the projection technique is justified. None of the investigated molecules posed any difficulties for LEDO-DFT with a value of c max = 4000 and we assume it to be reliable for other calculations as well. We therefore suggest to generally employ the projection technique in combination with the LEDO approximation Some Real Life Examples In quantum chemical methods, apart from structure determinations, it is frequently important to estimate energy differences with accuracy rather than the absolute quantities of reactant and product. We therefore have investigated the

113 4.2. Accuracy of LEDO-DFT 91 Table 4.7: Application of the projection technique with c max = 4000: Errors of total energies ( E 1 = E proj DFT E DFT and E 2 = E proj LEDO Eproj DFT in 10 3 au) and maximum absolute deviations of bond distances ( d 1 = d proj DFT d DFT and d 2 = d proj LEDO dproj DFT in pm) for conventional DFT and LEDO-DFT. M is the number of canonically orthogonalized orbitals eliminated by the projection. DFT LEDO-DFT M E 1 d 1 E 2 d 2 Naphthalene Anthracene C 6 H C 8 H C 10 H C 12 H C 14 H C 16 H C 18 H C 20 H C 20 (ring) C 20 (bowl) C 20 (cage) performance of LEDO-DFT for some representative energy differences of three different orders of magnitude. We have computed the dissociation energy of benzene into three acetylene molecules as the prototype for a reaction in which the size of the reactant and product molecules differ. The rotational barrier of ethane and the relative stabilities of different isomers of C 2 H 6 O, C 4 H 4, and C 6 H 6 serve as test systems in which additional errors introduced by non-additivity are excluded. All calculations have been performed as described at the beginning of Sec. 4.2 and the results are listed in Tables 4.8 and 4.9.

114 92 Chapter 4. Numerical Results Table 4.8: Reaction energy for the dissociation of benzene into three acetylene molecules as calculated with conventional DFT and LEDO-DFT. Calculations in which the projection technique was employed are marked by an asterisk. All energies and energy differences are given in au. E DFT E LEDO Error C 6 H C 2 H Dissociation Table 4.9: Relative stabilities of isomers of C 2 H 6, C 2 H 6 O, C 4 H 4 and C 6 H 6 computed using conventional DFT and LEDO-DFT. Calculations in which the projection technique was employed are marked by an asterisk. All energies and energy differences are given in au. E and E DFT LEDO-DFT Error C 2 H 6 staggered ethane eclipsed ethane C 2 H 6 O trans-ethanol cis-ethanol Dimethylether C 4 H 4 Cyclobutadiene

115 4.2. Accuracy of LEDO-DFT 93 E and E DFT LEDO-DFT Error Tetrahedrane C 6 H 6 Benzene Benzvalene Dewar benzene Prismane Bicyclopropenyl Although the relative errors do increase as the quantity being computed decreases, the smallest energies the rotational barrier of ethane and the relative stability of cis-ethanol are in error by only 7.4 % and 4.9 %, respectively. All other relative errors range from the excellent values of 0.02 % for the relative stability of Dewar benzene to 1.8 % for the relative stability of Benzvalene. According to these results and the data collected in the previous Secs., the relative error in energy differences calculated with LEDO-DFT can be expected to be in between 1 % and 10% for values of the order of 10 3 au and to be 1% or smaller for larger energy differences. It should furthermore be taken into account that the error arising from the use of approximate XC functionals and incomplete AO basis sets can be expected to be much larger than the consistently small errors introduced by the LEDO approximation. Despite the fact that the LEDO-DFT energy is not a bound to the exact energy, which means that the error in total energies can be both positive and negative, we conclude from our results that the prediction of energy differences such as relative stabilities or reaction energies with LEDO-DFT does not suffer from serious drawbacks.

116 94 Chapter 4. Numerical Results The molecular systems considered so far have been chosen on the basis of a systematical assessment of LEDO-DFT. Most of these are not of great interest in the present chemical or biochemical research. We therefore finally should like to present some test calculations on a set of molecules better reflecting the potential interests of, e.g., life sciences. The calculations have been performed as described at the beginning of Sec. 4.2, however, the change of the total energy between two structure optimization steps was only required to be smaller than 10 6 au and the projection technique with c max = 4000 as recommended in Sec was employed for LEDO-DFT. Results for the following molecules (in alphabetic order) are presented in Table 4.10: Acyclovir (C 8 H 11 N 5 O 3, an antiviral agent for treating herpes), Adrenaline (C 9 H 13 NO 3, a hormone and neurotransmitter), Caffeine (C 8 H 10 N 4 O 2, an alkaloid found in coffee beans and other plants), Capillin (C 12 H 8 O, an acetylenic ketone with fungicidal activity), (+)-trans-chrysanthemic acid (C 10 H 16 O 2, a terpene constituent of the flower Chrysanthemum cinerariaefolium), S-Ibuprofene (C 13 H 18 O 2, an anti-inflammatory and analgetic drug), LSD (C 20 H 25 N 3 O, a potent hallucinogen), Nicotine (C 10 H 14 N 2, a carcinogenic alkaloid found, e.g., in tobacco), α-pinene (C 10 H 16, a major constituent of turpentine), Valium (C 16 H 13 ClN 2 O, a sedative of the benzodiazepine family) and Vitamin C (C 6 H 8 O 6, an acid with antioxidant properties essential for life). Caffeine, capillin and acyclovir possess C s symmetry. All other molecules are not symmetric, as is typical for most compounds of natural origin or of relevance for pharmaceutical or biochemical applications. While the maximum error in bond distances is as high as 0.59 pm in the case of LSD, the RMS deviations clearly show that the overall agreement of the structures is actually very good. For

