he GW approximation of many-body perturbation theory (MBPT) is a widespread tool for the prediction of electronic excitations in materials and chemical compounds. It has been particularly effective for the calculation of band gaps of extended systems, and has recently been shown to well describe binding energies of molecules at large, even down to core levels. However, GW does not come without its own share of limitations: self-consistency (or lack thereof), prediction of photoemission satellites and description of strong correlations are all known issues of GW. In an attempt to address these issues, the work presented in this thesis revolves around the assessment of GW and beyond-GW theories. Spherical atoms were chosen as a test set due to not only being thoroughly characterized experimentally, but also requiring a relatively modest effort when it comes to implementing MBPT without introducing any further approximate technique (e.g. pseudopotentials, frequency representations, infinite basis set extrapolations). Depending on their end goal, the theories being studied present different formulations within MBPT. For the computation of quasi-particle (QP) properties, the framework of vertex corrections prescribed by Hedin's equations has proven ideal. Conversely, the incoherent part of photoemission spectra, populated by the so-called satellite peaks, sees the failure of not only GW, but also of many standard approached based on simple self-energy approximations in conjunction with the many-body Dyson equation. Within this broader context, one initial chunk of work presented in this thesis deals with the field of vertex corrections for the computation of QP properties (and Green's functions at large). The GW, 2nd Born and GW+second-order screened exchange (GW+SOSEX) self-energies were considered and benchmarked. These self-energies are simple enough to be written as a sum of just a few Feynman diagrams, each having a straightforward translation into analytical formulae. The full time-dependent Hartree-Fock (TDHF) vertex was also studied, which entails the inclusion of ladder diagrams to infinite order in the self-energy. This required the more involved solution of a Bethe-Salpeter equation (BSE)-like problem. These self-energies were used to solve perturbatively the Dyson equation of MBPT, yielding atomic ionization potentials (and valence states) which were compared against experimental data. Furthermore, the self-energies were employed for the self-consistent solution of the linearized Sham-Schlueter (LSSE) equation, producing exchange-correlation (xc) potentials within the Kohn-Sham density functional theory (KS-DFT) framework; providing a further benchmark, these potentials were compared against exact ones available in the literature. This thesis collects the details of the implementation of these methods for spherical atoms, and presents the results of the benchmark calculations. The starting point dependence is addressed for each of the self-energy approximations, and is found to still be a relevant issue. Nonetheless, an increased accuracy with respect to GW can indeed be attained with the inclusion of different sets of diagrams in the self-energy. Moving away from QP properties, a second chunk of work presented in this thesis has instead been concerned with satellite prediction in photoemission spectra. This work is the product of a joint effort together with the Theoretical Spectroscopy Group of the Laboratoire des Solides Irradiés of the École Polytechnique (Palaiseau, France) led by Dr. Lucia Reining. Acknowledging the inadequacy of the Dyson equation with simple self-energy approximations in this context, we tackled this problem by opting for a less traditional route involving a direct expansion of the interacting Green's function of MBPT.
L'approssimazione GW della many-body perturbation theory (MBPT) è uno strumento largamente diffuso per lo studio delle eccitazioni elettroniche. È stata particolarmente efficace per il calcolo dei gap di banda di sistemi estesi, e recentemente ha dimostrato di poter ben descrivere le energie di legame delle molecole in generale, addirittura fino ai livelli di core. L'approssimazione GW soffre però di alcune limitazioni: la self-consistenza (o la sua mancanza), la predizione dei satelliti di fotoemissione e la descrizione delle strong correlations sono tutti problemi noti di GW. Con lo scopo di superare questi limiti, il lavoro presentato in questa tesi riguarda il benchmark di GW e di approssimazioni al di là di esso. Sono stati utilizzati atomi sferici come test set, sia per la loro ampia caratterizzazione sperimentale, sia perché richiedono uno sforzo relativamente modesto nell'implementazione della MBPT senza dover introdurre ulteriori tecniche approssimate (es.: pseudopotenziali, rappresentazioni in frequenza, estrapolazioni al set di base infinito). A seconda del loro scopo, le teorie studiate presentano diverse formulazioni all'interno della MBPT. Per il calcolo di proprietà di quasi-particella (QP), lo schema delle correzioni di vertice prescritto dalle equazioni di Hedin si è rivelato ideale. La parte incoerente dello spettro di fotoemissione, popolata dai cosiddetti picchi satellite, vede invece il fallimento non solo di GW, ma anche di tanti approcci tradizionali basati su semplici approssimazioni della self-energia congiuntamente all'equazione di Dyson. In questo contesto, una parte iniziale di lavoro presentato in questa tesi riguarda il campo delle correzioni di vertice per il calcolo di QP. Innanzitutto, sono state considerate e sottoposte a benchmark le self-energie GW, 2nd Born (2B) e GW+second-order screened exchange (GW+SOSEX). Queste self-energie sono sufficientemente semplici da essere espresse come una somma di pochi diagrammi di Feynman, ciascuno avente una diretta traduzione in formule analitiche. È stato anche