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Th`Ese : Stefano EVANGELISTI Et Thierry LEININGER Rapporteurs : Carmen J

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Th`Ese : Stefano EVANGELISTI Et Thierry LEININGER Rapporteurs : Carmen J THTHESEESE`` En vue de l’obtention du DOCTORAT DE L’UNIVERSITE´ DE TOULOUSE D´elivr´e par : l’Universit´eToulouse 3 Paul Sabatier (UT3 Paul Sabatier) Prt ´esen ´ee et soutenue le 07/07/2015 par : Muammar EL KHATIB Characterization of metallic and insulating properties of low-dimensional systems JURY Carmen J. CALZADO Professeur Universidad de Sevilla Janos´ G. ANGY´ AN´ Directeur de Recherches Universit´ede Lorraine et CNRS Dirk ANDRAE Privatdozent docteur Freie Universit¨atBerlin Romuald POTEAU Professeur Universit´ePaul Sabatier Stefano EVANGELISTI Professeur Universit´ePaul Sabatier Thierry LEININGER Professeur Universit´ePaul Sabatier E´cole doctorale et sp´ecialit´e : SDM : Physicochimie th´eorique- COP 01 Unit´e de Recherche : Laboratoire de Chimie et Physique Quantiques (UMR-5626) Directeur(s) de Th`ese : Stefano EVANGELISTI et Thierry LEININGER Rapporteurs : Carmen J. CALZADO et J´anosG. ANGY´ AN´ To the memory of my father. To my mother, and my siblings. To my wife. Because without you I am nothing. Acknowledgments No matter what I believe or think about life, I would like to thank the energy we call God and the Universe for this opportunity. I am also very thankful to France, because it has been my country during these last five years and it has made me feel like being at home. I would like to thank my mother (Luz Marina Rodr´ıguez El Khatib), my sister (Yauhara El Katib) and my brother (Nohad El Khatib) for all their support since the moment I left Venezuela. I couldn’t have arrived to France and to this moment of my life without all of your help. Los amo. It is with a deep sadness that I would also like to thank my father (Anwar El Khatib), even if he is not in this world anymore to read this. Thank you very much father, for all the time you told me that I never had to surrender and for your advices. Te amo y te extrano˜ mucho. I’m infinitely thankful to my wife, because she has stand by my side fighting in in this long journey. She has been listening to me and supporting me in several times. She knows very well how difficult this has been for me. I think without her I couldn’t have made it. Thanks for your patience, being friends, and for encouraging me to go up to the end of my Ph.D. Indeed, a Ph.D. Acknowledgments is a question of being determined and persistent to accomplish the goal. Te amo. I thank my supervisors Stefano Evangelisti and Thierry Leininger for giving me the opportu- nity of doing a doctorate. They were not only my supervisors but also my friends. We have had nice discussions, thanks for giving me this chance to accomplish a big step in my life and for all your advices. I will be forever thankful. I would like to thank my friends in the different labs of the building 3R1 − B4: Adrien Gottle,¨ Simon Derouault, Mina Bionta, Oriana Brea, Franc¸ois Damon, Annael¨ Brunet, Aela´ Fortun, Gabriel Condon, Medha Soni, Lionel Lacombe, Isabelle Braud, Citlali Cabrera, Julien Alibert, Boris Decamps, Christophe Iftner, Avijit Shee, Adel Almoukhalalati, Luiz Oliveira, Juan Sanz, and Cecilia Gomes´ Pech. Thanks for making life fun, for all the beers at Dubliners, for the nice discussions about life/comics/movies/politics/stuff, and science. I thank Prof. Gian Luigi Bendazzoli from the Universita` di Bologna for all the discussions and insights about the main theory I employed in my dissertation and for the active development of the NEPTUNUS program widely used in this work. I thank Prof. Beate Paulus, and Edoardo Fertitta from the Freie Universitat¨ Berlin for all the discussions and work we have done together. I also thank Miguel Morales for his friendship and advices even in the distance. I thank all my fellows at the Laboratoire de Chimique et Physique Quantiques, I have been very glad to be part of it. I would like to acknowledge all members of the groupe de methodes´ et outils de la Chimique Quantique (GMO). Discussions during the GMO meetings were very Acknowledgments fruitful to improve my work. I also thank all people behind the European Master on Theoretical Chemistry and Computa- tional Modelling (TCCM) within the Eramus Mundus program. With special thanks to Professors Manuel Yanez˜ and Otilia Mo. I acknowledge the University of Toulouse and the Centre National de la Recherche Scien- tifique (CNRS), L’Agence Nationale de la Recherche (ANR) under the action ANR-11-INTB- 1009 MITLOW in collaboration with the Deutsche Forschungsgemeinschaft (DFG, Germany), and the Programme Investissements d’Avenir under the program ANR-11-IDEX-0002-02, refer- ence ANR-10-LABX-0037-NEXT for fundings to follow my Ph.D. studies. It is always delicate to write acknowledgments, especially because it is certain that I have forgotten some people, and I apologize for all the ones I forgot to acknowledge. Thank you very much for your help. Contents Introduction 1 1 Metals and insulators 7 1.1 General remarks . 