117 4.2. Accuracy of LEDO-DFT 95 Table 4.10: Accuracy of LEDO-DFT calculations: Errors of computed total energies ( E = E LEDO E DFT in 10 3 au) and dipole moments ( µ = µ LEDO µ DFT in 10 3 au) and maximum absolute and root-mean-square (RMS) deviations of bond distances ( d in pm) and bond angles ( γ in degrees) for some representative molecules. d γ Molecule E max RMS max RMS µ Acyclovir Adrenaline Caffeine Capillin Chrys. acid S-Ibuprofen LSD Nicotine α-pinene Valium Vitamin C the errors introduced by the LEDO approximation into the bond angles we can observe a similar tendency as for the linear alkanes (cf. Sec ). The maximum errors are around 0.5 degrees or smaller with exception of LSD, which shows a relatively large deviation of 1.3 degrees. The RMS values, however, are again encouragingly low. Taking into account the errors in the structure parameters, all dipole moments are reproduced with very good accuracy. The largest relative error of 11.7 % is actually observed for α-pinene, because the dipole moment as calculated without LEDO approximation is as low as au. We conclude that LEDO-DFT indeed seems to be reliable for investigations of systems as complex as presented in this section.

118 96 Chapter 4. Numerical Results t / min 100 conventional DFT (semi-direct) RI-DFT (incore) LEDO-DFT (incore) XC XC (LEDO) diagonalization t / min 20 conventional DFT RI-DFT LEDO-DFT XC XC (LEDO) Chain length Chain length Figure 4.8: CPU times (in minutes on a 2 GHz Intel Xeon processor) for DFT single point (left) and analytical gradient (right) calculations on linear alkanes with conventional DFT, conventional density fitting (RI-DFT) and LEDO approximation. 4.3 Efficiency of LEDO-DFT Finally, we should like to give a proof of the efficiency of LEDO-DFT. Table 4.11 contains CPU times of single point and analytical gradient calculations for linear alkanes, the chain length again being limited to a maximum of 30 carbon atoms. We compare LEDO-DFT to RI-J [127] (also termed RI-DFT) calculations, both of which have been performed fully incore as described at the beginning of Sec CPU times for RI-DFT and LEDO-DFT as compared to DFT calculations without further approximations (termed conventional in the following) are shown in Fig For an objective and fair comparison, the conventional DFT calculations have been performed in a semi-direct fashion with the same amount of core memory as required for the corresponding LEDO-DFT incore calculation. The most striking observation from Fig. 4.8 is the pronounced outperformance of conventional DFT both by RI-DFT and LEDO-DFT. The speedup increases with system size. Values of 10.5 and 11.1 are observed for single point calculations on n-c 30 H 62 with RI-DFT and LEDO-DFT, respectively. The speedup is less pronounced for the calculation of the analytical gradients. However, for n-c 30 H 62, values of 4.4 and 3.4 are still obtained with RI-DFT and LEDO-DFT, respectively. While the work required for the numerical quadrature of the XC potential contributes only a small fraction to the total cost of a single point or

119 4.3. Efficiency of LEDO-DFT 97 Table 4.11: CPU times (in seconds on a 2 GHz Intel Xeon processor) for incore DFT single point and analytical gradient calculations on linear alkanes with conventional density fitting (RI-J) and LEDO approximation. RI-J and LEDO-J denote the time spent for the evaluation of the Hartree contribution to the KS matrix and the gradients. For LEDO-DFT the time spent for the computation of the ERIs and the determination of the LEDO expansion coefficients (or their derivatives) is given separately (indicated as ERIs and coeffs). XC denotes the time spent during the numerical quadrature of the XC potential. Values in parentheses are extrapolated since for n-c 30 H 62 the SCF took fewer steps to converge with LEDO-DFT. n-c 10 H 22 n-c 20 H 42 n-c 30 H 62 Single point Conventional density fitting Total RI-J XC LEDO approximation Total (873) LEDO-J/ERIs/coeffs 36/12/15 102/39/36 194(201)/80/61 XC (402) Analytical gradients Conventional density fitting Total RI-J XC LEDO approximation Total LEDO-J/ERIs/coeffs 54/25/28 153/88/62 279/175/98 XC

