Development of Frozen-Density Embedding theory methods with correlated wavefunctions

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(1)Thesis. Development of Frozen-Density Embedding theory methods with correlated wavefunctions. ZECH, Alexander. Abstract Due to the unfavorable scaling of quantum chemical methods one usually has to compromise between accuracy and computational effort with growing system size. Finding approximations that overcome this constraint sparked the interest of researchers which eventually led to the development of so-called multiscale methods. This thesis pertains to a multiscale method called Frozen-Density Embedding theory (FDET), in which the system is described by means of two independent quantum mechanical descriptors, the wavefunction of the embedded species and the charge density of the environment. This work examined the effect of the non-linearity of FDET equations. The second topic of this thesis was the development and implementation of FDET-based methods for ground and excited states. For this purpose the fdeman module, which manages all steps of an FDET calculation, was implemented into the quantum chemistry software package Q-Chem.. Reference ZECH, Alexander. Development of Frozen-Density Embedding theory methods with correlated wavefunctions. Thèse de doctorat : Univ. Genève, 2019, no. Sc. 5398. DOI : 10.13097/archive-ouverte/unige:125820 URN : urn:nbn:ch:unige-1258203. Available at: http://archive-ouverte.unige.ch/unige:125820 Disclaimer: layout of this document may differ from the published version..

(2) Université de Genève. Faculté des sciences. Section de chimie et biochimie Département de chimie physique. Professeur T. A. Wesolowski. Development of Frozen-Density Embedding Theory Methods with Correlated Wavefunctions. THÈSE présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention chimie. par. Alexander Zech de Heidelberg, Allemagne. Thèse N◦ 5398. GENÈVE Ateliers d’impression de l’Université de Genève 2019.

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(4) For my parents.

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(6) ”Bildung ist das, was übrig bleibt, wenn man alles vergessen hat, was man gelernt hat.”. Werner Karl Heisenberg.

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(8) Abstract. Computational simulations by means of electronic structure methods have become a valuable tool for investigations in all branches of chemistry. Due to the unfavorable scaling of these methods one usually has to compromise between accuracy and computational effort with growing system size. Finding approximations that overcome this constraint sparked the interest of researchers which eventually led to the development of so-called multiscale methods, which was awarded with a Nobel prize in 2013. This class of methods is built on the assumption that only a small part of the system is chemically relevant, which calls for an accurate quantum mechanical treatment. In this way the residual part of the system, the environment, may be approximated in order to reduce the computational cost. Frozen-density embedding theory (FDET), a multiscale method being developed in the Wesolowski group, provides a formal framework in which the system is described by means of two independent quantum mechanical descriptors, the wavefunction of the embedded species and the charge density of the environment. This thesis comprises three parts pertaining to the variant of FDET, whereby the embedded species is represented by a correlated wavefunction deriving from ab initio wavefunction theory. The first part concerns the inhomogeneity of the non-additive kinetic functional Tsnad [ρA , ρB ] and the non-additive exchange-correlation functional nad [ρ , ρ ], which render FDET equations non-linear. Conventional FDET therefore Exc A B employs an iterative scheme in order to obtain self-consistency between the embedded wavefunction and the embedding potential. In this thesis the linearized FDET approximation is presented, which introduces an approximate homogenous form of these functionals avoiding the necessity of an iterative scheme. The approximation was tested on five bimolecular complexes featuring embedded species with local excitations, intramolecular charge-transfer excitations or a high polarizability. It was shown that self-consistent FDET energies can be reliably approximated by linearization of the non-additive functionals with negligible loss of accuracy and at reduced computational cost. In addition, the extent of inhomogeneity was analyzed quantitatively for three different combinations of kinetic- and exchange-correlation density functional approximations. Furthermore, we studied to what extent interstate properties such as wavefunction overlap and transition dipole moments are affected by the self-consistent optimization of the embedded wavefunction. The second topic of this thesis was the development and implementation of FDET-based methods for ground and excited states. For this purpose the fdeman module, which manages all steps of an FDET calculation, was implemented into the quantum chemistry software package Q-Chem. Concerning the ground state, a new approach for the total FDET energy was derived in which the embedded wavefunction stems from a non-variational method. This formalism was applied to the case of embedded MP2 (FDE-MP2) and its performance was assessed by means of interaction energies for eight hydrogen-bonded complexes. The FDE-MP2 interaction energies show satisfactory accuracy, however, they also exhibit a strong I.

(9) dependence on the choice of environment density and density functional approximation nad [ρ , ρ ]. Additionally errors in FDE-MP2 interaction energies are discussed for Exc A B relating to descriptors from energy decomposition analysis methods. In this part of the thesis, also the derivation of analytical nuclear gradients for embedded Hartree-Fock theory employing the linearized FDET approximation is presented. The final part of the thesis concerns the description of the embedded species in the electronically excited state. To this end we combined FDET with the algebraic diagrammatic concstruction (ADC) scheme for the polarization propagator resulting in the FDE-ADC method. We present a thorough investigation of the errors in excitation energies obtained with FDE-ADC(2) compared to the ADC(2) reference of the full system. The study was carried out on a set of 52 intermolecular complexes with varying interaction strength ranging from dispersion interaction to multiple hydrogen bonds. The analysis of errors in excitation energies was performed in view of parameters relevant in the context of FDET, for instance the overlap of subsystem densities and the degree of delocalization. It was found that the excitation energy error in π π ∗ states correlates with the magnitude of the complexation-induced shift, while n π ∗ states are not correlated. Additionally, different concepts of environment polarization are discussed and numerical examples obtained with the FDE-ADC(2) method are provided. Finally, the FDE-ADC method was applied to six biologically relevant systems, which involve a retinal chromophore enclosed in a protein binding site. Protein-induced shifts were computed with FDE-ADC(2) and compared to an ADC(2) reference. Although FDE-ADC(2) shifts are overestimated, they reproduce the relative trend of the reference shifts.. II.

(10) Résumé. La simulation numérique au moyen de méthodes de structure électronique est devenue un outil indispensable pour les recherches dans toutes les branches de la chimie. En raison de l’échelle défavorable de ces méthodes, il faut généralement faire un compromis entre la précision et le coût de calcul avec la taille croissante du système. La recherche d’approximations permettant de surmonter cette contrainte a suscité l’intérêt des chercheurs, ce qui a finalement conduit au développement de méthodes dites multiéchelles et a été récompensé par un prix Nobel en 2013. Cette classe de méthode repose sur l’hypothèse que seule une petite partie du système est chimiquement active, ce qui nécessite un traitement précis en mécanique quantique. De cette façon, la partie résiduelle du système, l’environnement, peut être approximée afin de réduire le coût computationnel. Une méthode multi-échelles, frozen-density embedding theory (FDET), est en cours de développement dans le groupe du Professeur Wesolowski et fournit un cadre formel dans lequel le système est décrit par deux descripteurs quantiques indépendants: la fonction d’onde du système d’intérêt et la densité de charge de l’environnement. Cette thèse comprend trois parties relatives à la variante du FDET, dans laquelle l’espèce d’intérêt est représentée par une fonction d’onde corrélée. La première partie concerne l’inhomogénéité de la fonctionnelle cinétique non-additive et de la fonctionnelle échangecorrélation non-additive, ceci rend les équations FDET non-linéaires. La méthode FDET conventionnelle utilise alors un schéma itératif afin d’obtenir une auto-cohérence entre la fonction d’onde et le potentiel dans lequel cette dernière est encastrée. Dans cette thèse, l’approximation FDET linéarisée est présentée. Cette approximation introduit une forme homogène approximative de ces fonctionnelles évitant la nécessité d’un schéma itératif. L’approximation a été testée numériquement et il a été démontré que les énergies FDET linéarisées sont pratiquement identiques aux énergies autocohérentes, tout en conservant l’orthogonalité des fonctions d’onde et en réduisant le coût computationnel. De plus, l’ampleur de l’inhomogénéité a été analysée quantitativement pour trois combinaisons différentes d’approximations fonctionnelles de l’énergie cinétique et de l’énergie d’échange-corrélation. Nous avons étudié aussi dans quelle mesure les propriétés inter-états telles que le recouvrement des fonctions d’onde et les moments dipolaires de transition sont affectées par l’optimisation auto-cohérente de la fonction d’onde encastrée. Le deuxième sujet de cette thèse est le développement et l’implémentation de méthodes basées sur FDET pour les états fondamentaux et les états excités. A cet effet, un module, qui gère toutes les étapes d’un calcul FDET, a été implémenté dans le logiciel de chimie quantique Q-Chem. En ce qui concerne l’état fondamental, une nouvelle approche de l’énergie totale FDET a été dérivée. Dans cette approche, la fonction d’onde encastrée découle d’une méthode non-variationnelle. Ce formalisme a été appliqué au cas de la théorie de la perturbation de Møller-Plesset au second ordre (MP2) et la performance de la méthode résultante III.