8 1.2 Quantitative approaches . 16 1.2.1 Cumulants . 17 1.2.2 Localization tensor . 21 1.3 Total position-spread tensor . 25 1.3.1 Second quantization . 27 1.3.2 The total position-spread tensor in a CAS-SCF wave function . 28 1.3.2.1 Evaluation of operators in density matrices . 31 1.3.2.2 Slater’s rules . 33 1.4 Spin-partitioned total position spread . 35 1.5 Conclusions . 36 2 Monitoring the wave function, and bond characterization 39 2.1 Behavior of the total position-spread tensor in diatomic systems . 40 2.1.1 Covalent bonds . 41 2.1.2 Ionic bonds . 42 ii CONTENTS 2.1.3 Weakly bonded molecules . 43 3 Mixed-valence compounds 57 + 3.1 The total position spread in mixed-valence compounds: A study on the H4 model system . 60 3.2 Computing the position-spread tensor in the CAS-SCF formalism . 71 4 One-dimensional systems 81 4.1 Introduction . 82 4.1.1 Peierls insulators . 82 4.1.2 Anderson insulators . 84 4.2 Mott insulators: theoretical concepts . 85 4.2.1 Slater insulator . 94 4.2.2 Mott insulator . 95 4.2.2.1 Mott-Hubbard insulator . 96 4.2.2.2 Mott-Heisenberg insulator . 98 4.3 The total position-spread tensor: spin partition . 99 4.3.1 Hubbard Hamiltonian . 100 4.3.2 Equally spaced hydrogen chains . 103 4.4 Spin delocalization in hydrogen chains described with the spin-partitioned total- position spread tensor . 121 4.4.1 Fixed-bond dimerized chains . 121 4.4.2 Homothetic dimerized chains . 122 4.5 Application to Heisenberg spin chains . 137 4.5.1 Heisenberg Hamiltonian . 137 CONTENTS iii 4.5.2 Equally-J spin chains . 138 4.5.3 Spin-paired chains . 139 5 Polycyclic aromatic hydrocarbons 173 5.1 Total position-spread tensor applied to polycyclic aromatic hydrocarbons . 207 Conclusions 213 Bibliography 219 Annexes 247 6.1 Phase transitions . 247 6.1.1 Thermodynamic phase transition . 247 6.1.2 Quantum phase transition . 247 6.2 Statistical correlation from the spin-partitioned total position-spread tensor . 248 6.3 Beryllium dimer: a bond based on non-dynamical correlation. 251 6.3.1 Introduction . 251 6.3.1.1 Dynamical correlation . 255 6.3.1.2 Non-Dynamical correlation . 256 French Abstract 271 iv CONTENTS List of Figures 1.1.1 Band fillings for: (a) insulators; (b) semi-conductors; (c) two-band semi-metal; and (d) one-band metal. 11 1.1.2 Metal-insulator transition phase diagram based on the Hubbard model. 13 1.1.3 Wave function of localized particles on a ring. Figure taken from Ref. [1]. 15 1.1.4 Kohn disconnectedness in a one-dimensional system composed by two electrons. Note that for the sake of simplicity the particles are spinless. 17 2.1.1 Schematic representation of F2 breathing orbitals in the valence bond framework. 42 + 3.1.1 Average geometries for the H4 . .......................... 61 3.1.2 Perpendicular components of the TPS tensor for the ground- and excited-state of + the H4 mixed-valence system. 61 3.2.1 Canonical molecular orbitals at ROHF/VDZP level representing the behavior of the hole at the three limit ξ values of interest. 72 4.1.1 Pictorial representation of the doubling of periodicity caused by a Peierls distortion. 83 4.2.1 Schematic representation of the pair correlation function. 90 4.2.2 Electrons moving as double occupancies in the Hubbard-upper band (a), and as holes in the Hubbard-lower band (b). Hubbard bands are considered charge bands. 98 vi LIST OF FIGURES 4.4.1 Coefficients of a linear fit to the position spreads for any fixed inter-nuclear dis- tance of the equally-spaced hydrogen chains. 125 4.4.2 Coefficients of a linear fit to the position spreads for any fixed inter-nuclear dis- tance of the fixed-bond dimerized hydrogen chains. 126 4.4.3 Coefficients of a linear fit to the position spreads for any fixed inter-nuclear dis- tance of the homothetic dimerized hydrogen chains. 127 5.0.1 General schema of the zig-zag and armchair linear polyacenes. 174 5.0.2 Sublattices for the different polygoacenes considered in this investigation. Atoms shown in blue are for A type, and red ones are for B type. The squares refer to the singly occupied molecular orbitals (SOMO). 175 5.0.3 (a) Different sublattices in partly saturated polygoacenes. (b) Schematic topolog- ical representation of hydrogens. 177 6.2.1 Statistical correlation as a function of the inter-nuclear distances for the H chains taking SP-TPS values computed at FCI level with a STO-12G basis set.
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