studiato il vertice time-dependent Hartree-Fock (TDHF) completo, il quale comporta l'inclusione di diagrammi ladder di ordine infinito nella self-energia. Esso ha richiesto la risoluzione, più involuta, di un problema di tipo Bethe-Salpeter equation (BSE). Queste self-energie sono state utilizzate per risolvere perturbativamente l'equazione di Dyson della MBPT, producendo potenziali di ionizzazione (e stati di valenza) atomici che sono stati confrontati con dati sperimentali. Inoltre, le self-energie sono state impiegati per la soluzione dell'equazione Sham-Schlueter linearizzata (LSSE), la quale ha prodotto potenziali di scambio e correlazione per lo schema Kohn-Sham (KS) della density functional theory (DFT); questi potenziali sono stati confrontati con quelli esatti presenti in letterature, fornendo un ulteriore benchmark. Questa tesi raccoglie i dettagli dell'implementazione di questi metodi per gli atomi sferici, e presenta i risultati dei benchmark. La dipendenza dal calcolo di partenza è considerata per ciascuna delle approssimazioni della self-energia, e si è trovata essere ancora rilevante. Nondimeno, una maggiore accuratezza rispetto a GW può essere raggiunta con l'inclusione di diversi insiemi di diagrammi nella self-energia. Andando oltre le QP, una seconda parte di lavoro presentata in questa tesi è invece incentrata sulla predizione dei satelliti. Questo lavoro è il prodotto di uno sforzo condiviso con il Theoretical Spectroscopy group del Laboratoire des Solides Irradiés dell'École Polytechnique (Palaiseau, France) diretto dalla Dott.ssa Lucia Reining. Abbiamo affrontato questo problema optando per un approccio meno tradizionale basato, su uno sviluppo in serie direttamente della funzione di Green many-body.
Studio di approssimazioni Many-body perturbation theory oltre il GW / Simone Vacondio , 2023 Oct 30. 35. ciclo, Anno Accademico 2021/2022.
Studio di approssimazioni Many-body perturbation theory oltre il GW
VACONDIO, SIMONE
2023
Abstract
he GW approximation of many-body perturbation theory (MBPT) is a widespread tool for the prediction of electronic excitations in materials and chemical compounds. It has been particularly effective for the calculation of band gaps of extended systems, and has recently been shown to well describe binding energies of molecules at large, even down to core levels. However, GW does not come without its own share of limitations: self-consistency (or lack thereof), prediction of photoemission satellites and description of strong correlations are all known issues of GW. In an attempt to address these issues, the work presented in this thesis revolves around the assessment of GW and beyond-GW theories. Spherical atoms were chosen as a test set due to not only being thoroughly characterized experimentally, but also requiring a relatively modest effort when it comes to implementing MBPT without introducing any further approximate technique (e.g. pseudopotentials, frequency representations, infinite basis set extrapolations). Depending on their end goal, the theories being studied present different formulations within MBPT. For the computation of quasi-particle (QP) properties, the framework of vertex corrections prescribed by Hedin's equations has proven ideal. Conversely, the incoherent part of photoemission spectra, populated by the so-called satellite peaks, sees the failure of not only GW, but also of many standard approached based on simple self-energy approximations in conjunction with the many-body Dyson equation. Within this broader context, one initial chunk of work presented in this thesis deals with the field of vertex corrections for the computation of QP properties (and Green's functions at large). The GW, 2nd Born and GW+second-order screened exchange (GW+SOSEX) self-energies were considered and benchmarked. These self-energies are simple enough to be written as a sum of just a few Feynman diagrams, each having a straightforward translation into analytical formulae. The full time-dependent Hartree-Fock (TDHF) vertex was also studied, which entails the inclusion of ladder diagrams to infinite order in the self-energy. This required the more involved solution of a Bethe-Salpeter equation (BSE)-like problem. These self-energies were used to solve perturbatively the Dyson equation of MBPT, yielding atomic ionization potentials (and valence states) which were compared against experimental data. Furthermore, the self-energies were employed for the self-consistent solution of the linearized Sham-Schlueter (LSSE) equation, producing exchange-correlation (xc) potentials within the Kohn-Sham density functional theory (KS-DFT) framework; providing a further benchmark, these potentials were compared against exact ones available in the literature. This thesis collects the details of the implementation of these methods for spherical atoms, and presents the results of the benchmark calculations. The starting point dependence is addressed for each of the self-energy approximations, and is found to still be a relevant issue. Nonetheless, an increased accuracy with respect to GW can indeed be attained with the inclusion of different sets of diagrams in the self-energy. Moving away from QP properties, a second chunk of work presented in this thesis has instead been concerned with satellite prediction in photoemission spectra. This work is the product of a joint effort together with the Theoretical Spectroscopy Group of the Laboratoire des Solides Irradiés of the École Polytechnique (Palaiseau, France) led by Dr. Lucia Reining. Acknowledging the inadequacy of the Dyson equation with simple self-energy approximations in this context, we tackled this problem by opting for a less traditional route involving a direct expansion of the interacting Green's function of MBPT.File | Dimensione | Formato | |
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