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Fabio Caruso Caruso CharacterizationSelf-consistent of ironGW oxideapproach thin for films the unifiedas a support description for catalytically of ground active and excited statesnanoparticles of finite systems Wednesday, June 26, 2013 DissertationDissertation zur Erlangungzur Erlangung des Grades des Grades DoktorDoktor der Naturwissenschaften der Naturwissenschaften (Dr. rer. (Dr. nat.) rer. nat.) eingereichteingereicht im Fachbereich im Fachbereich Physik Physik der Freieder UniversitFreie Universitat¨ Berlinat¨ Berlin Vorgelegtvorgelegt in Juli in 2013 Juli von 2013 von FabioFabio Caruso Caruso 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 Scheffler angefertigt. 1. Gutachter: Prof. Dr. Matthias Scheffler 2. Gutachter: Prof. Dr. Felix von Oppen Tag der Disputation: 21. Oktober 2013 Hierdurch versichere ich, dass ich in meiner Dissertation alle Hilfsmittel und Hilfen angegeben habe und versichere gleichfalls auf dieser Grundlage die Arbeit selbststandig¨ verfasst zu haben. Die Arbeit habe ich nicht schon einmal in einem fruheren¨ Promotionsverfahren verwen- det und ist nicht als ungenugend¨ beurteilt worden. Berlin, 31. Juli 2013 per Maria Rosa, Alberto, e Silvia ABSTRACT Ab initio methods provide useful tools for the prediction and characterization of material properties, as they allow to obtain information complementary to purely experimental investigations. 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 in molecules and extended systems. Its application to the characterization of the electronic properties of technologically relevant materials – such as, e.g., dyes for photovoltaic applications 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 also introduces a severe initial-state dependence that affects the G0W0 solution. Due to its non-perturbative nature, self-consistent GW (sc-GW ) ameliorates several shortcomings of the G0W0 scheme, such as the violation of the particle-number conservation and the dependence on the initial reference ground state. Nevertheless, the suitability and overall accuracy of sc-GW has often been questioned, mostly because of numerical problems with previous implementations and the lack of systematic studies for a wide range of systems. 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 simulations code (FHI-aims) in an all-electron numeric atom-centered orbital framework. With this implementation I then performed a systematic assessment of ground- and excited-state properties of atoms and molecules. In this work, sc-GW has been employed to study the excitation spectra of organic molecules, molecules of interest for photovoltaic applications, and prototypical donor- acceptor systems. At self-consistency, the quasi-particle energies are in good agreement with experiment and, on average, more accurate than G0W0 based on HF or semi-local DFT. For covalently bonded dimers, the capability of DFT- and MBPT-based approaches to describe the correlated electronic ground state at dissociation was investigated. Static and local approximations of exchange-correlation potentials – as opposed to non-local, frequency dependent self-energy approximations – are shown to be more effective in describing 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 comparable performance 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 in prototypical donor-acceptor compounds suggests that sc-GW is a promising method for the description of electronic excitations in charge-transfer systems. ZUSAMMENFASSUNG Im Vergleich zu einer rein experimentellen Vorgehensweise bieten ab initio Methoden die Moglichkeit¨ Ergebnisse auf fundamental anderem Weg zu erlangen. Eine vielver- sprechende Wahl fur¨ die Bestimmung von Ein-Teilchen Anregungen in Molekulen¨ und in ausgedehnten Systemen ist der GW -Ansatz in der Vielteilchenstorungstheorie.¨ Seine Verwendung bei der Analyse technologisch interessanter Materialien nimmt stetig zu, z.B. bei halbleitenden Farbstoffen fur¨ die Anwendung in der Photovoltaik oder bei transparenten leitenden Oxiden. Ublicherweise¨ wird die GW -Naherung¨ als eine Korrektur in erster Ordnung (G0W0) auf die Ergebnisse der Dichtefunktionaltheo- rie (DFT) oder Hartree-Fock-Theorie (HF) angewandt. Dies wiederum fuhrt¨ zu einer starken Abhangig¨ vom Startzustand der G0W0-Losung.