(11) FDE-MP2 a été évaluée au moyen d’énergies d’interaction pour huit complexes avec des liaisons hydrogène. Les énergies d’interaction FDE-MP2 montrent une précision satisfaisante, mais elles dépendent fortement du choix de la densité de l’environnement et nad [ρ , ρ ]. De plus, les erreurs dans les énergies de l’approximation fonctionnelle pour ExcT A B d’interaction FDE-MP2 sont discutées en relation avec les descripteurs des méthodes d’analyse de décomposition d’énergie. Dans cette partie de la thèse, nous présentons également la dérivation des gradients nucléaires analytiques pour la théorie Hartree-Fock encastrée utilisant l’approximation FDET linéarisée. La dernière partie de la thèse porte sur la description des systèmes d’intérêt à l’état excité électroniquement. À cette fin, nous avons combiné la méthode FDET avec algebraic diagrammatic construction scheme for the polarization propagator of nth order (ADC(n)), ce qui a donné la méthode FDE-ADC(n). Nous présentons une étude approfondie des erreurs dans les énergies d’excitation obtenues avec FDE-ADC(2) par rapport à la référence ADC(2) du système entier. L’étude a été menée sur un ensemble de 52 complexes intermoléculaires dont la force d’interaction varie de l’interaction de dispersion jusqu’aux liaisons hydrogène multiples. L’analyse des erreurs dans les énergies d’excitation a été effectuée en tenant compte des paramètres pertinents dans le contexte de la méthode FDET, par exemple le recouvrement des densités des sous-systèmes et le degré de délocalisation. On a constaté que l’erreur d’énergie d’excitation dans les états π π ∗ est corrélée avec l’ampleur du déplacement induit par la complexation, alors que les états n π ∗ ne sont pas corrélés. De plus, différents concepts de polarisation de l’environnement sont discutés et des exemples numériques obtenus avec la méthode FDE-ADC(2) sont fournis. Enfin, la méthode FDE-ADC(2) a été appliquée à six systèmes biologiques, qui impliquent un chromophore rétinal encastré dans un site de liaison protéique. Les déplacements induits par les protéines ont été calculés avec FDE-ADC(2) et comparés à une référence ADC(2). Bien que les déplacements FDE-ADC(2) soient surestimés, ils reproduisent la tendance relative des déplacements de référence.. IV.

(12) Contents Abstract. I. Résumé. III. Abbreviations. IX. 1 Models and methods in quantum chemistry 1.1 Introduction . . . . . . . . . . . . . . . . . . . . 1.1.1 Fundamentals . . . . . . . . . . . . . . . 1.2 Hartree-Fock Theory . . . . . . . . . . . . . . . 1.2.1 Hartree Method . . . . . . . . . . . . . 1.2.2 Hartree-Fock Equations . . . . . . . . . 1.2.3 Derivation of Hartree-Fock equations . . 1.2.4 Roothaan-Hall Equations . . . . . . . . 1.3 Density Functional Theory . . . . . . . . . . . 1.3.1 The Electron Density . . . . . . . . . . 1.3.2 The Thomas-Fermi Model . . . . . . . . 1.3.3 Mathematical Properties of Functionals 1.3.4 Hohenberg-Kohn Theorems . . . . . . . 1.3.5 Constrained Search . . . . . . . . . . . . 1.3.6 Kohn-Sham Approach . . . . . . . . . . 1.4 Second quantization . . . . . . . . . . . . . . . 1.4.1 Creation and annihilation operators . . 1.4.2 Second quantized operators . . . . . . . 1.5 Configuration Interaction . . . . . . . . . . . . 1.6 Perturbation Theory . . . . . . . . . . . . . . . 1.6.1 Møller-Plesset perturbation theory . . . 1.6.2 Relaxed versus unrelaxed density . . . . 1.7 Green’s function theory . . . . . . . . . . . . . 1.7.1 One particle Green’s function . . . . . . 1.7.2 Polarization propagator . . . . . . . . . 1.7.3 Perturbation expansion . . . . . . . . . 1.8 Algebraic Diagrammatic Construction Scheme . 1.8.1 Original derivation . . . . . . . . . . . . 1.8.2 Intermediate State Representation . . .. 1 1 1 3 3 5 7 10 13 13 13 14 16 19 21 29 29 30 31 33 34 36 37 38 40 41 42 43 45. V. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

(13) 2 Frozen-Density Embedding Theory 2.1 Introduction . . . . . . . . . . . . . . . . . . 2.1.1 Embedded Kohn-Sham system . . . 2.1.2 Embedded interacting wavefunction 2.2 Practical considerations . . . . . . . . . . . 2.2.1 Conventional FDET . . . . . . . . . 2.2.2 Choice of the environment density . 2.2.3 Basis expansion . . . . . . . . . . . . 2.3 Related embedding strategies . . . . . . . . 2.3.1 Projection-based embedding . . . . . 2.3.2 ONIOM strategy . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. 49 49 51 53 58 58 59 60 62 62 64. 3 Inhomogeneity of non-additive bifunctionals 3.1 Introduction . . . . . . . . . . . . . . . . . . . 3.1.1 Investigated systems . . . . . . . . . . 3.2 Linearized FDET . . . . . . . . . . . . . . . . 3.2.1 Total energies . . . . . . . . . . . . . . 3.2.2 First order term . . . . . . . . . . . . 3.3 Orthogonality . . . . . . . . . . . . . . . . . . 3.3.1 Wavefunction overlap . . . . . . . . . 3.3.2 Transition-dipole moment . . . . . . . 3.4 Extent of inhomogeneity . . . . . . . . . . . . 3.5 Conclusion and outlook . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. 67 67 69 72 74 75 77 78 79 80 84. . . . . . . . . . . . . . . . .. 87 87 88 88 90 91 92 94 97 98 102 107 109 111 111 112 116. 4 Embedded Species in the electronic ground state 4.1 Embedding of a non-variational wavefunction . . . . . . . . . . . . 4.1.1 The variational case revisited . . . . . . . . . . . . . . . . . 4.1.2 The non-variational case . . . . . . . . . . . . . . . . . . . . 4.1.3 Embedded Møller-Plesset perturbation theory . . . . . . . . 4.2 Interaction energies . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Symmetry-adapted perturbation theory . . . . . . . . . . . 4.2.2 Absolutely localized molecular orbital energy decomposition 4.2.3 Interaction energies from embedded MP2 . . . . . . . . . . 4.2.4 Comparison of interaction energies . . . . . . . . . . . . . . 4.2.5 Comparison with energy decomposition schemes . . . . . . 4.2.6 Effect of environment density . . . . . . . . . . . . . . . . . 4.2.7 Effect of density functional approximation . . . . . . . . . . 4.3 Nuclear gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Hartree-Fock gradients . . . . . . . . . . . . . . . . . . . . . 4.3.2 Embedded Hartree-Fock gradients . . . . . . . . . . . . . . 4.4 Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. 5 Embedded species in the electronically excited state 121 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.2 Benchmark study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 VI.

(14) 5.3. 5.4. 5.2.2 Molecular test set . . . . . . . . . . . . . . . . . . 5.2.3 Assignment of the excitations . . . . . . . . . . . . 5.2.4 Measures of the overlap . . . . . . . . . . . . . . . 5.2.5 Computational Details . . . . . . . . . . . . . . . . 5.2.6 Errors in Excitation Energies . . . . . . . . . . . . 5.2.7 Overlap parameter . . . . . . . . . . . . . . . . . . 5.2.8 Reduced weight ratio . . . . . . . . . . . . . . . . . 5.2.9 Other approximations for the embedding potential 5.2.10 Monomer expansion . . . . . . . . . . . . . . . . . 5.2.11 Effect of the density functional approximation . . . 5.2.12 Effect of environment density . . . . . . . . . . . . Biomolecular application: Rhodopsin mimics . . . . . . . 5.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . 5.3.2 Systems . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Excitation energies . . . . . . . . . . . . . . . . . . 5.3.4 Sources of errors . . . . . . . . . . . . . . . . . . . Conclusion and outlook . . . . . . . . . . . . . . . . . . .. A Project data A.1 Inhomogeneity . . . . . . . . . . . . . A.1.1 Difference densities . . . . . . . A.1.2 Non-orthogonality . . . . . . . A.1.3 Extent of inhomogeneity . . . . A.2 Ground state interaction energies . . . A.2.1 Interaction energy components A.2.2 Energy decomposition analysis A.3 Excitation energies . . . . . . . . . . . A.3.1 Difference density plots . . . . A.3.2 Errors in excitation energies . . A.3.3 Systems . . . . . . . . . . . . . A.3.4 Monomer expansion . . . . . . A.3.5 Effect of environment density . A.3.6 Rhodopsin mimics . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. 124 128 130 131 133 137 138 140 142 145 147 154 154 155 157 161 164. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. 171 171 173 176 179 180 183 187 191 191 195 197 199 200 203. Bibliography. 205. List of Publications. 229. List of Figures. 231. List of Tables. 235. Acknowledgments. 239 VII.

