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  • Technische Universitt Mnchen

    Department Chemie

    Fachgebiet Theoretische Chemie

    Self-Interaction, Delocalization, and Static Correlation Artifacts

    in Density Functional Theory: Studies with the Program ParaGauss

    Thomas Martin Soini

    Vollstndiger Abdruck der von der Fakultt fr Chemie der Technischen Universitt

    Mnchen zur Erlangung des akademischen Grades eines

    Doktors der Naturwissenschaften (Dr. rer. nat.)

    genehmigten Dissertation.

    Vorsitzender: Univ.-Prof. Dr. Ville R. I. Kaila

    Prfer der Dissertation: 1. Univ.-Prof. Dr. Dr. h.c. Notker Rsch (i.R.)

    2. Univ.-Prof. Dr. Andreas Grling

    (Friedrich-Alexander Universitt Erlangen-Nrnberg)

    Die Dissertation wurde am 08.01.2015 bei der Technischen Universitt Mnchen eingereicht

    und durch die Fakultt fr Chemie am 18.02.2015 angenommen.

  • i

    Acknowledgements

    The scientific work of this thesis was carried out at the Fachgebiet fr Theoretische Chemie

    of the Technische Universitt Mnchen under the guidance of Prof. Dr. Dr. h.c. Notker

    Rsch. To him I want to express my gratitude for providing me with the opportunity to study

    this interesting topic in his group as well as for his supervision and his interest in my projects.

    I am also very indebted to Dr. Sven Krger for numerous scientific discussions as well as

    for his continuous support over the last years, especially in the last phase of this thesis. My

    special thanks also go to Dr. Alexei Matveev for his help in improving my programming

    skills as well as to Dr. Alexander Genest for many valuable suggestions and discussions.

    I especially want to thank my colleague and friend Cheng-chau Chiu for his help in

    various aspects of my live. I also thank Dr. Astrid Nikodem for the good collaboration during

    the completion of the parallelized exact-exchange implementation.

    I further want to thank all my past and present colleagues Dr. Duygu Baaran, Dr. Ion

    Chiorescu, Dr. Konstantina Damianos, Dr. Wilhelm Eger, Ralph Koitz, Dr. Alena Kremleva,

    Bo Li, Dr. Remi Marchal, Dr. Raghunathan Ramakrishnan, Dr. Yin Wu and Dr. Zhijian Zhao

    for providing a friendly working atmosphere.

    I thank the International Graduate School of Science and Engineering at the Technische

    Universitt Mnchen for the generous scholarship and the Leibniz-Rechenzentrum of the

    Bayerische Akademie der Wissenschaften for providing the computing resources used to

    complete my scientific work.

    Last but not least I thank my family for their love, support, and encouragement, which

    enabled me to complete this work.

  • ii

  • iii

    Content

    1. Introduction

    1.1. Quantum Chemistry 1

    1.2 Thesis Outline 4

    2. Theory

    2.1. Aspects of Wave Function Theory 5

    2.1.1. Exact-Exchange and HartreeFock Theory 5

    2.1.2. Post-HF Methods and Correlation Effects 7

    2.2. KohnSham Density Functional Theory 10

    2.2.1. Fundamental Concepts 10

    2.2.2. Exchange-Correlation Holes 15

    2.2.3. Adiabatic Connection 17

    2.2.4. Local and Semi-Local Density Functional Approximations 18

    2.2.5. Self-Interaction Error 21

    2.2.6. Static Correlation Error 28

    2.2.7. Non-Covalent Interaction Error 31

    2.3. Hybrid Density Functional Theory 34

    2.3.1. Rationale for Exact-Exchange Mixing 34

    2.3.2. Exact-Exchange Potential 36

    2.3.3. Hybrid Density Functionals 37

    2.4. The DFT+U Method 40

    3. Algorithms and Implementation

    3.1. Exact-Exchange 45

    3.1.1. Electron-Repulsion Integrals 45

    3.1.2. Integral Processing and Symmetry Treatment 60

    3.1.3. Integral Screening 65

    3.1.4. Gradients 69

    3.1.5. Parallelization and Run Time Aspects 71

    3.2. Generalized DFT+U Method 76

    3.2.1. Projector Generation 76

    3.2.2. DFT+Umol Energy 79

    3.2.3. DFT+Umol Gradients 79

  • iv

    4. Applications

    4.1. General Computational Details 81

    4.2. DFT+Umol Analysis of the Self-Interaction Error in Ni(CO)m, m = 1 4 83

    4.2.1 Introduction 83

    4.2.2 Molecular Geometries 84

    4.2.3. Dissociation Energies 86

    4.2.4. Electronic Structure Aspects 89

    4.2.5. Summary and Conclusions 95

    4.3. Transition Metal Cluster Scaling Study with Hybrid DFT 97

    4.3.1 Introduction 97

    4.3.2 Cluster Scaling Procedure and Computational Models 98

    4.3.3. Structural, Energetic, and Ionization Properties 100

    4.3.4. Electronic Structure Aspects 114

    4.3.5. Conclusions 117

    4.4. CO Adsorption on Platinum Model Clusters 118

    4.4.1. The CO Puzzle 118

    4.4.2. Adsorption Site Models 122

    4.4.3 Structural Aspects 126

    4.4.4. CO Adsorption Energies 127

    4.4.5. Electronic Structure Aspects 134

    4.4.6. Conclusions 139

    5. Summary 143

  • v

    List of Abbreviations

    ACE Accompanying Coordinate Expansion (method)