120 98 Chapter 4. Numerical Results gradient calculation in the case of conventional DFT, this step consumes roughly 50% of the total CPU time for RI-DFT and LEDO-DFT. It should be furthermore mentioned that both for RI-DFT and LEDO-DFT, the diagonalization step constitutes a considerable fraction of the total cost of the single point calculations for the larger systems. Total CPU times for single point calculations are comparable for LEDO-DFT and RI-J calculations. Table 4.11 shows that the evaluation of the ERIs and the determination of the expansion coefficients (coeffs) which take place before the SCF process still take a large fraction of the total time for LEDO-DFT. A moderate speedup of the numerical quadrature of the XC potential is observed for LEDO-DFT. The a priori elimination of near-linear dependences of the onecenter overlap densities of the LEDO expansion basis is fully exploited for the determination of the expansion coefficients which is necessary to be competitive with RI-J. However, we should like to point out that these near-linear dependences cannot be efficiently exploited during all stages of a LEDO-DFT calculation. This is the case, e.g., for the evaluation of the ERIs, which means that a huge amount of ERIs is computed which actually are dispensable (cf. Sec ). This could be circumvented if the LEDO expansion would be based on auxiliary functions instead of one-center overlap densities, as will be discussed in the concluding chapter. Total CPU times for the evaluation of analytical gradients are higher for LEDO-DFT than for conventional density fitting. Again, a large fraction of the time is spent for the computation of the derivatives of the ERIs and the determination of the derivatives of the expansion coefficients (coeffs) as needed for the LEDO contribution [Eq. (3.14)] to the gradient. As for single point calculations, the a priori elimination can only be fully exploited during the determination of the derivatives of the expansion coefficients. Note that the ratio LEDO-J/RI-J decreases with increasing system size both for single point and gradient calculations. We conclude by observing that the cost/accuracy ratio of LEDO-DFT is not yet as good as for conventional density fitting. However, LEDO-DFT is competitive and there is reason to hope that future improvements will change this ratio in favor of LEDO-DFT.

121 Chapter 5 Summary In this work the foundations have been laid for maturing LEDO-DFT as a generally applicable, fast and reliable DFT method which could enable exciting applications of electronic structure calculations in various areas of contemporary research. Many different aspects concerning LEDO-DFT and its applicability have been tackled. The most important achievements can be summarized as follows: Theoretical investigations Density fitting methods are in wide use in electronic structure calculations nowadays. During the course of this work it has been worked out how the LEDO expansion fits into this general framework. This allows for a deeper understanding of the advantages and drawbacks of LEDO-DFT as compared to the various other approaches. A review of density fitting methods focusing on applications in DFT has been included in chapter 2 of this work. Emphasis has been put on the distinction of the different norms that can be applied during the density fitting procedure and especially the consequences of using global or local expansions. A prerequisite for the successful application of any density fitting method is the availability of accurate fit bases. Therefore, a proper norm for the optimization of auxiliary basis sets for the LEDO expansion has been derived. Furthermore, with the a priori elimination, the most efficient method available to deal with near-linear dependences in the LEDO expansion basis has been introduced. Finally, a projection technique suggested by Löwdin has been presented in 99

122 100 Chapter 5. Summary the framework of LEDO-DFT, which shall guarantee for the SCF convergence of critical cases. Practical implementation Efficient algorithms for energy and analytical gradient calculations with LEDO- DFT have been developed and successfully implemented into the TURBO- MOLE [10, 11] program package. The main features and algorithms of the implementation are thoroughly discussed in chapter 3 of this work. A preliminary implementation of LEDO-DFT for energy calculations by Kollmar [1] had already been available before in the program system MOLPRO 2000 [9]. However, that implementation had not been conceptualized with efficiency in mind and no objective assessment of the potential advantages of LEDO-DFT had been possible at that time. The situation is different now and additionally, for the first time, an implementation of LEDO-DFT featuring analytical gradients is available. This finally allows for efficient structure optimizations and semi-numerical harmonic frequency analyses. Optimization of auxiliary orbitals Using the norm [Eq. (2.111) in Sec ] for the systematic improvement of the LEDO expansion basis {Ω p } presented in this work, auxiliary orbitals {χ µ } for the Ahlrichs SVP [190] basis set for the first three rows of atoms, rare gases excepted, have been optimized. It has been shown that a set of two uncontracted Gaussians is sufficient for most purposes. The exponents of these auxiliary orbitals can be optimized consecutively and there is no strong atom pair-dependence of the optimal exponents which justifies their optimization for homonuclear atom pairs. It has been demonstrated that, although the best exponents depend on the interatomic distance, fixed exponents improve the quality of the LEDO fit by two orders of magnitude at each interatomic distance if these are chosen properly. It has furthermore been shown that an a priori elimination of near-linear dependences in the LEDO expansion basis is feasible which leads to a drastic reduction of the dimension of the systems of linear equations to be solved.

123 101 Proof of accuracy The accuracy of LEDO-DFT employing the optimized auxiliary orbitals in combination with the a priori elimination has been demonstrated by structure optimizations and frequency analyses for a test set of 142 small molecules. Total energies, structural parameters, dipole moments and harmonic frequencies are reproduced with very small errors for most molecules of the test set. Application of an extended expansion basis leads to very good results for critical test molecules as well. Test calculations on linear alkanes with a chain length of up to 30 carbon atoms have shown that the errors in the structure parameters grow only very slowly with increasing system size. Furthermore it has been proven that the Löwdin projection technique can indeed be successfully applied in the framework of LEDO-DFT in order to guarantee for SCF convergence in critical cases without significantly deteriorating the results. Finally, it has been demonstrated that LEDO-DFT can be expected to yield reliable results for typical applications of DFT like, e.g., the computation of energy differences such as reaction energies and relative stabilities. Results obtained for a variety of more complex molecules as might be of interest in contemporary research are also very encouraging. Proof of efficiency The computational efficiency of LEDO-DFT has been investigated for linear alkanes with a maximum chain length of 30 carbon atoms. A speedup of up to 11.1 for the SCF and 3.4 for the analytical gradients has been observed as compared to DFT without further approximations. The timings furthermore show that LEDO-DFT is competitive with conventional density fitting (RI-J approximation), although the latter is still slightly in advantage. The use of the a priori elimination of near-linear dependences of the one-center overlap densities of the LEDO expansion basis for the determination of the expansion coefficients has been essential in achieving these positive results.