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(16) Abbreviations ADC. algebraic diagrammatic construction. ALMO. absolutely localized MO. AO. atomic orbital. BLA. bond length alternation. BO. Born-Oppenheimer. BSSE. basis set superposition error. CAS. complete active space. CC. coupled cluster. CI. configuration interaction. CSF. configuration state function. CT. charge transfer. DFA. density functional approximation. DFT. density functional theory. DH. double hybrid. DZ. double zeta. FCI. full CI. FDET. frozen-density embedding theory. GF. Green’s function. GGA. general gradient approximation. GTO. Gaussian-type orbital. HF. Hartree-Fock. HK. Hohenberg-Kohn. ISR. intermediate state representation. KSCED. Kohn-Sham equations with constrained electron density IX.

(17) LCAO. linear combination of atomic orbitals. LDA. local density approximation. MAE. mean absolute error. MCSCF. multi-configurational SCF. ME. monomer expansion (FDET). MeURE. median unsigned relative error. ML. machine learning. MO. molecular orbital. MPPT. Møller-Plesset perturbation theory. MURE. mean unsigned relative error. OO. orbital-optimized. PES. potential energy surface. PSB. protonated Schiff base. PT2. second order perturbation theory. QZ. quadruple zeta. RPA. random phase approximation. RSPT. Rayleigh-Schrödinger perturbation theory. SAPT. symmetry-adapted perturbation theory. SCF. self-consistent field. SE. supermolecular expansion (FDET). TF. Thomas-Fermi (functional). TZ. triple zeta. UEG. uniform electron gas. WF. wavefunction. WFT. wavefunction theory. XC. exchange-correlation. X.

(18) Chapter 1. Models and methods in quantum chemistry. 1.1. Introduction. 1.1.1. Fundamentals. The Schrödinger equation [1–3] is one of most fundamental equations in quantum mechanics. In quantum chemistry it is applied to a system of nuclei and electrons, whereby one often deals with the non-relativistic time-independent Schrödinger equation ĤΨ = EΨ.. (1.1). The solution to this equation is the molecular wavefunction Ψ, which depends on the positions of electrons r and nuclei R. The molecular Hamiltonian Ĥ for N electrons and M nuclei in atomic units is given by Ĥ = −. N X 1 i=1. 2. ∇2i. −. M X A=1. N X M N N M X M X X 1 ZA X X 1 ZA ZB 2 ∇ − + + 2mA A riA rij RAB i=1 A=1. = T̂e + T̂n + V̂ne + V̂ee + Vnn .. i=1 j>i. (1.2). A=1 B>A. (1.3). Since the Schrödinger equation cannot be solved analytically (except for a hydrogen or hydrogen-like atom), approximations have to be introduced in order to reduce the complexity of the calculation. The most important approximation is the Born-Oppenheimer 1.

(19) 2. 1. MODELS AND METHODS IN QUANTUM CHEMISTRY. (BO) approximation, which is based on the assumption that the motion of the electrons and the nuclei can be considered separately in view of the large mass differences. For the consideration of the electronic problem all terms with electronic contributions including the nuclear Coulomb repulsion term are combined to the electronic Hamilton operator Ĥel = T̂e + V̂ne + V̂ee + Vnn .. (1.4). The solution to the electronic Schrödinger equation Ĥel Ψel (r; R) = Eel (R)Ψel (r; R). (1.5). is the electronic wavefunction Ψel , which only depends parametrically on the positions of the nuclei. The separation of nuclear and electronic variables also introduces the concept of potential energy surfaces (PES) via Eel (R). The Born-Oppenheimer approximation is a good approximation as long as the electronic states are well separated. As the energy difference of two electronic states approaches zero, the coupling term T̂n Ψel becomes non-negligible and the BO approximation breaks down. Electronic structure methods in quantum chemistry aim to find solutions to Eq. 1.5 by either approximating the wavefunction or the Hamiltonian. The vast majority of wavefunction approximation thereby relies on expressing the molecular electronic wavefunction in terms of one-electron wavefunctions {ψi (r)} (orbitals). However, also other avenues have been explored, for instance two-electron functions (geminals). [4]. 1.1.1.1. A comment on notation. Unless explicitly specified otherwise, the el subscript of the electronic Hamiltonian (Eq. 1.5) will be dropped. Moreover, since this work and electronic structure methods in general concern the electronic energy, the nuclear repulsion term Vnn will be omitted whenever unimportant. Regarding the integrals of many-electron or one-electron wavefunction the Dirac notation will be used, i.e. hΨ|Ψi =. Z. Ψ∗ (r)Ψ(r)dr. (1.6).

(20) Hartree-Fock Theory. 3. The one- and two-electron integrals∗ in the Dirac notation is given by [5] Z. ψi∗ (r1 )ĥ(r1 )ψj (r1 )dr1 Z Z −1 hij|iji = hψi ψj |ψi ψj i = ψi∗ (r1 )ψj∗ (r2 )r12 ψi (r1 )ψj (r2 )dr1 dr2 .. hi|ĥ|ji = hψi |ĥ|ψj i =. 1.2. (1.7) (1.8). Hartree-Fock Theory. 1.2.1. Hartree Method. A starting point for simplifications to Eq. 1.5 is to assume a system of non-interacting electrons, i.e. V̂ee = 0. The electronic Hamiltonian then becomes separable and can be expressed as a sum of one-electron Hamiltonians. The molecular wavefunction that solves this uncorrelated system exactly takes the form of a product of independent one-particle wavefunctions, referred to as the Hartree product: ΨHP (r1 , r2 , ..., rN ) = ψ1 (r1 ) · ψ2 (r2 ) · ... · ψN (rN ) =. N Y. ψi (ri ).. (1.9). i=1. Usually it is assumed that the spatial orbitals {ϕi } and therefore also the spin orbitals {ψi } form an orthonormal set. Z. ψi∗ (r)ψj (r)dr = δij .. (1.10). Although a system of non-interacting electrons lacks a physical foundation, the Hartree product can still be employed as a model wavefunction to solve Eq. 1.5 including V̂ee . The total electronic energy in the Hartree model thus reads:. EHP =. N Z X i. ψi∗ (ri )ĥi ψi (ri )dri. N X N Z Z X |ψi (ri )|2 · |ψj (rj )|2 dri drj , + rij i. (1.11). j>i. where ĥ denotes the core-Hamiltonian: M. X ZA 1 ĥi = − ∇2i − . 2 riA. (1.12). A=1. ∗. By convention the integrat/ion dummy variables are chosen to be space and spin coordinates of electron one and electron two, respectively..

(21) 4. 1. MODELS AND METHODS IN QUANTUM CHEMISTRY. According to the variational principle the set of solutions of an energy functional (expressed in terms of an expectation value) are bounded from below E[Ψ] =. hΨ|Ĥ|Ψi ≥ E0 . hΨ|Ψi. (1.13). One then obtains a set of N eigenvalue equations (Hartree equations): .  N Z ∗ X ψj (rj )ψj (rj ) ĥi + drj  ψi = εi ψi . rij. (1.14). j>i. The second term in Eq. 1.14 describes the Coulomb repulsion to all other electrons. However, by integration over the spatial and spin coordinates of a given ψj (r), the explicit interaction between electron i and j is replaced by an averaged electrostatic field (mean field ). Although the Hartree method is surprisingly effective in some cases, it nevertheless exhibits fundamental flaws. The form of the wavefunction implicates that the probability density of one electron is independent from another, which is clearly unphysical as they repel each other instantaneously. |ΨHP |2 = |ψ1 |2 · |ψ2 |2 · ... · |ψN |2. (1.15). Furthermore the Hartree product has (with the exception of a Helium atom in the ground state) the wrong symmetry properties. From theoretical and experimental studies in the 1920s, e.g. the Stern-Gerlach experiment [6] , it is established that electrons are fermions. According to the Pauli exclusion principle a fermionic wavefunction must be antisymmetric with respect to the interchange of two particles, i.e.: Ψ(1, 2, ..., k, l, ..., N ) = −Ψ(1, 2, ..., l, k, ..., N ). (1.16). A wavefunction obeying the Pauli exclusion principle is obtained by a linear combination of Hartree products which can conveniently be expressed in the form of a Slater determinant [7] (also realized by Heisenberg [8] and Dirac [9] ) ψ1 (r1 ). ψ2 (r1 ). .... ψN (r1 ). ψ1 (r2 ) 1 Φ(r1 , r2 , ..., rN ) = √ N! .... ψ2 (r2 ). .... ψN (r2 ). .... .... .... ψ1 (rN ) ψ2 (rN ) ... ψN (rN ). ,. (1.17).

