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<ul><li><p>Self-consistent GW approach for theunified description of ground and excited</p><p>states of finite systems</p><p>Characterization of iron oxide thin filmsas a support for catalytically active</p><p>nanoparticles</p><p>Dissertation</p><p>zur Erlangung des Grades</p><p>Doktor der Naturwissenschaften (Dr. rer. nat.)</p><p>eingereicht im Fachbereich Physikder Freie Universitat Berlin</p><p>Vorgelegt in Juli 2013 von</p><p>Fabio Caruso</p><p>Wednesday, June 26, 2013</p><p>Dissertation</p><p>zur Erlangung des Grades</p><p>Doktor der Naturwissenschaften (Dr. rer. nat.)</p><p>eingereicht im Fachbereich Physikder Freie Universitat Berlin</p><p>vorgelegt in Juli 2013 von</p><p>Fabio Caruso</p></li><li><p>Diese Arbeit wurde von Juni 2009 bis Juni 2013 am Fritz-Haber-Institut der Max-Planck-Gesellschaft in der Abteilung Theorie unter Anleitung von Prof. Dr. Matthias Schefflerangefertigt.</p><p>1. Gutachter: Prof. Dr. Matthias Scheffler2. Gutachter: Prof. Dr. Felix von Oppen</p><p>Tag der Disputation: 21. Oktober 2013</p></li><li><p>Hierdurch versichere ich, dass ich in meiner Dissertation alle Hilfsmittel und Hilfenangegeben habe und versichere gleichfalls auf dieser Grundlage die Arbeit selbststandigverfasst zu haben.Die Arbeit habe ich nicht schon einmal in einem fruheren Promotionsverfahren verwen-det und ist nicht als ungenugend beurteilt worden.</p><p>Berlin, 31. Juli 2013</p></li><li><p>per Maria Rosa, Alberto, e Silvia</p></li><li><p>ABSTRACT</p><p>Ab initio methods provide useful tools for the prediction and characterization of materialproperties, as they allow to obtain information complementary to purely experimentalinvestigations. The GW approach to many-body perturbation theory (MBPT) has re-cently emerged as the method of choice for the evaluation of single-particle excitations inmolecules and extended systems. Its application to the characterization of the electronicproperties of technologically relevant materials such as, e.g., dyes for photovoltaicapplications or transparent conducting oxides is steadily increasing over the last years.The GW approximation is typically introduced as a first-order perturbative correction(G0W0) to density-functional theory (DFT) or Hartree-Fock (HF). However, this alsointroduces a severe initial-state dependence that affects the G0W0 solution. Due to itsnon-perturbative nature, self-consistent GW (sc-GW ) ameliorates several shortcomingsof the G0W0 scheme, such as the violation of the particle-number conservation andthe dependence on the initial reference ground state. Nevertheless, the suitability andoverall accuracy of sc-GW has often been questioned, mostly because of numericalproblems with previous implementations and the lack of systematic studies for a widerange of systems.</p><p>In this doctoral work, the sc-GW approach for total energies and spectroscopic prop-erties has been implemented in the Fritz Haber Institute ab initio molecular simulationscode (FHI-aims) in an all-electron numeric atom-centered orbital framework. With thisimplementation I then performed a systematic assessment of ground- and excited-stateproperties of atoms and molecules.</p><p>In this work, sc-GW has been employed to study the excitation spectra of organicmolecules, molecules of interest for photovoltaic applications, and prototypical donor-acceptor systems. At self-consistency, the quasi-particle energies are in good agreementwith experiment and, on average, more accurate than G0W0 based on HF or semi-localDFT. For covalently bonded dimers, the capability of DFT- and MBPT-based approachesto describe the correlated electronic ground state at dissociation was investigated. Staticand local approximations of exchange-correlation potentials as opposed to non-local,frequency dependent self-energy approximations are shown to be more effective indescribing the dissociation regime. sc-GW calculations for ground-state properties(as, e.g., binding energies, bond lengths, vibrational frequencies, and dipole moments)are presented. For ground-state properties, I show that sc-GW achieves a comparableperformance to exact-exchange plus correlation in the random-phase approximation(EX+cRPA) which is, however, not as good as that of renormalized second-order per-turbation theory (rPT2). Finally, the correct distribution of the electron density inprototypical donor-acceptor compounds suggests that sc-GW is a promising methodfor the description of electronic excitations in charge-transfer systems.