124 102 Chapter 5. Summary

125 Chapter 6 Outlook This work has demonstrated that accurate structure optimizations with an implementation of LEDO-DFT including the determination of analytical gradients are possible. The results of the benchmark calculations carried out demonstrate that the optimized auxiliary orbitals meet the requirements of the LEDO expansion and that a reasonable accuracy in the final DFT results is obtained. These auxiliary orbitals can be considered as the best compromise between accuracy and computational demand. However, in order to outperform conventional density fitting methods, the cost/accuracy ratio still has to be improved. Possible directions for future work shall be outlined in what follows. Auxiliary functions There is still room for improvement and at this point we should like to comment on the option to base the LEDO expansion exclusively on a properly chosen auxiliary basis {Λ r } instead of the one-center overlap densities between AOs {φ µ } and auxiliary orbitals {χ µ } (cf. Sec ). First of all, it is very likely that better fits can be achieved with less expansion functions because the quality of the expansion basis could be tuned in a more efficient way. It is clear that this would have a positive impact both on the accuracy and on the efficiency of LEDO-DFT calculations. While the asymptotic scaling behavior will always remain O(N 2 ), the prefactor would be reduced. The LEDO expansion basis presented in this contribution contains two parameters per atom (the exponents) which, e.g. for carbon, lead to 376 additional expansion functions out of which 103

126 104 Chapter 6. Outlook 190 are linear dependent. A change of one parameter thus simultaneously affects a large amount of expansion functions. Furthermore, not all of the non-linear dependent expansion functions lead necessarily to a significant improvement of the fit. The problem of linear dependences would not occur at all with auxiliary functions and the flexibility would be much higher since the exponents of the auxiliary functions are not coupled. The knowledge gained in this work with auxiliary orbitals can be transferred to auxiliary functions for which actually good starting points are available. The one-center overlap densities presently used for the LEDO expansion basis which have large contributions to the fit of diatomic overlap densities will give a good hint on how to choose the angular momentum quantum numbers and exponents of auxiliary functions. It is furthermore likely that auxiliary basis sets employed in RI methods can be used as a starting point for the optimization of auxiliary basis sets for LEDO-DFT as well. Especially the auxiliary functions employed in ab initio methods as RI-MP2 and RI-CC2 [140, 144] are likely to yield good results since the demands to these fitting bases are higher than to bases fitting the complete electron density as in RI-J methods. The optimization of such auxiliary basis sets for the LEDO expansion can then be carried out efficiently based on the norms introduced in this work (cf. Sec ). From the point of view of the practical implementation, auxiliary functions certainly are also advantageous. The memory management can be organized in a more transparent way because one has not to deal with compound indices as is the case for one-center orbital products. Especially appealing is also that the cumbersome treatment of near-linear dependences becomes no longer necessary. An implementation of LEDO-DFT based on auxiliary functions is therefore highly desirable. Building on the infrastructure created during the course of this work this should not pose any significant problems. A population analysis based on the LEDO expansion Finally, the possible use of the LEDO expansion in a population analysis shall be outlined. The LEDO approximation is based on a projection of all diatomic overlap densities onto the two atoms giving rise to them. Thus, all two-center contributions to the electron density are distributed in a unique and natural way onto the individual atoms. This suggests to base a population analysis on the

127 105 LEDO decomposition of the complete electron density. The quality of the fit, and consequently of the approximate one-center densities obtained, will depend on the LEDO expansion basis. Therefore the electron counts of the individual atoms in a molecule will eventually converge to an optimal result with increasing size (quality) of the LEDO expansion basis. Whether or not the LEDO partitioning of the two-center part of the electron density will deliver physically meaningful results regarding this electron count will have to be seen in future investigations. Certainly it might be a promising way leading to an unbiased method for a population analysis.

128 106 Chapter 6. Outlook

129 Chapter 7 Zusammenfassung Diese Dissertation beschreibt die Entwicklung und Implementierung von effizienten Algorithmen für die limited expansion of diatomic overlap Dichtefunktionaltheorie (LEDO-DFT) [1] in das Programmpaket TURBOMOLE [10, 11]. Ausführliche Testrechnungen mit dieser Implementierung unter Verwendung von in dieser Arbeit optimierten Auxiliarbasissätzen dokumentieren die Genauigkeit und Leistungsfähigkeit von LEDO-DFT. LEDO-DFT ist ein neuer, von Kollmar und Hess entworfener Formalismus [1] im Rahmen der Kohn Sham Dichtefunktionaltheorie (KS-DFT) [2, 3], welcher ein formales Skalierungsverhalten von O(N 2 ) für die Berechnung der Säkularmatrix aufweist. Hierbei sei die Systemgröße durch N (z. B. die Anzahl der verwendeten Basisfunktionen) charakterisiert. Herkömmliche Implementierungen der KS- DFT weisen ein formales Skalierungsverhalten von O(N 4 ) auf. Dieses kann mit Hilfe konventioneller Dichtefitmethoden auf O(N 3 ) gesenkt werden. Entscheidend für den tatsächlichen Aufwand einer Rechnung für große Moleküle ist allerdings das asymptotische (oder effektive) Skalierungsverhalten zusammen mit dem daran gekoppelten Vorfaktor. Das asymptotische Skalierungsverhalten ist für alle KS-DFT-Verfahren O(N 2 ). Durch das günstigere formale Skalierungsverhalten eröffnet LEDO-DFT jedoch die Möglichkeit, den Vorfaktor für das asymptotische Skalierungsverhalten weiter zu senken. Die Genauigkeit von LEDO-DFT für Energien und Strukturparameter wurde durch Kollmar und Hess mit einer Implementierung von LEDO-DFT in das Programmpaket MOLPRO [9] demonstriert [1]. Die Autoren beschränkten sich hierbei auf wenige, H-, C-, N- und O- 107