(22) Hartree-Fock Theory. 5. where (N !)−1/2 is a normalization factor such that hΦ|Φi = 1. In comparison to Eq. 1.15 the probability density of this type of wavefunction contains cross-terms that correlate the motion of electrons with same spin as can be easily shown for a two-electron system: |Φ(r1 , r2 )|2 =. 1 |ψ1 (r1 )|2 · |ψ2 (r2 )|2 + |ψ1 (r2 )|2 · |ψ2 (r1 )|2 2 − ψ1∗ (r1 )ψ2∗ (r2 )ψ1 (r2 )ψ2 (r1 ) − ψ1∗ (r2 )ψ2∗ (r1 )ψ1 (r1 )ψ2 (r2 ). . (1.18). This extension to Hartree’s theory, i.e. the application of a single Slater determinant to solve the electronic Schrödinger equation, was reported by Fock [10] and represents the arguably most important wavefunction model in quantum chemistry, known as Hartree-Fock theory.. 1.2.2. Hartree-Fock Equations. Solving Eq. 1.5 using a Slater determinant to approximate the wavefunction leads to the following expression for the total electronic energy: EHF [Φ] = hΦ|Ĥ|Φi =. N X i. =. N X N N X N X X hi|ĥ|ii + hij|iji − hij|jii i. N X i. (1.19). j>i. i. (1.20). j>i. N. 1X hi|ĥ|ii + hij||iji, 2. (1.21). ij. where we have introduced the antisymmetrized two-electron integral hij||iji = hij|iji − hij|jii.. (1.22). In comparison to Eq. 1.11 a new two-electron integral, the exchange integral, emerges in the total energy equation due to the antisymmetry of the wavefunction. The exchange contribution to the total energy is negative therefore reducing the repulsion between electrons. Another implication of the exchange term concerns the so-called self-interaction, a spurious interaction of an electron with itself. So far the summation indices j were limited by the condition j > i in order to avoid an electron interacting with its own.

(23) 6. 1. MODELS AND METHODS IN QUANTUM CHEMISTRY. electrostatic field. For the case i = j the exchange and Coulomb integral will be identical, so that these two integrals cancel out and do not enter the total energy. The two-electron integrals can be re-expressed in terms of the following operators ψj∗ (rj )ψj (rj ) drj |ψi i rij # "Z ψj∗ (rj )P̂ij ψj (rj ) K̂j |ψi i = drj |ψi i rij Z ∗ ψj (rj )ψi (rj ) = drj |ψj i. rij Jˆj |ψi i =. Z. (1.23) (1.24) (1.25). Whereas the Coulomb operator Jˆj functions as a linear operator, the exchange operator K̂j represents a non-local operator due to the permutation operator P̂ij that interchanges spatial and spin coordinates of electron i and j. Both operators take the form of a potential and are often combined to a effective one-electron potential operator, the Hartree-Fock potential v̂HF (ri ) =. N X j. Jˆj (ri ) − K̂j (ri ).. (1.26). Just as in the Hartree method the explicit electron-electron interaction is reduced to an average interaction due to this effective potential. Due to its average treatment of the electron-electron interaction the Hartree-Fock method is classified as an uncorrelated method. Although HF theory usually recovers about 99% of the total energy, the remaining part, called the correlation energy, is vital in order to explain chemical phenomena such as dispersion interaction. The correlation energy is defined as the difference between the exact energy and the Hartree-Fock limit energy Ec = E0 − EHF. (1.27). and therefore always negative. In practical calculations where finite basis sets are employed to solve the Hartree-Fock equations the correlation energy is generally interpreted as the difference between the lowest possible energy in this basis and the HF energy. Since Hartree-Fock yields most of the total energy it is often used as a starting point for so-called post-HF methods that are designed to retrieve the correlation energy, Even though the Hartree-Fock method is generally labeled as an uncorrelated method, it does contain a certain type of electron correlation, spin correlation. In practice the missing electron correlation manifests itself in various molecular properties, for instance bond lengths in HF-optimized structures are generally too short and electron.

(24) Hartree-Fock Theory. 7. densities usually too compact, especially in centers of bonds. In some cases even the dipole moment is affected, a famous case is carbon monoxide for which HF predicts a dipole moment that points in the opposite direction as compared to a correlated method.. 1.2.3. Derivation of Hartree-Fock equations. The form of Eqs. 1.23 to 1.26 implies that given a trial wavefunction the Hartree-Fock energy can be obtained as the expectation value of true and effective one-electron operators, i.e. through a linear eigenvalue equation. However, since the operators Jˆj and K̂j themselves depend on the solution {ψi } the Hartree-Fock equations are nonlinear and require an iterative procedure for its solution. By application of the variational. principle (Eq. 1.13) to the wavefunction optimization problem, one determines the optimal wavefunction as the one that yields the lowest energy solution. Following the rules of differential calculus, the solution is given by stationary points of the energy functional with respect to infinitesimal changes in the spin orbitals ψi ⇒ ψi + δψi . Inserting the variation ψi + δψi into Eq. 1.20 yields: EHF [Φ + δΦ] =. N X i. h(ψi + δψi )|ĥ|(ψi + δψi )i N. 1X + h(ψi + δψi )(ψj + δψj )|(ψi + δψi )(ψj + δψj )i 2 ij. −. 1 2. N X ij. h(ψi + δψi )(ψj + δψj )|(ψj + δψj )(ψi + δψi )i. (1.28). For a stationary point the first order variation in the energy must satisfy δ (1) EHF = 0. One obtains it by collecting the terms linear in δψ and rearranging the two-electron integrals (cf. exercise 3.3 in Ref. 5): δ (1) EHF [Φ] =. N N X X hδψi |ĥ|ψi i + hδψi |Jˆj − K̂j |ψi i + c.c., i. (1.29). ij. where c.c. denotes the complex conjugate. For compactness Eqs. 1.12, 1.23 and 1.24 are merged into an effective one-electron operator termed Fock operator F̂ = ĥ +. X j. Jˆj − K̂j ,. (1.30).

(25) 8. 1. MODELS AND METHODS IN QUANTUM CHEMISTRY. which is also a linear and Hermitian operator [5,11] . Therefore, the first order change in energy follows as. δ. (1). N n o X EHF [Φ] = hδψi |F̂ |ψi i + hψi |F̂ |δψi i ,. (1.31). i. which implies that the Hartree-Fock energy functional can only reach a stationary point, if hδψi |F̂ |ψi i = 0.. (1.32). This is only an intermediate result as one needs to consider that the variations δψ are not independent but coupled through the condition of orthonormality of the spin-orbitals Z. ψp∗ (r)ψq (r)dr = hψp |ψq i = hp|qi = δpq .. (1.33). In order to introduce this constraint we need to apply the Lagrange method of multipliers and inspect the first order variation in the Lagrangian δ (1) L = δ (1) EHF −. N X ij. {λij hδψi |ψj i + λij hψi |δψj i} = 0.. (1.34). From Eqs. 1.34 and 1.31 one obtains two equivalent sets of equations F̂ |ψi i = F̂ |ψi i =. N X j N X j. λij |ψj i. (1.35). λ∗ji |ψj i,. (1.36). from which it is evident that λij must be elements of a Hermitian matrix. The sets of equations obtained so far can be conveniently expressed in matrix notation: Fψ = ψλ.. (1.37). In principle, Eq. 1.35 represents already the final result. It also shows that the orbitals for which the Hartree-Fock energy is stationary are not unique. However, since λ is a Hermitian matrix one can always find a basis {ψ 0 } in which it is diagonal. This change of basis can be expressed in terms of a unitary transformation, i.e. |ψi0 i =. X k. Uki |ψk i. (1.38).

