Figure 0.2: Organic solar cells can be not only light and semitransparent, but also exible. Innovative applications, such as organic solar cell sheets adhered at vitreous building fronts or car roofs, come into reach. Figure reproduced by courtesy of Fraunhofer Center for Organic

Acknowledgement

Concerning my colleagues at the Néel Institut, I want to thank Valerio Olevano, who is always ready to pull out a pencil for a derivation. Moreover, there is also Claudio Attaccalite, who is not only of great help in physical questions, but who is also more than talented in connecting people. For really interesting discussions in the L_sim group, special thanks goes to my co-supervisor Thierry Deutsch and Ivan Duchemin at CEA. Most importantly, I want to thank my supervisor Xavier Blase, who accompanied and guided my work for the last three years in a very responsible and reassuring manner. I am thankful that he gave me the opportunity to work on this highly interesting topic and to be involved in the FIESTA project since the beginning. It was a really multifaceted subject and it was great to have an insight into programming, code testing and physics applications, all at the same time. It was a big pleasure to work with him on this project and his enthusiastic and motivating manner resulted in very fruitful discussions. There is just one major issue with him, namely that he refused eating my daily prepared cakes at the end of the rst year. Just as Paul Boulanger pointed out so tellingly today: A mind is such a terrible thing to waste, a waist is a terrible thing to mind. This is just one of Paul's pieces of wisdom and I will surely miss our lunch breaks. There, on a toujours craint le pire and we discussed everything from black holes to shamanism (which is, in the end, probably the same). In this context, special thanks for his support, helpfulness and most didactic explanations.

The three years of my doctorate in Grenoble represent one of the periods, which inuenced me the most in my life. It has been a challenge, but at the same time a great possibility to get in touch with another cultural background and to integrate into the French system (it counts probably as a kind of success that the carefully collected bumps and scrapes in my car do not bother me anymore). Moreover, with Grenoble being a small, but quite international melting pot, I had the great opportunity to meet very interesting people, coming as far as Argentina (even though mainly from Italy). This group of internationals (and some few courageous French) has grown to be a family and is, due to its openness and cordiality, a real enrichment for me. I do not want to pick out specic persons, I just want to thank all of you for the nice moments we spent together. With your diverse and fascinating ways of life, you denitely inuenced mine. I learned a lot about me and gained important insights (the recipe of spaghetti carbonara -where cream and even onions are strictly forbidden! -is just one example).

Apart from the Grenoble maa, which is now establishing as a well-connected European network with headquarters in Paris, Barcelona and London, I want to devote special thanks to my landlords and friends, the Hottebart family. It is a real pleasure to see you growing up, Romane, Eliette and Gabrielle, and you are an important factor making my stay in Grenoble so enjoyable (even though I am a bit jealous that Gabrielle now speaks better French than me and that you all have more etoiles in skiing). I was always bienvenue chez les Schti and even though hundreds of kilometers away from my family, I felt at home. Thank you very much for the exceptional experiences and your warm and caring manner. Also, I do not want to forget Franzi and our weakly Abendbroten. I will denitely miss these evenings, full of laughter, happiness and funny ideas. You even went so far as to have a surgery in order to bear me company this makes it nearly impossible to express my gratefulness in words. Last, but not least, I want to thank, along with Christoph and Martina, of course my parents and my grandmother who have always been a great support. You surely know me the best and as such, you also know that I am a blink away from the deadline (and a Abendbrot with Franzi). There is thus not enough time left for a profound contemplation, however, this last sentence of a three years work is devoted to you.

For job oers in the Grenoble region, please contact me at (0,1,1,1,0,0,1).

[strangers to second quantization call the hotline for redirection]

Résumé

Cette thèse se propose d'explorer les mérites d'une famille d'approches de simulation quantique ab initio, les théories de perturbation à N-corps, pour l'exploration des propriétés électroniques et optiques de systèmes organiques [START_REF] Hedin | New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas Problem[END_REF][START_REF] Strinati | Application of the Green's functions method to the study of the optical properties of semiconductors[END_REF]. Nous avons étudié en particulier l'approximation dite de GW et l'équation de Bethe-Salpeter, très largement utilisées dès les années soixante pour les semiconducteurs de volume [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF][START_REF] Aryasetiawan | The GW method[END_REF][START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF], mais dont l'utilisation pour les systèmes organiques moléculaires est très limitée. L'étude de quelques cas d'intérêt pour le photovoltaïque organique, et en particulier de petites molécules pour lesquelles sont disponibles des données expérimentales ou des résultats issus d'approches de chimie quantique, nous ont permis de valider ces approches issues de la physique du solide.

Ce doctorat s'inscrit dans le cadre du développement d'un outil de simulation quantique spécique (le projet FIESTA) dont l'objectif est de combiner les formalismes GW et Bethe-Salpeter avec les techniques de la chimie quantique, c'est-à-dire en particulier l'utilisation de bases localisées analytiques (bases gaussiennes) et des approches de type résolution de l'identité pour le traitement des intégrales Coulombiennes [START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF][START_REF] Faber | First-principles GW calculations for DNA and RNA nucleobases[END_REF][START_REF] Blase | Charge-transfer excitations in molecular donor-acceptor complexes within the many-body Bethe-Salpeter approach[END_REF]. Ce code est aujourd'hui massivement parallélisé, permettant, au delà des études de validation présentées dans ce travail de thèse, l'étude de systèmes complexes comprenant plusieurs centaines d'atomes. En cours de développement, l'incorporation d'approches hybrides combinant mécanique quantique et écrantage à longue portée par des approches modèles de milieu polarisable m'a permis d'une part de me familiariser avec le code et le développement méthodologique, et permet d'autre part d'envisager l'étude de systèmes réalistes en couplage avec leur environnement.

Le manuscrit s`ouvre sur une introduction au photovoltaïque organique an de mettre en lumière les questionnements spéciques qui requièrent le développement de nouveaux outils théoriques à la fois ables en terme de précision et susamment ecaces pour traiter des systèmes de grande taille. Le premier chapitre est d'ordre méthodologique et rappelle les fondements des techniques ab initio de type champ-moyen (Hartree, Hartree-Fock et théorie de la fonctionnelle de la densité). En partant des principes de la photoémission, les théories de perturbation à N-corps et la notion de quasi-particule sont ensuite introduites, conduisant aux équations de Hedin et aux approximations GW et COHSEX [START_REF] Hedin | On correlation eects in electron spectroscopies and the GW approximation[END_REF]. De même, à partir de la compréhension d'une expérience d'optique, le traitement des interactions électron-trou est présenté, menant à l'équation de Bethe-Salpeter. Le chapitre 2 introduit brièvement les spécicités techniques liées à l'implémentation des formalismes GW et Bethe-Salpeter.

Les propriétés analytiques des bases gaussiennes et les principes mathématiques derrière les techniques de type résolution de l'identité et déformation de contour, sont brièvement décrites. Le troisième chapitre présente les résultats scientiques obtenus durant cette thèse.

Le cas paradigmatique d'un polypeptide model nous permettra de discuter des spécicités de l'approche GW appliquée à des systèmes moléculaires an d'obtenir des énergies de quasiparticule de bonne qualité [START_REF] Faber | Many-body Green's function GW and Bethe-Salpeter study of the optical excitations in a paradigmatic model dipeptide[END_REF]. De même, l'utilisation de l'équation de Bethe-Salpeter pour l'obtention du spectre optique de ce système sera présentée, ainsi que le cas d'une famille de colorants [START_REF] Faber | Many-body Green's function study of coumarins for dye-sensitized solar cells[END_REF] d'importance pour les cellules de Graetzel (les coumarines). Finalement, nous explorons dans le cas du fullerène C 60 [12] et du graphène le calcul des termes de couplage électron-phonon dans le cadre de l'approche GW, c'est-à-dire au delà des approches standards de type théorie de la fonctionnelle de la densité. Notre étude vise à vérier si une approximation statique et à écrantage constant au premier ordre permet de garder la qualité des résultats GW pour un coût numérique réduit. Après la conclusion, les appendices donnent le détail de certaines dérivations.

Abstract

The present thesis aims at exploring the properties and merits of the ab initio Green's function many-body perturbation theory (MBPT) GW and Bethe-Salpeter formalisms [START_REF] Hedin | New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas Problem[END_REF][START_REF] Strinati | Application of the Green's functions method to the study of the optical properties of semiconductors[END_REF], in order to provide a well-grounded and accurate description of the electronic and optical properties of condensed matter systems. While these approaches have been developed for extended inorganic semiconductors and extensively tested on this class of systems since the 60s [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF][START_REF] Aryasetiawan | The GW method[END_REF][START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF], the present work wants to assess their quality for gas phase organic molecules, where systematic studies still remain scarce. By means of small isolated study case molecules, we want to progress in the development of a theoretical framework, allowing an accurate description of complex organic systems of interest for organic photovoltaic devices. This represents the main motivation of this scientic project and we prot here from the wealth of experimental or high-level quantum chemistry reference data, which is available for these small, but paradigmatic study cases.

This doctoral thesis came along with the development of a specic tool, the FIESTA package [START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF][START_REF] Faber | First-principles GW calculations for DNA and RNA nucleobases[END_REF][START_REF] Blase | Charge-transfer excitations in molecular donor-acceptor complexes within the many-body Bethe-Salpeter approach[END_REF], which is a Gaussian basis implementation of the GW and Bethe-Salpeter formalisms applying resolution of the identity techniques with auxiliary bases and a contour deformation approach to dynamical correlations. Initially conceived as a serial GW code, with limited basis sets and functionalities, the code is now massively parallel and includes the Bethe-Salpeter formalism. The capacity to perform calculations on several hundreds of atoms to moderate costs clearly paves the way to enlarge our studies from simple model molecules to more realistic organic systems. An ongoing project related to the development of discrete polarizable models accounting for the molecular environment allowed me further to become more familiar with the actual implementation and code structure.

The manuscript at hand is organized as follows. In an introductory chapter, we briey present the basic mechanisms characterizing organic solar cells, accentuating the properties which seek for an accurate theoretical description in order to provide some insight into the factors determining solar cell eciencies. The rst chapter of the main part is methodological, including a discussion of the principle features and approximations behind standard mean-eld techniques (Hartree, Hartree-Fock, density functional theory). Starting from a description of photoemission experiments, the MBPT and quasiparticle ideas are introduced, leading to the so-called Hedin's equations, the GW method and the COHSEX approach [START_REF] Hedin | On correlation eects in electron spectroscopies and the GW approximation[END_REF]. In order to properly describe optical experiments, electron-hole interactions are included on top of the description of inter-electronic correlations. In this context, the Bethe-Salpeter formalism is introduced, along with an excursus on time-dependent density functional theory. Chapter 2 briey presents the technical specications of the GW and Bethe-Salpeter implementation in the FIESTA package. The properties of Gaussian basis sets, the ideas behind the resolution of the identity techniques and nally the contour deformation approach to dynamical correlations are discussed. The third chapter deals with the results obtained during this doctoral thesis.

On the electronic structure level, a recent study on a paradigmatic dipeptide molecule [START_REF] Faber | Many-body Green's function GW and Bethe-Salpeter study of the optical excitations in a paradigmatic model dipeptide[END_REF] will be presented. Further, also its optical properties will be explored, together with an in-depth discussion of charge-transfer excitations in a family of coumarin molecules [START_REF] Faber | Many-body Green's function study of coumarins for dye-sensitized solar cells[END_REF]. Finally, by means of the Buckminster fullerene C 60 [12] and the two-dimensional semi-metal graphene, we will analyze the reliability of two many-body formalisms, the so-called static COHSEX and constant-screening approximation, for an ecient calculation of electron-phonon interactions in organic systems at the MBPT level. After a short conclusion, the Appendix containing details and derivations of the formalisms presented before closes this work.

Conclusion & Perspectives

A. Appendix: Details and derivations Many-body perturbation theory: towards organic photovoltaics

The present doctoral thesis was initiated having in mind a computational framework which would allow to simulate properties important for organic photovoltaics and to steer experimental research in this eld. As such, in order to understand the challenges organic photovoltaics represents to computational physicists, we rst briey discuss the basic principles of organic photovoltaic cells. For a comprehensive overview, the reader is referred to Refs. [START_REF] Benanti | Organic solar cells: An overview focusing on active layer morphology[END_REF][START_REF] Kippelen | Organic photovoltaics[END_REF][START_REF] Su | Organic photovoltaics[END_REF]. This is followed by a short comparison of standard ab initio methods, where we focus on approaches available to calculate electronic structure and optical absorption properties. The presented computational techniques will be compared with special emphasis on accuracy and eciency. In this context, we dene accuracy through the comparison to experimental or correlated quantum chemistry results. Concerning eciency, we take a pragmatic view, where we attempt to nd a proper ratio between accuracy and computing time in the light of the complexity of the targeted systems. Subsequent to the paragraph on computational methods, the organization of the present manuscript follows.

Organic photovoltaics: basic concepts Photovoltaics: a promising renewable energy source

The search for sustainable and unlimited energy sources is one of the key challenges of the 21st century. Among diverse renewable energy sources, such as wind power, biomass and hydroelectricity, photovoltaics takes a prominent place. In photovoltaic cells, sun light is directly converted into electricity. This is basically achieved through the following working principle: a semiconducting material absorbs photons, which have an energy exceeding the semiconductor's gap. This creates pairs of bound positive and negative charges, so-called excitons. These bound pairs then dissociate, yielding free charges that drift to the respective electrodes through a built-in potential. This provides a closed circuit with direct current ow.

Solar cells of the rst generation utilize bulk crystalline silicon as absorbing semiconducting material. Most commercial solar cells to date belong to this type and power conversion eciencies of over 20% are provided. However, the production process is complicated and material costs of perfectly crystalline silicon are high. Therefore, much research eort is spent on the exploration of novel solar cell concepts. Among the most promising ones are e.g. low-cost thin-lm solar cells, where the use of very eciently absorbing direct band gap semiconductors, such as e.g. amorphous silicon or chalcopyrite compound semiconductors (CIS, CIGS), allows for thinner layers and thus production time and material savings. Another example are multijunction concepts, which harvest, through the stacking of several singlejunction sub-cells with varying gaps, a broader range of the solar spectrum and thereby obtain record power conversion eciencies of up to 45%. A compilation of to date power conversion eciencies for various types of solar cells is provided by Fig. (0.1). Among these trend-setting directions which have risen much attention recently are organic solar cells. It is these types of cells, which we will focus on in the following.

1

Organic photovoltaic cells: functional principle and characteristics Perspectives and challenges In organic photovoltaic cells, the traditional silicon absorbing layer is replaced by organic semiconducting materials. In particular, thin-lms of polymers or molecules, mainly composed of carbon, hydrogen, oxygen and nitrogen, are applied. The utilization of organic materials as absorbers oers dierent advantages compared to standard silicon cells. First of all, a low-cost mass production seems possible. This is due to the fact that the needed raw materials are abundantly available and that material costs are low. In 1 For a comprehensive introduction into solar cells concepts and technology, see Ref. [START_REF] Mertens | Photovoltaik -Lehrbuch zu Grundlagen, Technologie und Praxis[END_REF].

Contents

Functional principle and characteristics The utilization of organic semiconductors as active absorbing layers in organic photovoltaic cells provides the possibility of cheap, light-weight and exible end products, entering new emerging consumer markets as compared to standard inorganic solar cells. Apart from these application related advantages, the organic materials used represent a highly interesting challenge for fundamental research. Consisting of weakly interacting molecular units rather than strongly covalently bound atoms, organic semiconductor crystals are at the interface between solid state physics and quantum chemistry theories, urging for new concepts. Organic semiconductors, which are characterized by delocalized πconjugated electrons, possess inherent characteristics distinguishing them from their inorganic analogues. In order to mention only a small excerpt of properties, one deals with: narrow electronic bands, high eective masses, strong electron-phonon coupling, a high degree of structural disorder, discrete and narrow absorption peaks, and strongly bound electron-hole pairs. These aspects signicantly inuence the semiconductor properties. By way of example, the strong electron-phonon coupling in these materials, i.e. the interplay between the electronic and the atomic structure [START_REF] Kippelen | Organic photovoltaics[END_REF], is one of the reasons leading to charge localizations. As a consequence, free charge carrier transport may rather be described in terms of a hopping of polarons than by standard band models of nearly free electrons. Likewise, the strong electronhole binding energy necessitates new concepts in order to eciently separate the electron and the hole and to create free charge carriers. For the sake of comparison, the electronhole binding energy in inorganic semiconductors 3 is usually of the order of some few meV and consequently room temperature, corresponding to 25 meV, is sucient to immediately dissociate the exciton after its creation. In organic semiconductors, however, the excitonic binding energy is one order of magnitude larger, typically around 0.5-1 eV [START_REF] Su | Organic photovoltaics[END_REF]. This reects the fact that organic semiconductors usually have a low dielectric constant [START_REF] Su | Organic photovoltaics[END_REF], namely ε = 3-4, and thus the electron-hole interaction is not as eciently screened by the surrounding medium as in inorganic semiconductors. Moreover, due to weak intermolecular interactions, the excited electron and hole have a tendency to stay on the molecule they have been created

on. This results in short distances between the charges and consequently a strong Coulomb interaction. As a consequence, an additional driving force is needed to separate the electron and the hole. To date, organic solar cells therefore apply donor-acceptor interfaces consisting of two dierent materials. One of these materials, the so-called donor, is characterized by a low ionization energy (IE), i.e. it easily gives electrons and stabilizes holes. The second material, the so-called acceptor, has a high electron anity (EA) and thus eciently takes up electrons. A simplied scheme of such a bilayer organic solar cell is provided by Fig. (0.3), 3 By way of example, the excitonic binding energy of silicon is around 15 meV [START_REF] Hull | Properties of crystalline silicon[END_REF].

∆

Contents to understand the associated challenges for computational methods, a discussion of these two steps is provided in the following.

Absorption In general, there are two main losses related to the absorption step in solar cells, both directly connected to the optical gap. First, energy is lost through the thermalization of excess photon energy. Instead of creating electrons and holes in the highest occupied (HOMO) and the lowest unoccupied molecular orbital (LUMO), radiation with an energy overcoming the optical gap excites electrons from lower occupied to higher unoccupied states.

These higher excited states usually relax extremely fast in the lowest excitonic state, releasing the excess energy as heat through lattice vibrations. Consequently, the maximum electrical energy which one can gain per electron corresponds to the optical gap, favoring wide band gap semiconductors. This condition, however, disagrees with the second source of energy losses, namely the non-absorption of low-energy photons. Since only photons with an energy higher than the optical gap contribute to the electron-hole pair generation, small gaps would be desirable in order to maximize the amount of created electron-hole pairs. One sees that the choice of an appropriate gap size is very complicated and a careful balance between thermalization and absorption losses has to be found. As a rough estimate, the Shockley-Queisser limit predicts for single-junction cells an optimum gap of 1.4 eV, yielding a maximum theoretical eciency of around 30% [START_REF] Mertens | Photovoltaik -Lehrbuch zu Grundlagen, Technologie und Praxis[END_REF][START_REF] Shockley | Detailed balance limit of eciency of p-n junction solar cells[END_REF]. Since the maximum is rather broad, materials with gaps ranging from 0.8-1.7 eV are usually suitable for single-junction solar cells [START_REF] Falk | Physik und Technologie der Solarzellen[END_REF].

Organic semiconductors have fundamental gaps which are typically signicantly larger and, even though the electron-hole binding energy can amount to some hundreds of meV, the optical gaps are often too large to capture the low-energy part of the visible radiation. Therefore, much eort is put in the reduction of the optical gap. From a theoretical point of view, these considerations call for an accurate calculation of the fundamental gap and optical absorption spectra, in order to discriminate more or less suited organic systems.

Exciton dissociation For an eective charge separation, the energy which is gained by transferring the electron from the donor to the acceptor should be as high as possible. [START_REF] Aryasetiawan | The GW method[END_REF] This can be favored by maximizing the band oset ∆ between the LUMO (HOMO) of the donor and the LUMO (HOMO) of the acceptor (see Fig. 0.3b). Consequently, an ecient exciton dissociation necessitates a small energy dierence between the HOMO of the donor and the LUMO of the acceptor. However, it is exactly this dierence which determines the open circuit voltage V oc , which in turn is directly related to the power conversion eciency. An opening of the two gaps could cure the problem and increase both and V oc . This, however, would imply that a smaller part of the solar spectrum is absorbed.

Clearly, the presented energy level diagram is a strong simplication of real photovoltaic devices, since one should compare excitonic energy osets. Nevertheless, this simple model already points out the importance of band osets and gaps, and the diculty to nd an ideal trade-o between the dierent requirements. One sees that the design of ecient organic solar 4 Of course, energy has to be conserved in the system. The exact role of the electronic excess energy in driving the dissociation is still very debated [START_REF] Bakulin | The Role of Driving Energy and Delocalized States for Charge Separation in Organic Semiconductors[END_REF][START_REF] Grancini | Hot exciton dissociation in polymer solar cells[END_REF][START_REF]Measuring internal quantum eciency to demonstrate hot exciton dissociation[END_REF].

Contents cells represents a very complex optimization problem and that the choice of high-performance material combinations is a major challenge. Here, calculations could oer a guideline and help experiment to sort out appropriate semiconductors without actually testing every material in the laboratory. This calls for an accurate theoretical description of the electronic structure and optical absorption properties and is, as it will be discussed below, a demanding objective for ab initio theories.

Organic solar cells: devices

In order to concretize the diculties for theory arising from organic photovoltaic cells, two of the most successful realizations of organic-based solar cells are shortly presented in the following: all-organic bulk heterojunction cells and dye-sensitized solar cells (DSSCs). For a complete overview, the reader is referred to Refs. [START_REF] Benanti | Organic solar cells: An overview focusing on active layer morphology[END_REF][START_REF] Kippelen | Organic photovoltaics[END_REF][START_REF] Su | Organic photovoltaics[END_REF][START_REF] Chidichimo | Organic solar cells: problems and perspectives[END_REF]. In the case of all-organic bulk heterojunction cells, the organic semiconductor serves for light absorption and charge carrier transport, whereas for DSSCs the organic material is only used as absorber. The cell structures are quite dierent, however, they have in common that light absorption creates bound electron-hole pairs, which have to be dissociated. Therefore, diverse types of organic solar cells are often grouped together and called exciton solar cells.