    ACM3, ACM1, Adiabatic Connection Method (different variants)

    AO Atomic Orbital

    B88, B3, B97, Becke Functionals (different variants)

    CAS Complete Active Space (method)

    CC Coupled Cluster

    CGTO Contracted Gaussian Type Orbital

    CPU Central Processing Unit

    DFT Density Functional Theory

    DLB Dynamic Load Balancing (library)

    FCI Full Configuration Interaction (method)

    ERI Electron-Repulsion Integral

    EXX Exact-Exchange

    FDO Functional Derivative with respect to Orbitals

    FEN Fractional Electron Number

    FLL Fully Localized Limit

    FLOP Floating Point Operation

    FMO Fragment Molecular Orbital

    FON Fractional Occupation Number (technique)

    GGA Generalized Gradient Approximation

    GKS Generalized KohnSham (formalism)

    HEG Homogeneous Electron Gas

    HF HartreeFock (method)

    HFS HartreeFockSlater (model)

    HK HohenbergKohn

    HLG HOMO-LUMO gap

    HOMO Highest Occupied Molecular Orbital

    HRR Horizontal Recursion Relation

    KED Kinetic Energy Density

    KS KohnSham (formalism)

    LDA Local Density Approximation (method)

    LSDA Local Spin Density Approximation (method)

    LUMO Lowest Unoccupied Molecular Orbital

    LYP LeeYangParr

    M06, M06L, Minnesota Functionals (different variants)

    MBPT Many-Body Perturbation Theory

    MCSCF Multi-Configuration Self-Consistent-Field (method)

  • vi

    MD McMurchieDavidson

    MGGA Meta Generalized Gradient Approximation

    MO Molecular Orbital

    MP MllerPlesset (method)

    MPI Message Passing Interface (library)

    MSIE Many-electron Self-Interaction Error

    NCIE Non Covalent Interaction Error

    NGA Non-Separable Gradient Approximation

    OEP Optimized Effective Potential (method)

    OER One-Electron Region

    OPTX Optimized LDA Exchange Functionals (different variants)

    OS ObaraSaika

    PBE PerdewBurkeErnzerhof

    PGTO Primitive Gaussian Type Orbital

    PH PopleHehre

    PKZB PerdewKurthZupanBlaha

    PW PerdewWang GGA

    PWLDA PerdewWang LDA

    PZ PerdewZunger

    RKS Restricted KohnSham (formalism)

    SCE Static Correlation Error

    SCF Self-Consistent-Field (method)

    SE Schrdinger Equation

    SIC Self-Interaction Correction

    SIE Self-Interaction Error

    TPSS TaoPerdewStaroverovScuseria

    UKS Unrestricted KohnSham (formalism)

    vdW van der Waals

    VRR Vertical Recursion Relation

    VSXC van VoorhisScuseria Functional

    VWN VoskoWilkNusair

    WFT Wave Function Theory

    XC Exchange-Correlation

  • 1

    1. Introduction

    1.1. Quantum Chemistry

    Electronic structure theory[1-7] of materials and molecules aims to obtain accurate

    computational descriptions of such systems at an atomic length scale. Predictions of physical

    observables of such quantum mechanical systems can then be computed from this

    description. The fields of quantum chemistry and computational chemistry apply electronic

    structure theory to chemical problems.[8-10] The studied chemical entities range from

    individual atoms over common molecules to larger biomolecules, nanoparticles and extended

    systems, like solids and their surfaces.

    The electronic structure description of such systems is determined by the underlying

    Schrdinger equation[11] (SE) which can be solved analytically only for a few one-electron

    cases.[12,13] Thus, quantum chemistry needs to rely on approximate solution techniques for the

    many-electron SE. To obtain useful predictions it is desirable to compute for example

    reaction energies with a precision of ~2 kcal/mol (~8 kJ/mol, chemical precision). These

    results are usually obtained from total energies of much larger values which therefore need to

    be computed with a high relative accuracy. Except for high level quantum chemical

    approximations, most methods do not reliably deliver chemical precision and their accuracy

    usually varies depending on the type of systems at hand. While in the case of main group

    compounds an accuracy of a few kcal/mol is feasible, a precision of 10 kcal/mol or more may

    still be reasonable for reaction energies involving systems with transition metal elements.

    The HartreeFock (HF) method[2,14-16] is one of the earliest electronic structure

    approximations and the simplest meaningful approach based on wave function theory (WFT).

    The HF ansatz for the many-electron wave function as Slater-determinant fulfills the

    requirements of electronic non-distinguishability and