(26) Hartree-Fock Theory. 9. If U is a real matrix, the original and transformed wavefunction will only differ in a phase factor ±1. The Hartree-Fock wavefunction is therefore invariant with respect to. unitary transformation (”orbital rotations”) within the occupied space. Since the Fock operator depends on the spin orbitals it is important to assess the impact of the unitary transformation on the Fock operator. It is easy to show [5] that the Fock operator is also invariant to any arbitrary unitary transformation of spin orbitals, i.e. F̂0 = F̂. Inserting ψ = ψ 0 U† into Eq. 1.37 and multiplying with U from the right results in: F̂ψ 0 = ψ 0 U† λU. (1.39). F̂ψ 0 = ψ 0 ε,. (1.40). where ε denotes the matrix of Lagrange multipliers in the diagonal representation. Consequently, Eq. 1.40 represents a set of pseudo-eigenvalue equations, termed canonical Hartree-Fock equations: F̂ |ψi0 i = εi |ψi0 i.. (1.41). The eigenvalues {εi } are referred to as orbital energies and the basis {ψi0 } forms the set of so-called canonical spin orbitals. In the further course only the canonical form of the. Hartree-Fock equations will be considered and hence the prime notation will be dropped.. εi = hi|ĥ|ii +. N X k. hik||iki. (1.42). Once the final set of spin orbitals is found, one can express the total energy in terms of orbital energies. However, a compensation for the double counting of the electron-electron interaction needs to be applied:. EHF =. N X i. εi −. 1X hij||iji. 2. (1.43). ij. Let us review the derivation of the Hartree-Fock equations from a different perspective. Assuming a complete basis of occupied and unoccupied, so-called virtual MOs, the infinitesimal change δψi can be expressed in said basis: δψi =. occ X j. κij ψj +. virt X a. κia ψa. (1.44).

(27) 10. 1. MODELS AND METHODS IN QUANTUM CHEMISTRY. The first term can be neglected since it corresponds to a transformation within the occupied space, which changes neither the wavefunction nor the total energy. From the remaining second term and Eq. 1.32 we obtain, hψa |F̂ |ψi i = 0,. (1.45). which is termed the general Hartree-Fock equation. From Eq. 1.45 we can conclude that the elements of the occupied-virtual block of the Fock matrix correspond to the gradients of the energy with respect to the molecular orbitals. In order to determine a stationary point of the energy functional these elements need to vanish.. 1.2.4. Roothaan-Hall Equations. Obtaining numerically exact solutions to the HF equations, i.e. molecular orbitals, is a mathematically difficult task. Generally there are two strategies to construct the molecular orbitals. The numerical HF methods map the MOs onto a set of real-space grid points and usually exploit symmetry to simplify the problem. [12–14] These methods are very flexible, but intractable for most molecular systems. The second and most widely used approach formulates MOs algebraically as an expansion in simple one-electron basis functions, sometimes also referred to as atomic orbitals. Many different forms of atomic basis functions are conceivable, the most common choice for nonperiodic molecular systems is a Gaussian-type orbital (GTO) due to its computationally efficient evaluation of 4-center-2-electron integrals. [15] Other choices for basis functions are extensively covered in the literature [15–17] and will not be discussed at this point. Each molecular orbital ψi is then expressed as a linear combination of atomic orbitals χµ (LCAO): [11] ψi (r) =. K X. Cµi χµ (r).. (1.46). µ. If the basis set was complete, the LCAO expansion would be an exact expansion. In practical computational calculations, this step constitutes an approximation as one is limited to a finite basis. The great advantage of Roothaan’s approach is that the non-linear integro-differential equations are cast into an algebraic form that allows the application of efficient linear algebra operations..

(28) Hartree-Fock Theory. 11. Using Eq. 1.46 the Hartree-Fock equations (Eq. 1.41) may be written as:. F̂i. K X µ. Cµi |µi = εi. K X µ. Cµi |µi.. (1.47). By multiplying from the left with basis functions χν and integration over the electronic coordinates one obtains a set of K Hartree-Fock equations in the atomic orbital basis, which in matrix notation are concisely summarized to FC = SCε.. (1.48). where F is the Fock matrix with consisting of elements Fµν = hµ|F̂ |νi. (1.49). and S the overlap matrix due to non-vanishing overlap integrals of the generally nonorthogonal atomic basis functions Sµν = hµ|νi.. (1.50). Analogous to the derivation above, the Roothaan-Hall equations also need to fulfill the orthonormality constraint (Eq. 1.33), which in AO basis is expressed as C† SC = 1.. (1.51). The orthogonalization can be achieved by a non-unitary transformation basis. For any non-orthogonal basis it is always possible to determine a non-unitary transformation matrix X χ̃µ (r) =. X. Xνµ χν (r),. (1.52). ν. such that hχ̃µ |χ̃ν i = δµν . The non-orthogonal C and orthogonal basis C̃ are then related via C = XC̃.. (1.53).

(29) 12. 1. MODELS AND METHODS IN QUANTUM CHEMISTRY. Inserting Eq. 1.53 into Eq. 1.48 and multiplying from the left with X† we bring the Roothaan-Hall equations to the conventional matrix eigenvalue form: (X† FX)C̃ = (X† SX)C̃ε F̃C̃ = C̃ε. (1.54) (1.55). The eigenvalue problem can then be solved for C̃ by diagonalization of F̃. The form of Eq. 1.55 suggests that the Roothaan-Hall equations are solved by simply diagonalizing the Fock matrix. The non-linear character of the Hartree-Fock equations, however, did not change and the Fock matrix remains a function of the MO coefficients, i.e. F = F(C).. 1.2.4.1. Fock Matrix. In the following we shall examine the Fock matrix more closely where we assume a closed α = C β = C . Using the definition of the Fock operator (Eq. 1.30) shell system, i.e. Cµi µi µi. the expression for the Fock matrix is easily obtained as: Fµν = hµν +. X λσ. 1 Pλσ hµλ|νσi − hµλ|σνi . 2. (1.56). where hµν is the core-Hamiltonian (Eq. 1.12) in the atomic orbital basis and Pλσ represents the 1-particle density matrix, which for a closed-shell system is defined as:. Pµν = 2. N/2 X. ∗ Cµi Cνi .. (1.57). i=1. The 1-particle density matrix is the representation of the electronic charge density ρ(r) in the AO basis: ρ(r) = hΦ|ρ̂|Φi = where ρ̂(r) =. PN i. K X. Pµν χ∗µ (r)χν (r),. (1.58). µν. δ(ri − r) is the density operator. The dependence of the Fock matrix on. the density matrix and in turn on the MO coefficients showcases the non-linear character of the Hartree-Fock equations and is the reason why these equations need to be solved iteratively..

(30) Density Functional Theory. 1.3 1.3.1. 13. Density Functional Theory The Electron Density. The probability of finding electron 1 in a volume element dr1 while the other electrons are anywhere is obtained by integrating over all electronic (spin and spatial) coordinates x of electrons 2, 3, .., N and the spin coordinate s of electron 1: Z. Z .... |Ψ(x1 , x2 , ..., xN )|2 ds1 dx2 ...dxN. dr1. (1.59). Due to the indistinguishability of electrons, the probability ρ(r) of finding any electron in dr1 is given by N times the probability for one electron: Z ρ(r) = N. Z .... |Ψ(x1 , x2 , ..., xN )|2 ds1 dx2 ...dxN .. (1.60). Thereby, the electron density is normalized to the number of electrons Z. ρ(r)dr = N hΨ|Ψi = N. (1.61). The complexity of an electronic wavefunction rapidly increases with the number of electrons (4N variables) whereas the electron density depends on only three variables regardless of the system size. This in comparison reduced complexity offers many practical advantages which is why models based on the electron density attracted great interest and remain to be an important area of research in quantum chemistry.. 1.3.2. The Thomas-Fermi Model. The key concept in density functional theory (DFT) is to express the total electronic energy using the electron density as the principal variable. A first attempt to this concept was realized in the late 1920s by Thomas and Fermi (TF). Using a statistical approach to describe the distributions of electrons they devised a simple model for atoms [18,19] : ETF [ρ] = TTF [ρ] + J[ρ] − Z. Z. ρ(r) dr |r − R|. (1.62). where the first term represents the Thomas-Fermi kinetic energy functional [18,19] Z TTF [ρ] = CF. ρ5/3 (r)dr,. CF =. 3 (3π 2 )2/3 10. (1.63).