All-organic bulk heterojunction cells The rst all-organic solar cells, which applied the concept of an active donor-acceptor interface for the exciton dissociation, were so-called bilayer cells. Even though their device structure is of course more complex, it nevertheless resembles very much the extremely simplied scheme presented in As discussed in the following, bilayer structures are not ideal to provide an ecient light absorption and exciton dissociation at the same time. Instead, all-organic bulk heterostructure solar cells are most popular device structures at present (see Fig. 0.4). The number of excitons which actually diuse to the donor-acceptor interface before recombination is crucial for the power conversion eciency. In organic materials, the diusion length, i.e. the average distance between excitation and recombination, is only around 10 nm, while usual device structures apply 100-200 nm thick donor/acceptor layers in order to avoid an incomplete absorption [START_REF] Su | Organic photovoltaics[END_REF].

As a result, light is indeed absorbed in the whole layer, however, only absorption in a thin region around the interface actually contributes to the photon-to-electron conversion, whereas most of the incident light remains unused. Bulk heterojunction cells tackle this problem by both allowing for an eective absorption through nm thick absorption layers and by reducing the average distance an exciton has to travel in order to reach the donor-acceptor interface. This is achieved by replacing the two at donor/acceptor layers by a single bulk absorption layer consisting of a blend of the donor and the acceptor material (see Fig. 0.4b).

That way, not only the interfacial area is greatly enlarged, but also the average exciton travel distance to the interface is of the order of the diusion length. A percolation pathway for is used as acceptor, combined with a semiconducting polymer as donor. The added side group of PC 61 BM enhances its miscibility compared to C 60 [13, 17].

Figures taken from Ref. [START_REF] Zeman | Solar Cells[END_REF].

date, Grätzel's approach is commonly applied and DSSCs are therefore often termed Grätzel cells. Eciencies of up to 12% are reached (see Fig. 0.1), where mainly ruthenium-based organometallic complexes are used as absorbers (see Fig. 0.5a). Nevertheless, there is a trend towards pure organic absorbers, which are cheaper, easier to synthesize and free from resource limitations. Intense research is conducted in this direction, since solar cell eciencies are to date usually smaller as compared to cells based on organo-metallic complexes [START_REF] Hardin | The renaissance of dye-sensitized solar cells[END_REF]. Later in this work, a study on the optical properties of all-organic coumarin dyes is presented, a family of molecules which recently led to DSSCs with very promising eciencies [START_REF] Hara | Molecular Design of Coumarin Dyes for Ecient Dye-Sensitized Solar Cells[END_REF][START_REF] Hara | Design of new coumarin dyes having thiophene moieties for highly ecient organic-dye-sensitized solar cells[END_REF][START_REF] Hara | Oligothiophene-Containing Coumarin Dyes for Ecient Dye-Sensitized Solar Cells[END_REF][START_REF] Wang | A High-Light-Harvesting-Eciency Coumarin Dye for Stable Dye-Sensitized Solar Cells[END_REF].

This sections clearly demonstrates that organic photovoltaics makes high demands on computational methods. Apart from the usually experimentally targeted accuracy of 0.1-0.2 eV for electronic levels and optical excitation energies, one has to tackle: a huge number of atoms, hybrid systems (e.g. TiO 2 nanoparticles/dye molecules/liquid electrolyte), complex bulk and interface morphologies showing non-negligible disorder, and a complex interplay between electron-electron, electron-hole and electron-vibrations coupling with similar magnitude.

In the following, the merits and limitations of ab initio computational method available to predict electronic and absorption properties are briey described.

Computational aspects

The photon-to-electron conversion process in organic solar cells is very complex and includes a variety of physical aspects. In this work, and as a rst step into the eld of organic photovoltaics, we concentrate on the calculation of the electronic structure and optical properties of Contents small molecules in the gas phase. We seek an ab initio computational method able to reliably predict these properties, where only fundamental physical constants and the molecular structures are given as input, whereas no reference data from experiment enters. The above listed aspects imply that the sought computational method responds to the following requirements, by being: accurate (within an 0.1-0.2 eV error range as compared to experimental or high level quantum chemistry reference data), ecient (both molecules with up to 100 atoms on a standard computer within one day and several hundreds of atoms on large-scale computers should be feasible), parameter-free, and system-independent.

The two last issues ask for an universal formalism, which works equally well for extended, nite, semiconducting or metallic systems. Universalism is a necessary condition having in mind hybrid systems, such as e.g. inherent to Grätzel cells, or donor-acceptor interfaces in general.

Status report: ab initio electronic structure theories Concerning electronic structure calculations, density functional theory (DF T ), particularly in combination with standard (semi)local exchange-correlation functionals, is an ecient and widely applied tool [START_REF] Hohenberg | Inhomogeneous Electron Gas[END_REF][START_REF] Kohn | Self-Consistent Equations Including Exchange and Correlation Eects[END_REF].

However, errors of the order of several eVs on the electronic energy levels, namely as large as the visible range of the solar spectrum, are not unusual. On the contrary, correlated quantum chemistry methods [START_REF] Jensen | Introduction to Computational Chemistry[END_REF] yield an excellent agreement with experiment, though they are computationally too demanding to treat systems with more than a few tens of atoms.

In this work, we rely on the many-body perturbation theory GW method [START_REF] Hedin | New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas Problem[END_REF], which has proven to be a reliable electronic structure theory formalism for extended systems. Fornite systems, however, systematic studies remain scarce. This is due to the high computational costs related to the straightforward application of periodic boundary condition codes to molecules, necessitating the development of nite-system-specic packages. The FIESTA code [START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF] attempts to address this issue and is conceived as an ecient mean to treat the electronic structure of molecular systems at the many-body perturbation theory GW level.

It provides a real-space atom-centered Gaussian function basis set implementation, which is very suited for the description of molecular systems. Further, Gaussian basis sets are most popular in computational quantum chemistry and thus allow a direct comparison to higher level quantum chemistry reference data. Finally, even though this is beyond the scope of the present thesis, the proposed real-space basis approach straightforwardly opens the way to embedding techniques, where the electronic structure of a single molecule is evaluated by taking into account the surrounding molecules of the organic crystal. Here, space is divided in an active region around the molecule of interest, which is treated with the highest available accuracy, whereas the ambient medium is included at a lower level of theory to reduce the Contents computational eort. This allows to go beyond the gas phase description and to simulate organic semiconductors in a more realistic way.

In this context, we want to point out that a trade-o between accuracy and eciency is inevitable. Conceiving the FIESTA package and choosing input parameters such as e.g. a specic basis set, we deliberately accept certain approximations in order to reduce computational costs, however, always adhering to the targeted maximum 0.1-0.2 eV error range.

Status report: optical absorption formalisms Concerning the calculation of optical absorption properties from rst principles, TDDFT [START_REF] Runge | Density-Functional Theory for Time-Dependent Systems[END_REF][START_REF] Marques | Time-Dependent Density Functional Theory[END_REF], the time-dependent extension to DF T , is very popular. TDDFT provides a computationally feasible scheme and yields, for standard (local) optical excitations, energies and oscillator strengths in very close agreement with experiment and correlated quantum chemistry methods. In the present work, however, we want to particularly address the problem of non-local charge-transfer optical excitations. The latter are neutral excitations of great conceptual importance, where the promoted electron and the respective hole are spatially separated, but still interacting. Even though the exact microscopic mechanisms of the exciton dissociation at the donor/acceptor interface are still strongly debated [START_REF] Bakulin | The Role of Driving Energy and Delocalized States for Charge Separation in Organic Semiconductors[END_REF][START_REF] Caruso | Long-range exciton dissociation in organic solar cells[END_REF][START_REF] Yost | Electrostatic Eects at Organic Semiconductor Interfaces: A Mechanism for "Cold" Exciton Breakup[END_REF], the charge separation process from bound to free carriers is supposed to take place through an intermediate charge-transfer excited state [START_REF] Sariciftci | Photoinduced Electron Transfer from a Conducting Polymer to Buckminsterfullerene[END_REF][START_REF] Schmidt-Mende | Self-Organized Discotic Liquid Crystals for High-Eciency Organic Photovoltaics[END_REF][START_REF] Hardin | The renaissance of dye-sensitized solar cells[END_REF]. There is still no clear picture about the exact process, hence calling for the accurate calculation of the excited state properties with respect to the electron-hole distance. Unfortunately, standard TDDFT approaches with (semi)local exchange-correlation functionals fail in correctly describing these excitations by largely underestimating the electron-hole binding energy with increasing separation. Most recently, it has been suggested that a combined GW/BSE approach [START_REF] Strinati | Application of the Green's functions method to the study of the optical properties of semiconductors[END_REF][START_REF] Onida | Electronic excitations: density-functional versus many-body Green's-function approaches[END_REF][START_REF] Bussi | Eects of the Electron-Hole Interaction on the Optical Properties of Materials: the Bethe-Salpeter Equation[END_REF], where rst the underlying GW electronic structure is calculated and then electron-hole interactions are included through the Bethe-Salpeter equations, cures the problem. In this context, we recently obtained, together with other groups, promising results for intermolecular charge-transfer systems, where the excited electron and the hole are located on dierent molecules [START_REF] Blase | Charge-transfer excitations in molecular donor-acceptor complexes within the many-body Bethe-Salpeter approach[END_REF][START_REF] Duchemin | Short-Range to Long-Range Charge-Transfer Excitations in the Zincbacteriochlorin-Bacteriochlorin Complex: A Bethe-Salpeter Study[END_REF]. In this work, we go further and explore the most common case of intramolecular charge-transfer excitations by means of the GW/BSE approach [START_REF] Faber | Many-body Green's function GW and Bethe-Salpeter study of the optical excitations in a paradigmatic model dipeptide[END_REF][START_REF] Faber | Many-body Green's function study of coumarins for dye-sensitized solar cells[END_REF].

Organization of the present thesis

The present thesis is organized as follows. First, in order to set up a common framework, concepts and denitions are briey recapitulated in Chapter 1. The latter is divided into two parts, where the rst one is devoted to electronic structure theory and the second one to optical absorption. We try to establish a clear connection between experimentally and theoretically accessible quantities in order to clarify which properties are calculable with the respective formalisms and which features are not captured.

The chapter starts with a short introduction to photoemission spectroscopy (PES), an experimental technique directly accessing the electronic structure [START_REF] Hüfner | Photoelectron Spectroscopy -Principles and Applications[END_REF][START_REF] Reinert | Photoemission spectroscopy -from early days to recent applications[END_REF]. The latter is a complex many-particle quantity, where the particle number of the interacting-particle system changes through the ejection/injection of single electrons during the PES measurement. As Contents it will be pointed out in the subsequent section of Chapter 1, the electronic structure can in principle be exactly treated in terms of the many-body Schrödinger equation. Due to the enormous number of correlated particles in solids or molecules, an analytic or exact numerical solution is out of reach and approximating electronic structure theories have to be introduced. The key problem is the inclusion of the Coulomb interaction, which correlates the particles on a long-range scale. As important representatives of ab initio electronic structure methods, the Hartree, Hartree-Fock and density-functional theory (DF T ) formalisms will be briey discussed. These are mean-eld approaches, i.e. the Coulomb interaction is taken into account only in an averaged way. The many-body problem of interacting particles is reduced to the description of a single particle moving in an eective Coulomb eld created by the others. Particularly DF T is an ecient and widely applied tool to calculate structural properties. However, as it will be demonstrated many times in this work, it can lead to nonnegligible errors of the order of some eV on the energy of electronic levels. In order to go beyond the mentioned mean-eld approaches, we present in detail the electronic many-body problem from the view point of many-body perturbation theory, where lies the main emphasis of the present thesis. The central quantity of this formalism are Green's functions G, which describe the propagation of electrons and holes in the environment of an interacting manybody system and which are thus perfectly suited to model the PES process. To be precise, the added charges are considered as quasiparticles, i.e. bare electrons/holes surrounded by a positively/negatively charged screening cloud created by interactions with the system. Due to this screening, quasiparticles only weakly interact via the screened Coulomb potential W , rather than via the bare Coulomb potential. Similar to mean-eld approaches, one arrives at an eective single-particle problem, the so-called quasiparticle eigenvalue equation, whose solution gives access to the electronic energies measured in PES experiments. The main ingredient of this equation is the self-energy, an operator which accounts for all interactions beyond Hartree and which is energy-dependent and non-local. In principle, the self-energy can be exactly calculated through the self-consistent solution of a closed set of ve integrodierential equations, namely the Hedin's equations [START_REF] Hedin | New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas Problem[END_REF]. In practice, however, this is not feasible. Following perturbation theory, where the screened Coulomb potential W is taken as perturbation, one retrieves the so-called GW method and the static COHSEX approximation [START_REF] Hedin | On correlation eects in electron spectroscopies and the GW approximation[END_REF].

The second part of the Chapter 1 deals with optical absorption. We rst shortly comment on optical absorption experiments and then focus on the fundamental physical dierences between electronic structure and optical absorption measurements. Whereas in PES, one considers charged excitations, electrons are promoted from occupied to bound unoccupied states in optical absorption experiments. The excited electron and the created hole attract each other through Coulomb forces and, as it will be demonstrated in detail by means of simple absorption models, it is crucial to take into account this interaction in order to accurately describe the absorption process. This implies that, within many-body perturbation theory, one has to go beyond the single quasiparticle (GW ) picture towards an interacting quasielectron-quasihole description. This is achieved by rst calculating the underlying electronic (GW ) structure and then solving the Bethe-Salpeter equations to add electron-hole interactions. Within Contents the domain of DF T , time-dependent DF T is a popular method to directly access optical absorption properties. Since it is, in addition to correlated quantum chemistry approaches, the most commonly used technique, a short review of this formalism closes Chapter 1.

Chapter 2 describes the technical specications of the FIESTA code. General details on Gaussian basis sets with emphasis on auxiliary basis sets are given, along with the introduction of contour deformation techniques. Beyond formalisms and theories, the accuracy of actual calculations may suer from e.g. convergency issues related to the chosen basis sets or e.g. from an inadequate treatment of dynamical correlations through plasmon-pole models.

Such technical concerns have taken an important part in the present doctoral thesis. In the subsequent Chapter 3, namely GW and BSE in practice, we will constantly refer to Chapter 2 in regard to technical terms.

Chapter 3 is divided in three parts. First, we will comment on recent GW results which we obtained for a model dipeptide molecule, a paradigmatic system for which large discrepancies on the optical absorption level arise between the dierent quantum chemistry and TDDFT approaches [START_REF] Faber | Many-body Green's function GW and Bethe-Salpeter study of the optical excitations in a paradigmatic model dipeptide[END_REF][START_REF] Serrano-Andres | Theoretical Study of the Electronic Spectroscopy of Peptides. III. Charge-Transfer Transitions in Polypeptides[END_REF]. Already at the electronic structure level, the chosen molecule is a highly interesting study case. We will demonstrate that the standard GW Scissor approach of calculating the many-body GW correction to the underlying DF T -LDA electronic structure for the HOMO and LUMO only, while correspondingly shifting the remaining states, is not enough. It will be pointed out that for a correct ordering and spacing of the frontier orbitals, it is indispensable to go beyond the Scissor approach by explicitly correcting several states around the gap. This is similar to results we previously obtained within the scope of my Master thesis for the DNA/RNA nucleobases (for details see Ref. [START_REF] Faber | First-principles GW calculations for DNA and RNA nucleobases[END_REF]). In addition, we will propose a reliable alternative to the standard single-shot G 0 W 0 approach, which suers from nonnegligible starting point dependencies [START_REF] Bruneval | Benchmarking the Starting Points of the GW Approximation for Molecules[END_REF]. Within G 0 W 0 , the many-body correction to the DF T eigenvalues is calculated in a single step. This implies much lower computational costs than self-consistent schemes, however, the choice of an appropriate starting point has a large impact on the resulting many-body electronic structure. In the present work, we carry out inexpensive self-consistent COHSEX calculations on top of DF T -LDA electronic structure.

We will show that, even though overestimating fundamental gaps, the self-consistent COHSEX approach readily yields the correct level spacing and ordering. This makes self-consistent COHSEX calculations a reliable starting point for G 0 W 0 (Scissor) calculations, oering the possibility to treat also large systems on the many-body level.

The second section of Chapter 3 deals with the study of optical absorption properties by means of a combined GW/BSE approach, where we mainly focus on the problem of intramolecular charge-transfer excitations. We will present two interesting systems characterized by low-lying charge-transfer excitons, one of them being the already introduced model dipeptide. The other one is a family of coumarin dyes [START_REF] Faber | Many-body Green's function study of coumarins for dye-sensitized solar cells[END_REF], which are both from the fundamental, but also from the applicatory point of view, highly attractive. Apart from the possibility to study charge-transfer excitations, these molecules recently earned much attention due to the high eciencies obtained when applied as all-organic, transition metal free absorbers in Grätzel cells. Moreover, this family of molecules is an impressive example of molecular design through chemical engineering, where the stacking properties and the gap Contents have been systematically optimized [START_REF] Hara | Molecular Design of Coumarin Dyes for Ecient Dye-Sensitized Solar Cells[END_REF][START_REF] Hara | Design of new coumarin dyes having thiophene moieties for highly ecient organic-dye-sensitized solar cells[END_REF]. Both for the dipeptide and for the coumarin molecules, TDDFT with inexpensive (semi)local exchange-correlation functionals has problems to reproduce charge-transfer excitation energies and only parametrized range-separated exchange-correlation functionals cure the problem [START_REF] Yanai | A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)[END_REF][START_REF] Stein | Prediction of charge-transfer excitations in coumarin-based dyes using a range-separated functional tuned from rst principles[END_REF]. In this work, we will show that our combined GW/BSE approach turns out to be a reliable alternative. It succeeds in describing intramolecular charge-transfer excitations in good agreement with the quantum chemistry reference data, while being parameter-free and system independent.

The third issue we address in Chapter 3 is the accurate calculation of electron-phonon coupling strengths from rst principles. The latter takes a prominent place in organic semiconductors, as in various elds of condensed matter physics, and for a realistic modeling of these materials a reliable estimation is indispensable. As this quantity is sensitive to the quality of the underlying electronic structure, it necessitates the accurate calculation of the latter. DF T and especially density functional perturbation theory (DF P T ) provide a most ecient way to access electron-phonon coupling matrix elements. However, recent studies show a signicant underestimation of up to 50% as compared to experiment when using (semi)local exchange-correlation functionals. Further, it has been demonstrated that a manybody treatment on the GW level cures the problem and yields results in close agreement with experiment [START_REF] Faber | Electron-phonon coupling in the C 60 fullerene within the many-body GW approach[END_REF][START_REF] Lazzeri | Impact of the electron-electron correlation on phonon dispersion: Failure of LDA and GGA DFT functionals in graphene and graphite[END_REF][START_REF] Antonius | Many-Body Eects on the Zero-Point Renormalization of the Band Structure[END_REF]. However, due to the lack of ecient techniques as in the case of DF P T , a frozen-phonon approach with step-wise atomic displacements along the phonon modes has to be carried out. This makes the evaluation of electron-phonon coupling matrix elements within GW very expensive. In this work, we propose two alternative many-body approaches, namely the static COHSEX and the constant screening approximation, yielding a much less demanding frozen-phonon framework as compared to a full GW treatment. We will assess their accuracy by means of the Buckminster fullerene C 60 and the most popular two- dimensional semi-metal graphene. Concerning the static COHSEX approach, we will show that it leads, especially in the case of graphene, to non-negligible discrepancies as compared to the GW reference. However, the constant screening approach, where we neglect the variation of the screened Coulomb potential with respect to the atomic displacements, yields results in excellent agreement with the corresponding GW and static COHSEX reference. Even though this approach has still to be validated for a larger variety of systems, the obtained results

are promising and open the door to an inexpensive and reliable man-body treatment of the electron-phonon coupling.

After a short summary of the presented results and the discussion of future perspectives in Chapter 4, the Appendix, comprising detailed information on derivations and theorems, and a list of publications and conference contributions, closes the present thesis. The microscopic processes underlying PES experiments constitute a manifold of complicated many-particle interactions. The ejected photoelectron leaves the sample with a maximum kinetic energy, if the remaining electronic system is completely relaxed. However, in the case where the ionized many-body system is left in an excited state, with neutral excitations such as e.g. electron-hole pairs, the kinetic energy of the photoelectron is lowered and so-called satellite peaks occur in the measured spectrum. This will be discussed in more detail later in this chapter. Apart from these intrinsic losses, the photoelectron can also loose energy on its way to the surface by creating secondary electrons through inelastic scattering processes [START_REF] Onida | Electronic excitations: density-functional versus many-body Green's-function approaches[END_REF][START_REF] Hedin | Electron correlation: Keeping close to an orbital description[END_REF]. In order to facilitate the theoretical description of the photoemission process, one usually neglects these extrinsic losses and assumes the sudden approximation, where the interaction between the escaping photoelectron and the remaining electrons is not taken into account. In other words, the photoelectron and the ionized system are completely decoupled. Moreover, the response of the system to the ionization, i.e. to the creation of a positive potential or photohole, is assumed to be instantaneous. An additional approximation consists of the neglect of frequency dependent eects in the PES spectrum, i.e. the energy dependence of the electron-photon scattering cross section and the more eective scattering between lowenergy photoelectrons and the system's electrons is not taken into account (see Fig. 1.1). In consideration of the assumptions made, a direct comparison of theory and experimental data is complicated and one has to choose wisely the data one compares with. By way of example, one should prefer e.g. spectra measured at suciently high frequencies, expecting here the sudden approximation to be more likely valid (see Fig. 1.1).