</p></li><li><p>ZUSAMMENFASSUNG</p><p>Im Vergleich zu einer rein experimentellen Vorgehensweise bieten ab initio Methodendie Moglichkeit Ergebnisse auf fundamental anderem Weg zu erlangen. Eine vielver-sprechende Wahl fur die Bestimmung von Ein-Teilchen Anregungen in Molekulenund in ausgedehnten Systemen ist der GW -Ansatz in der Vielteilchenstorungstheorie.Seine Verwendung bei der Analyse technologisch interessanter Materialien nimmtstetig zu, z.B. bei halbleitenden Farbstoffen fur die Anwendung in der Photovoltaikoder bei transparenten leitenden Oxiden. Ublicherweise wird die GW -Naherung alseine Korrektur in erster Ordnung (G0W0) auf die Ergebnisse der Dichtefunktionaltheo-rie (DFT) oder Hartree-Fock-Theorie (HF) angewandt. Dies wiederum fuhrt zu einerstarken Abhangig vom Startzustand der G0W0-Losung. Dank seines nicht-storungs-theoretischen Charakters vermag es der selbstkonsistente GW -Ansatz (sc-GW ) einigeder Nachteile von G0W0, z.B. die Nichterhaltung der Teilchenzahl oder die Abhangigvom Referenzzustand, aufzuheben. Auf Grund von numerischen Problemen in ex-istierenden Implementationen und der fehlenden systematischen Analyse einer Vielzahlvon Systemen wurde die Eignung und Genauigkeit von sc-GW allerdings oft in Fragegestellt.</p><p>In der vorliegenden Doktorarbeit wurde die sc-GW Methode fur Gesamtenergienund spektroskopische Eigenschaften im Rahmen eines alle Elektronen umfassendennumerischen, atom-zentrierten Basis Ansatzes in das Fritz Haber Institute ab initiomolecular simulations package (FHI-aims) implementiert. Mittels dieser Implementa-tion wurde eine systematische Untersuchung der Grund- und Anregungszustande vonAtomen und Molekulen durchgefuhrt.</p><p>Anschliessend wurde die sc-GW -Methode fur die Untersuchung von Anregungsspek-tren organischer Molekule, Molekule fur die Verwendung in der Photovoltaik und proto-typische Donor/Akzeptor System angewandt. Bei erreichter Selbstkonsistenz stimmendie Quasiteilchen Energien gut mit experimentellen Ergebnissen uberein und sind imDurchschnitt deutlich genauer als bei G0W0-Rechnungen, die auf HF oder semi-lokalerDFT basieren. Weiterhin wurde der korrelierten elektronischen Grundzustandes beider Dissoziation von kovalent gebundenen Dimeren auf seine Beschreibung durchDFT und Storungstheoretischen Methoden untersucht. Im Vergleich zu nicht-lokalen,frequenzabhangigen Selbstenergienaherungen wie z.B. GW beschreibt die statische undlokale Naherung des Austausch-Korrelationspotentials in DFT die Dissoziation besser,</p></li><li><p>vorausgesetzt man verwendet moderne Funktionale wie z.B. exakter Austausch plusKorrelation in der Random Phase Approximation (Ex+cRPA). Abschliessend werdendie Grundzustandseigenschaften (Bindungsenergien, Bindungslangen, Schwingungs-frequenzen und Dipolmomente) untersucht. sc-GW ist ahnlich leistungsfahig wieEx+cRPA, aber weniger leistungsfahig als renormalized second-order pertubation the-ory (rPT2). Aus der korrekten Verteilung der Elektronendichte in prototypischen Donor-Akzeptor Systemen lasst sich schlieen, dass sc-GW eine viel versprechende Methodezur Beschreibung dieser Systeme ist.</p></li><li><p>CONTENTS</p><p>1 Introduction 1</p><p>I Theoretical Background 5</p><p>2 The Many-Body Problem 7</p><p>2.1 The Many-Body Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 7</p><p>2.2 The Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . . 8</p><p>2.3 Ab-initio electronic-structure approaches . . . . . . . . . . . . . . . . . . 9</p><p>2.4 Wave-function-based methods . . . . . . . . . . . . . . . . . . . . . . . . 10</p><p>2.5 Density-functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 13</p><p>3 Green-Function Methods 23</p><p>3.1 Definition of the Green function . . . . . . . . . . . . . . . . . . . . . . . . 23</p><p>3.2 Connection to DFT: The Sham-Schluter equation . . . . . . . . . . . . . . 28</p><p>3.3 Relation to physical properties . . . . . . . . . . . . . . . . . . . . . . . . 29</p><p>4 Hedins equations: the GW approximation 37</p><p>4.1 Hedins equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37</p><p>4.2 The GW approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41</p><p>4.3 Perturbative G0W0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42</p><p>4.4 Partially self-consistent GW . . . . . . . . . . . . . . . . . . . . . . . . . . 46</p><p>4.5 Fully self-consistent GW . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47</p><p>4.6 Connection of GW and RPA . . . . . . . . . . . . . . . . . . . . . . . . . . 49</p><p>II Implementation 53</p><p>5 Self-consistent GW equations in a numeric orbital basis 55</p><p>5.1 Numeric atom-centered orbitals . . . . . . . . . . . . . . . . . . . . . . . . 55</p><p>5.2 The resolution of the identity . . . . . . . . . . . . . . . . . . . . . . . . . 59</p><p>5.3 Working equations for self-consistent GW . . . . . . . . . . . . . . . . . . 60</p><p>xiii</p></li><li><p>xiv CONTENTS</p><p>6 Frequency and time dependence in self-consistent GW 696.1 Imaginary time and frequency formalism . . . . . . . . . . . . . . . . . . 696.2 Evaluation of Fourier integrals . . . . . . . . . . . . . . . . . . . . . . . . 71</p><p>7 Numerical evaluation of physical quantities 777.1 Spectral function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.2 Galitskii-Migdal total energy . . . . . . . . . . . . . . . . . . . . . . . . . 81</p><p>III Applications to atoms and molecules 85</p><p>8 Unified description of ground and excited states 878.1 Independence of the starting point and conservation laws . . . . . . . . 878.2 Total energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908.3 Electron density and dipole moment . . . . . . . . . . . . . . . . . . . . . 958.