(31) 14. 1. MODELS AND METHODS IN QUANTUM CHEMISTRY. and the second term J[ρ] denotes the classical Coulomb repulsion of the density with itself 1 J[ρ] = 2. Z Z. ρ(r)ρ(r0 ) drdr0 . |r − r0 |. (1.64). Whereas TTF [ρ] is an approximate functional, the remaining terms in Eq. 1.62 originate from classical electrostatics and do not constitute functional approximations. The TF model was later extended to include an approximate exchange contribution to the total energy using the Dirac exchange functional [20] (TFD model): Ex = −Cx. Z ρ. 4/3. (r)dr,. 9α Cx = 8. 1/3 3 π. (1.65). where α = 2/3. A closely related approximate exchange functional is obtained for α = 1, the Slater exchange functional. [21] Although pioneering for its time, the TF and TFD model suffer from severe shortcomings. Most notably, the both models fail to describe interatomic bonds and thus molecules. [22]. 1.3.3 1.3.3.1. Mathematical Properties of Functionals Functional Derivative. For problems involving functionals one frequently needs to determine a function f ∗ for which the functional F [f ] is stationary. The space of functions f is explored by infinitesimal variations δf which can be expressed in terms of δf (x) = φ(x),. (1.66). where is an infinitesimal factor and φ(x) an arbitrary test function. The variation δF of a functional F [f ] due to a variation of the function δf is defined by: δF := F [f + δf ] − F [f ].. (1.67).

(32) Density Functional Theory. 15. In order to evaluate δF one may develop a Taylor series of the functional F [f + δf ] = F [f + φ] in powers of δf and , respectively. [23] dF [f + φ] 1 d2 F [f + φ] F [f + φ] = F [f ] + ·+ · 2 + · · · 2 d 2 d =0 =0 n K X 1 d F [f + φ] = · n + O(K+1 ) n! dn =0 . (1.68) (1.69). n=0. From the first order term, the (first order) functional derivative δF [f ]/δf (x) is given by Z dF [f + φ] δF [f ] F [f + φ] − F [f ] = φ(x) dx = lim Ñ0 d δf (x) =0 . (1.70). Note that in order for the second equality in Eq. 1.70 to hold, the functional derivative δF [f ]/δf (x) must exist. In other words, a functional is said to be differentiable if Eq. 1.70 is satisfied. Assuming differentiability up to Kth order, the Taylor expansion of F [f + δf ] reads: Z K X 1 δ n F [f ] F [f + φ] = δf (x1 )...δf (xn ) dx1 ...dxn + O(K+1 ) n! δf (x1 )...δf (xn ). (1.71). n=0. 1.3.3.2. Homogeneity. Scaling relations are a useful tool in DFT to extract properties from functionals and establish exact conditions that guide the design of new functionals. There exist several scaling relations, two of which shall be presented here that involve the electronic coordinate and the density. Other scaling relations rely on KS orbitals [24] , a grand canonical ensemble picture [25] or combined scaling of the electronic coordinate and the density [26] . The most commonly known scaling technique is uniform coordinate scaling: ρλ (r) = λ3 ρ(λr),. (1.72). where λ3 maintains the normalization of the density to N electrons. A functional is then said to be homogeneous of degree k under coordinate scaling if X[ρλ ] = λk X[ρ].. (1.73).

(33) 16. 1. MODELS AND METHODS IN QUANTUM CHEMISTRY. To give an example, the exact exchange energy functional (from HF) is homogeneous of degree one ExHF [ρλ ] = λExHF [ρ]. (1.74). and the non-interacting kinetic energy functional (see Eq. 1.101) is homogeneous of degree two under coordinate scaling Ts [ρλ ] = λ2 Ts [ρ].. (1.75). Ensuring normalization is not a requirement for a scaling relation, for instance with density scaling the normalization varies linearly with the scaling parameter. ργ (r) = γρ(r). (1.76). Analogous to the scaling relation above, a functional is called homogeneous of degree k under density scaling if it satisfies [27–29] X[ργ ] = γ k X[ρ]. (1.77). or Z. δX[ρ] ρ(r)dr = kX[ρ]. δρ(r). (1.78). In the context of the integral expression (Eq. 1.78), which can be obtained by differentiation of Eq. 1.77 with respect to γ and evaluation at γ = 1, it is considered homogeneous of degree k in ρ. [27] Examples of homogeneous functionals under density scaling are the Thomas-Fermi kinetic energy functional TT F [ρ] (k = 5/3) or the classical functionals describing Coulomb repulsion J[ρ] (k = 2) and attraction Vne [ρ] (k = 1). Inhomogeneity occurs, if for a functional X[ρ] there does not exist a unique k that satisfies Eqs. 1.73, 1.77 or other devised scaling relationships.. 1.3.4 1.3.4.1. Hohenberg-Kohn Theorems Hohenberg-Kohn Theorem I. The theorems presented by Hohenberg and Kohn in 1964 laid the foundations of modern density functional theory. Up until this landmark publication no formal proof was presented that the total energy could be expressed in terms of density functionals..

(34) Density Functional Theory. 17. Let us first revisit the operator describing the interaction of electrons and nuclei (Eq. 1.2). Without loss of generality we can express the operator in terms of the external potential vext (r).. V̂ne =. N M X X i. =. A. N Z X i. ZA − |RA − ri |. !. vext (r)δ(ri − r)dr. (1.79). (1.80). Z =. vext (r)ρ̂(r)dr,. (1.81). where in the last line the definition of the density operator was used (see Sec. 1.2.4.1). The expectation value of this operator can then be expressed as a density functional: hΨ|V̂ne |Ψi =. Z vext (r)ρ(r)dr = Vne [ρ]. (1.82). Considering Eqs. 1.5 and 1.81, the electronic Hamiltonian is then completely determined by the number of electrons N and the external potential vext (r). Since Ĥ determines the wavefunction through the variational principle also all ground state properties are defined by N and vext (r), which is schematically presented as {vext , N } ⇒ Ĥ ⇒ Ψ ⇒ properties.. (1.83). The first Hohenberg-Kohn (HK) theorem [30] establishes the electron density as a fundamental variable by showing that this cascade of determination can be reversed, in particular, that the ground state density uniquely determines the ground state wavefunction and all other electronic properties. The theorem states that “vext (r) is (to within a constant) a unique functional of ρ(r); since, in turn, vext (r) fixes Ĥ we see that the full many-particle ground state is a unique functional of ρ(r)”. [30] This statement is based on the following mappings [27,31] 1. 2. ρ(r) ⇐⇒ Ψ(x1 , ..., xN ) ⇐⇒ vext (r).. (1.84). The original proof of the first HK theorem is fairly simple and proceeds by reductio ad absurdum. In short, the proof shows that there cannot be two external potentials (differing by more than a constant) associated with the same ground state density. [30] This uniqueness of vext (r) for a given ρ(r), however, does not pertain to any density. An electron density, for which the mapping between vext (r) and ρ(r) exists, is called v-representable..

(35) 18. 1. MODELS AND METHODS IN QUANTUM CHEMISTRY. Since the original proof involves the variational principle the theorem is restricted to ground states. A further limitation to non-degenerate states occurs as the wavefunction is assumed to be a unique functional of the density (mapping 1 in Eq. 1.84). Later approaches in DFT partially lifted these restrictions as will be discussed below (Sec. 1.3.5). The relations established by the first HK theorem allow us to write the ground state energy purely in terms of density functionals Ev [ρ] = T [ρ] + Vee [ρ] + Vne [ρ].. (1.85). The subscript v here was added to emphasize the restriction to v-representable densities. At this point it is convenient to group together the kinetic energy T [ρ] and the electronelectron interaction energy Vee [ρ] introducing a new functional, the Hohenberg-Kohn functional [30] FHK [ρ] = T [ρ] + Vee [ρ].. (1.86). This universal functional of ρ(r) is system-independent and exists for any external potential and any number of electrons (as N is already defined by ρ(r)), but only for v-representable densities. The HK functional lies at the center of density functional theory as it would enable us to solve the Schrödinger equation exactly. Unfortunately, the explicit form of its components T [ρ] and Vee [ρ] is unknown.. Z Ev [ρ] = FHK [ρ] + vext (r)ρ(r)dr | {z } | {z } universal. 1.3.4.2. (1.87). system-dependent. Hohenberg-Kohn Theorem II. The second Hohenberg-Kohn theorem [30] provides an analogue of the variational principle based on the density and thus a method for the optimization of ρ(r). Following the first HK theorem, a trial density ρ̃(r) will determine its own external potential ṽext (r) and wavefunction Ψ̃(x1 , ..., xN ). Invoking the variational principle for this wavefunction one obtains hΨ̃|Ĥ|Ψ̃i = FHK [ρ̃] +. Z. vext (r)ρ̃(r)dr = Ev [ρ̃] ≥ Ev [ρ].. (1.88).