In order to get a rst rough idea of the microscopic mechanisms, one can further simplify the scheme by considering a single electron only to be involved in the PES process. Within this single-particle picture (see Fig. 1.2a), an electron is promoted from a bound single-particle state with energy ε i into an unbound continuum state with energy ε f by the absorption of a photon of energy ω. The electronic structure of the initial N -particle ground state and the nal (ionized) system are taken to be equal. In other words, the ejection of the electron and the related creation of a positive potential is assumed to have no inuence on the remaining electrons. According to energy conservation, simple relations between the measured kinetic energy E kin of the photoelectron and the orbital energy ε i can be established (see Fig. 1.2a):

-ε i + E kin = ω, E b = ω -E kin ≡ -ε i .
This is known as Koopmans or frozen-orbital approximation and the binding energy E b , i.e.

E b ε i ε f E v E b ≈ -ε i (N -1) ε i N (N -1) N E N 0 E N -1 i (N -1) i ω + E N 0 = E kin + E N -1 i ,
E b = ω -E kin ≡ E N -1 i -E N 0 .
The binding energy E b for occupied states is thus the energy needed to eject an electron from a system of N interacting electrons, where the creation of a positive potential and its inuence on the electronic structure of the remaining electrons is taken into account. By way of example, the ionization energy (I E), i.e. the energy needed to eject an electron from the highest occupied (ho) state is given by:

IE ≡ E N -1 ho -E N 0 .
The measured binding energy E b , i.e. the electronic structure, is consequently a complicated many-body quantity, necessitating to go beyond single particle orbital energies.

In order to obtain information about unoccupied states, i.e. to get a complete view of the electronic structure of the system, a complementary method to PES is usually used, called inverse PES (IPES). The measurement principle of IPES is the following: an electron with a

xed energy E kin is inserted into the i-th unoccupied state of the N electron system, which in turn relaxes to the charged (N + 1) ground state under photon emission (see Fig. (1.2b)).

The energy E b needed to insert an electron in state i of a system of N interacting electrons is given by:

E kin + E N 0 = ω + E N +1 i , E b = ω -E kin ≡ E N 0 -E N +1 i ,
and can consequently easily be obtained by measuring the energy distribution of the outgoing photons. By way of example, the energy needed to insert an electron into the lowest unoccupied (lu) state of the N electron system, i.e. the so-called electron anity (EA), is dened as:

EA = ω -E kin ≡ E N 0 -E N +1 lu .
To conclude, the described underlying microscopic mechanisms of PES/IPES are complicated many-body processes involving not only the ejection/insertion of a single electron, but also the associated creation of an additional positive/negative potential. The obtained electronic structure represents thus an excited state property, where an added charge interacts with the whole many-body system. This explains why the electronic structure is very dicult to calculate and usually approximations are needed. In the subsequent sections, we explore dierent levels of electronic structure theory and discuss both microscopic processes they describe and their range of validity. We focus on ab initio theories, i.e. approaches where no adjustable parameters enter. We further do not account for extrinsic eects, i.e. we limit our considerations on the intrinsic spectrum related to the excitation of a photoelectron and the response of the (N -1) system, whereas energy losses the photoelectron suers from on its way out of the target are neglected. Including extrinsic eects clearly goes beyond the scope of this work and the reader is referred to Ref. [START_REF] Hedin | Electron correlation: Keeping close to an orbital description[END_REF] for a comprehensive discussion.

Electronic structure theory: a many-body problem

The many-body problem Atoms, molecules and solids are systems composed of positively charged atomic cores and the respective electrons. In quantum mechanics, assuming nonrelativistic and time-independent problems, the behavior of interacting electrons and nuclei is governed by the following Schrödinger equation:

ĤΨ (x, R) = EΨ (x, R) ,
where Ψ (x, R) is the many-particle wave function depending on the spatial coordinates R ≡ (R 1 . . . R n ) of the K ion cores with nuclear charge Z I . The spatial coordinates r ≡ (r 1 . . . r N ) and the spin coordinates {s i } of the N electrons are regrouped in the generalized coordinate x, while the total energy of the system is represented by E. The Hamiltonian Ĥ is composed as follows:

Ĥ = - 1 2 N i=1 ∇ 2 i - 1 2 K I=1 ∇ 2 I M I + N i=1 N j>i 1 |r i -r j | + K I=1 K J>i Z I Z J |R I -R J | - N i=1 K I=1 Z I |r i -R I | ,
where the rst two terms are kinetic energy contributions. The remaining terms originate from the electron-electron, core-core and electron-core Coulomb interaction, respectively. Here and in the following, atomic units are used, i.e. the mass m e and the charge e of an electron, the Planck constant and the permittivity of the vacuum 4πε 0 are set to unity.

The electronic problem Since the mass M of the nuclei is essentially larger than the electronic mass m e , the velocity of the ion cores is much smaller and one supposes that electrons adjust immediately to changes in the core positions. Following Born and Oppenheimer [START_REF] Born | Zur Quantentheorie der Molekeln[END_REF][START_REF] Born | Theoretical investigations on the relation between crystal dynamics and x-ray scattering[END_REF],

the Hamiltonian can be expanded in terms of the mass ratio (m e /M ) 1/4 , resulting in a decoupling of the Schrödinger equation in an electronic and an ionic part. For now, we focus on the electronic Schrödinger equation, where the ion cores are kept frozen at xed positions R 0 . As a result, their kinetic energy contribution vanishes and the repulsive ion-ion Coulomb potential becomes a constant:

Ĥe R 0 Ψ e x, R 0 = E R 0 Ψ e x, R 0 , (1.1) Ĥe R 0 = - 1 2 N i=1 ∇ 2 i + N i=1 N j>i 1 |r i -r j | - N i=1 K I=1 Z I r i -R 0 I . (1.2)
The last term in equation (1.2), a constant potential originating from the interaction of the electrons with the xed ion cores, is from now on termed external potential Vext , even though Vext is not necessarily restricted to the electron-core potential and can in principle contain any applied external eld. In the following, except for some special cases, the electronic spin degree of freedom is not explicitly considered, i.e. we limit ourselves to systems with spin paired electrons and thus no magnetization. Moreover, we omit the explicit notation of the nuclear positions R 0 , only representing parameters. The electronic kinetic energy contribution and the external potential are grouped together in ĥ0 , ĥ0 = -

1 2 ∇ 2 + Vext ,
which is the so-called single-particle Hamiltonian, only containing operators acting on a single electron. On the contrary, the electron-electron Coulomb interaction term in equation (1.2) is a two-particle operator, correlating the motion of the electrons and making the electronic problem very dicult to solve for a larger number of particles.

Correlation means that the behavior of a single electron is aected by the motion of all the other electrons. In crystals, where a 1 cm 3 volume contains roughly 10 23 electrons, the enormous complexity of solving the Schrödinger equation becomes obvious. Calculating total energies, such as E (N ) and E (N ± 1), in order to obtain the electronic structure amounts thus to a complex many-body problem, too dicult to be solved exactly. For realistic systems, approximations are consequently inevitable. Since the Coulomb interaction constitutes a clearly non-negligible contribution, low-order perturbation theories have to be manipulated with care. 1 In principle, there is no controlled way to treat these correlations and one usually chooses approximations including the most important features of the strong Coulomb interaction. Only in very few cases mathematical error estimates can be made and approximations have rather to be justied through systematic comparisons with experiments.

An often used starting point to deal with correlations is to nd an appropriate set of one- electron orbitals, obtained using reasonable approximative single-particle expressions for the Coulomb term. Among these approaches are so-called mean-eld theories, where the manybody system is reduced to the problem of a single electron moving in an averaged (eective) interaction eld of the other electrons. Two important representatives, the Hartree-Fock approach (HFA) and density functional theory (DFT) are introduced in the following.

E trial = Ψ trial | Ĥe |Ψ trial , E 0 = Ψ 0 | Ĥe |Ψ 0 , E trial ≥ E 0 .
Consequently, searching through all admissible many-body wave functions represents a way to systematically minimize the total energy and thus to nd the ground state energy and wave function:

E 0 = min E [Ψ trial ] .
Possible trial wave functions have to fulll certain conditions, characteristic to fermionic wave functions. An important criterion is anti-symmetry, i.e.

Ψ (r 1 . . . r i r j . . . r N ) = -Ψ (r 1 . . . r j r i . . . r N ) .

This is nothing else than a generalization of the Pauli exclusion principle, which states that two fermions are not allowed to occupy the same state. In addition, a physical meaning can only be associated to the absolute square of a wave function. In particular, one is usually interested in averaged quantities, such as the charge density n (r) and the pair density (r, r ):

n (r 1 ) = N ¯|Ψ (r 1 . . . r N )| 2 dr 2 . . . dr N , (r 1 , r 2 ) = N (N -1) ¯|Ψ (r 1 . . . r N )| 2 dr 3 . . . dr N .
The former represents the probability to nd one of the N electrons at r, while the latter describes the probability that one electron is at r and another one at r . 2 Since the probability 2 Following Ref. [START_REF] Koch | A Chemist's Guide to Density Functional Theory[END_REF], we introduced a normalization factor of N (N -1) for the pair density, corresponding to the total number of non-distinct pairs. However, also the prefactor N (N -1) /2 is common, corresponding to the total number of distinct pairs [START_REF] Parr | Density-Functional Theory of Atoms and Molecules[END_REF].

of nding the N electrons anywhere in space has to be 1, one imposes ˙|Ψ (r

1 . . . r N )| 2 d 3 r 1 . . . d 3 r N = 1
as a normalization condition to the wave function.

From the mentioned physical properties, important mathematical characteristics of wave functions can be derived, such as continuity and quadratic integrability. Nevertheless, the manifold of possible functions is enormous and in practice a complete search through all elements is, except for very small systems, not feasible. Instead, one has to limit the search on subsets which are both physically meaningful and manageable.

The Hartree approximation The many-electron wave function Ψ depends on 3N spatial electronic coordinates and is consequently highly non-trivial. Several approaches exist, where functions depending on the known single-electron orbitals are taken as a subset. One of the most straightforward ways is to approximate the many-body wave function by a simple product of orthonormal single-particle orbitals:

Ψ (r 1 . . . r N ) ≈ Ψ H = φ 1 (r 1 ) . . . φ N (r N ) .
This denotes a drastic approximation and e.g. the anti-symmetry condition is not fullled.

Instead, the Hartree wave function Ψ H represents the exact solution for a system of N noninteracting bosons. Nevertheless, the simplicity of the approximated many-body wave function invites to apply the mentioned minimization scheme. The degrees of freedom to vary are the N single-electron orbitals, under the constraint to remain orthonormal. This boundary condition introduces Lagrange multipliers ε i , resulting in:

ĥ0 + ṼH,i φ i (r) = ε i φ i (r) .
(1.

3)

The complex many-body problem thus reduced, by the introduction of an approximated Hartree wave function, to an eective single-particle eigenvalue equation. Here, the statedependent Hartree operator is introduced,

ṼH,i (r) = ˆd3 r ρ i (r ) |r -r | ,
corresponding to a classical electrostatic potential at point r generated by a charge distribution ρ i (r ). The latter is obtained by ρ i (r ) = j =i |φ j (r )| 2 , where the absolute square of the wave function for the particle i under study is explicitly not considered in order to avoid an unphysical self-interaction. Since only the N states lowest in energy are occupied in the ground state, only these contribute to ρ i (r ).

To conclude, the many-body problem expressed in terms of the many-body Schrödinger equation is reduced to a subproblem, where a single particle moves in an averaged repulsive eld, compensated by the attractive background of the positive ion cores. Concerning the total ground state energy of the system, it equals the sum of the eigenvalues ε i , where the Coulomb repulsion energy is counted twice for each (ij) pair and thus has to be subtracted:

E 0 = N i ε i - N i ˆd3 r φ * i (r) ṼH,i φ i (r) .
The self-consistent eld scheme A priori, the Hartree potential is not known, since it depends through ρ i (r ) on the single-particle orbitals φ j , i.e. on the solutions itself. Consequently, the problem has to be solved in an iterative way, where one starts with an initial guess for the orbitals. From the latter, the Hartree potential, the missing ingredient to set up the Hartree equations, is constructed. As a solution, updated single-particle orbitals φ j and thus a corrected Hartree potential are obtained. This procedure is continued until reaching a dened convergency limit for the input and the updated potential. Since the resulting Hartree potential is nally consistent with the generating orbitals, this scheme is called self-consistent eld approach. The mathematical algorithms associated with the fast convergence to the xed point solution, such as Lanczos method [START_REF] Lanczos | An Iteration Method for the Solution of the Eigenvalue Problem of Linear Dierential and Integral Operators[END_REF] or direct inversion of the iterative subspace (DIIS) [START_REF] Pulay | Convergence acceleration of iterative sequences. The case of SCF iteration[END_REF], will be not detailed here.

The Hartree-Fock approximation Even though the Hartree approach denotes a strong simplication to the many-body problem, some important physical trends are already covered by the eective electrostatic potential. However, in order to also account for antisymmetry, the many-body wave function is approximated as an antisymmetric product of one-electron orbitals:

Ψ (x 1 . . . x N ) ≈ Ψ SD = 1 √ N ! φ 1 (x 1 ) . . . φ N (x 1 ) . . . . . . . . . φ 1 (x N ) . . . φ N (x N )
. This is the popular Hartree-Fock approach (H F A) and the resulting wave functions are socalled Slater determinant Ψ SD [START_REF] Fock | Näherungsmethode zur Lösung des quanten-mechanischen Mehrkörperproblems[END_REF][START_REF] Slater | Note on Hartree's Method[END_REF]. Anti-symmetry and hence the Pauli exclusion principle are respected, since Ψ SD changes sign under the exchange of two rows or columns and vanishes for two equal rows/columns. Again, the introduced variational scheme can be applied, where the total energy is minimized with respect to the one-electron orbitals. This leads to singleparticle eective equations, the Hartree-Fock equations:

ĥ0 + VHF φ i (x) = ε i φ i (x) .
(1.4)

The above relation is very similar to the Hartree result, however, due to the anti-symmetry of the wave function, a more complex operator occurs, the Hartree-Fock operator VHF . It is composed of two contributions, whose physical meaning can be made clear considering the corresponding total Hartree-Fock energy:

E HF = N i φ i | ĥ0,i |φ i + 1 2 N i N j (ii|jj) -(ij|ji) , where (ii|jj) = ¨drdr |φ i (x)| 2 |φ j (x )| 2 |r -r | and (ij|ji) = ¨drdr φ i (x) φ * j (x) φ j (x ) φ * i (x ) |r -r | .
The rst term in the Hartree-Fock total energy arises from ĥ0 , i.e. from the kinetic energy and the external potential. The second one, the so-called Hartree energy, plays the role of a classical averaged electrostatic repulsion. It corresponds to the energy φ i | ṼH,i |φ i introduced in Hartree theory, diering only in the fact that here the considered particle can be included in the sum. The Hartree operator within HF theory is dened as:

V H (r) = ˆdr n (r ) |r -r | ,
where the charge density n (r) is expressed in terms of single-particle orbitals:

n (r) = N ´|Ψ SD (r . . . r N )| 2 dr 2 . . . dr N , = N j |φ j (r)| 2 .
In the following, it is always this potential we refer to as Hartree potential and not the statedependent potential ṼH,i from Hartree theory. Analogue, we introduce the so-called Hartree energy contribution, corresponding to 1 2 (ii|jj), as follows:

E H = 1 2 ˆdrdr n (r) n (r ) |r -r | .
The third contribution in equation (1.4), (ij|ji), is governed by the Fock operator VF , which is dened by its eect on an orbital φ i :

V F,j (r) φ i (r) = ˆφ * j r φ i r 1 |r -r | dr φ j (r) .
It is called exchange contribution and has no classical analogue, since it emanates from the fermionic nature of electrons. It cancels for electrons with opposite spin and only electrons with the same spin are subject to it. Since the result of the application of VF on a orbital depends on the value of the latter in the entire space, this operator is called non-local.

For the case (i = j), the Hartree and the exchange term cancel ((ii|jj) = (ij|ji)), i.e. an unphysical self-interaction of the electron with itself is automatically avoided. Analogue to the Hartree equations, the Hartree-Fock equations (1.4) have to be solved self-consistently.

The obtained eigenvalues have an important physical interpretation. It can be easily demonstrated that the orbital energies ε i obtained in the presented single determinant approximation correspond to the already introduced Koopmans binding energies, i.e. to a frozen orbital approximation.

Electron correlation Even though a single Slater determinant captures important physical eects present in many-body systems, it does not represent an exact solution to the many-body wave function. As a consequence, this approach covers only part of the electron correlation, i.e. the eect an electron has on the others. In this context, the so-called correlation energy is introduced as the dierence between the exact ground state energy and the Hartree-Fock energy. However, some correlations are already governed within the Hartree-Fock approach, namely exchange interactions originating from the anti-symmetric nature of the fermionic wave function. 3 On the contrary, correlations due to the charge of the electrons are completely neglected within Hartree-Fock theory. By way of example, the probability (r, r ) of nding one electron at r and another one at r is not simply cl (r, r ) = [N/ (N -1)] n (r) n (r ), as it would be the case for classical non-interacting charge distributions, but the electrons try to avoid each other.

4

The region around an electron is thus depleted of other electrons and a screening hole is formed around, reducing the interaction between electrons and thus the Coulomb energy. Within Hartree-Fock, (rs; r s ) diers from a simple product only for equal spin electrons through the introduction of an exchange correlation f x :

HF ↑↑ r, r = n (r, ↑) n r , ↑ + f x r, r .
However, electrons with opposite spins move completely uncorrelated, following: 5

HF ↓↑ r, ↑; r , ↓ = n (r, ↑) n r , ↓ .
As a result, two electrons of opposite spin are even allowed to be simultaneously at r. Consequently, in average electrons come too close together, giving rise to an overestimation of the electron-electron repulsion term.

In order to account for correlation eects beyond the exchange interaction f x , an approach based on a single ground state Slater determinant is not sucient. Instead, the exact solution for an interacting electron system requires an innite sum of Slater determinants, including also determinants with excitations of one, two, three etc. electrons:

Ψ (x 1 . . . x N ) = ∞ i=0 Ψ SD,i .
This approach is known as Full Conguration Interaction Method (Full CI). Since one works with the exact Hamiltonian and the full wave function space, correlation is in principle completely taken into account [START_REF] Boys | Electronic Wave Functions. I. A General Method of Calculation for the Stationary States of Any Molecular System[END_REF][START_REF] Dykstra | Ab initio calculation of the structures and properties of molecules[END_REF]. However, the number of determinants included in the sum scales exponentially with the system's size and therefore, one usually limits the sum to single or double excitations (C ISD). Thereby, the scaling reduces to N Here, the gradient of the density has a discontinuity, resulting from the singularity of the electron-ion potential for r → R 0 .

The cusp can be directly related to the nuclear charge Z i . The actual physical proof of these plausibility arguments is provided by the Hohenberg-Kohn theorems.

The Hohenberg-Kohn theorems The basis of DF T has been set in 1964 by Hohenberg and Kohn who introduced, in two theorems, the charge density as main ingredient to describe a many-body system [START_REF] Hohenberg | Inhomogeneous Electron Gas[END_REF]. The rst theorem states that the external potential V ext (r), the only system dependent ingredient of the Hamiltonian, is uniquely dened within an additive constant by the ground state density n 0 (r). Two external potentials, diering by more than a constant and providing the same ground state density, do thus not exist. Accordingly, n 0 (r) contains all important information to uniquely determine the full many-body Hamiltonian and hence the true total ground state energy:

n 0 (r) → V ext (r) → Ĥ → E 0 .
The latter is a functional of n 0 (r) and can be expressed as follows:

E [n 0 ] = F HK [n 0 (r)] + ´d3 r V ext (r) n 0 (r) , F HK [n 0 ] = T [n 0 ] + E ee [n 0 ] .
E ee [n 0 ] = 1 2 ˆd3 rd 3 r n 0 (r) n 0 (r ) |r -r | + E ncl .
The second Hohenberg-Kohn theorem introduces the variational principle for the energy functional with respect to the charge density. It states that E [n] has its minimum at the ground state equilibrium density n 0 (r),

n 0 (r) = N ´|Ψ 0 (r . . . r N )| 2 d 3 r 2 . . . d 3 r N , E 0 = min n E [n] = E [n 0 ] ,
and provides thus a systematic way to obtain the ground state energy from trial densities. It is important to mention that the variational principle only holds for the exact Hohenberg-Kohn functional. As soon as approximations to this functional are made, total energies lower than the true ground state energy can be obtained.

The Kohn-Sham approach The rst Hohenberg-Kohn theorem proves that the the ground state density provides all necessary information of the system, while the variational principle provides a systematic way to obtain it. However, the explicit form of the Hohenberg-Kohn functional F HK [n] is not known. After many trails and errors, the Kohn-Sham ansatz turns DF T into a widely used practical method [START_REF] Kohn | Self-Consistent Equations Including Exchange and Correlation Eects[END_REF]. The idea is to introduce an auxiliary reference system of N non-interacting electrons whose density n s (r) equals the density n (r) of the corresponding interacting system. Since the many-body wave function of the non-interacting electron gas is a single Slater determinant, the non-interacting density n s (r) is completely determined by single-electron orbitals, the so-called Kohn-Sham wave functions φ KS,i (r):

n s (r) = N i |φ KS,i (r)| 2 ≡ n (r) .
Also the kinetic energy T s of the reference system can be readily obtained from the Kohn-Sham orbitals:

T s [n] = - N i ˆd3 r φ * KS,i (r) ∇ 2 2 φ KS,i (r) .
The latter is supposed to constitute the major fraction of the kinetic energy of the interacting system. As a consequence, the Hohenberg-Kohn functional can be rewritten as follows:

F HK [n] = T s [n] + E H [n] + E xc [n] ,
where the so-called exchange-correlation functional E xc contains all unknown contributions, i.e. the residual part of the kinetic energy, repulsive exchange interactions due to Pauli's exclusion principle and correlation eects due to repulsive Coulomb interactions between charged particles:

E xc = (T -T s ) + (E ee -E H ) .
A priori, its exact form is not known and nding appropriate approximative functionals represents a challenging eld of research.