130 108 Chapter 7. Zusammenfassung Atome enthaltende kleine Moleküle. Weitherhin wurde für alle Rechnungen eine vorläufige, empirisch bestimmte Auxiliarbasis eingesetzt und numerische Gradienten für die Strukturoptimierungen verwendet. Später wurde vom Autor der vorliegenden Dissertation im Rahmen einer Diplomarbeit [12] eine rudimentäre Implementierung von LEDO-DFT für Einzelpunktrechnungen in das Programmpaket TURBOMOLE realisiert. Diese Dissertation baut auf der Arbeit von Kollmar und Hess [1] auf und erweitert die oben erwähnte Implementierung in TURBOMOLE. Die Zielsetzungen zu Beginn dieser Arbeit waren: 1. einen Weg zur systematischen Optimierung von Auxiliarbasissätzen für LEDO-DFT zu finden 2. Auxiliarbasissätze für einen größeren Satz von Atomtypen zu optimieren 3. die Implementierung der analytischen Gradienten für LEDO-DFT 4. die Optimierung der Implementierung bezüglich Effizienz 5. die objektive Beurteilung von LEDO-DFT bezüglich (a) Genauigkeit und (b) tatsächlichem Rechenaufwand Die vorliegende Arbeit läßt sich im wesentlichen in drei Teile gliedern. Der erste Teil (Kapitel 2) befasst sich mit sämtlichen theoretischen Grundlagen, welche zum Verständnis von LEDO-DFT erforderlich sind. Eine kurze Einführung in die DFT steht hierbei am Anfang. Da LEDO-DFT als eine spezielle Dichtefitmethode angesehen werden kann, wurde eine ausführliche Übersicht über Dichtefitverfahren mit Schwerpunkt auf deren Anwendung in der KS-DFT ausgearbeitet. Hierbei wurde Wert auf eine Unterscheidung der verschiedenen Normen und insbesondere auf das Verständnis der Konsequenzen globaler oder lokaler Entwicklungen, welche bei Dichtefitverfahren zum Einsatz kommen können, gelegt. Dies erlaubt eine bessere Einschätzung der Vor- und Nachteile von LEDO-DFT im Vergleich zu anderen bestehenden Verfahren. Wesentlich für die erfolgreiche Anwendung von LEDO-DFT ist die Verfügbarkeit von guten

131 109 Fitbasissätzen. Im Theorieteil wird daher eine geeignete Norm vorgestellt, die es erstmals ermöglicht hat, Auxiliarbasissätze für die LEDO-Entwicklung systematisch zu optimieren. Es wird weiterhin auf mögliche Formen der Auxiliarbasis für die LEDO-Entwicklung eingegangen (welche direkt aus Auxiliarfunktionen oder Einzentrenprodukten von Atomorbitalen und Auxiliarorbitalen bestehen kann) und ein neues, hocheffizientes Verfahren zum Umgang mit linearen Abhängigkeiten in der Entwicklungsbasis, die sogenannte a priori Eliminierung, vorgestellt. Im Anschluss daran wird der LEDO-DFT Formalismus beschrieben und die Formeln zur Berechnung des analytischen Gradienten abgeleitet. Schließlich wird ein von Löwdin vorgeschlagener Projektionsformalismus [151] vorgestellt, welcher zur Verbesserung des Konvergenzverhaltens des SCF (self-consistent field) für kritische Moleküle im Rahmen von LEDO-DFT eingesetzt werden kann. Der zweite Teil (Kapitel 3) ist komplett der Beschreibung der Implementierung von LEDO-DFT in das Programmpaket TURBOMOLE gewidmet. Hierbei wird zuerst auf Details für Einzelpunktrechnungen eingegangen und danach die Implementierung der analytischen Gradienten geschildert. Es wurde dabei Wert darauf gelegt, dass klar ersichtlich wird, wie die einzelnen Schritte einer Rechnung mit LEDO-DFT ablaufen und wie hohe Effizienz, z. B. durch Integralabschätzungen und Einsatz der a priori Eliminierung, erreicht wurde. Im dritten Teil (Kapitel 4) werden numerische Ergebnisse präsentiert, welche mit der in Kapitel 3 vorgestellten Implementierung von LEDO-DFT erhalten wurden. Zunächst wird eine systematische Vorgehensweise zur Optimierung von Auxiliarorbitalen unter Verwendung der in Kapitel 2 vorgestellten Norm am Beispiel des SVP [190] Basissatzes beschrieben. Es zeigte sich, dass für die meisten Anwendungen ein Satz von zwei unkontrahierten Gaußfunktionen als Auxiliarorbitale ausreicht. Die Exponenten dieser Gaußfunktionen müssen nicht simultan, sondern können nacheinander optimiert werden. Des weiteren stellte sich heraus, dass keine starke Abhängigkeit der optimalen Exponenten vom Atompaartyp besteht. Es wurde gezeigt, dass feste Exponenten die Qualität des LEDO-Fits bei beliebigen interatomaren Abständen um zwei Größenordnungen verbessern, obwohl die besten Exponenten vom interatomaren Abstand abhängen. Es wurde weiterhin demonstriert, dass die a priori Eliminierung linearer Abhängigkeiten

