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Dissertation / Doctoral Thesis DISSERTATION / DOCTORAL THESIS Titel der Dissertation /Title of the Doctoral Thesis Low scaling GW method: implementation and applications verfasst von / submitted by Peitao Liu angestrebter akademischer Grad / in partial fulfilment of the requirements for the degree of Doktor der Naturwissenschaften (Dr. rer. nat.) Wien, 2017 / Vienna, 2017 Studienkennzahl lt. Studienblatt / A 796 605 411 degree programme code as it appears on the student record sheet: Dissertationsgebiet lt. Studienblatt / Physik field of study as it appears on the student record sheet: Betreut von / Supervisor: Univ.-Prof. Dipl.-Ing. Dr. Georg Kresse Univ.- Assoz. Prof. Dipl.-Ing. Dr. Cesare Franchini Vorwort Die vorliegende Arbeit setzt sich aus drei Teilen zusammen. Der erste Teil deckt die Grundlagen zum elektronischen Vielteilchenproblem in der Physik der kondensierten Materie ab und stellt den theoretischen Hintergrund der GW Methode vor. Nach einer Einfuhrung¨ zur Dichtefunk- tionaltheorie (DFT) werden spater¨ Methoden welche uber¨ konventionelle DFT hinausgehen, wie DFT+U und Hybridfunktionale vorgestellt. Dabei werden auch die jeweiligen Grenzen der Anwendbarkeit dieser Methoden diskutiert. Anschließend wird ein Uberblick¨ uber¨ die GW Ap- proximation gegeben, beginnend mit dem Bandluckenproblem¨ und experimenteller Photoemis- sionsspektroskopie. Die Herleitung der GW Approximation aus der Vielkorper-St¨ orungstheorie¨ wird danach ausfuhrlich¨ behandelt. Insbesondere werden die Ein-Teilchen-Green’s-Funktionen, die Dyson-Gleichung und die Hedin-Gleichungen eingefuhrt.¨ Zuletzt wird die praktische Um- setzung der GW Approximation besprochen, einschließlich der konventionellen GW Implemen- tierungen sowie der Space-Time-Methode. Hierbei werden auch die Nachteile der bestehenden GW Implementierungen aufgezeigt, wodurch die Notwendigkeit fur¨ einen niedrig skalierenden GW Algorithmus deutlich wird. Aus dieser Uberlegung¨ stammt die Motivation zu dieser Ar- beit. Abschließend kann gesagt werden, dass dieser Teil eine solide Grundlage bildet und eine konsequente Terminologie fur¨ den zweiten Teil der Arbeit einfuhrt.¨ Der zweite Teil prasentiert¨ unsere kurzlich¨ entwickelte Low-Scaling-GW Methode, die im ”Vienna ab-initio Simulation Package” (VASP), einem weit verbreiteten Programm zur Berech- nung der elektronischen Struktur, implementiert wurde. Diese neue Methode ermoglicht¨ schnelle Quasiteilchenberechnungen mit einer Skalierung, die sich in der Systemgroße¨ kubisch verhalt¨ und linear in der Anzahl der k-Punkte, die verwendet werden um die Brillouin-Zone zu sampeln. Dadurch werden sowohl die Nachteile der konventionellen GW Implementierungen, welche in der Regel auf kleine Systeme beschrankt¨ sind, als auch jene der speicherintensiven Space-Time- Methode uberwunden.¨ Es wird ausfuhrlich¨ erlautert,¨ wie eine so gunstige¨ Skalierung erreicht werden kann. Danach wird der Erfolg der Methode bei der Berechnung von Quasiteilchen- Energien und Spektralfunktionen typischer Isolatoren, Halbleiter und Metalle diskutiert. Die Ergebnisse werden mit konventionellen GW Berechnungen verglichen und es zeigt sich eine sehr hohe Ubereinstimmung.¨ Daruber¨ hinaus wurde die neue Methode auch erfolgreich zur Bes- timmung der Ionisierungspotentiale und der Elektronenaffinitaten¨ von hundert Closed-Shell- Molekulen¨ angewandt. Die Ergebnisse werden mit experimentellen Daten sowie Ergebnissen 3 aus anderen elektronischen Strukturpaketen verglichen. Insgesamt wurde eine gute Ubereinstimmung¨ erzielt. Um diese Ubereinstimmung¨ zu erreichen, korrigieren wir die Fehler fur¨ endliche Ba- sissatze¨ sowie Fehler, die durch periodisch wiederholte Bilder entstehen. Fur¨ die etwas großeren¨ Abweichungen der Elektronenaffinitaten¨ werden mogliche¨ Erklarungen¨ diskutiert. Wegen der geringen Skalierung unserer neuen Methode glauben wir, dass sie ein großes Anwendungspoten- zial besitzt, insbesondere fur¨ große Einheitszellen. Dies wird im Bereich der Materialmodel- lierung dazu beitragen, Methoden welche uber¨ DFT hinaus gehen zu etablieren. Die vorgestellte Low-Scaling-GW stellt zwar einen großen Fortschritt gegenuber¨ alterer¨ GW Methoden dar, jedoch wird in der aktuellen Implementierung die Spin-Orbit-Kopplung (SOC) nicht berucksichtigt,¨ wahrend¨ dies bei der herkommlichen¨ GW Implementierung in VASPmoglich¨ ist. In diesem Abschnitt werden daher einige Anderungen¨ an der herkommlichen¨ Implemen- tierung der GW besprochen, welche es erlauben maßig¨ große Systeme zu behandeln, bei denen SOC nicht vernachlassigt¨ werden darf. Dies wird durch optimierte imaginare¨ Zeit/Frequenz Gitter sowie außerst¨ effiziente diskrete Fourier-Transformationen erreicht, beides Hauptmerk- male der neuen Low-Scaling-GW Methode. Die modifizierte GW Methode wurde fur¨ ZnO und CdTe getestet und es zeigt sich eine ausgezeichnete Ubereinstimmung¨ mit der herkommlichen,¨ unmodifizierten GW Implementierung. Daruber¨ hinaus wird eine teilweise selbst-konsistente Low-Scaling-GW0r Methode vorgestellt und vorlaufige¨ Testergebnisse fur¨ Si werden prasentiert.