Minimizing the energy functional E [n] with respect to the density leads to a density-only Euler eigenvalue equation:

∂F [n] ∂n (r) + V ext (r) = µ,
where µ is the chemical potential. Taking the one-particle Kohn-Sham orbitals φ KS,i as variational parameters nally gives the Kohn-Sham (K S) equations:

-

1 2 ∇ 2 + Veff φ KS,i (r) = ε KS,i φ KS,i (r) .
(1.5)

The latter represent a single-particle problem, where an electron moves in an eective one-

particle potential V ef f , V ef f (r) = V H (r) + V ext (r) + V xc (r) .
In this context, the exchange-correlation potential V xc is introduced as:

V xc (r) = ∂E xc [n] ∂n (r)
.

The eective one-particle potential V ef f apparently depends on the density and thus on the solution of the problem itself. Therefore, a self-consistent eld scheme starting from an initial guess density has to be applied in order to solve the Kohn-Sham eigenvalue problem.

The DFT Kohn-Sham (1.5) and Hartree-Fock equations (1.4) appear very similar, however, important conceptual dierences have to be noted. First, in contrast to the Hartree-Fock scheme which is conceived to nd the best approximative many-body wave function, DF T -K S is in principle an exact theory, provided that the correct exchange-correlation energy is known.

Moreover, the DF T -K S eective potential and thus the exchange correlation functional are local in space, compared to the non-local Fock exchange term in Hartree-Fock. Formally, the DF T -K S scheme seems thus less complicated, however, the true exchange-correlation functional is supposed to show a very complicated non-local dependence on n. This points out the challenge in nding a suitable approximation for V xc , covering both as many correlation eects as possible and being at the same time computationally feasible.

A manifold of dierent exchange-correlation functionals have been elaborated and merits and limitations of the diverse functionals have been studied in detail for diverse materials [START_REF] Koch | A Chemist's Guide to Density Functional Theory[END_REF]. The Local Density Approximation (LDA) to V xc represents the pioneering exchangecorrelation functional [START_REF] Kohn | Self-Consistent Equations Including Exchange and Correlation Eects[END_REF][START_REF] Perdew | Self-interaction correction to density-functional approximations for many-electron systems[END_REF][START_REF] Vosko | Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis[END_REF]. It has been proposed by Kohn and Sham and is based on the assumption that the charge density of the system only slowly varies. As a consequence, each volume element is described by the density n (r) of a uniform electron gas. Based on the exchange-correlation energy xc (n) per electron in a homogeneous gas, a quantity that has been calculated using Quantum Monte Carlo techniques [START_REF] Ceperley | Ground State of the Electron Gas by a Stochastic Method[END_REF], one deduces for the exchange-correlation energy of the whole volume:

E LDA xc [n] = ˆd3 r xc (n) n (r) .
The LDA represents one of the simplest approaches, however, it proved to yield ground state properties in excellent agreement with experiment for a large class of systems, in particular due to the satisfaction of important sum rules.

A possible improvement can be achieved by also including information about the density gradient. This leads to semi-local generalized gradient approximations (GGA), among which the functional of Perdew, Burke and Ernzerhof (P BE) is the most popular [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF]. However, (semi)local functionals seem not appropriate for molecules, which are characterized by extremely inhomogeneous and localized densities. Moreover, from equation (1.5) one can readily see that self-interaction eects, which can be dramatic for localized orbitals, are not avoided within (semi)local functionals. In order to tackle these issues, so-called hybrid functionals have been designed. They are based on (semi)local functionals such as LDA or P BE, but also include a certain percentage of exact exchange from Hartree-Fock theory. These functionals, especially the so-called B3LY P functional [START_REF] Becke | A new mixing of Hartree-Fock and local density-functional theories[END_REF], have been shown to yield satisfying results and have been established as functionals of choice for e.g. the optimization of ground state molecular structures [START_REF] Raghavachari | Perspective on 'Density functional thermochemistry. III. The role of exact exchange[END_REF]. Within the DF T community, the development of exchangecorrelation functionals represents a vivid eld of research of its own. A detailed discussion is clearly beyond the scope of this work and the reader is referred to Ref. [START_REF] Koch | A Chemist's Guide to Density Functional Theory[END_REF].

Physical interpretation of the Kohn-Sham eigenvalues The quantities accessible through the DF T -K S scheme are rst of all ground state related properties, i.e. ground state densities, total energies or structural properties such as lattice constants. For various classes of systems, such as atoms [START_REF] Gunnarsson | Density Functional Calculations for Atoms, Molecules and Clusters[END_REF], molecules [START_REF] Gunnarsson | Density Functional Calculations for Atoms, Molecules and Clusters[END_REF], metals [START_REF] Moruzzi | Calculations of electronic properties of metals[END_REF] and insulators [START_REF] Yin | Theory of static structural properties, crystal stability, and phase transformations: Application to Si and Ge[END_REF], more than satisfactory result have been obtained at favorable computational costs.

Corresponding to the Hohenberg-Kohn theorem, the ground state density contains the entire information needed to construct the exact Hamiltonian. Consequently, also excited state properties should, in principle, be accessible. However, so far it is not known how to extract this information from the Kohn-Sham formalism. By way of example, Kohn-Sham eigenvalues enter the equations as Lagrange multipliers in order to ensure orthogonality of the Kohn-Sham wave functions during the minimization of the energy functional. This is analogue to the derivation of the Hartree-Fock equations, where the HF eigenvalues guarantee orthogonality of the Hartree-Fock single-particle wave functions. Provided that the electronic structure is unaected by changing the number of electrons in the system (frozen orbital approximation), the HF eigenvalues gain a direct physical meaning through Koopmans theorem, associating them with electron removal/addition energies ε HF i,j :

ε HF i,j =          E N (n 1 , . . . , n N ) -E N -1 (n 1 , . . . n i -1, . . . , n N ) E N +1 (n 1 , . . . , n N , . . . n j + 1, . . .) -E N (n 1 , . . . , n N ) . E E (N 0 ) N 0 ∈ N IE (N 0 ) ≡ E N 0 -1 ho -E N 0 EA (N 0 ) ≡ E N 0 -E N 0 +1 lu 1 IE (N 0 -1) > IE (N 0 ) ε ho N E N (N -1) (n i = 1) (n j = 0) DF T KS DF T KS ε DF T i = ∂E N 0 ∂ ñi ñi =1 , ε DF T i ñi ∈ [0, 1] E (N ≡ N 0 + α) N 0 ∈ N α ∈ [0, 1] E (N ) = (1 -α) E (N 0 ) + αE (N 0 + 1) . E (N ) DF T ho N IE occupied level: -IE (N 0 ) = E N 0 -E N 0 -1 ho = ε DF T ho=N 0 = ∂E N 0 ∂n ho n ho =1-η , (1.6)
where η = 0 + . This is known as the ionization potential theorem and is illustrated in Fig.

( -EA

(N 0 ) = E N 0 +1 lu -E N 0 = ε DF T ho=N 0 +1 = ∂E N 0 +1
∂n ho n ho =1-η .

(1.7)

In general, it follows from Janak's theorem that the energy of the highest occupied eigenvalue is a stair-step function with respect to the particle number N , jumping at the integer points (see Fig. 1.4b). This is physically sound, since e.g. the energy needed to eject an electron (I E) or to insert one (EA) are dierent, i.e. the fundamental gap is non-zero.

It is important to note that for the majority of approximate exchange-correlation functionals, the presented relations are not forced to hold. Moreover, only the highest occupied Kohn-Sham eigenvalue is directly associated with a physical quantity, whereas there is no conclusion for the remaining eigenvalue spectrum. Despite these discoveries, in practice, the whole Kohn-Sham eigenvalue spectrum is usually associated with excitation energies. This is due to the fact that more accurate and at the same time inexpensive ab initio alternatives are rare. The consequences on the quality of the resulting electronic structures will be discussed in detail in subsequent chapters. A comprehensive overview of excited states in DF T is provided by Refs. [START_REF] Kronik | Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals[END_REF][START_REF] Kraisler | Fundamental gaps with approximate density functionals: The derivative discontinuity revealed from ensemble considerations[END_REF][START_REF] Jones | The density functional formalism, its applications and prospects[END_REF].

Many-body perturbation theory

The key feature of the so far introduced electronic structure theories, namely the HF approximation and DF T , is to map the many-body system onto an eective single-particle problem. Concerning excitation energies, these approaches have the conceptual disadvantage that only the N particle ground state enters, whereas the ionized electronic conguration is not explicitly considered. The HF A is limited to the single particle picture (Koopmans binding energies) and standard DF T calculations on N particles access only the ionization energy IE. In order to model the microscopic processes occurring when removing (adding)

an electron from the system in PES/ IPES experiments, a formalism connecting a system of N interacting electrons to one with (N ± 1) would be desirable.

Many-particle physics [START_REF] Mahan | Many-Particle Physics[END_REF][START_REF] Gross | Many-particle theory[END_REF][START_REF] Bruus | Many-Body Quantum Theory in Condensed Matter Physics -An Introduction[END_REF] provides quantities which can be directly related to the mentioned process, namely single-particle Green's functions. As it will be discussed in the following, the latter are propagators, which describe the motion of an additional charge in an interacting many-body system. Here, the many-body problem can be also transformed into an eective one-particle problem. In contrast to the HF or Kohn-Sham equations, not a bare particle, but a so-called quasiparticle is considered. The latter behaves like a noninteracting electron or hole subject to an external potential, however, its mass and energy are renormalized due to Coulomb interactions with the remaining particles. Based on one-particle Green's functions, a closed set of equations, the Hedin's equations, will be set up. From these, spectroscopic quantities such as excitation energies are, in theory, exactly accessible from rst principles. Representing a set of ve integro-dierential equations which have to be solved self-consistently, this denotes a very demanding problem. Therefore, practical approaches to the Hedin's equations, namely the so-called GW and COHSEX approximation, will be discussed.

Real experimental measurements can not be carried out at absolute zero, but at very low temperatures. Since many physical quantities only weakly depend on temperature in this low temperature regime, calculations at absolute zero can often be helpful describing real systems.

Moreover, from a theoretical point of view, the ground state of an interacting system |Ψ 0 , a property at zero temperature or at small thermal energies not sucient to excite the system, is of great conceptual importance. Therefore, we limit the description to zero-temperature single-particle Green's functions in the following and refer the reader to Appendix A.2 for more detailed information.

Green's functions as propagators

Our starting point is a system of N interacting electrons in a static external potential V ext .

In order to study the electronic structure, we introduce the single-electron Green's function G e (rt, r t ), dened such that i G e (rt, r t ) is the probability amplitude to detect an electron at (rt) after the insertion of an electron to the interacting many-body system at (r t ).

Similarly, we can dene a single-hole Green's function G h (r t , rt), describing the probability amplitude to nd a hole at (rt) subsequent to its creation at (r t ) [START_REF] Friedrich | Many-Body Perturbation Theory: The GW Approximation[END_REF]. These quantities represent thus theoretical tools to describe electron injection (ejection) and the induced reaction of the system. It is important to note that despite their name, single-particle Green's functions are true many-body quantities, characterizing the propagation of a single particle in an interacting many-particle system.

Using the eld operator description in the Schrödinger picture, the one-electron Green's function is dened as follows:

G e (rt, r t ) = - i Ψ N 0 (t) Φ(r) S(t, t ) Φ † (r ) Ψ N 0 (t ) θ t -t . (1.8)
Hence, a process is described which adds an electron at time t and position r to the N-electron ground state Ψ N 0 (t ) , through the application of the eld operator Φ † (r):

Ψ N +1 (r t ) = Φ † (r ) Ψ N 0 (t ) . to t, S(t, t ) Φ † (r ) Ψ N 0 (t ) = e -i Ĥ(t-t ) Ψ N +1 (r t ) = Ψ N +1 (r t) ,
where interactions of the added electron and its environment are considered via the manybody Hamiltonian Ĥ. By means of the annihilation eld operator Φ(r), the process is nalized at time t by verifying whether there is an electron at position r:

Φ(r) S(t, t ) Φ † (r ) Ψ N 0 (t ) .
Except the case where an electron is at r, the above expression vanishes. The probability amplitude is then obtained by calculating the overlap of the nal state with the N-particle ground state Ψ N 0 (t) at time t. The right time order is ensured by including the step function θ:

θ t -t =    1 for t > t 0 for t < t ,
i.e. an electron is rst added and then annihilated. Transforming equation (1.8) from the Schrödinger representation into the Heisenberg picture, one obtains an equivalent description.

Here, the time-dependence is governed by eld operators instead of wave functions, following:

Φ( †) (r, t) = e i Ĥt Φ( †) (r) e -i Ĥt .
This leads to:

G e (rt, r t ) = -i Ψ N 0 (0) e i Ĥt Φ(r) e -i Ĥ(t-t ) Φ † (r ) e -i Ĥt Ψ N 0 (0) θ (t -t ) = -i Ψ N 0 Φ(r, t) Φ † (r , t ) Ψ N 0 θ (t -t ) .
(1.9)

The inverse process of rst ejecting and then reinjecting an electron to the system can be formulated in a similar way. It is considered as the propagation of a hole from (rt) to (r t ),

i.e. of a particle possessing the same mass and opposite charge than an electron and moving opposite in time:

G h (r t , rt) = - i Ψ N 0 Φ † (r , t ) Φ(r, t) Ψ N 0 θ t -t .
(1.10)

For the sake of convenience, equations (1.9) and (1.10) are combined to one time-ordered Green's function G T ≡ G e (rt, r t ) -G h (r t , rt) using Wick's time-ordering operator T . The latter arranges time-dependent operators Ô with earliest times to the right giving rise to a factor (-1) for each permutation in the fermionic case:

T Ô1 (t 1 ) Ô2 (t 2 ) = Ô1 (t 1 ) Ô2 (t 2 ) θ (t 1 -t 2 ) -Ô2 (t 2 ) Ô1 (t 1 ) θ (t 2 -t 1 ) , G T (rt, r t ) = - i Ψ N 0 T Φ(r, t) Φ † (r , t ) Ψ N 0 .
(1.11)

Depending on the time order, this equation describes either the propagation of an electron (t > t ) or a hole (t < t ).

Besides the time-ordered Green's function, it is very useful to introduce the retarded Green's

function G R , G R (rt, r t ) = - i Ψ N 0 Φ(r, t) Φ † (r , t ) Ψ N 0 θ t -t ,
where {. . .} denotes the anticommutator. Analogously, the advanced Green's function G A is dened as:

G A (rt, r t ) = i Ψ N 0 Φ(r, t) Φ † (r , t ) Ψ N 0 θ t -t .
G T (r, r , τ ) = - i unocc m e -iεmτ f * m (r)f m (r ) Θ (τ ) + i occ l
e -iε l τ g * l (r )g l (r) Θ (-τ ) , (1.12) where we dened the Lehmann amplitudes for the (N + 1) system,

f m (r) = Ψ N +1 m Φ † (r) Ψ N 0 f * m (r) = Ψ N 0 Φ(r) Ψ N +1 m ,
and the (N -1) system:

g l (r) = Ψ N -1 l Φ(r) Ψ N 0 g * l (r) = Ψ N 0 Φ † (r) Ψ N -1 l .
The excitation energy needed to insert an electron into an unoccupied state of an interacting N particle system in its ground state with total energy E N 0 , creating a (N + 1) system with total energy E N +1 m , is here dened as ε m = E N +1 m -E N 0 . The excitation energy needed to remove an electron from an occupied state of the N particle system, creating a (N -1) system with total energy E

N -1 l , is ε l = E N 0 -E N -1 l [55]
. The notation (l-occupied, m-unoccupied)

in the above equations serves to distinguish these two cases. The excitation energies ε m,l correspond to the negative value of the binding energy E b as dened before in the discussion of PES experiments. Since the Hamiltonian does not explicitly depend on time, the problem is translationally invariant and the Green's function only depends on the time dierence τ = t -t . Consequently, a Fourier transformation from time to frequency space is suitable, following:

G(r, r , ω) = ˆ∞ -∞ dτ e iωτ G(r, r , τ ), G(r, r , τ ) = 1 2π ˆ∞ -∞ dω e -iωτ G(r, r , ω).
For the time-ordered Green's function, the Fourier transformation yields:

G T (r, r , ω) = 1 unocc m f * m (r)f m (r ) ω -ε m + iη + 1 occ l g * l (r )g l (r) ω -ε l -iη , (1.13)
where the small imaginary part η appears to ensure convergence (see Appendix A.2). When not explicitly needed in the following, i.e. except for integrals, we set η to zero. Likewise, we can derive an expression for the retarded Green's function:

G R (r, r , ω) = 1 unocc m f * m (r)f m (r ) ω -ε m + iη + 1 occ l g * l (r )g l (r) ω -ε l + iη . (1.14)
In this representation, one can easily see that G T and G R have poles at the excitation energies ε m/l measured as binding energies in PES/IPES experiments. This suggests that these quantities are, in principle, accessible. However, this is a highly non-trivial problem, since all many-body interactions are taken into account.

A suitable starting point from which the interacting system can be modeled is a system of N non-interacting particles, governed by ĥ0 = -1 2 ∇ 2 r + Vext . In this case, the excitation energies are simply the single-particle energies ε 0 m/l and the many-electron states are single Slater determinants constructed from single-particle wave functions φ m/l , simplifying the Lehmann amplitudes to φ m/l . Thus, the Green's function G T 0 for a system of N non-interacting particles denotes in the Lehmann representation:

G T 0 (r, r , ω) = n φ * n (r) φ n (r ) ω -ε 0 n + iη sgn (µ -ε 0 n ) , (1.15)
where we introduced the chemical potential µ to distinguish between occupied and unoccupied states regrouped in the index n. For the retarded Green's function one nds:

G R 0 (r , r, ω) = n φ n (r)φ n (r ) ω -ε 0 n + iη . (1.16)
Besides one-particle excitation spectra, important ground state properties are accessible through the single-particle Green's function. By way of example, the ground state expectation value of any one-particle operator, such as e.g. the density operator n(r, t) = Φ † (r, t) Φ(r, t), can be obtained:

n (r, t) = -iG r, r, t, t + .

Moreover, also the exact total ground state energy can be calculated by means of the one-particle Green's function using e.g. the Galitskii-Migdal relation [START_REF] Galitskii | Application of quantum eld theory methods to the many body problem[END_REF]:

E N 0 = - i 2 ˆd3 r lim t →t + lim r →r i∂ ∂t -h 0 (r) G r,
i ∂G(rt,r ,t ) ∂t = δ (r -r ) δ (t, t ) + Ψ N 0 T ∂ Φ(r,t) ∂t Φ † (r , t ) Ψ N 0 .
The time-derivative of the annihilation operator Φ(r, t) reads: i ∂ Φ(r, t) ∂t = Φ(r, t), Ĥ = h 0 (r) Φ(r, t) + ˆd3 r v r, r Φ † (r , t) Φ(r , t) Φ(r, t), (1.17) where v (r, r ) = 1 |r-r | is the bare Coulomb potential and where h 0 (r) accounts for the kinetic and the ionic potential. Inserting the time-derivative results in:

i ∂ ∂t -h 0 (r) G(rt, r t ) + 1 i ´d3 r v (r, r ) Ψ N 0 T Φ(r, t) Φ(r , t) Φ † (r , t) Φ † (r , t ) Ψ N 0 = δ (r -r ) δ (t, t ) .
The above equation consists of a single-particle expression and an expectation value containing four eld operators accounting for electron-electron interactions. This expectation value can be associated with a two-particle Green's function G 2 at zero temperature, in general dened as [START_REF] Strinati | Application of the Green's functions method to the study of the optical properties of semiconductors[END_REF][START_REF] Bruus | Many-Body Quantum Theory in Condensed Matter Physics -An Introduction[END_REF]:

G 2 r 1 t 1 ; r 2 t 2 ; r 1 t 1 ; r 2 t 2 = 1 (i ) 2 Ψ N 0 T Φ(r 1 , t 1 ) Φ(r 2 , t 2 ) Φ † (r 2 t 2 ) Φ † (r 1 t 1 ) Ψ N 0 .
Analogue to the one-particle Green's function G 1 , it represents the probability amplitude of nding simultaneously two particles at (r 1 , t 1 ) and (r 2 , t 2 ), which have been added to the system at (r 1 t 1 ) and (r 2 t 2 ). It thus contains information about two-particle processes and the involved interactions. Depending on the time order it either describes the propagation of two electrons, two holes or an electron-hole pair. In terms of the two-particle Green's function, the EOM in its dierential form is represented by:

i ∂ ∂t -h 0 (r 1 ) G(11 ) + i ˆd2 v (r 1 , r 2 ) G 2 12, 1 2 + = δ 1, 1 . (1.18)
Concerning the introduced shorthand notation, natural numbers regroup both space and time arguments. In this specic case, we dened 1 ≡ (r 1 , t), 1 ≡ (r 1 , t 1 ), 2 ≡ (r 2 , t) and 2 + ≡ (r 2 , t + η), where the innitesimal parameter η ensures the right time order.

Provided that the Hamiltonian does not explicitly depend on time, the Green's function G only depends on the time dierence τ = t=t . Hence, it is very useful to transform equation (

ω -h 0 (r 1 )) G(r 1 , r 1 ; ω) + i ˆd3 r 2 v (r 1 , r 2 ) G 2 r 1 r 2 , r 1 r 2 ; ω = δ r 1 , r 1 . (1.19)
Equations (1.18) and (1.19) provide a possibility to calculate the single-particle Green's function from the known single-particle potential h 0 (r) and the two-body Green's function G 2 .