132 110 Chapter 7. Zusammenfassung in der LEDO-Entwicklungsbasis ohne Qualitätsverlust durchführbar ist. Dies führt zu einer drastischen Reduktion der Dimension des linearen Gleichungssystems, welches zur Bestimmung der LEDO-Entwicklungskoeffizienten gelöst werden muss. Im Anschluss daran werden Ergebnisse von Testrechnungen an einem Testsatz von 142 kleinen Molekülen vorgestellt, welche die Genauigkeit von LEDO-DFT unter Verwendung der optimierten Auxiliarorbitale und der a priori Eliminierung dokumentieren. Die Abweichung der Gesamtenergien, Strukturparameter, Dipolmomente und harmonischen Schwingungsfrequenzen (berechnet durch Ableitung der analytischen Gradienten) ist für die meisten Moleküle des Testsatzes im Vergleich zu konventionellen DFT-Rechnungen klein. Die Verwendung eines sytematisch erweiterten Auxiliarorbitalsatzes führte auch für die kritischeren Fälle zu sehr guten Resultaten. Testrechnungen an linearen Alkanen mit einer Kettenlänge von bis zu 30 Kohlenstoffatomen zeigten, dass die Fehler in den Strukturparametern nur sehr langsam mit der Systemgröße ansteigen. Es wurde weiterhin demonstriert, dass die Projektionstechnik nach Löwdin tatsächlich mit Erfolg und ohne signifikante Einbußen an Genauigkeit eingesetzt werden kann, um SCF-Konvergenz in besonders schwierigen Fällen zu erreichen. Schießlich wurde gezeigt, dass LEDO-DFT auch zur Vorhersage von Energiedifferenzen wie Reaktionsenergien und relativen Stabilitäten von Isomeren geeignet ist. Auch die Ergebnisse von Testrechnungen an einigen komplexeren Molekülen wie sie typischerweise in der aktuellen Forschung von Interesse sein könnten, sind sehr ermutigend. Im letzten Abschnitt dieses Kapitels wird die Effizienz von LEDO- DFT untersucht. Hierzu wurden Zeitmessungen für Einzelpunkt- und Gradientenrechnungen an linearen Alkanen mit einer Kettenlänge von bis zu 30 Kohlenstoffatomen durchgeführt. Eine Beschleunigung der Rechnungen um einem Faktor von bis zu 11,1 bzw. 3,4 für den SCF-Teil bzw. die analytischen Gradienten wurde im Vergleich zu DFT ohne Dichtefitting beobachtet. Die Zeitmessungen zeigen weiterhin, dass LEDO-DFT in der Tat konkurrenzfähig zu herkömmlichen Dichtefitverfahren (RI-DFT) ist, obwohl die letzteren immer noch leicht im Vorteil sind. Um diese positiven Ergebnisse zu erhalten, war der Einsatz der a priori Eliminierung bei der Bestimmung der LEDO-Entwicklungskoeffizienten unerlässlich. In dieser Arbeit wurde gezeigt, dass mit analytischen Gradienten für

133 111 LEDO-DFT im allgemeinen Strukturoptimierungen ohne relevante Einbußen an Genauigkeit möglich sind. Die Ergebnisse der durchgeführten Testrechnungen ergaben, dass die in dieser Arbeit optimierten Auxiliarorbitale für den SVP [190] Basissatz die Anforderungen der LEDO-Entwicklung erfüllen und zu guten Ergebnissen von KS-DFT-Rechnungen mit LEDO-Näherung führen. Die Ergebnisse dieser Arbeit können als bester unter Verwendung von Auxiliarorbitalen erreichbarer Kompromiss zwischen Genauigkeit und Rechenaufwand angesehen werden. Um herkömmliche Dichtefitmethoden (RI-DFT) zu übertreffen, sind bezüglich der Auxiliarbasis jedoch noch Verbesserungen nötig. Diese sind nur durch den Einsatz von Auxiliarfunktionen an Stelle der Auxiliarorbitale zu erwarten. Aus diesem Grund ist eine Implementierung von LEDO-DFT basierend auf Auxiliarfunktionen höchst wünschenswert. Durch die hier geleisteten Vorarbeiten wird dies ohne größere Probleme zu realisieren sein. Zusammenfassend kann gesagt werden, dass diese Arbeit die Grundlagen dafür geliefert hat, dass LEDO-DFT zu einem allgemein anwendbaren, schnellen, und verlässlichen DFT-Verfahren ausgebaut werden kann, welches interessante Anwendungen von Berechnungen der Elektronenstruktur in verschiedensten Bereichen der Chemie und verwandter Wissenschaften ermöglichen könnte.