(36) Density Functional Theory. 19. It should be noted that the expression above involves the exact Hamiltonian and thus vext (r) and not ṽext (r). Equality of Eq. 1.88 is reached if the exact density ρ(r) ⇒ Ψ[ρ] is used.. We have now attained a tool to optimize a trial density which proceeds by minimization of the energy with respect to infinitesimal variations in the density subject to the constraint that the number of electrons remains constant. Using Lagrange’s method of undetermined multipliers a new functional L[ρ, µ] is constructed that incorporates the R condition ρ(r)dr = N . The stationary point must therefore fulfill the following relation δL[ρ, µ] δ = δρ(r) δρ(r). Z Ev [ρ] − µ ρ(r)dr − N = 0,. (1.89). which leads to the following equation known as the Euler-Lagrange equation δEv [ρ] − µ = 0. δρ(r). (1.90). The Lagrange multiplier for the conservation of electrons µ = vext (r) + represents the chemical potential µ =. δFHK [ρ] δρ(r). (1.91). δE [27] δN .. Shortly after the HK theorems had been introduced the spectroscopist Bright Wilson made remarks on the validity of the density as a fundamental variable. [32] He noted that the number of electrons N could be trivially obtained by integration (Eq. 1.61). Additionally, the cusps of ρ(r) determine the positions and charge [33] of the nuclei. Therefore the density provides all necessary quantities that define the system.. 1.3.5. Constrained Search. The HK theorems are valid only for v-representable densities, which in some cases excludes physically reasonable densities and densities from a degenerate ground state. [34–36] Another inconvenience is that the conditions for v-representability are not known. [37] The constrained search formalism introduced by Levy [34] provides a generalization to the Hohenberg-Kohn theorems..

(37) 20. 1. MODELS AND METHODS IN QUANTUM CHEMISTRY. The starting point for this formalism is the variational principle E = minhΨ|Ĥ|Ψi.. (1.92). Ψ. The minimization procedure can be divided into two steps, an outer minimization with respect to a density that integrates to N electrons and an inner minimization† over all antisymmetrized wavefunctions that yield ρ(r) E = min. ρÑN. min hΨ|T̂ + V̂ee + V̂ne |Ψi .. ΨÑρ. (1.93). Recalling Eq. 1.82, the expectation value of V̂ne may be excluded from the wavefunction search, giving . E = min min hΨ|T̂ + V̂ee |Ψi + vext (r)ρ(r)dr ρÑN ΨÑρ Z = min F [ρ] + vext (r)ρ(r)dr . Z. ρÑN. (1.94) (1.95). One obtains a minimization over density only (Eq. 1.95) which is consistent with Eq. 1.88 by introducing an alternative definition of the universal functional: F [ρ] = min hΨ|T̂ + V̂ee |Ψi. ΨÑρ. (1.96). This definition partially loosens the restrictions implied in Eq. 1.86 as it extends the domain of valid densities to any density that is obtained from some antisymmetric wavefunction. Such densities are called N-representable. The N-representability condition encompasses v-representability, i.e. for any density ρv (r) that is v-representable, the functionals become identical F [ρv ] = FHK [ρv ] providing further proof of the first HK theorem. In more mathematical terms, the conditions [35,38] for N -representability apart from Eq. 1.61 are. Z. ρ(r) ≥ 0 |∇ρ(r)1/2 |2 dr < ∞. (non-negativity),. (1.97). (finite kinetic energy).. (1.98). Due to the softer conditions the restriction to non-degenerate ground states is removed. In summary, the constrained search formalism circumvents the v-representability problem †. The minimization of Ψ is performed on the set of functions S := {Ψ ∈ L2 (R3 ) | hΨ|ρ̂|Ψi = ρ(r)}. The commonly used notation Ψ Ñ ρ is therefore an abbreviation for doing the constrained minimization on S..

(38) Density Functional Theory. 21. elegantly and provides explicit conditions for densities for which the universal functional F [ρ] exists. The theorems presented in Secs. 1.3.4 and 1.3.5 provide the theoretical foundation of density functional theory. Complications arise, however, as one puts that theory into practice. The existence of the universal functional is proven by the first HK theorem, but its explicit form or a recipe how to construct it is unknown. Also the constrained search formulation (Eq. 1.96) has only theoretical value as a search among all wavefunctions is unfeasible in practice.. 1.3.6. Kohn-Sham Approach. Although the HK theorems set the framework for modern DFT, its practical realization stands and falls with the universal functional. Especially for the kinetic component T [ρ], a high accuracy is paramount since its value is usually in the same order of magnitude as the total energy itself (cf. virial theorem). Going beyond simple kinetic energy functionals relying on pure functionals, such as Eq. 1.63, poses an almost insurmountable problem and remains to be a desired target of DFT. In 1965, Kohn and Sham (KS) proposed a formalism [39] remedying this dilemma by restating the problem in such a way that large parts of T [ρ] and Vee [ρ] are evaluated exactly, therefore leaving only a small part of the total energy which has to be approximated. Centerpiece of the KS formalism is the introduction of an artificial reference system of non-interacting electrons moving in an effective potential. [39] The Hamiltonian for such a system does not contain electron-electron repulsion terms and reads Ĥs = −. N X 1 i. 2. ∇2i. +. N X. veff (ri ).. (1.99). i. The exact wavefunction in such a system takes the form of a single Slater determinant |Φi = |ψ1 , ..., ψN i (see Eq. 1.17). It is further assumed that the density associated with the non-interacting reference wavefunction is identical to the density of the interacting system: ρ(r) =. N X i. |ψi (r)|2 .. (1.100).

(39) 22. 1. MODELS AND METHODS IN QUANTUM CHEMISTRY. The kinetic energy of the non-interacting reference system is then exactly given by Ts [ρ] = hΦ|T̂ |Φi =. N X 1 hψi | − ∇2i |ψi i. 2. (1.101). i. In analogy to the first HK theorem (Sec. 1.3.4.1), this definition only holds if there exists a density ρ(r) associated with a non-interacting ground state. Such densities are called non-interacting v-representable. This constraint, however, is weakened by defining Ts [ρ] via the constrained search formalism (Sec. 1.3.5). Ts [ρ] = min hΦ|T̂ |Φi = hΦ[ρ]|T̂ |Φ[ρ]i ΦÑρ. (1.102). Then, for Ts [ρ] to exist, the density must derive from an antisymmetric wavefunction. Note that Eq. 1.102 holds for all N -representable densities since such densities can always be decomposed into set of N orthonormal orbitals. [27,35,38] With Ts [ρ] properly defined we are now able to reformulate the total energy (Eq. 1.85) for the Kohn-Sham case: E[ρ] = Ts [ρ] + J[ρ] + Vne [ρ] + Exc [ρ].. (1.103). With the formulation of Eq. 1.103, a new functional, the exchange-correlation (XC) functional, is introduced, which collects the difference to the exact kinetic energy and the non-classical part of Vee [ρ]: Exc [ρ] ≡ (T [ρ] − Ts [ρ]) + (Vee [ρ] − J[ρ]).. (1.104). The first parentheses in Eq. 1.104 adds the remaining contribution to the kinetic energy not recovered by Eq. 1.101. Its magnitude is small since Ts [ρ] typically makes up ca. 99% of the total kinetic energy. [40] We can also deduce [27] that Ts [ρ] ≤ T [ρ], which renders the difference T [ρ] − Ts [ρ] a positive contribution to the total energy.. The Euler-Lagrange equation (Eq. 1.90) for the Kohn-Sham energy expression in Eq. 1.103 reads: µ = veff (r) +. δTs [ρ] δρ(r). (1.105). with the effective potential as veff (r) = vext (r) +. δJ[ρ] δExc [ρ] + δρ(r) δρ(r). (1.106).