This is an important result, however, of little practical use. In order to obtain the two-particle Green's function, a solution to an EOM containing a three particle Green's function is needed.

The latter in turn is obtained by solving an EOM involving a four-particle Green's function.

That means, in order to calculate the single-particle Green's function, a hierarchy of equations of motions for higher order Green's functions has to be solved. This is not astonishing regarding the fact that equation (1.18) is an exact reformulation of the many-body problem implying all kinds of complex many-particle correlations.

The Dyson equation A useful reformulation of the the single-particle Green's function EOM is to map the complicated many-body system onto an eective one-particle problem. This can be achieved by introducing a one-particle quantity, the so-called self-energy Σ, pushing the two-particle Green's function out of the EOM. Following Ref. [START_REF] Aryasetiawan | The GW method[END_REF], we rst introduce the mass operator M , accounting for electron-electron interactions both at the classical Hartree level and beyond:

ˆd2 M (12) G(21 ) = -i ˆd2 v (r 1 , r 2 ) G 2 12, 1 2 + .
Inverting the above equation using the inverse Green's function

G -1 , ˆd2 G(12)G -1 (21 ) = ˆd2 G -1 (12)G(21 ) = δ(1, 1 ),
yields the explicit denition of the mass operator:

M (12) = -i ˆd4 ˆd3 v (r 1 , r 3 ) G 2 13, 43 + G -1 (42).
(1.20)

Consequently, the equation of motion (1.18) can be written as follows:

i ∂ ∂t -h 0 (r 1 ) G(11 ) -ˆd2 M (12) G(21 ) = δ 1, 1 , (1.21)
i.e. we reformulated the many-body problem in terms of the mass operator. In frequency space, we consequently obtain:

( ω -h 0 (r 1 )) G(r 1 , r 1 ; ω) -ˆdω ˆd3 r 2 M r 1 r 2 ; ω G r 2 , r 1 ; ω + ω = δ r 1 , r 1 .
Even though representing a one-particle quantity, the latter is a highly non-trivial object being not only non-local and non-Hermitian, but also implying a self-consistent solution scheme for the EOM due to its dependence on G. Nevertheless, it constitutes a suited starting point for perturbative approaches.

Tackling the equation of motion by means of perturbation theory (see Appendix A.3) can be achieved by starting from an exactly solvable problem, such as the case of non-interacting particles subject to a classical Hartree potential V H . For the latter, the non-interacting singleparticle Green's function G 0 can be readily calculated following equation (1.21) via:

i ∂ ∂t -H 0 (r 1 ) G 0 (11 ) = δ 1, 1 ,
where we introduced the Hamiltonian Ĥ0 :

Ĥ0 = ĥ0 + VH = - 1 2 ∇ 2 + Vext + VH .
Due to the choice to include the operator of the classical Hartree potential in the latter, this kind of electron-electron interaction has to be removed from the mass operator in order to not counting it twice. This leads to the denition of the self-energy operator Σ, Σ = M -VH , governing all interactions beyond Hartree. Reformulating the EOM (1.21) in terms of G 0 and Σ leads to:

i ∂ ∂t 1 -H 0 (r 1 ) G(11 ) -ˆd3 Σ (13) G 31 = δ 1, 1 , (1.22)
which can be reformulated to the so-called Dyson equation for the single-particle Green's function [START_REF] Dyson | The Radiation Theories of Tomonaga, Schwinger, and Feynman[END_REF][START_REF] Dyson | The S Matrix in Quantum Electrodynamics[END_REF]:

G(11 ) = G 0 (11 ) + ˆd2 ˆd3 G 0 (12)Σ(23)G( 3 1).
(1.23)

Rewriting the latter in a separated form with respect to the particular contributions, following:

G -1 (54) = G -1 0 (54) -Σ (54) ,
facilitates the Fourier transformation of equation (1.22) to frequency space:

[ ω -H 0 (r)] G r, r , ω -ˆdr Σ r, r , ω G r , r , ω = δ r, r .

Dyson's equation (1.23) represents an innite series in the perturbation Σ based on the known building blocks G 0 and can be symbolically written as:

G = G 0 + G 0 ΣG 0 + G 0 ΣG 0 ΣG 0 + . . . .
Truncating the series at zero order simply results in the propagation of a non-interacting particle. Including higher order terms provides more and more interactions and nally the innite series describes the full many-body problem correctly. Equation (1.23) oers thus a systematic way to include correlation eects with the challenge to nd both physically meaningful and feasible truncations.

It is often of high practical use and repeatedly applied in the subsequent sections to express the Dyson equation within a non-interacting Hartree basis φ i , in which G 0 is diagonal:

G T 0,ii ≡ φ i | G 0 |φ i = ˜d3 r d 3 r n φ * n (r)φ i (r)φ * i (r )φn(r ) ω-ε 0 n ±iη = n δ in ω-ε 0 n ±iη = 1 ω-ε 0 i ±iη .
Assuming that the self-energy is diagonal in this basis, namely that Hartree and quasiparticle wave functions strongly overlap, it follows for the interacting Green's function:

G T ii ≡ φ i | G |φ i = φ i | G -1 0 -Σ -1 |φ i = 1 ω -ε 0 i -Σ ii , (1.24)
with Σ ii = φ i | Σ |φ i .

The spectral function and the quasiparticle picture

The spectral function Coming back to the actual problem of accurately calculating PES/ IPES spectra, we search a formalism which directly connects the Green's function as presented in the preceding sections to experimentally accessible quantities such as the measured photocurrent. As already commented on, the accurate theoretical description of PES/IPES experiments requires many-body concepts, however, approximations are inevitable.

One of these is the sudden approximation, where the measured photocurrent I directly results from the excitation of an electron from an initial state Ψ i to a nal state Ψ f due to the interaction of the electronic system with a photon eld. Losses on its way out to the surface are not considered. Within Fermi's Golden Rule and the dipole approximation, the transition rate Ω, i.e. the probability of a transition per unit time, can be written as follows [START_REF] Dirac | The Quantum Theory of the Emission and Absorption of Radiation[END_REF][START_REF] Fermi | Nuclear Physics[END_REF]:

Ω ≈ 2π | Ψ f | r |Ψ i | 2 δ (E f -E i -ω) .
The main ingredients are the matrix element of the perturbation and a δ-function ensuring energy conservation. The matrix element is further simplied within a single-electron picture, where the initial state is described by a single-particle orbital φ i,m , from which the electron is ejected, and the decoupled remaining electronic system Ψ N -1 i,m :

Ψ N i ≈ Ĉ φ i,m Ψ N -1 i,m .
Antisymmetry of the state is guaranteed by the operator Ĉ. Likewise, the nal state is characterized by the single-particle orbital of the free photoelectron φ f,Ekin and the left over

electrons Ψ N -1 f,m : Ψ N f ≈ Ĉ φ f,Ekin Ψ N -1 f,m .
Within the single-electron approximation, the matrix element thus reduces to:

Ψ f | r |Ψ i = φ f,Ekin | r |φ i,m Ψ N -1 f,m Ψ N -1 i,m ,
i.e. it is given by the overlap of the initial and nal state of the (N -1) electron system and the matrix element consisting of the perturbation between the single-electron orbitals. Within Koopmans approximation, these states are supposed to be equal,

Ψ N -1 f,m = Ψ N -1 i,m , yielding: Ψ f | r |Ψ i = φ f,Ekin | r |φ i,m .
The relaxation of the remaining electronic system is thus completely neglected and the nal state Ψ N -1 f,m corresponds to the frozen ground state electronic conguration, where one electron in state m is missing.

In order to go beyond Koopmans approximation, rst one has to allow the electronic system to react on the creation of a charge in state m. This results in a relaxed many-body state labeled Ψ N -1 f,m,0 , where an electron is still missing in state m, but where the remaining energy levels are renormalized. Moreover, one also has to consider nal state congurations, where the electronic system not only relaxes, but where the system also reacts in form of neutral excitations, such as the creation of e.g. electron-hole pairs or plasmon excitations. The manifold of all possible nal excited states, including Ψ N -1 f,m,0 , is labeled with the index s, Ψ N -1 f,m,s , and assigned the energy E N -1 f,m,s . The electronic system can thus react in many dierent ways to the ionization, provided that the particle number (N -1) remains xed and that energy conservation is respected, i.e. the sum of the kinetic energy of the outgoing electron and the energy of the nal state is constant. The above matrix element then reads:

Ψ f | r |Ψ i = φ f,Ekin | r |φ i,m s Ψ N -1 f,m,s Ψ N -1 i,m
.

The measured photocurrent I, which is proportional to the transition rate Ω, becomes:

I ≈ f im | φ f,Ekin | r |φ i,m | 2 s Ψ N -1 f,m,s Ψ N -1 i,m 2 δ ( ω -E f,kin -E b ) , (1.25)
where E N i,0 is the total energy of the initial (neutral) electronic system. E b represents the binding energy, which has been already dened as

E b ≡ E N -1 f,m,s -E N i,0 , i.e. it is always N m Ψ N -1 f,m,0 m Ψ N -1 f,m,s s = 0 m m m A A m (ω) ≡ s Ψ N -1 f,m,s Ψ N -1 i,m 2 δ ( ω -E b ) . E kin = 0 ĉ † m ĉm m A m (ω) = s | N -1, m, s| ĉm |N, 0 | 2 δ ( ω -E b ) = s N, 0| ĉ † m |N -1, m, s N -1, m, s| ĉm |N, 0 δ ( ω -E b ) = s g * m,s g m,s δ ( ω -|ε m,s |) , g m,s ε m,s ≡ E N i,0 -E N -1 f,m,s A A A
to the imaginary part of the Green's function [START_REF] Bruus | Many-Body Quantum Theory in Condensed Matter Physics -An Introduction[END_REF]:

A(r, r , ω) ≡ - 1 π G T (r, r , ω) sgn (ω -µ) ,
where µ represents the chemical potential. The other way around, one can also express the Green's function G in terms of A, yielding its so-called spectral representation:

G T (r, r , ω) = ˆ∞ -∞ dω A(r, r , ω ) ω -ω + iη sgn (ω -µ)
.

Here, A is introduced as position resolved function by replacing the creation ĉ † m and annihi- lation operators ĉm in equation (1.26) by eld operators Φ( †) (r). However, any basis can be chosen for A and G without loss of generality.

Important properties of the spectral function can be easily deduced by considering the single-particle Green's function G, expressed in a diagonal single-particle Hartree basis (1.24):

G T ii (ω) = 1 ω -ε 0 i -Σ ii (ω)
.

In the case of non-interacting particles in an eective Hartree potential V H , one nds:

G T 0,ii (ω) = 1 ω -ε 0 i ± iη .
Using the Dirac identity,

lim η→0 + 1 x + iη = lim η→0 + - η x 2 + η 2 = -πδ(x)
for x ∈ R, we nd for the spectral function:

A 0,ii (ω) = δ ω -ε 0 i .
In the non-interacting case, the spectral function thus consists of δ-peaks at the Hartree energies ε 0 i . Since we used the time-ordered Green's function for the derivation, A is generalized compared to equation (1.26) and accounts both for the ejection (ε 0 i < µ) and the insertion (ε 0 i > µ) of an electron to the system. For the interacting case, A becomes more dicult to evaluate due to the non-local, non-Hermitian and frequency dependent nature of the self-energy Σ:

A ii (ω) = 1 π {Σ ii (ω)} ω -ε 0 i -{Σ ii (ω)} 2 + [ {Σ ii (ω)}] 2 .
Due to the frequency-dependent self-energy, it is dicult to make predictions of photoemission spectra from the above formula. However, in general, one can distinguish between at least two kinds of excitations. When slowly turning on interactions in the system, the former δ-

peaks in A are shifted from ω = ε 0 i to ω = ε 0 i + {Σ ii (ω)}
. Moreover, they broaden to accounting for all kinds of many-body interactions.

Lorentzian functions, whose full width at half maximum (FWHM) is proportional to

{Σ ii }.
These peaks, which are in close correspondence to the original non-interacting peaks, are attributed to so-called quasiparticle excitations, whose lifetime is inversely proportional to the peak width. This point will be discussed in detail below. In addition, satellite peaks occur in the spectrum. The latter are caused by excitations, where due to the creation of a photohole dierent kinds of neutral excitations such as plasmons, spin waves, phonons or electron-hole pairs are provoked. These elementary excitations related to bosons are usually termed collective excitations. In the following, we treat collective excitations only sketchily and focus on the former case, the quasiparticle excitations.

The quasiparticle picture Quasiparticles are approximate low-lying elementary excitations of macroscopic systems [START_REF] Landau | Statistical physics[END_REF]. They are not real particles, but mathematical tools which oer the possibility to describe the motion of particles in an interacting many-body system. Many kinds of dierent quasiparticles exist, in the following we focus on electron/hole quasiparticles.

In this limit, a quasiparticle can be thought of as an electron/hole injected into the system and a surrounding positively/negatively charged cloud, see Fig. (1.6). The latter is formed due to repulsive interactions with the remaining electrons/holes through Coulomb interactions and the Pauli exclusion principle. The many-body problem consequently reduces to a single particle picture, where not a bare electron/hole, but a particle encircled by a screening cloud moves through the system. As a result, two quasiparticles are not interacting via the bare, but via the much weaker screened Coulomb interaction W . Screening is thus supposed to be the main physical eect when adding a particle to a system. It is governed by the inverse of the material specic dielectric function ε:

W r, r , ω = ˆd3 r v r -r ε -1 r, r , ω = ˆd3 r ε -1 (r, r , ω) |r -r | .
Due to the supposed weak interaction, we assume a one-to-one correspondence of the interacting and the non-interacting system and label quasiparticles by the same quantum numbers k or σ as their non-interacting counterparts. However, these dressed particles possess a renor-malized energy ε qp and mass, and a nite life-time τ k , since they are not eigenstates of the system. In this context, we want to point out that quasiparticle energies, and the quasiparticle picture in general, are mathematical tools, introduced to make the description of the manybody system feasible. The corresponding observables are the already presented excitation Reducing the motion of bare electrons interacting with each other through the bare Coulomb potential v to the picture of weakly interacting quasiparticles denotes an eective way to simplify the many-body problem. However, this approach is not generally valid and it breaks down in strongly correlated materials, where the elementary excitations signicantly depend on each other.

energies ε m = E N +1 m -E N 0 and ε l = E N 0 -E N -1 l ,
The spectral function within the quasiparticle picture Within the quasiparticle picture, the Green's function and hence the spectral function are strongly simplied by supposing weak correlations and thus excitations with a long lifetime τ , i.e. {Σ ii (ω)} → 0. Moreover, one is interested in the position of poles at:

ω -ε 0 i -{Σ ii (ω)} = 0,
where ε 0 i is the mean-eld Hartree solution. The energies fullling the above relation are labeled quasiparticle (qp) energies:

ε qp,i ≡ ε 0 i + {Σ ii (ε qp,i )} .
Instead of accounting for the complete frequency dependence of the self-energy, we limit our considerations on frequencies near the quasiparticle poles and expand the real part of the self-energy around these poles in a Taylor series:

{Σ ii (ω)} = {Σ ii (ε qp,i )} + ∂ {Σ ii (ω)} ∂ω ω=ε qp,i (ω -ε qp,i ) + . . . .
As it will be detailed in the following and corresponding to Fig. (1.7), this can be interpreted as a scan of the spectral function in a narrow frequency window around ε 0 i , where δ-function like excitation peaks are grouped together to a single quasiparticle peak at ε qp,i .

That way, broad satellite peak structures, which are farther apart from ε 0 i , are not included.

As demonstrated below, the quasiparticle peak has a Lorentzian shape associated with anite peak width. Inserting the above expansion of the self-energy in the one-particle Green's function G ii yields:

G ii (ω) = Z qp 1 ω-ε 0 i -{Σ ii (ε qp,i )} + (1 -Z qp ) G inc = G coh + (1 -Z qp ) G inc ,
where the expansion terms up to linear order are grouped together in the coherent and higher order terms in the incoherent part of the Green's function (see Appendix A.4). The coherent part yields for non-interacting particles ( {Σ ii (ε qp,i )} → 0) poles at the Hartree energies ε 0 i .

Therefore, this part can be associated with the creation of a quasiparticle. The introduced factor Z qp is called quasiparticle weight and dened as:

Z qp ≡ 1 - ∂ {Σ ii (ω)} ∂ω ω=ε qp,i -1
.

For the spectral function, we consequently retrieve a similar decomposition:

A ii (ω) = 1 π Z qp {Σ ii (ε qp,i )} ω -ε 0 i -{Σ ii (ε qp,i )} 2 + [ {Σ ii (ε qp,i )}] 2 + (1 -Z qp ) A inc ,
providing an intuitive physical interpretation of A. In the non-interacting case, the coherent part of A reduces to a δ-function centered at ε 0 i , but for interacting particles it represents a

Lorentzian curve centered at the renormalized energy

ε qp = ε 0 i -{Σ ii (ε qp,i )} with a width related to {Σ ii (ε qp,i )}.
This can be associated with the creation of a quasiparticle, whose lifetime τ is inversely proportional to the peak width. Thus, the broadening of the curve with respect to the non-interacting particle picture contains information about many-particle interactions. The incoherent part A inc covers satellite excitations, i.e. all kinds of excitations which can not be described within a weakly correlated quasiparticle picture. Since the spectral function has to fulll the following sum rule [START_REF] Bruus | Many-Body Quantum Theory in Condensed Matter Physics -An Introduction[END_REF]:

1 2π ˆdω A (ω) = 1,
the renormalization factor or quasiparticle weight Z qp is a measure of correlations in the system. Z qp is obtained by integrating the coherent (quasiparticle) part of the spectral function, whereas the weight of the incoherent part is (1 -Z qp ). Consequently, for non-interacting problems Z qp equals 1. However, in systems, where correlations become important it is diminished and the quasiparticle picture becomes questionable (see Fig. 1.7). Indeed, the quasiparticle concept is only applicable to systems, where quasiparticles have a suciently long lifetime, i.e. where little scattering occurs. Fermion systems in which the picture of almost non-interacting quasiparticles holds are called Fermi liquids [START_REF] Bruus | Many-Body Quantum Theory in Condensed Matter Physics -An Introduction[END_REF][START_REF] Landau | Statistical physics[END_REF]. Quasiparticles are approximate excitations of these systems, only making sense for timescales shorter than the quasiparticle lifetime. Contrary, the incoherent part of A accounts for all interactions which are not describable by a particle and its screening cloud, such as the already mentioned collective excitations. In cases, where the quasiparticle picture breaks down and incoherent contributions become dominant, one has to consider other levels of theory than Fermi liquid theory. Due to the Peierls divergence and charge-spin separation, this is for example the case for 1D systems, where one has to pass on to Luttinger liquid theory [START_REF] Bruus | Many-Body Quantum Theory in Condensed Matter Physics -An Introduction[END_REF].

A(ω)

A(ω)

ε qp ε qp δ ε 0 ε 0 A A coh G coh H 0 (r) ψ qp ν (r) + ˆdr Σ r, r , ε qp ψ qp ν r = ε qp ψ qp ν (r) . ε qp ψ qp ν ε qp

DF T KS

state on which they are applied, the eigenvalue problem is non-linear. Nevertheless, important conclusions can be drawn with respect to the introduced mean-eld equations. Comparing the latter to the quasiparticle equation strictly derived within a Green's functions approach, the eective elds in Hartree, HF and DF T -K S can be interpreted as approximate selfenergies of dierent level of theory. By way of example, the DF T -K S exchange-correlation functional can be regarded as a (semi)local and energy-independent self-energy neglecting lifetime eects. In this spirit, the often good quantitative agreement of DF T -K S band structures with experimental dispersion relations seems plausible.

In order to nd a practical solution to the quasiparticle equation, further approximations are inevitable. The main dierence between the introduced mean-eld approaches and the quasiparticle formalism is supposed to manifest in diering eigenenergies, whereas the eect on the wave functions is said to be small [START_REF] Hedin | Electron correlation: Keeping close to an orbital description[END_REF]. Therefore, as a starting point, one assumes

ε qp,ν = ψ KS ν Ĥ0 ψ KS ν + ψ KS ν Σ (ε qp,ν ) ψ KS ν .
In this case, H 0 is chosen to be the Kohn-Sham Hamiltonian,

Ĥ0 ≡ ĤKS = - 1 2 ∇ 2 + Vext + VH + Vxc .
Consequently, the already covered eects in the exchange-correlation functional have to be removed from the self-energy in order to avoid a double counting:

ε qp,ν = ψ KS ν ĤKS ψ KS ν + ψ KS ν Σ (ε qp,ν ) -Vxc ψ KS ν .
The quasiparticle energy ε qp is thus obtained in rst-order perturbation theory from Kohn-Sham eigenvalues and wave functions following:

ε qp,ν = ε KS ν + ψ KS ν Σ (ε qp,ν ) -V xc ψ KS ν .
(1.28)

In order to circumvent the state-dependence of the above equation, the self-energy can be expanded up to linear order in a Taylor series around the known Kohn-Sham eigenvalues (see Appendix A.4). This yields:

ε qp,ν ≈ ε KS ν + Z KS ν ψ KS ν Σ ε KS ν -V xc ψ KS ν , (1.29)
where Z KS ν is a renormalization factor:

Z KS ν =   1 - ∂Σ νν (ω) ∂ω ω=ε KS ν   -1 with Σ νν (ω) = ψ KS ν |Σ (ω)| ψ KS ν .
Provided that the chosen linear approximation is valid, Z KS should be very similar to the introduced quasiparticle weight Z qp , i.e. the rst derivatives (∂Σ νν /∂ω) at ε KS . Containing all many-body eects beyond Hartree, the latter is a highly complex quantity still to be determined.