¨ Die Auswirkungen der Selbstkonsis- tenz werden in Bezug auf Teilchenzahlerhaltung und Bandlucken¨ diskutiert. Abschließend wird noch eine praktikable Moglichkeit¨ zur Kombination unserer Low-Scaling-GW Methode mit der Dynamical-Mean-Field-Theorie (DMFT), d.h. GW+DMFT, vorgestellt. Besonders hervorge- hoben und ausfuhrlich¨ besprochen wird die frequenzabhangige,¨ partiell abgeschirmte Wechsel- wirkung U(i!), welche mittels der Constrained-Random-Phase-Approximation (cRPA) berech- net wird sowie das Double-Counting, das bei GW+DMFT auftritt. Vorlaufige¨ Ergebnisse von GW+DMFT fur¨ SrVO3 werden prasentiert.¨ Am Ende der Arbeit befindet sich eine Conclusio in welcher die Ergebnisse und Schlussfolgerungen zusammengefasst werden und ein Ausblick uber¨ die weitere Entwicklung skizziert ist. Es sei noch angemerkt, dass einige Gleichungen und Herleitungen separat in den Anhangen¨ platziert wurden. Obwohl viele dieser Gleichungen zum besseren Verstandnis¨ des Haupttextes beitragen, wurde zum Zwecke der besseren Lesbarkeit und Ubersicht¨ diese Aufteilung gewahlt.¨ Der interessierte Leser wird daher in den Anhangen¨ nutzliche¨ und ausfuhrliche¨ Details finden. 4 Preface The thesis is divided into three main parts. The first part covers basic textbook and literature knowledge about the electronic structure problem in condensed matter physics. It introduces the theoretical background of the GW method. Starting from density functional theory (DFT), methods beyond DFT, such as DFT+U and hybrid functionals are then introduced, followed by a discussion of their limitations. Afterwards, an overview of the GW approximation is given, setting out from the band gap problem and experimental photoemission spectroscopy. Then, the GW approximation is established starting from many-body perturbation theory. In particular, the one-particle Green’s functions, Dyson equation, and Hedin’s equations are introduced. At the end, practical implementations of the GW approximation including the conventional GW implementations and the space-time method are discussed. Also, the drawbacks of existing GW implementations are raised, highlighting the need for a low-scaling GW scheme. This provides the motivation for the thesis. The first part lays a solid foundation and builds a consistent terminology for the second part of the thesis. The second part presents our recently developed low-scaling GW method, which is im- plemented in the widely used electronic structure code Vienna ab-initio simulation package (VASP). This new implementation allows for fast quasiparticle calculations with a scaling that is cubic in the system size and linear in the number of k-points used to sample the Brillouin zone. This overcomes the shortcomings of conventional GW implementations that are usually restricted to small systems. It also addresses some of the drawbacks of the first implementations of the space-time method such as the large memory requirements. An exhaustive explanation regarding how such low scaling can be achieved is given. Then, the success of this method in predicting the quasiparticle energies and spectral functions of typical insulators, semiconductors and metals is discussed. The results are compared to conventional GW calculations, showing very good agreement. In addition, the new method is also successfully applied to predict the ionization potentials and electron affinities of 100 closed shell molecules. The results are com- pared with the ones from experiments and other electronic structure packages. Overall, good agreement is obtained. In order to achieve this agreement, we correct for finite basis set errors as well as errors introduced by periodically repeated images. For the slightly larger deviations on the electron affinities, tentative explanations are given. Due to the low scaling of our new method, we believe that this method has great potential for applications, in particular for large 5 unit cells. This will greatly help to establish methods beyond DFT in the realm of materials modeling. However, the spin-orbit coupling (SOC) has not yet been implemented in our low-scaling GW method, which is instead covered by the conventional GW implementation in the VASP code. Hence, some modifications are made to the conventional GW implementation. This makes possible the applications of the conventional GW method to moderately large systems where SOC is non-negligible. This is achieved by employing the optimized imaginary time/frequency grids and efficient discrete Fourier transformations which are the key features
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