(40) Density Functional Theory. 23. The variational problem can also be cast into an orbital-dependent form by rewriting the total energy as a functional of N spin-orbitals subject to the constraint of orthonormal orbitals: Ω[{ψi }] = E[{ψi }] −. N X ij. λij hψi |ψj i. (1.107). As stationary points must satisfy δΩ[{ψi }] = 0 with respect to infinitesimal variations in the orbitals, one obtains a set of N Euler-Lagrange equations of the form [27] 1 2 δJ[ρ] δExc [ρ] ψi (r) = εi ψi (r). + vext (r) + − ∇i + 2 δρ(r) δρ(r). (1.108). Parts of the effective potential, in particular the Coulomb potential and the exchangecorrelation potential, depend through Eq. 1.100 on the solution {ψi } of the KS equations.. These equations are therefore nonlinear and need to be solved iteratively (self-consistent field), just as in the case of Hartree-Fock theory.. Equally, after determining the optimal orbitals the total KS energy does not simply emerge as the sum of eigenvalues {εi }, but from the following equation EKS [ρ] =. N X i. 1 hψi | − ∇2i |ψi i + J[ρ] + Vne [ρ] + Exc [ρ]. 2. (1.109). The similarities between KS-DFT and HF theory derive from the use of a single determinant, although it fulfills a different role in each case. Whereas HF theory approximates the wavefunction of the interacting system by a single Slater determinant, in KS-DFT it represents the exact wavefunction for the auxiliary reference system of non-interacting electrons. The success of KS-DFT is owed to the smallness of the difference T [ρ] − Ts [ρ].. 1.3.6.1. Practical Considerations. In the previous section we have derived the KS equations (Eq. 1.108) and pointed out the common elements compared to HF theory (Eq. 1.41). It is therefore not surprising that the practical solution of the KS equations proceeds by LCAO expansion of the molecular orbitals (Eq. 1.46). Following the Roothaan-Hall formalism, the nonlinear KS equations are effectively cast into the language of linear algebra. The matrix equation analogous to (Eq. 1.48) thus reads FKS C = SCε,. (1.110).

(41) 24. 1. MODELS AND METHODS IN QUANTUM CHEMISTRY. where FKS represents the Kohn-Sham matrix. In comparison to the Fock matrix, the exact exchange contribution is replaced here by corresponding matrix elements of the exchange-correlation potential: Z (vxc )µν =. χµ (r). δExc [ρ] χν (r)dr. δρ(r). (1.111). As the exact exchange-correlation functional Exc [ρ] has to be approximated in practice, the corresponding exchange contribution is also approximated. Consequently the Coulomb interaction of an electron with itself is not completely cancelled, which leads to the so-called self-interaction error. Another issue relevant in practical calculations are three-dimensional integrals in real space Z I=. F (r)dr,. (1.112). which are ubiquitous in (KS-)DFT and in most cases cannot be evaluated analytically. In practical calculations one must resort to approximating the integral I numerically by a weighted sum of integrand values F (ri ), referred to as quadrature: Ngrid. X. I≈. wi F (ri ).. (1.113). i. Due to the behavior of the electron density special care must be taken in the distribution of grid points. As the electron density is quickly varying in vicinity of the nucleus and is slowly varying at longer distances, integration by quadrature in Cartesian coordinates is less suitable. The most widely adopted integration scheme is instead the Becke partitioning [41] , wherein the molecular integral is subdivided into atomic contributions Z. M X. Z F (r)dr =. ! wA (r) F (r)dr =. A. M Z X. FA (r)dr. (1.114). A. using nuclear weight functions that are normalized to. PM A. wA (r) = 1. Each atomic. integral (in spherical polar coordinates) Z. ∞Z π. Z. 2π. IA = 0. 0. FA (r, θ, φ)r2 dr sin θ dθ dφ. (1.115). 0. is then approximated by quadrature according to IA ≈. Nr X i. wir. Nω X j. wjω FA (ri , θj , φj ),. (1.116).

(42) Density Functional Theory. 25. where wir and wjω are weights for the Nr radial and Nω angular grid points, respectively. The angular integration is usually performed over θ and φ simultaneously as highly efficient integration algorithms render a separate integration over the two angular coordinates unnecessary. For the radial integration several schemes are employed, for instance Euler-Maclaurin [42] or MultiExp [43] , whereas the two-dimensional angular integration relies for the vast majority of KS-DFT implementations on angular grids developed by Lebedev [44,45] . These so-called Lebedev grids are designed in such a way that they exactly integrate all spherical harmonics up to a certain order.. 1.3.6.2. Density Functional Approximations. With the introduction of the non-interacting wavefunction, KS-DFT only leaves a small but important part of the total energy, the exchange-correlation (XC) energy (Eq. 1.104), to be approximated. Whereas the accuracy of correlated wavefunction methods, such as configuration interaction, is systematically improvable (see sections below), there is no comparable rigorous procedure to improve density functional approximations. The development of new approximations is rather guided by the intention to ameliorate known deficiencies of existing XC functionals, e.g. inability to predict dispersion bound complexes. Nevertheless, the different types of density functional approximations form a hierarchy, for which the ”Jacob’s ladder” termed by John Perdew [46] is a well-fitting metaphor (see Fig. 1.1). Each rung of this ladder represents a different level of approximation that build upon each other. As one climbs the ladder the level of accuracy is expected to increase although there is no guarantee and an increased accuracy may only be valid for some properties. The following listing outlines the key properties for each rung. A detailed review of functionals is instead provided by Refs. 37, 15 and 47.. (1) The local density approximation (LDA) constitutes the lowest rung and contains functionals that only depend - as the name implies - on the local values of ρ(r). Z Ẽxc [ρ] =. f (ρ)dr. (1.117). These XC functionals are exact for the infinite uniform electron gas (UEG), which can be physically interpreted as a model for an idealized metal. It is for that reason why it has frequently been employed in solid-state physics. As molecular systems on the other hand are finite and exhibit rapidly varying charge densities, LDA yields insufficient accuracy for chemical applications. To give an example, SVWN is an well-known LDA functional, wherein the exchange contribution is given by the.

(43) 26. 1. MODELS AND METHODS IN QUANTUM CHEMISTRY. Chemical Accuracy. (5). Double Hybrid. (4). Hybrid GGA/meta-GGA. (3). meta-GGA. (2). GGA. (1). LDA. Hartree World - no correlation, no exchange. Figure 1.1: The Jacob’s ladder of density functional approximations. The quantities on the left represent components used for the construction of an approximation (explained in the text). The acronyms LDA and GGA stand for Local Density Approximation and Generalized Gradient Approximation, respectively.. Slater (S) exchange functional (Eq. 1.65) and the correlation functional developed Vosko, Wilk and Nusair (VWN) [48] . (2) In order to account for the non-uniformity of the electron density, the generalized gradient approximation (GGA) extends the information in a given point r by the gradient of the density. Z Ẽxc [ρ] =. f (ρ, ∇ρ)dr. (1.118). The functionals are usually not formulated explicitly in terms of ∇ρ but rather in terms of one of the following dimensionless quantities x(r) =. |∇ρ(r)| ρ4/3 (r). (1.119). s(r) =. x(r) . 2(3π 2 )1/3. (1.120). The latter is called reduced density gradient and can be interpreted as a local inhomogeneity parameter for the density, for instance s(r) adopts large values.

(44) Density Functional Theory. 27. near the nucleus (large gradient) and smaller values in a bonding region while the uniform electron gas would yield s(r) = 0. In general, the accuracy of GGA functionals is significantly improved as compared to LDA and due to their computational efficiency GGA functionals are still widely used. Popular examples of GGA functionals encompass PBE [49] , BLYP [50,51] , BP86 [50,52] or PW91 [53,54] . (3) Following the strategy of GGA the next natural step is to include higher derivatives of the density such as the Laplacian ∇2 ρ(r) and/or the Kohn-Sham kinetic energy density. τ (r) =. N X i. |∇ψi (r)|2. (1.121). A density functional approximations of this type is referred to as meta-GGA Z Ẽxc [ρ] =. f (ρ, ∇ρ, ∇2 ρ, τ )dr. (1.122). Examples for meta-GGAs include TPSS [55] , SCAN [56] or B97M-rV [57] . (4) Typically the exchange contributions comprise most of the exchange-correlation energy. Therefore, the exchange contribution constitutes a suitable starting point for further improvement of the accuracy. So-called hybrid functionals [58] address this issue by adding a fraction of exact exchange Ex [ψi ] from HF theory evaluated with KS orbitals: Ẽxc [ρ] = cx Ex [ψi ] + (1 − cx )ExDFT [ρ] + EcDFT [ρ]. (1.123). The theoretical justification for hybrid functionals is provided by the adiabatic connection formula [59–61] Z Exc [ρ] = 0. 1. hΨλ |V̂ee |Ψλ i − J[ρ] dλ =. Z. 1. Wλ dλ,. (1.124). 0. where Ψλ is the wavefunction that minimizes a Hamiltonian with scaled electronelectron repulsion hΨ|T̂ + λV̂ee |Ψi. The inclusion of exact exchange can then be. inferred from the non-interacting limit (λ = 0) for which the integrand is simply equal to the exact exchange energy W0 = Ex . Eq. 1.124 is in principle an exact relation, although W1 and the explicit dependence on λ are unknown in practice. Although hybrid functionals typically exhibit a great performance for properties that can be attributed to short-range electron-electron interactions, they are less successful at describing long-range interactions. To give an example, charge-transfer.