Hedin's equations

Linear response theory In principle, the self-energy can be obtained by self-consistently solving a cycle of integro-dierential equations. Before going into that, it is necessary to introduce several auxiliary quantities. In order to connect experiment and theory, it is very interesting to study the system's reaction in equilibrium to a small external disturbance Ĥ applied at a certain time t 0 . In terms of the Hamiltonian, the system is governed by an equilibrium and a perturbation part:

Ĥ = Ĥ0 + Ĥ .
Provided that the applied external potential is weak, the response of the system linearly depends on it. Within this so-called linear response regime, the system's reaction in terms of the expectation value of any operator Ô is governed by response/correlation functions C:

Ô (t) = Ô 0 -i ˆ∞ t 0 dt C t -t , (1.30)
with

C t -t = -i Ô (t) , Ĥ t 0 Θ t -t .
Equation (1.30) is known as the general Kubo formula [START_REF] Bruus | Many-Body Quantum Theory in Condensed Matter Physics -An Introduction[END_REF]. It denotes an important relation, since it connects out-of-equilibrium quantities Ô (t) to equilibrium averages represented by . . . 0 . The step function ensures that C is a retarded quantity, i.e. respects causality. Strictly speaking, only retarded response functions are physically meaningful and can be compared to experiment, however, for mathematical reasons, we are going to work with time-ordered quantities only in the following.

In the case of PES or optical absorption experiments, the incoming light represents an external electromagnetic perturbation causing charge redistributions and thus a polarization of the system. In addition to the applied perturbative potential Ûext , one also has to take into account the potential created by the induced charge density n ind . Therefore, the total eld sums up to:

Vtot = Ûext + Vind ,
where the induced potential denotes an induced Hartree potential [START_REF] Bruus | Many-Body Quantum Theory in Condensed Matter Physics -An Introduction[END_REF]:

V ind (r) = ˆdr v rr n ind r .

(1.31)

It is important not to confound the applied time-dependent potential Ûext with the static potential Vext , accounting for the core ion potential. The relation between the total and the external potential denes the inverse of the dielectric function ε -1 , which is a measure for the polarizability of the system:

V tot (1) = ˆd2 ε -1 (1, 2) U ext (2) .
We already encountered ε -1 for the denition of the screened Coulomb potential W :

W (1, 2) = ˆd3 -1 (13) v(32),
which characterizes how strongly a bare charge is screened. Assuming linear response theory and a perturbation of Ĥ = ´dr n (r) U ext (r, t), the induced charge density, n (1) ≡ n ind (1) , can be readily determined from the Kubo formula:

n ind (1) = ˆd3 χ(1, 3)U ext (3).
(

Here, we assumed that the equilibrium system is charge neutral, i.e.

n (1) 0 = 0. This introduces the corresponding response function, the reducible polarizability χ:

χ R rt, r t = -i N, 0| n (rt) , n r t |N, 0 Θ t -t .
The latter is a retarded quantity and represents a density-density correlation function. Its time-ordered analogue, which is used in the following, reads:

χ T rt, r t = -i N, 0| T n (rt) , n r t |N, 0 .
χ can also be expressed as derivative of the induced density with respect to the external perturbing potential Ûext :

∂n ind (1) ∂U ext (2) = ˆd3 ∂χ(1, 3) ∂U ext (2) 0 U ext (3) + ˆd3 χ(1, 3) ∂U ext (3) ∂U ext (2) δ(2,3) → χ(1, 2) ≡ ∂n ind (1) ∂U ext (2) . (1.33)
It is a non-local quantity, since in an interacting system a perturbation at [START_REF] Hedin | New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas Problem[END_REF] can induce modications at [START_REF] Strinati | Application of the Green's functions method to the study of the optical properties of semiconductors[END_REF]. A related response function is the irreducible polarizability P which gives the change in the density with respect to the total potential V tot :

P (1, 2) ≡ ∂n ind (1)
∂V tot (2) .

(1.34)

P accounts for the fact that the application of an external eld not only aects the electron density, but consequently also induces an additional Hartree potential. It is closely related to the reducible polarizability through a Dyson-like equation:

χ(1, 2) = P (1, 2) + ˆd34 P (1, 3)v (3, 4) χ(4, 2) (1.35)
and represents the irreducible part with respect to the bare Coulomb potential v, singled out from χ.

Hedin's equations As already discussed before, screening is supposed to be the main physical eect of the system's reaction to an additional charge. Hedin's approach consists of expanding the self-energy in terms of the screened Coulomb potential W , while supposing that Σ is a functional of the single-particle Green's function G. That way, the problem is assumed to converge much faster as compared to expansions in the bare Coulomb potential v and eventually only very few expansion terms have to be considered. The expansion in W can be derived following Schwinger's functional derivative approach [START_REF] Schwinger | On the Green's functions of quantized elds. I[END_REF][START_REF] Martin | Theory of Many-Particle Systems. I[END_REF], where a small timevarying external perturbation U ext is rst introduced as mathematical tool and then set to zero, as soon as the self-energy is obtained. In the course of the derivation, one last quantity is needed in addition to the ones introduced above, the so-called irreducible vertex function

Γ: Γ(1, 2; 3) ≡ - ∂G -1 (1, 2) ∂V tot (3) .
As detailed in Appendix A.3, the outcome of the derivation is that the self-energy Σ, the singleparticle Green's function G, the irreducible vertex function Γ, the irreducible polarizability P and the screened Coulomb potential W are closely related to each other. They form Hedin's equations, an exact closed set of ve integro-dierential equations [START_REF] Hedin | New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas Problem[END_REF]:

Σ(1, 2) = i ´d34 G(1, 3) Γ(3, 2; 4) W (1, 4) G(1, 2) = G 0 (1, 2) + ´d34 G 0 (1, 3) Σ(3, 4) G(4, 2) Γ(1, 2; 3) = δ(1, 2) δ(2, 3) + ´d4567 δΣ(1,2) δG(4,5) G(4, 6) Γ(6, 7; 3) G(7, 5) Σ Γ Γ P W Σ. W Σ = GW + GW GW G + . . . , G W W χ W (1, 2) = v(1, 2) + ˆd34 v(1, 3) χ(3, 4) v(4, 2). P Γ G Σ Γ CHAPTER 1. METHODOLOGY
and to suppose that the vertex function is diagonal in space and time:

Γ GW (1, 2; 3) ≡ δ (1, 2) δ (2, 3) .
This assumption signicantly simplies Hedin's equations (see Fig. 1.9):

Σ GW (1, 2) = i G(1, 2) W (1, 2) G(1, 2) = G 0 (1, 2) + ´d34 G 0 (1, 3) Σ GW (3, 4) G(4, 2) Γ GW (1, 2; 3) = δ(1, 2) δ(2, 3) P GW (1, 2) = -i G(1, 2) G(2, 1) W (1, 2) = v(1, 2) + ´d34 v(1, 3) P GW (3, 4) W (4, 2),
where the self-energy is taken to rst order in the perturbation W . Both the self-energy Σ and the irreducible polarizability P reduce to convolution products when Fourier transformed into frequency space:

P GW (r, r , ω) = -i 2π ´∞ -∞ dω G(r, r , ω + ω ) G(r, r , ω ), Σ GW (r, r , ω) = i 2π ´∞ -∞ dω e iω η G(r, r , ω -ω ) W (r, r , ω ).
Since the screened Coulomb potential is a symmetric function of the frequency ω (see equation 1.36), the self-energy can equivalently be expressed as:

Σ GW (r, r , ω) = i 2π ˆ∞ -∞ dω e iω η G(r, r , ω + ω ) W (r, r , ω ).
The factor e iηω with η → 0 + enters to ensure convergency. The GW approximation [START_REF] Aryasetiawan | The GW method[END_REF][START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF][START_REF] Hedin | On correlation eects in electron spectroscopies and the GW approximation[END_REF][START_REF] Friedrich | Many-Body Perturbation Theory: The GW Approximation[END_REF] is not a rigorous approach in terms of perturbation theory, but its validity range is much larger than one could expect. That will be pointed out in subsequent chapters, where we will see that GW gives signicantly improved results for excitation energies compared to Hartree-Fock or DF T -K S, yielding a good agreement with experiment. As a hand wave justication, it seems physically sound to express Σ in terms of the dynamically screened potential W . That way, energy-dependent correlations are included, which are not present in simple one-particle approaches based on mean-elds of the bare (static) Coulomb potential. One thus relies on the assumption that interactions are covered to a large amount by W . However, due to their screened nature, the system almost behaves like a non-interacting system and W is assumed to be small, justifying the use of the lowest order approximation Σ = iGW . From a practical point of view, the GW approximation opens the way to calculate quasiparticle energies from rst principles for real-size systems.

Σ

GW Σ P G Σ GW Σ GW Σ x Σ c Σ GW r, r , ω = i 2π ˆdω e iηω G(r, r , ω ) W (r, r , ω -ω ) ≡ Σ x + Σ c . V m W (r, r , ω) = ´d3 r v (r , r ) ε -1 (r, r , ω) = v (r, r ) + m 2ωmVm(r)V * m (r ) ω 2 -ω 2 m , V m (r) = ´d3 r v (r, r ) N, m| n (r ) |N, 0 . (ω m = E N,m -E N,0 -iη) E N N perturbation.
For the self-energy, this leads to:

Σ GW r, r , ω = i 2π v r, r ˆdω e iηω G(r, r , ω )+ i 2π ˆdω e iηω G(r, r , ω ) W r, r , ω -ω , where we dened W ≡ [W (1, 2) -v (1, 2)].
As it is demonstrated in Appendix A.6, assuming a non-interacting Green's function G 0 , the rst part, i.e. the exchange operator Σ x , reduces to the Fock exchange:

φ k,0 | Σ x |φ k,0 = - occ n ˆdrdr φ k,0 (r) φ n,0 (r ) φ * k,0 (r ) φ * n,0 (r) |r -r | .
(1.37)

Using again the non-interacting Green's function G 0 , the second part of the self-energy can be analytically integrated by means of contour deformation techniques, yielding:

Σ c r, r , ω = i m =0 V m (r) V * m (r ) φ i,0 (r) φ * i,0 (r ) ω + ω m -ε i,0 + j m =0 V m (r) V * m (r ) φ j,0 (r) φ * j,0 (r ) ω -ω m -ε j,0 ,
where the index i labels occupied and j unoccupied states. Taking the expectation value of Σ c with respect to the same one-particle basis and considering diagonal elements only results in:

k| Σ c (ω) |k = i m =0 | k| V m |i | 2 ω + ω m -ε i,0 + j m =0 | k| V m |i | 2 ω -ω m -ε j,0 . (1.38)
Finally, the quasiparticle energy can be evaluated within a Hartree-Fock basis ({φ i,0 } ≡ {φ i,HF } , ε i,0 ≡ ε i,HF ) analogue to equation (1.29) as follows:

ε qp,k ≈ ψ HF k h 0 + V H + Σ x ψ HF k + Z HF k ψ HF k Σ c (ε k,HF ) ψ HF k = ε HF k + Z HF k ψ HF k Σ c (ε k,HF ) ψ HF k .
From the above equation and equation (1.38), the eect of interactions beyond Hartree-Fock on the quasiparticle energy is obvious. Calculating the expectation value for an electron in the highest valence band, vb| Σ c (ε vb,HF ) |vb , the sum over unoccupied states provides only a small contribution, whereas the major part stems from the sum over occupied states. In the case of small intra-spacings between occupied/unoccupied states, the neutral excitation energy ω m , which is on the scale of the Hartree-Fock gap, is much larger than the dierence |ε vb,HF -ε i,HF |. Consequently, we obtain a positive contribution to the quasiparticle energy, i.e. the highest valence band is shifted upwards in energy with respect to the Hartree-Fock value. Analyzing the expectation value for the lowest conduction band, cb| Σ c (ε cb,HF ) |cb , we nd that a large negative contribution originates from the sum over unoccupied states, shifting the lowest conduction band down in energy. This explains the systematic overestimation of the gap within Hartree-Fock and elucidates that energy-dependent correlations within GW tend to close the bare exchange gap.

The COHSEX reformulation of the self-energy

Another way of reformulating the self-energy Σ within the GW approximation is the COH-SEX formulation, i.e. the separation of Σ GW into two terms, the Coulomb-hole (COH) and screened exchange (SEX) contribution: Σ GW = Σ COH + Σ SEX . Since COHSEX calculations on molecules will be discussed later in this chapter, we briey introduce the relevant terminology. In order to arrive at the COHSEX formulation, the frequency integral of the self-energy,

Σ(r, r , E) = i 2π ˆ∞ -∞ dω e iηω G(r, r , E + ω)W r, r , ω , (1.39
G T (r, r , ω) = l φ * l (r)φ l (r ) ω-ε l +iη sgn(µ-ε l ) , W (r, r , ω) = v (r, r ) + m 2ωmVm(r)V * m (r ) ω 2 -ω 2 m ,
where (ω m = E N,m -E N,0 -iη) are neutral excitation energies. φ l and ε l should be actually Lehmann amplitudes and excitation energies, however, without loss of generality in the COH-SEX derivation, single-particle wave functions and eigenvalues as obtained from HF or DF T can be equally used. The outcome of the derivation is that the GW self-energy can be split into two terms, where the screened exchange contribution reads:

Σ SEX r, r , ω = - occ l φ * l r φ l (r) W r, r , ω -ε l . (1.40)
Its main ingredient is the screened Coulomb potential W , shifted by the poles of the Green's function for occupied states ε l . Comparing Σ SEX to the bare exchange operator Σ x in equation (1.37) illustrates that Σ SEX represents a screened exchange interaction depending on the energy of the particular state. The Coulomb hole contribution reads:

Σ COH (r, r , ω) = m,l φ * l (r) φ l (r ) Vm(r)V * m (r )
ω-ε l +ωm .

(1.41)

The static COHSEX approximation

In general, the separation of the self-energy in Σ COH and Σ SEX is exact, representing an ideal starting point for approximations. Within the so-called static COHSEX approximation, one assumes that the interaction via the screened exchange interaction W is constant. Instead of a frequency dependent screened Coulomb potential, we consider W for (ω -ε l ) = 0 only.

This corresponds to the response of a system to a weak static external potential, where the ground state density is reorganized due to the modied total potential, however, where the system is not excited. This seems a plausible approximation for both the low-frequency and the high-frequency limit, where the system density can not follow the rapid perturbation.

Within the static limit, the screened exchange contribution reads:

Σ SEX static r, r , 0 = - occ l φ l (r) φ * l r W r, r , 0 ,
while the Coulomb hole term becomes:

Σ COH static r, r , 0 = 1 2 δ r -r W r, r , 0 -v r -r .
Here, we used that

l φ * l (r) φ l (r ) = δ (r -r ) and W (r, r , 0) = v (r -r )+2 m Vm(r)V * m (r ) ωm .
The static COHSEX approximation provides an intuitive understanding of the self-energy.

It diers from directly setting W to its static limit in the original Σ GW expression, which would only yield the Σ SEX static term. The Σ SEX static contribution resembles the bare exchange Σ x , however, static screening eects due to the system's polarizability are considered. These weaken the eect of the bare exchange. Therefore, in polarizable materials, Σ SEX static is expected to improve excitation energies with respect to the Hartree-Fock approach. Compared to a full evaluation of Σ SEX , computational costs are reduced tremendously, since W has only to be evaluated once instead of calculating it as a function of the frequency.

The Coulomb hole contribution not only becomes static, but also local. It corresponds to the interaction energy of a classical point charge with an induced charge distribution.

The latter can be calculated within linear response theory, assuming an external potential U ext (r, r 0 ) = ´dr v (r, r ) (±δ (r , r 0 )) created by a (±)point charge adiabatically added at r 0 . U ext causes an induced potential, which in turn acts on the point charge. Following equations (1.31) and (1.32), it is given by:

V ind (r, r 0 ) = ˆdr dr dr v r, r χ r , r v r , r δ r , r 0 .

The above expression can be rewritten in terms of the screened Coulomb potential,

W r, r = W -v = ˆdr dr v r, r χ r , r v r , r , resulting in: V ind (r, r 0 ) = W (r, r 0 ) .
The energy needed to insert a point charge into a polarizable medium is then given by:

E ind = 1 2 ˆdr (±δ (r, r 0 )) V ind (r, r 0 ) = 1 2 W (r 0 , r 0 ) ,
where the factor 1 2 enters, because the charge is added adiabatically to the system [9]. Includ- ing also the probability to nd a charge i at r 0 , we arrive at:

E ii ind = 1 2 ˆdr 0 W (r 0 , r 0 ) |φ i (r 0 )| 2 ,
which is nothing else than the static Coulomb hole self-energy, i| Σ COH static |i , for the state i.

Neglecting dynamical eects, the static COHSEX approximation denotes a rather strong approximation. However, in cases where the added charge can be considered as a classical point charge, e.g. in the case of core level ionization, it is expected to work rather well.

Moreover, it represents a very interesting approximation to the GW self-energy from the computational point of view. A detailed discussion about its accuracy and limitations is provided in subsequent chapters.

Optical absorption spectroscopy

As already pointed out in the introductory chapter, two physical properties are of main interest for our study on organic solar cells. First, we focus on an accurate calculation of the electronic structure. The latter determines for instance band osets which are crucial for organic solar cell eciencies. As discussed in detail in the preceding sections, it can be obtained from rst principles by a Green's function formalism, where PES excitation energies are calculated within the many-body perturbation theory framework of GW . Second, we are interested in optical absorption quantities, such as electron-hole binding energies, since they are decisive for the optimization of the photoabsorption process with respect to the amount of sunlight absorbed by the solar cell. In the present section, a many-body perturbation theory approach based on the Bethe-Salpeter equations (BSE) is presented, where electron-hole interactions are explicitly included in order to accurately determine absorption spectra.

Optical absorption experiments When exposing a material to electromagnetic radiation, the latter is partially absorbed on its way through the medium. This is attributed to neutral excitations which occur if the photon energy ω matches the neutral excitation energies

( m = E N,m -E N,0
). The latter are the dierence between the total energy of the equilibrium system E N,0 and the excited one E N,m . This implies that we need to properly describe the excited states of the N particle system. Depending on the energy range, dierent kinds of neutral excitations exist. Within the microwave regime, rotational excitations are most likely for molecules. Changes in the vibrational state (phonons) often occur in the infrared, whereas electronic excitations such as electron-hole pairs or plasmonic oscillations are typical for the optical range. In addition, mixtures of dierent types of excitations with a combined excitation energy are also possible. In the following, we focus on the optical regime, i.e. we consider excitations of the electronic system only.

In a typical optical absorption spectroscopy experiment, a probe is exposed to visible radiation with intensity I from one side, while a detector on the facing side measures the amount of radiation passing through. Since absorption depends on a manifold of material specic quantities, absorption spectroscopy provides important information about the medium under study. It is a widely applied tool and, e.g. in analytical chemistry, used to distinguish between dierent kinds of substances present in a sample. Measuring the absorption with respect to ω and subtracting the reference spectrum of the light source provides the material specic absorption spectrum. The related observable is the absorption coecient σ ( ω), which is dened as the ratio of the absorbed energy and the incoming intensity I:

σ ( ω) ≡ ω • U ( ω) I ,
where U ( ω) represents the number of absorbed photons per unit volume and time. The absorption coecient is closely related to the macroscopic dielectric function εM . The latter determines the linear response of a system with respect to a small external electromagnetic perturbation and describes its optical properties. It is a complex quantity,

εM ( ω) = ε 1 ( ω) + iε 2 ( ω) , (1.42)
where real and imaginary part can be deduced from each other through the Kramers-Kronig relations:

ε 1 ( ω) -1 = 2 π P ´∞ 0 dω ω ε 2 ( ω ) (ω ) 2 -ω 2 , ε 2 ( ω) = -2ω π P ´∞ 0 dω ε 1 ( ω ) (ω ) 2 -ω 2 .
In the above equations, P represents the Cauchy principle value. The imaginary part ε 2 ( ω)

is directly accessible through experiment through its relation to the absorption coecient:

ε 2 ( ω) = n r c 0 ω σ ( ω) ,
where c 0 is the speed of light in vacuum and where we assume a constant refraction index n r .

The dielectric function within a single-particle picture Neutral excitations can be treated on dierent levels of sophistication. In order to get a rough overview of the most important features, we only consider interband electron-hole excitations where an electron is promoted from an occupied (v) to an unoccupied (c) state leaving behind its corresponding hole. In semiconductors and isolators, these type of excitations are lowest in energy. In addition, plasmons can be excited, i.e. collective oscillations relative to the positive background of the ion cores. In semiconductors, the plasmon excitation energies are usually higher than the lowest electron-hole excitations and meanwhile they are not included in our considerations.

On the simplest level of theory, we work in a single-particle picture, i.e. we decouple the excited electron from the remaining electronic system and approximate the neutral excitation energy by single-particle energies:

m = E N,m -E N,0 ≈ ε c -ε v .
The number of absorbed photons per unit volume and time, U ( ω), is obtained by summing up the probabilities Ω of all possible transitions from occupied to unoccupied states:

U ( ω) ∝ vc Ω v→c ( ω) .
Within rst-order time-dependent perturbation theory, Ω is given by Fermi's golden rule:

Ω v→c ( ω) ∝ | v| Ap |c | 2 δ (ε c -ε v -ω) , (1.43)
where A denotes the vector potential of the radiation and p = -i ∇ the electronic momentum operator. Equation (1.43) is analogue to the transition rate introduced for PES experiments, but here one is interested in a neutral nal state, i.e. the electron is not supposed to leave the system. Moreover, only vertical transitions are considered, where the photon momentum is not explicitly taken into account, since it is small as compared to electronic momenta in solids.