¨ Dank seines nicht-storungs-¨ theoretischen Charakters vermag es der selbstkonsistente GW -Ansatz (sc-GW ) einige der Nachteile von G0W0, z.B. die Nichterhaltung der Teilchenzahl oder die Abhangig¨ vom Referenzzustand, aufzuheben. Auf Grund von numerischen Problemen in ex- istierenden Implementationen und der fehlenden systematischen Analyse einer Vielzahl von Systemen wurde die Eignung und Genauigkeit von sc-GW allerdings oft in Frage gestellt. In der vorliegenden Doktorarbeit wurde die sc-GW Methode fur¨ Gesamtenergien und spektroskopische Eigenschaften im Rahmen eines alle Elektronen umfassenden numerischen, atom-zentrierten Basis Ansatzes in das Fritz Haber Institute ab initio molecular simulations package (FHI-aims) implementiert. Mittels dieser Implementa- tion wurde eine systematische Untersuchung der Grund- und Anregungszustande¨ von Atomen und Molekulen¨ durchgefuhrt.¨ 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 stimmen die Quasiteilchen Energien gut mit experimentellen Ergebnissen uberein¨ und sind im Durchschnitt deutlich genauer als bei G0W0-Rechnungen, die auf HF oder semi-lokaler DFT basieren. Weiterhin wurde der korrelierten elektronischen Grundzustandes bei der Dissoziation von kovalent gebundenen Dimeren auf seine Beschreibung durch DFT und Storungstheoretischen¨ Methoden untersucht. Im Vergleich zu nicht-lokalen, frequenzabhangigen¨ Selbstenergienaherungen¨ wie z.B. GW beschreibt die statische und lokale Naherung¨ des Austausch-Korrelationspotentials in DFT die Dissoziation besser, vorausgesetzt man verwendet moderne Funktionale wie z.B. exakter Austausch plus Korrelation in der Random Phase Approximation (Ex+cRPA). Abschliessend werden die Grundzustandseigenschaften (Bindungsenergien, Bindungslangen,¨ Schwingungs- frequenzen und Dipolmomente) untersucht. sc-GW ist ahnlich¨ leistungsfahig¨ wie Ex+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 schließen, dass sc-GW eine viel versprechende Methode zur Beschreibung dieser Systeme ist. CONTENTS 1 Introduction1 I Theoretical Background5 2 The Many-Body Problem7 2.1 The Many-Body Hamiltonian.........................7 2.2 The Born-Oppenheimer approximation...................8 2.3 Ab-initio electronic-structure approaches..................9 2.4 Wave-function-based methods........................ 10 2.5 Density-functional theory........................... 13 3 Green-Function Methods 23 3.1 Definition of the Green function........................ 23 3.2 Connection to DFT: The Sham-Schluter¨ equation.............. 28 3.3 Relation to physical properties........................ 29 4 Hedin’s equations: the GW approximation 37 4.1 Hedin’s equations................................ 37 4.2 The GW approximation............................. 41 4.3 Perturbative G0W0 ............................... 42 4.4 Partially self-consistent GW .......................... 46 4.5 Fully self-consistent GW ............................ 47 4.6 Connection of GW and RPA.......................... 49 II Implementation 53 5 Self-consistent GW equations in a numeric orbital basis 55 5.1 Numeric atom-centered orbitals........................ 55 5.2 The resolution of the identity......................... 59 5.3 Working equations for self-consistent GW .................. 60 xiii xiv CONTENTS 6 Frequency and time dependence in self-consistent GW 69 6.1 Imaginary time and frequency formalism.................. 69 6.2 Evaluation of Fourier integrals........................ 71 7 Numerical evaluation of physical quantities 77 7.1 Spectral function................................ 77 7.2 Galitskii-Migdal total energy......................... 81 III Applications to atoms and molecules 85 8 Unified description of ground and excited states 87 8.1 Independence of the starting point and conservation laws........ 87 8.2 Total energies.................................. 90 8.3 Electron density and dipole moment..................... 95 8.4 Summary..................................... 99 9 The bond-breaking puzzle: many-body versus density-functional theory 101 9.1 Evaluation of the sc-RPA total energy.................... 102 9.2 Potential-energy
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