134 112 Chapter 7. Zusammenfassung

135 Appendix A Auxiliary Orbitals for the atom pairs CF, CH and CP In this appendix, we present additional results which justify the use of auxiliary orbitals optimized for homonuclear atom pairs as described in Sec. 4.1 also for heteronuclear atom pairs. Figs. A.1 to A.3 contain contour plots of the norms for the heteronuclear atom pairs CF, CH and CP at typical bond distances under variation of the exponents of the two sets of auxiliary orbitals (cf. Sec ) ζ(d ) of F ζ( f ) of F ζ(d ) of C ζ( f ) of C Figure A.1: CF at an interatomic distance of 133 pm: Contour plots of the norm 5p5p under variation of the exponents ζ(d) (left, isovalues between and ) and CF under variation of the exponents ζ(f) (right, isovalues between and ) of carbon and fluorine. The minima for the exponents optimized for the homonuclear atom pairs CC and FF are indicated by a cross. 113

136 114 Appendix A. Auxiliary Orbitals for the atom pairs CF, CH and CP ζ( p) of H ζ(d ) of H ζ(d ) of C ζ( f ) of C Figure A.2: CH at an interatomic distance of 109 pm: Contour plots of the norm 5p2s under variation of the exponents ζ(d)/ζ(p) (left, isovalues between and ) and CH under variation of the exponents ζ(f)/ζ(d) (right, isovalues between and ) of carbon and hydrogen. The minima for the exponents optimized for the homonuclear atom pairs CC and HH are indicated by a cross ζ(d ) of P ζ( f ) of P ζ(d ) of C ζ( f ) of C Figure A.3: CP at an interatomic distance of 185 pm: Contour plots of the norm 5p7p under variation of the exponents ζ(d) (left, isovalues between and ) and CP under variation of the exponents ζ(f) (right, isovalues between and ) of carbon and phosphorus. The minima for the exponents optimized for the homonuclear atom pairs CC and PP are indicated by a cross. In Table A.1 we compare the values of the norms without auxiliary orbitals, using auxiliary orbitals with fixed exponents (as optimized for homonuclear atom pairs), and using auxiliary orbitals with the best exponents (as optimized for the

137 115 Table A.1: Quality of the LEDO fit for different heteronuclear atom pairs at typical bond distances without auxiliary orbitals, using auxiliary orbitals with fixed exponents (as optimized for homonuclear atom pairs), and using auxiliary orbitals with the best exponents (as optimized for the heteronuclear atom pairs). no aux. orbs. fixed ζ best ζ CF CH CP 5p5p CF p2s CH p7p CP heteronuclear atom pairs). The plots and the tabulated data show that the use of auxiliary orbitals leads to a reduction of the norm AB by two orders of magnitude in all instances. Furthermore, the exponents optimized for homonuclear atom pairs are very close to the minima.

138 116 Appendix A. Auxiliary Orbitals for the atom pairs CF, CH and CP

139 Appendix B Accuracy of LEDO-DFT for a test set of 142 small molecules In this appendix, we present a comprehensive list of the results of benchmark calculations on a test set of 142 small molecules containing the atoms H, Li F and Na Cl as discussed in Sec All calculations have been performed with the SVP [190] basis set and the BP86 [30, 33, 35] XC functional as described in Sec The auxiliary orbitals given in Table 4.1 have been employed for the LEDO expansion. Table B.1: Accuracy of LEDO-DFT calculations: Errors of computed total energies ( E = E LEDO E exact in 10 3 au), bond distances ( d in pm), bond angles ( γ in degrees), dipole moments ( µ = µ LEDO µ exact in 10 3 au) and harmonic frequencies ( ν in cm 1 ) for a test set of small molecules. Maximum absolute and root-mean-square (denoted with R) errors are given for d, γ and ν. Molecule E d R( d) γ R( γ) µ ν R( ν) H N F HF HOF

140 118 Appendix B. Accuracy of LEDO-DFT for a test set of 142 small molecules Molecule E d R( d) γ R( γ) µ ν R( ν) CH C 2 H C 2 H C 2 H C 3 H Allene Cyclopropene C 3 H Cyclopropane C 3 H C 4 H Benzene NH cis-n 2 H trans-n 2 H N 2 H N 4 H H 2 O H 2 O HCN CH 3 NH CH 3 CN CH 3 NC CH 2 CHCN HCCCN CH 2 N CH 3 N 2 CH

141 119 Molecule E d R( d) γ R( γ) µ ν R( ν) CH 3 N CO CO H 2 CO CH 3 OH CH 2 CO CH 3 OCH CH 2 CHCHO HCOOH HCOOCH HCONH Glycine N 2 O N 2 O CF C 2 F NF N 2 F N 2 F F 2 O F 2 O LiLi LiH LiF LiCl Li 2 O LiOH

142 120 Appendix B. Accuracy of LEDO-DFT for a test set of 142 small molecules Molecule E d R( d) γ R( γ) µ ν R( ν) LiCH NaNa NaH NaF NaCl Na 2 O NaOH NaCH BeH BeO Be(OH) BeF BeCl Be 2 Cl Be(CH 3 ) MgH MgO Mg(OH) MgF MgCl Mg(CH 3 ) MgCH 3 Cl BH B 2 H BF BCl BH 3 NH

143 121 Molecule E d R( d) γ R( γ) µ ν R( ν) BH 3 PH B(OH) BHNH BH 2 NH B(CH 3 ) AlH Al 2 H AlF AlCl AlF AlCl Al(OH) AlH 3 NH Si SiH Si 2 H Si(CH 3 ) SiCl Si(CH 3 ) 3 Cl P P PH cis-p 2 H trans-p 2 H anti-p 2 H PF PF

144 122 Appendix B. Accuracy of LEDO-DFT for a test set of 142 small molecules Molecule E d R( d) γ R( γ) µ ν R( ν) PCl P(CH 3 ) PO(CH 3 ) H 3 PO P 4 O S H 2 S H 2 S CS CS CH 3 SCH SO SO H 2 SO H 2 SO SF SF SF Cysteine Cl HCl ClF HOCl HCOCl CH 3 Cl CH 3 COCl CCl

145 123 Molecule E d R( d) γ R( γ) µ ν R( ν) COCl CSCl Note, that 10 3 au corresponds to 219 cm 1.