Again, the initial and nal state of the decoupled remaining electronic system are taken to be equal, i.e. we assume a frozen-orbital (Koopmans) picture. In order to arrive at an expression for ε 2 , the matrix element in equation (1.43) has to be evaluated. The derivation is somewhat lengthy 6 and therefore only the nal result is presented. For non-degenerate parabolic bands and excitations at the direct gap, ε 2 can be derived to [START_REF] Chuang | Physics of Photonic Devices[END_REF]:

ε 2 ( ω) ∝ 1 2 ω 2 ω -E g Θ ( ω -E g ) , (1.44)
where E g represents the fundamental electronic energy gap. The latter is dened by charged excitations, namely as the energy dierence between the rst ionization energy, i.e. the energy needed to eject an electron from the system, and the rst electron anity, which is the necessary energy to insert an electron. The above equation is valid for singlet transitions without spin ips. For photon energies smaller than E g , there is no absorption and the material is transparent. For photon energies greater than E g , the function 2 ω 2 • ε 2 increases like the square root of ( ω -E g ) with increasing ω (see Fig. This suggests that the single-particle approximation seems to miss important physical eects.

The dielectric function including excitonic eects One crucial eect that has to be included in the independent particle model is the electron-hole interaction. Indeed, the promoted electron and the created hole are not uncorrelated, but they inuence each other through an attractive Coulomb potential. We rather have to consider a quasiparticle in the form of a bound electron-hole pair instead of two independent particles. Due to the attractive potential, the energy needed to excite this so-called exciton is smaller than the excitation energy needed to create two bare particles. Therefore, in contrast to equation (1.44), optical absorption is

ε 2 2 ω 2 ε 2 E g ( ω < E g ) E opt E e-h B E opt E e-h B
eV meV some nanometer apart from each other.

For the sake of pedagogy, we focus on the description of Wannier-Mott excitons in the following. Here, due to ecient screening, we can straightforwardly treat the electron and the corresponding hole within the quasiparticle picture. The latter is also called eective-mass approximation, since the electron and the hole are regarded as bare particles, however, they are assigned a renormalized energy and mass m * . Within this approximation, the exciton is governed by the excitonic wave function Φ exc ,

Φ exc (r h , r e ) = vc a vc φ v (r h ) φ c (r e ) .
It is a linear combination of products from quasielectron φ c (r e ) and quasihole φ v (r h ) wave functions with the coecients a vc . That implies that one needs to go beyond a single-particle picture and instead considers a two-particle problem. The corresponding Schrödinger equation reads [START_REF] Schley | Optische Eigenschaften von InN und InN-basierten Halbleitern[END_REF][START_REF] Elliott | Intensity of Optical Absorption by Excitons[END_REF]:

- 2 2m * e ∇ 2 re - 2 2m * h ∇ 2 r h - e 2 4πε 0 ε r |r e -r h | + E g Φ exc (r h , r e ) = E exc Φ exc (r h , r e ) .
It models the absorption process by a quasielectron and a quasihole in a potential equal to the gap energy E g . They have eective masses m * and are coupled to each other via a screened Coulomb potential expressed in terms of the permittivity of the vacuum 4πε 0 and the relative permittivity ε r . The above equation can be solved in terms of a separation ansatz, where the motion of the center of gravity and the relative motion of the electron and the hole are decoupled. The latter can be treated analogously to the hydrogen problem or the positronium atom (for details see Ref. [START_REF] Yu | Fundamentals of Semiconductors -Physics and Materials Properties[END_REF][START_REF] Schley | Optische Eigenschaften von InN und InN-basierten Halbleitern[END_REF][START_REF] Elliott | Intensity of Optical Absorption by Excitons[END_REF][START_REF] Klingshirn | Semiconductor Optics[END_REF]). As a result, one obtains bound excitonic states for exciton energies (E exc < 0) and continuum states for (E exc > 0). The exciton energy of bound states at the Γ-point of the band structure is given by:

E exc n = E g - Ry * n 2 ,
where

Ry * = m * r e 4 2 (4πε 0 ε r ) 2 = m * r m e 1 ε 2 r
Ry is an eective Rydberg energy accounting for screening eects in the medium. The latter is determined by the ratio of the reduced eective mass m * r = m * e m * h / (m * e + m * h ) and the free electron mass, the relative permittivity ε r and the Rydberg constant Ry = 13.6 eV. By way of example, in indium nitride (InN), with m * e ≈ 0.06m e [START_REF] Wu | Eects of the narrow band gap on the properties of InN[END_REF][START_REF] Fu | Eective mass of InN epilayers[END_REF] and m * h ≈ 0.5m e [START_REF] Vurgaftman | Band parameters for nitrogen-containing semiconductors[END_REF], and a relative permittivity of ε r = 9.5 [START_REF] Persson | First-principle calculations of the dielectric function of zinc-blende and wurtzite InN[END_REF], one obtains for the rst bound exciton (n = 1) an eective Rydberg energy of Ry * ≈ 8 meV. This is one order of magnitude smaller than the Rydberg constant Ry, which elucidates the importance of screening. Moreover, room temperature, which corresponds to a thermal energy of around 25 meV, is sucient to dissociate the bound electron-hole pair, whose binding energy can be calculated following (E e-h b,n = Ry * /n 2 ). Contrary, the binding energy of organic semiconductors is usually one order of magnitude larger, as it will be explicitly shown for the coumarin family later in this work.

The consideration of electron-hole interactions signicantly modies the imaginary part of the dielectric function compared to the already discussed single-particle picture, yielding a hydrogen-like Rydberg series [START_REF] Yu | Fundamentals of Semiconductors -Physics and Materials Properties[END_REF][START_REF] Schley | Optische Eigenschaften von InN und InN-basierten Halbleitern[END_REF][START_REF] Winzer | Optische und elektronische Eigenschaften von AlGaN/GaN-Heterostrukturen[END_REF][START_REF] Grahn | Introduction to Semiconductor Physics[END_REF]: For excitation frequencies higher than E g , the electron and the hole are not bound anymore, but still somewhat correlated by the interaction with the medium. The imaginary part of the dielectric function ε 2 for the exciton continuum is given by [START_REF] Yu | Fundamentals of Semiconductors -Physics and Materials Properties[END_REF][START_REF] Schley | Optische Eigenschaften von InN und InN-basierten Halbleitern[END_REF][START_REF] Grahn | Introduction to Semiconductor Physics[END_REF]:

ε exc,b 2 ( ω) ∝ 1 2 ω 2 n Ry * n 3 δ ( ω -E exc n ) .
ε exc,f 2 ( ω) ∝ 1 2 ω 2 Θ ( ω -E g ) 1 1 -exp -Ry * / | ω -E g | . (1.46)
The latter is expected to coincide with equation (1.44) of the single-particle approximation.

However, even though the curves are very similar for high energies ω, they strongly dier close to ( ω = E g ). In contrast to equation (1.44), ε exc,f 2 is non-zero and has a peak at ( ω = E g ), see Fig. (1.11). This is due to the fact that the limit of both ε exc,b -→ E g ) are identical and nite. Moreover, one observes a signicant enhancement of absorption above E g with respect to equation (1.44), which is described by the so-called Sommerfeld enhancement factor. Combining the results for bound and continuum states and considering in addition broadening eects through a temperature dependent parameter Γ, one nally arrives at the so-called Elliott's model for the imaginary part of the dielectric function [START_REF] Elliott | Intensity of Optical Absorption by Excitons[END_REF][START_REF] Grahn | Introduction to Semiconductor Physics[END_REF]:

ε exc 2 ( ω) ∝ 1 2 ω 2 n Ry * Γn 3 exp - ω -E g + Ry * /n 2 2 Γ 2 + 1 2 ω 2 1 + erf [( ω -E g ) /Γ] 1 -exp -Ry * / | ω -E g | .
One thus obtains an equation to model optical absorption spectra, which contains three tting input parameters: the fundamental gap E g , the broadening parameter Γ and the eective Rydberg constant Ry * .

Including excitonic eects, using even simple models such as the presented eective-mass approximation, drastically changes the characteristics of ε 2 . The agreement with respect to experimental absorption spectra can be signicantly improved (see Fig. 1.10). We therefore conclude that electron-hole interactions play an important role in absorption experiments.

In the following, we transfer the gained insights to the already presented many-body perturbation theory formalism. That means we go beyond the single-quasiparticle description

ε 2 ω ≤ E g 2 ω 2 -1 ω -E g ω > E g ω < E g ω → E g ω → E g ω = E g GW ε V tot = ˆd2 ε -1 (1, 2) U ext (2) . χ CHAPTER 1. METHODOLOGY irreducible analogue P through: ε (1, 2) = δ (1, 2) -ˆd3 v (1, 3) P (3, 2) and ε -1 (1, 2) = δ (1, 2) + ˆd3 v (1, 3) χ (3, 2) .
If the system is not polarizable, χ is zero and consequently the total potential V tot equals the external potential U ext . However, for polarizable media, induced internal elds are created.

In this case, χ is in average usually negative, thus it weakens the eect of the external potential. As already introduced in preceding chapters, χ is a density-density correlation function following:

χ rt, r t ≡ ∂n ind (rt) ∂U ext (r t ) = -i N, 0| T n (rt) , n r t |N, 0 .
Analogue to the single-particle Green's function, it reads in the Lehmann representation [START_REF] Strinati | Application of the Green's functions method to the study of the optical properties of semiconductors[END_REF]:

χ (r, r , ω) = m =0 ρm(r)ρ * m (r ) ω+(E N,m -E N,0) -iη - ρ * m (r)ρm(r ) ω-(E N,m -E N,0) +iη , ρ m (r) = N, m| n (r) |N, 0 ,
where ρ m are charge uctuations. The reducible polarizability thus has poles at the neutral excitation energies ( m = E N,m -E N,0 ) of the created electron-hole pair like excitations.

From microscopic to macroscopic Up to now, two dierent kinds of dielectric functions have been presented. The microscopic dielectric function ε (1.47) as introduced within linear response theory and its macroscopic counterpart ε M (1.42), which is accessible through optical experiments. However, no relation between these two quantities has been established so far.

Following Refs. [START_REF] Adler | Quantum Theory of the Dielectric Constant in Real Solids[END_REF][START_REF] Wiser | Dielectric Constant with Local Field Eects Included[END_REF][START_REF] Ehrenreich | The Optical Properties of Solids[END_REF], the macroscopic dielectric function can be expressed in terms of the microscopic dielectric function for periodic structures through:

ε M (ω) ≡ lim k→0 1 ε -1 G=0,G =0 (k, ω) . Here, ε GG (k, ω) ≡ ε (k + G, k + G , ω) is the Fourier transform of ε (r, r , ω)
, where G is the reciprocal lattice vector and where the electronic wave vector k is limited to the rst Brillouin zone. The dielectric constant ε 0 corresponds to the static limit of the macroscopic dielectric function: ε 0 ≡ ε M (ω = 0). For crystals, the absorption spectrum, i.e. the imaginary part of the macroscopic dielectric function ε 2 , is thus obtained by calculating the microscopic dielectric function in reciprocal space followed by an inversion. In general, ε (r, r , ω) is non-local, i.e. it depends on (r, r ) and not on the dierence (r -r ). As a consequence, in reciprocal space ε GG (k, ω) is a non-diagonal matrix and thus an inversion couples the matrix elements with each other. This is known as local eld eects, arising in systems with microscopic inhomogeneities.

In nite systems, the photoabsorption cross section η can directly be accessed through the imaginary part of the dynamical polarizability α (ω):

η (ω) ≡ 4πω c {α (ω)} .
The dynamical polarizability in turn depends solely on the applied external eld and the reducible polarizability χ [START_REF] Onida | Electronic excitations: density-functional versus many-body Green's-function approaches[END_REF]:

α ij (ω) = -ˆdrdr U ext,i (r, ω) χ r, r , ω U ext,j r , ω .
By way of example, for dipolar external elds along the x and z direction, respectively, one obtains the non-diagonal α xz polarizability [START_REF] Onida | Electronic excitations: density-functional versus many-body Green's-function approaches[END_REF]:

α xz (ω) = -ˆdrdr x χ r, r , ω z .
Consequently, absorption is directly accessible through the introduced Green's function techniques. Nevertheless, it is important to note that the previously considered quantities, such as the Green's function G or the polarizability χ, are time-ordered objects. This implies that they include the knowledge of future interactions and hence they are no observables.

Instead, their retarded analogues only refer to bygone interactions and are thus measured in experiments. Contrary to the introduced time-ordered quantities, it is, however, not possible to establish a set of coupled integral equations for their calculation, since certain identities in the derivation do not hold. One has to pass by the time-ordered functions instead, which afterwards are transformed into their retarded counterparts. For the reducible polarizability, and correlation functions in general, simple transformation rules exist [START_REF] Strinati | Application of the Green's functions method to the study of the optical properties of semiconductors[END_REF][START_REF] Sottile | Response functions of semiconductors and insulators: from the Bethe-Salpeter equation to time-dependent density functional theory[END_REF], e.g.:

χ R (r, r , ω) = χ T (r, r , ω) χ R (r, r , ω) = sign (ω) χ T (r, r , ω)
For frequencies ω > 0, the retarded polarizability χ R equals thus the time-ordered polarizability. In subsequent sections, only time-ordered quantities are considered. Therefore, the superscript T will be dropped.

Common approximations to the polarizability Analogue to the one-particle Green's function, applying the equation of motion (EOM) technique to the polarizability yields a Dysonlike equation for χ in terms of the irreducible polarizability P and the bare Coulomb potential v:

χ(1, 2) = P (1, 2) + ˆd34 P (1, 3)v (3, 4) χ(4, 2).
However, the building blocks P from which χ is constructed are complicated objects themselves, given by Hedin's equation:

P (12) = -i ˆd34 G(1, 3) Γ(3, 4; 2) G(4, 1). (1.48)
This is in contrast to the Dyson equation for the single particle Green's function,

G(11 ) = G 0 (11 ) + ˆd2 ˆd3 G 0 (12)Σ(23)G(3 1), (1.49)
where the non-interacting Green's function G 0 represents known building blocks and where the complexity of the problem is fully governed by the self-energy Σ.

The simplest approximation to P consists of considering non-interacting particles, yielding the independent irreducible polarizability P IP :

P IP (12) ≡ -iG 0 (12) G 0 (21) .
The latter describes the absorption process as the uncorrelated motion of an excited bare electron and the created hole. These are completely decoupled from each other and the remaining electronic system. In the Lehmann representation (see Appendix A.7), it reads:

P IP (r 1 , r 2 ; ω) = m,l φ l (r 1 ) φ * l (r 2 ) φ m (r 2 ) φ * m (r 1 ) ω -ε 0 l -ε 0 m + iη - φ m (r 1 ) φ * m (r 2 ) φ l (r 2 ) φ * l (r 1 ) ω + ε 0 l -ε 0 m -iη , (1.50)
i.e. it is constructed from single-particle wave functions φ. Its poles correspond to the transition of an electron from an occupied single-particle state with energy ε 0 m to an unoccupied state with energy ε 0 l . This approximation is known as the random-phase approximation (RPA).

Within the RP A, χ and ε read:

χ RP A (1, 2) = P IP (1, 2) + ´d34 P IP (1, 3)v (3, 4) χ RP A (4, 2) ε RP A (1, 2) = δ (1, 2) -´d3 v (1, 3) P IP (3, 2) .
(1.51)

In order to go beyond the RP A, one considers quasiparticles instead of bare electrons and holes:

P QIP ≡ -iG (12) G (21)
.

(1.52)

The subscript QIP refers to quasi-independent particles. This highlights the fact that the two particles propagate without interacting with each other, however, both are quasiparticles and interact with the surrounding medium. Analogue to equation (1.50), the Lehmann representation is obtained by:

P QIP (r 1 , r 2 ; ω) = m,l f l (r 1 ) f * l (r 2 ) f m (r 2 ) f * m (r 1 ) ω -(ε l -ε m ) + iη - f m (r 1 ) f * m (r 2 ) f l (r 2 ) f * l (r 1 ) ω + (ε l -ε m ) -iη .
Contrary to the independent-particle polarizability, Lehmann amplitudes f l,m and quasiparticle energies ε l,m enter instead of single-particle quantities. P QIP corresponds to the irreducible polarizability calculated within the GW approximation, therefore this approach is called GW -RP A in the following. It represents the independent motion of a quasielectron and a quasihole. This is an important extension to P IP , however, as pointed out in the preceding section, it is crucial to explicitly include the interaction between the excited electron

P GW (Γ ≡ 1) P Γ Γ GW (1, 2; 3) = δ (1, 2) δ (2, 3) , P Γ Γ Γ(1, 2; 3) = δ(1, 2) δ(2, 3) + ˆd4567 δΣ(1, 2) δG(4, 5)
G(4, 6) Γ(6, 7; 3) G [START_REF] Faber | First-principles GW calculations for DNA and RNA nucleobases[END_REF][START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF].

[-iG(1, 3)G(4, 1)] ´d34

P (12) = -i ´d34 G(1, 3) Γ(3, 4; 2) G(4, 1) = -i ´d34 G(1, 3)G(4, 1)δ(3, 4) δ(4, 2)
-i ´d345678G(1, 3)G(4, 1) δΣ (3,4) δG [START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF][START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF] (3,4) δG [START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF][START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF]) P (5, 2, 6) .

G(5, 7) Γ(7, 8; 2) G(8, 6) = -i G(1, 2)G(2, 1) -i ´d3456G(1, 3)G(4, 1) δΣ
P (1, 2)

P (5, 2, 6) P (12) G 2 function L following: L(1, r t; 2, rt + ) = -G 2 (1, r t; 2, rt + ) + G 1 (1, 2)G 1 (r t, rt + ).
The latter contains both the independent propagation of two particles through the one-particle Green's functions G 1 and their coupled motion through the two-particle Green's function G 2 .

Comparing the identities for the two-point polarizability χ,

χ (1, 2) ≡ -iG 1 (11 + ) U ext (2) = i G 2 (1, 2; 1 + , 2 + ) -G 1 (1, 1 + )G 1 (2, 2 + ) ,
and the two-particle correlation function L, we realize that χ is nothing else than a degenerate form of L obtained from: χ (1, 2) = -iL(1, 2; 1 + , 2 + ).

(1.54)

From the following relation (see Appendix A.1),

G 2 1, r t; 2, rt + = G 1 (1, 2) G 1 r t, rt + - ∂G 1 (1, 2) ∂U ext (r, r , t) ,
we directly nd:

L(1, r t; 2, rt + ) = ∂G 1 (1, 2) ∂U ext (r, r , t)
.

Using the relation:

∂G 1 (12) ∂U ext (34) = -¨d5d6 G 1 (15) ∂G -1 1 (56) ∂U ext (34) G 1 (62) , L is transformed into: L(1, r t; 2, rt + ) = -´d3d4 G 1 (1, 3) ∂G -1 1 (3,4)
∂Uext(r,r ,t) G 1 (4, 2) .

(1.55)

The inverse of the one-particle Green's function is determined through the following Dyson equation:

G -1 1 = G 0 1 -1 -(M + U ext ) ,
where M is the mass operator (1.20) containing all possible interactions and where G 0 1 is the one-particle Green's function for a system of non-interacting particles. As it is demonstrated in Appendix A.8, inserting the above relation for G -1

1 into equation (1.55) yields a closed
Dyson-like equation:

L(1, 2; 1 , 2 ) = G 1 1, 2 G 1 2, 1 + ˆd3456 G 1 (1, 3) G 1 4, 1 K (3, 5; 4, 6
) L(6, 2; 5, 2 ).

(1.56)

The latter is known as Bethe-Salpeter equation for L [START_REF] Strinati | Application of the Green's functions method to the study of the optical properties of semiconductors[END_REF][START_REF] Onida | Electronic excitations: density-functional versus many-body Green's-function approaches[END_REF][START_REF] Bussi | Eects of the Electron-Hole Interaction on the Optical Properties of Materials: the Bethe-Salpeter Equation[END_REF][START_REF] Sottile | Response functions of semiconductors and insulators: from the Bethe-Salpeter equation to time-dependent density functional theory[END_REF][START_REF] Bruneval | Exchange and Correlation in the Electronic Structure of Solids, from Silicon to Cuprous Oxide: GW Approximation and beyond[END_REF][START_REF] Rebolini | Electronic excitation energies of molecular systems from the Bethe-Salpeter equation: Example of the H2 molecule[END_REF]. It describes the propagation of an electron and a hole through two terms. The rst one represents the propagation of two independent quasiparticles, whereas the second one describes their coupled motion through an eective two-particle interaction, the so-called kernel K:

K ≡ ∂M (3, 4) ∂G 1 (5, 6
) .

For the uncorrelated motion term, it is convenient to introduce a quasi-independent twoparticle correlation function L QIP analogue to equation 1.52:

L QIP (1, 2; 1 , 2 ) = G 1 1, 2 G 1 2, 1 .
Expressing the Bethe-Salpeter equation (1.56) in terms of L QIP yields:

L(1, 2; 1 , 2 ) = L QIP (1, 2; 1 , 2 ) + ˆd3456 L QIP (1, 4; 1 , 3) K (3, 5; 4, 6) L(6, 2; 5, 2 ).
One thus obtains an equation, where the quasi-independent quantity L QIP is connected to the full four-point polarizability L through the kernel K. This is completely analogue to the Dyson equation for the single-particle Green's function (1.49), where the self-energy Σ links the non-interacting to the interacting Green's function. To describe absorption experiments, the two-point quantities P and χ are in principle sucient, however, no closed equations can be set up for them within the Green's function approach. Therefore, one passes by the four-point quantity L. The latter contains more information than actually needed to describe absorption. Once the Bethe-Salpeter equation is solved, L is hence contracted to χ following equation (1.54). This yields:

-iL(1, (3, 5; 4, 6) [-iL(6, 2; 6, 2)]

2; 1, 2) = -iG 1 (1, 2) G 1 (2, 1) + ´d3456 [-iG 1 (1, 4) G 1 (4, 1)] iK
χ (1, 2) = P GW (1, 2) + ´d3456 P GW (1, 4) iK (3, 5; 4, 6) χ (6, 2) , (1.57)
where

P GW (1, 2) = -iG 1 (1, 2) G 1 (2, 1).
Reducible and irreducible quantities In order to single out the Hartree term from the eective two-particle interaction K, we split the mass operator into the Hartree contribution and the self-energy Σ:

M = V H + Σ, with V H (1, 2) = δ (1, 2) -i ´d3 v (1, 3) G 1 (3, 3 +
) . The latter follows from: n (1) = -iG (11 + ).