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99</p><p>9 The bond-breaking puzzle: many-body versus density-functional theory 1019.1 Evaluation of the sc-RPA total energy . . . . . . . . . . . . . . . . . . . . 1029.2 Potential-energy curve of H2: sc-RPA and sc-GW . . . . . . . . . . . . . . 1039.3 Natural occupations and the role of spin-symmetry . . . . . . . . . . . . 1069.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108</p><p>10 Assessment of GW methods for photoemission processes in molecules 11110.1 A hierarchy of theoretical consistency . . . . . . . . . . . . . . . . . . . . 11110.2 The azabenzenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11310.3 Ionization energies of closed-shell molecules . . . . . . . . . . . . . . . . 12310.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128</p><p>11 Conclusions and outlook 131</p><p>IV Backmatter 135</p><p>Appendices 137</p><p>A Functional identities 139</p><p>B Convergence with the grid parameters 141</p><p>C Ionization energies of closed-shell molecules 143</p><p>D Total energy of atoms 147</p><p>E Spectra of benzene, pyrazine, pyridazine, and pyrimidine 149</p><p>F Derivation of the Klein and Luttinger-Ward functionals 163</p></li><li><p>CONTENTS xv</p><p>Curriculum Vit 167</p><p>Publications 169</p><p>Bibliography 173</p><p>Acknowledgements 193</p></li><li><p>1 INTRODUCTION</p><p>In the XV century, the driving force of the first chemical and physical discoveries wasthe belief that base metals, such as plumb and iron, could be transformed into goldby bringing them in touch with the lapis philosophicus, the philosophers stone. Thisbelief led to many efforts towards the synthesis of the lapis philosophicus perpetratedby severals scholars over the centuries, including Isaac Newton. Nowadays, materialscience has moved towards less-esoteric goals but, in a certain sense, is still addressingthe problem of turning iron into gold, i.e., through the transformation of sun lightinto more controllable forms of energy.</p><p>Solar cells based on the photovoltaic effect, whereby electricity can be generatedthrough exposure of the cell to sunlight, constitute one out of many examples of solar-energy conversion techniques. In all solar-energy conversion devices, the light har-vesting is triggered by the absorption of a photon in a light absorbing material. Thephoton energy is thus transferred to the material and transformed, e.g., in electronicexcitations and/or nuclear vibrations. Providing the theoretical tools for predicting (notjust reproducing) these processes in complex materials is one of the roles of atomisticmodeling in the quest for new materials for energy conversion. The achievement ofthis goal is the key to go beyond conventional trial and error approaches to materialresearch as recently recognized and endorsed by large scale projects, such as theMaterial Genome Initiative.1 However, despite the omnipresence of electronic excita-tions in energy-conversion and energy-storage devices, the methods at hand for theirtheoretical description are at an earlier stage of development as compared to approachesthat address ground-state properties.</p><p>Density-functional theory (DFT) is the method of choice for the ab initio description(i.e., based on the fundamental equations of quantum mechanics) of the ground-stateproperties of atoms, molecules, and periodic solids. DFT is in principle exact for theground-state energy, and the electron density although in practice approximationsbecome necessary. On the whole, DFT contributed enormously to the developmentof condensed-matter physics and quantum chemistry, and is by far the most diffuseelectronic structure approach. However, when it comes to excited states, DFT and ourpresent techniques to deal with it falls short. For instance, it is well known that evenexact Kohn-Sham DFT would not correctly reproduce the fundamental band gap of a</p><p>1http://www.whitehouse.gov/mgi</p><p>1</p><p>http://www.whitehouse.gov/mgi</p></li><li><p>2 Introduction</p><p>semiconductor. To describe direct and inverse photoemission processes (i.e., processesinvolving the loss or gain of an electron), DFT is generally coupled to many-bodyperturbation theory (MBPT). One advantage of this synergy is the possibility to treatground- and excited-states at different levels of theory: The ground-state propertiescan be determined at the DFT level, whereas MBPT-based methods which are oftencomputationally more demanding than conventional density-functional approximations can be employed just for the evaluation of excited-state properties. Two commonframeworks for the evaluation of spectral properties are dynamical mean-field theory(DMFT) and the GW approximation. DMFT has gained popularity due to its capabilityto reproduce spectral features of materials generally classified as strongly correlated. Theapplicability of DMFT is restricted to systems with strongly localized frontier orbitals as,for instance, 3d- and 4f -electron systems. The GW approximation, on the other hand, ismore versatile and it can be in principle applied to any compound.</p><p>The popularity of the GW approach has steadily increased over the last years. In...</p></li></ul>