146 124 Appendix B. Accuracy of LEDO-DFT for a test set of 142 small molecules

147 List of Tables 2.1 Scaling behavior of LEDO-DFT Recommended exponents of auxiliary orbitals Size of the LEDO expansion basis Accuracy of LEDO-DFT for some selected small molecules Accuracy of LEDO-DFT for 142 small molecules: statistics Accuracy of LEDO-DFT with an extended expansion basis Accuracy of LEDO-DFT for linear alkanes Assessment of the projection technique Accuracy of LEDO-DFT for the dissociation energy of benzene Accuracy of LEDO-DFT for relative stabilities of various isomers Accuracy of LEDO-DFT for some representative molecules Comparison of CPU times for LEDO-DFT and RI-DFT A.1 Quality of the LEDO fit for heteronuclear atom pairs B.1 Accuracy of LEDO-DFT for 142 small molecules

148 126 List of Tables

149 List of Figures 3.1 Flow chart of the SCF program Flow chart of the gradient program Variation of 5p5p with ζ(d) and CC with ζ(f) for CC at 120pm Variation of 5p5p, CC with ζ(d), ζ(f) for CC at 120pm Best ζ(d), ζ(f) and norms 5p5p, CC for CC at varying distances Variation of 5p5p, CN with ζ(d), ζ(f) for CN at 122pm Fit quality for different thresholds in the a priori elimination Timings for the determination of the LEDO expansion coefficients IR spectrum of Cysteine Comparison of CPU times for DFT, RI-DFT and LEDO-DFT.. 96 A.1 Variation of 5p5p, CN with ζ(d), ζ(f) for CF at 133pm A.2 Variation of 5p2s, CH with ζ(d/p), ζ(f/d) for CH at 109pm A.3 Variation of 5p7p, CP with ζ(d), ζ(f) for CP at 185pm

150 128 List of Figures

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175 Danke Es gibt ein Leben neben der Forschung... There is a life beyond research and at this point I should like to apologize for all the moments during which I have neglected my social relations because of this thesis. This page is dedicated to all my friends and my family. They have been of great importance to carry me through these sometimes hard times. In this context I would like to thank Frank and Maria Lauderbach, Stefan Sperner, Michael Lutz, Micha l Cyrański, João Pêgo and Mira Pashtrapanska, Guido Marconi and Laura Bettinetti, Pilar Rodriguez, Emel Sümer and Hasan Pehlivanogullari. One of my major distractions from the daily sorrows and source for regeneration has been sports climbing. In this context it is a pleasure to mention Christian Kollmar again, as well as Stefan and Florian Seidel and Martin Weimer. I would like to express my sincere gratitude to my parents Fedora Götz-Amadesi and Walter Götz for their faith, their constant support and their love, without which I would never have got here in the first place. Furthermore I should like to mention my sister Silvia Götz, my lovely grandfather Alfred Götz, Heinz and Hanna Dinkel, and Sergio Amadesi. I also owe a debt of gratitude to Teresa and Nuno Carmo as well as Júlia and Nuno Carmo, who have always treated me as if I would already be part of their family. A very special thank you goes to Patrícia Pinto who joined me here in Erlangen to share her life with me. Only through her constant love, her unconditional support and her encouragement during difficult times has any of this been possible. She has had to put up with the stress and bad moods associated with completing a Ph.D. and she patiently beared all the hours I have spent in an absentminded state behind books and the computer. Thank you for all the wonderful moments we had together.

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177 Curriculum Vitae Andreas Walter Götz Personal Details Date of birth 6th of February, 1977 Place of birth Citizenship Marital Status Nürnberg, Germany German and Italian Unmarried Education since 11/2001 PhD studies in Theoretical Chemistry at the Friedrich Alexander-Universität Erlangen Nürnberg, Germany 10/ /2001 Chemistry studies at the Friedrich Alexander-Universität Erlangen Nürnberg, Germany 04/ /1998 Chemistry studies at the Università di Bologna, Italy 09/ /1995 Sigena-Gymnasium (high school) in Nürnberg, Germany 09/ /1987 Wiesenschule (elementary school) in Nürnberg, Germany Employment since 10/2004 Scientific employee at the Chair of Theoretical Chemistry of the Friedrich Alexander-Universität Erlangen Nürnberg, Germany 07/ /2004 Scientific employee at the Chair of Theoretical Chemistry of the Rheinische Friedrich Wilhelms-Universität Bonn, Germany 05/ /2004 Scientific employee at the Chair of Theoretical Chemistry of the Friedrich Alexander-Universität Erlangen Nürnberg, Germany