The kernel is consequently separated into: (5,6) ,

K = ∂V H (3,4) ∂G 1 (5,6) + ∂Σ(3,4) ∂G 1 (5,6) = -iδ (3, 4) δ (5, 6) v (3, 5) + ∂Σ(3,4) ∂G 1
L(1, 2; 1 , 2 ) = G 1 1, 2 G 1 2, 1 + ˆd3456 G 1 (1, 3) G 1 4, 1 K R (3, 5; 4, 6) L(6, 2; 5, 2 ).
From that, the reducible analogue L is obtained via:

L(1, 2; 1 , 2 ) = L(1, 2; 1 , 2 ) + ˆd3456 L(1, 4; 1 , 3) K H (3, 5; 4, 6) L(6, 2; 5, 2 ).
The choice of the kernel The kernel K accounts for all interactions and is hence a complicated many-body object. It can be strongly simplied depending on the chosen approximation for the self-energy Σ in equation (1.58). The roughest approximation consists of completely neglecting the self-energy (Σ ≡ 0). As a result, the Bethe-Salpeter equation reduces to a time-dependent Hartree approach: [START_REF] Aryasetiawan | The GW method[END_REF][START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF]] L(6, 2; 6, 2 ).

L T DH (1, 2; 1 , 2 ) = G 1 (1, 2 ) G 1 (2, 1 ) + ´d46 G 1 (1, 4) G 1 (4, 1 ) [-iv
(1.59)

Equation (1.59) can be contracted to χ following equation (1.57). This results in a closed equation for χ:

χ T DH (1, 2) = P GW (1, 2) + ´d46 P GW (1, 4) v (4, 6) χ T DH (6, 2), with P GW (1, 2) ≡ -iG 1 (1, 2) G 1 (2, 1
). The latter is nothing else than the already introduced GW -RP A for χ.

In order to go beyond GW -RP A, a possibility is to use the bare exchange part Σ x of the self-energy to construct the kernel K:

Σ ≈ Σ x = iG 1 (1, 2, τ ) v (1, 2) .
This results in a Hartree-Fock like kernel following:

K HF (3, 5; 4, 6) = -iδ (3, 4) δ (5, 6) v (3, 5) + i δ(3, 5)δ (4, 6) v (3, 4) .
As it is demonstrated in Appendix A.8, one obtains for the Bethe-Salpeter equation:

L T DHF (1, 2; 1 , 2 ) = G 1 (1, 2 ) G 1 (2, 1 ) + ´d46 G 1 (1, 4) G 1 (4, 1 ) [-iv (4, 6
)] L T DHF (6, 2; 6, 2 )

+ ´d56 G 1 (1, 5) G 1 (6, 1 ) [iv (5, 6)] L T DHF (6, 2; 5, 2 ), (1.60)
which represents a time-dependent Hartree-Fock approach (T DHF ). The independent motion of an electron and a hole is governed by the rst term in the above equation, whereas the second term includes the classical Hartree contribution and the third term the Fock exchange. The excited electron and the hole thus interact through exchange interactions, however, correlation is not taken into account. It is important to note that in contrast to equation (1.59), the terms can not be readily contracted to two-point quantities following (1 → 1, 2 → 2). Indeed, the exchange term remains a true four-point quantity. That implies that one is obliged to solve the Bethe-Salpeter equation for L, instead of a closed equation for χ as it is the case in the T DH approach. In general, including interactions beyond Hartree enforces a detour over four-point equations.

In order to go beyond the T DHF approach, one can include correlations by using Σ within the GW approximation. This yields for the kernel:

K GW (3, 5; 4, 6) = -iδ (3, 4) δ (5, 6) v (3, 5) + ∂iG(3,4)W (3,4) ∂G 1 (5,6) = -iδ (3, 4) δ (5, 6) v (3, 5) + iδ (3, 5) δ (4, 6) W (3, 4) ,
where we imposed (∂W/∂G ≡ 0), i.e. the change in the screening due to the excitation is neglected. This eect is usually supposed to be small, even though systematic studies on this approximation are scarce. As shown in Appendix A.8, the Bethe-Salpeter equation for the GW kernel reads:

L T DSHF (1, 2; 1 , 2 ) = G 1 (1, 2 ) G 1 (2, 1 ) + ´d46 G 1 (1, 4) G 1 (4, 1 ) [-iv (4, 6
)] L T DSHF (6, 2; 6, 2 ) [START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF][START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF]] L T DSHF (6, 2; 5, 2 ).

+ ´d56 G 1 (1, 5) G 1 (6, 1 ) [iW
(

The comparison to expression (1.60) points out that one obtains a time-dependent screened Hartree-Fock approach (T DSHF ), where the time-dependent screened Coulomb potential W

replaces the static bare Coulomb potential v in the third term. Consequently, the independent motion of two particles is not only correlated by the Hartree contribution of the second term, but also by a screened dynamical exchange contribution. In the following, a solution scheme for the time-dependent screened HF A is worked out.

Fourier transformation to frequency space and the static approximation Since L is a four-point quantity, it depends, in principle, on four dierent time arguments. However, the

Coulomb potential v is instantaneous, v (1, 2) = v (x 1 x 2 ) δ (t 1 -t 2 ) ,
and also the one-particle Green's functions in equation (1.61) only depend on time dierences.

This suggests that the time arguments can be contracted to a single one in order to permit a straightforward Fourier transformation to frequency space. However, as it is demonstrated in the Appendix, this is not possible without further ado and approximations are implied. First, one assumes translational time invariance and an isochronous propagation of the electron and the hole. 7 In addition, the dynamically screened interaction W is approximated within its static limit:

W (1, 2) ≈ W stat (x 1 x 2 ) δ (t 1 -t 2 ) ,
7 For L = L (1234), i.e. L = L (t1 -t4; t2 -t3), we set t1 = t3; t2 = t4.

an approximation we already encountered for the static COHSEX approach. This is a drastic assumption, however, as it is shown in detail in Appendix A.8, it greatly simplies equation (1.61) and one obtains:

L (τ ) = L QIP (τ ) -i ´dτ L QIP (τ ) L (τ -τ ) v (x 3 x 4 ) +i ´dτ L QIP (τ ) L (τ -τ ) W stat (x 3 x 4 ) ,
where τ = t 1 -t 2 and τ = t 4 -t 1 . In order to simplify the notation, space coordinates are omitted. As a result, a straightforward Fourier transformation using the convolution theorem can be performed, yielding:

L (ω) = L QIP (ω) -iL QIP (ω) L (ω) v (x 3 x 4 ) + iL QIP (ω) L (ω) W stat (x 3 x 4 ) .
Reintroducing space coordinates, L (ω) reads:

L x 1 x 2 ; x 1 x 2 ; ω = L QIP x 1 x 2 ; x 1 x 2 ; ω + ´dx 3 x 4 x 5 x 6 L QIP x 1 x 4 ; x 1 x 3 ; ω K (x 3 x 4 ; x 5 x 6 ) L x 6 x 2 ; x 5 x 2 ; ω , (1.62)
with

L QIP x 1 x 2 ; x 1 x 2 ; ω = 1 2π ˆdω G 1 x 1 x 2 ; ω + ω G 1 x 2 x 1 ; ω
being the Fourier transform of the already introduced quasi-independent two-particle correlation function L QIP . K is the many-body perturbation theory kernel within the GW approximation, following:

K (x 3 x 4 ; x 5 x 6 ) = iδ (x 5 , x 3 ) δ (x 6 , x 4 ) W stat (x 3 x 4 ) -iδ (x 3 , x 4 ) δ (x 5 , x 6 ) v (x 3 x 5 ) .
It is important to keep in mind that we used a static, i.e. strongly approximated, screened

Coulomb potential W in order to straightforwardly carry out the Fourier transformation.

The eective two-particle problem In principle, equation (1.62) can be solved by inversion for every frequency ω. However, it seems more handy to reformulate the problem in terms of an eective eigenvalue equation. This is analogue to the already introduced one-particle eective equations, such as the HF , DF T -K S or quasiparticle equation. However, in the case of absorption, an eective two-particle equation is needed. This equation then serves e.g.

to determine neutral excitation energies eciently.

Using the Lehmann representation of the Green's function in Fourier space, an explicit expression for the quasi-independent two-particle correlation function L QIP within the RP A or GW -RP A is obtained (see Appendix A.7):

-iL RP A QIP x 1 x 2 ; x 1 x 2 ; ω = m,l

φ l (x 1 )φm(x 2 )φ * m x 1 φ * l x 2 ω-(ε l -εm)+iη - φ * m x 2 φ l (x 2 )φ * l x 1 φm(x 1 ) ω+(ε l -εm)-iη , (1.63)
where φ l,m are single-particle wave functions and ε l,m are single-or quasiparticle energies.

Occupied states are labeled by the index m, unoccupied ones by l. Equation (1.63) suggests to work in transition space, i.e. to dene a two-particle excitonic basis {ψ exc },

ψ exc i (x 1 , x 2 ) ≡ n 1 n 2 c i,n 1 n 2 φ n 1 (x 1 ) φ * n 2 (x 2 ) ,
where one sums over all single-particle basis functions φ m,l appearing in G 0 and L RP A QIP . The transformation of any four-point quantity F (x 1 x 2 x 3 x 4 ) to this basis follows:

F (x 1 x 2 ; x 3 x 4 ) = n 1 n 2 n 3 n 4 φ n 1 (x 1 ) φ * n 2 (x 2 ) F n 1 n 2 n 3 n 4 φ n 3 (x 3 ) φ * n 4 (x 4 ) with F n 1 n 2 n 3 n 4 = ˆdx 1 x 2 x 3 x 4 φ n 1 (x 1 ) φ * n 2 (x 2 ) F (x 1 x 2 ; x 3 x 4 ; ω) φ n 3 (x 3 ) φ * n 4 (x 4 ) .
Transferred to the Bethe-Salpeter equation (1.62), this yields the following matrix elements:

L n 1 n 2 n 3 n 4 (ω) = L n 1 n 2 n 3 n 4 QIP (ω) + L n 1 n 2 n 5 n 6 QIP (ω) K n 5 n 6 n 7 n 8 L n 7 n 8 n 3 n 4 (ω) ,
or, represented in matrix notation:

[L (ω)] = [L QIP (ω)] + [L QIP (ω)] [K] [L (ω)] . (1.64)
At this point it is useful to multiply the Bethe-Salpeter equation with a factor (-i) following equation (1.57), since in the end of the calculation one is interested in contracting L to χ:

χ (1, 2) = -iL(1, 2; 1 + , 2 + ).
One thus nds for the matrix equation:

[-iL (ω)] = [-iL QIP (ω)] + [-iL QIP (ω)] [iK] [-iL (ω)] .
Evaluating the quasi-independent polarizability L RP A QIP within the single-particle basis starting from equation (1.63), results in (see Appendix A.8):

-iL

n 1 n 2 n 3 n 4 QIP (ω) = -i ´dx 1 x 1 x 2 x 2 φ n 1 x 1 φ * n 2 (x 1 ) L QIP x 1 x 2 ; x 1 x 2 ; ω φ * n 3 (x 2 ) φ n 4 x 2 = m,l δ(n 1 ,m)δ(n 2 ,l)δ(n 3 ,m)δ(n 4 ,l)
ω-(εn 2 -εn 1 )+iη -δ(n 1 ,l)δ(n 2 ,m)δ(n 3 ,l)δ(n 4 ,m) ω+(εn 1 -εn 2 )-iη , which points out that the matrix -iL RP A QIP is diagonal in the transition basis (n 1 = n 3 , n 2 = n 4 ). Moreover, only (occupied → unoccupied) or (unoccupied → occupied) transitions contribute, whereas (occupied → occupied) or (unoccupied → unoccupied) do not occur. This can be depicted in form of a (n 1 n 2 , n 3 n 4 ) matrix:

[-iL QIP (ω)] = (n 1 n 2 ) (n 3 n 4 ) → mm ll ml lm ↓ mm 0 0 0 0 ll 0 0 0 0 ml 0 0 -1 ∆εn 2 n 1 -ω 0 lm 0 0 0 1 ∆εn 2 n 1 -ω (1.65)
where ∆ε n 2 n 1 = (ε n 2 -ε n 1 ). Consequently, the quasi-independent polarizability L RP A QIP can be compactly written in terms of occupation factors f i (f m = 1, f l = 0):

-iL n 1 n 2 n 3 n 4 QIP (ω) = (f n 2 -f n 1 ) δ (n 1 , n 3 ) δ (n 2 , n 4 ) ∆ε n 2 n 1 -ω ,
where zeros on the diagonal appear for f n 2 = f n 1 . From now on, we restrict ourselves to the physical meaningful (ml, lm) subspace, where [-iL QIP ] has only non-zero diagonal elements and thus is invertible. Moreover, we introduce an occupation matrix [F ],

F n 1 n 2 n 3 n 4 = (f n 2 -f n 1 ) δ (n 1 , n 3 ) δ (n 2 , n 4 ) ,
which also has only non-zero elements in the chosen subspace. It follows:

[-iL QIP (ω)] = -i LQIP (ω) [F ] with -i LQIP (ω) [F ] = 1 ∆εn 2 n 1 -ω 0 0 1 ∆εn 2 n 1 -ω      f (n 2 -n 1 ) -1 0 0 f (n 2 -n 1 ) 1     
.

(1.66)

The Bethe-Salpeter matrix equation within the (ml, lm) subspace then reads:

[-iL (ω)] = -i LQIP (ω) -1 -[F ] [iK]
-1

[F ] .

(1.67)

Since [-iL QIP ] is a diagonal matrix, its inverse is readily calculated from equation (1.66) to:

-i LQIP (ω)

-1 = ∆ε n 2 n 1 0 0 ∆ε n 2 n 1 -ω 1 0 0 1 ,
where the frequency dependence has been singled out. This permits to introduce a frequencyindependent two-particle eective Hamiltonian H 2p , following:

[-iL (ω)] = H 2p -1ω -1 [F ] . (1.68)
with

H 2p = ∆ε lm 0 0 ∆ε ml + i K ml,ml K ml,lm
-K lm,ml -K lm,lm .

In general, the two-particle eective Hamiltonian H 2p is a non-Hermitian block matrix, reading in a shorthand notation: For the sake of convenience, we focus on the resonant part of the two-particle eective Hamiltonian in the following and search for suited solution schemes. The polarizability L for the resonant part can be obtained within the T DA analogue to equation (1.68) by:

H 2p = H res K coupl -K coupl * -(H res ) * . (1.
[-iL (ω)] = ([H res ] -1ω) -1 [F ] .
This implies that, in principle, a matrix with elements stemming from four-point quantities has to be inverted for every frequency ω. As demonstrated in the following, this laborious approach can be circumvented by making use of some mathematical transformations. In the so-called spectral representation, any Hermitian matrix [M ] can be expressed in terms of their eigenvalues ε λ and eigenvectors |λ , following:

[M ] = λ ε λ |λ λ| . If [M ] is invertible, its inverse [M ] -1 is given by: [M ] -1 = λ 1 ε λ |λ λ| .
For our actual problem, we thus obtain:

[-iL (ω)] = ([H res ] -ω) -1 = λ |λ λ| ε λ -ω ,
where |λ and ε λ are the eigenvectors and eigenvalues of [H res ]. The problem of inverting a four-point equation for every frequency ω is thus reduced to an eigenvalue problem for the eective two-particle Hamiltonian:

[H res ] |λ = ε λ |λ .
This implies that H res has to be diagonalized once in order to obtain its eigenvectors and eigenvalues, the latter corresponding to neutral excitation energies. The presented solution scheme is equivalently valid for the full two-particle Hamiltonian H 2p , however, due to its non-Hermitian nature, a generalized eigenvalue equation distinguishing between left and right eigenvectors has to be solved.

As it is detailed in Appendix A.8, the matrix elements H res ml,m l of the resonant part read:

H res ml,m l = H diag ml,ml + H exch ml,m l + H scr ml,m l ,

i.e. it is split into a diagonal, an electron-hole exchange and a screened electron-hole interaction part with:

H diag ml,ml = ∆ε lm δ mm δ ll , H exch ml,m l = 2η ´dx 1 x 2 φ m (x 1 ) φ * l (x 1 ) v (x 1 x 2 ) φ * m (x 2 ) φ l (x 2 ) , H scr ml,m l = -´dx 1 x 2 φ m (x 1 ) φ * l (x 2 ) W stat (x 2 x 1 ) φ * m (x 1 ) φ l (x 2 ) .
This implies that the neutral excitation energies which are obtained by diagonalizing [H res ] consist of three contributions. The rst one stems from the diagonal part H diag ml,ml and is simply the energy dierence between an occupied and an unoccupied single-or quasiparticle level. Further, electron-hole interactions are taken into account through the exchange and the static screened exchange term. The spin is not explicitly included in the derivation, however, the factor 2η in the second term implicitly accounts for it. For singlet excitations, where the promoted electron and the corresponding hole have the same spin, one has (η = 1), whereas for triplet excitations one sets (η = 0). Note that the standard notation 'exchange' is misleading, since H exch ml,m l is clearly a Hartree and not a Hartree-Fock contribution.

In conclusion, the many-body perturbation theory BSE formalism oers the possibility to access optical absorption properties from rst principles. Usually, it is applied in its time-dependent screened Hartree-Fock approximation, see equation (1.60). In practice, it goes along with an underlying GW calculation, from which electronic excitation energies ε l,m are drawn out to construct the quasi-independent polarizability L RP A QIP . In addition, also the bare Coulomb potential v and the static screened Coulomb potential W stat needed for the (screened) exchange term of the two-particle Hamiltonian H 2p can be directly reused from the GW calculation. Once the GW electronic structure is calculated, electron-hole interactions can thus be included in a relatively inexpensive fashion. The BSE approach enjoys great popularity and its reliability has been shown for diverse systems, such as clusters [START_REF] Onida | Ab Initio Calculations of the Quasiparticle and Absorption Spectra of Clusters: The Sodium Tetramer[END_REF], surfaces [START_REF] Rohlng | Excitons and Optical Spectrum of the Si(111) -(2 × 1) Surface[END_REF] and solids [START_REF] Albrecht | Ab Initio Calculation of Excitonic Eects in the Optical Spectra of Semiconductors[END_REF][START_REF] Benedict | Optical Absorption of Insulators and the Electron-Hole Interaction: An Ab Initio Calculation[END_REF]. In subsequent chapters, its applicability to molecular systems will be discussed in detail.

Excursus: time-dependent DFT

A widely used alternative to the GW/BSE formalism to calculate optical excitation properties for nite systems is time-dependent density functional theory (TDDFT) [START_REF] Runge | Density-Functional Theory for Time-Dependent Systems[END_REF][START_REF] Marques | Time-Dependent Density Functional Theory[END_REF][START_REF] Casida | Recent Advances in Density Functional Methods, Part I[END_REF][START_REF] Casida | Time-dependent density-functional theory for molecules and molecular solids[END_REF],

where one obtains the entire excitation spectrum of relatively large systems at reasonable costs. Analogue to DF T , the time-dependent many-body Schrödinger equation is mapped onto an eective one-particle time-dependent Kohn-Sham equation in TDDFT, through a one-to-one correspondence of the time-dependent one-body external potential U ext (r, t) and the time-dependent one-body density n (r, t) [START_REF] Runge | Density-Functional Theory for Time-Dependent Systems[END_REF][START_REF] Marques | Time-Dependent Density Functional Theory[END_REF]:

Ĥ (t) Ψ (r 1 . . . r N , t) = i dΨ (r 1 . . . r N , t) dt -→ - ∇ 2 2 + V ef f [n] (r, t) φ i (r, t) = i dφ i (r, t) dt .
The ctitious Kohn-Sham system with density n (r, t) = N i |φ i (r, t)| 2 yields by construction the density of the real system and is governed by a time-dependent one-particle eective Kohn-Sham potential V ef f (r, t), consisting of a time-dependent external U ext (r, t), a timedependent Hartree,

V H (r, t) = ˆd3 r n (r , t) |r -r | ,
and a time-dependent exchange-correlation contribution V xc (r, t). This is completely similar to the time-independent DF T formalism, however, the exchange-correlation potential is now much more complex, since it is a functional of the entire history of the density n (r, t), involving the solution of all time-dependent Coulomb-interacting problems [START_REF] Marques | Time-Dependent Density Functional Theory[END_REF]. One therefore not only has to make assumptions on its spatial, but also on its temporal form. In practice, the adiabatic approach is most common, where V xc (r, t) is assumed to not depend on the history of the density, but only on its present form [START_REF] Kronik | Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals[END_REF]. In principle, the TDDFT framework can be useful, whenever time-dependent electrons are involved. In practice, TDDFT is widely applied within the linear response regime in order to access the system's response to a weak time-dependent external perturbation. Within this regime, one only has to consider the eective potential for densities close to the ground state density n GS (r, t), i.e. n (r, t) = n GS (r, t)+n ind (r, t), instead of densities strongly varying in time. Dierent TDDFT methods to calculate excitation energies within the linear response regime exist, such as the real-timeevolution scheme, and the reader is referred to Ref. [START_REF] Marques | Time-Dependent Density Functional Theory[END_REF] for a comprehensive overview. Here, we focus on one of the most popular approaches to access excitation energies for nite systems, namely the eigenvalue problem approach in transition space proposed by Casida [START_REF] Casida | Recent Advances in Density Functional Methods, Part I[END_REF].

In general, assuming that the time-dependent external perturbation is a weak electric eld, the response of the ground state charge density n GS (r, t) is to induce small changes n ind (r, t)

via the already introduced reducible polarizability χ [START_REF] Onida | Electronic excitations: density-functional versus many-body Green's-function approaches[END_REF][START_REF] Tiago | Optical excitations in organic molecules, clusters, and defects studied by rst-principles Green's function methods[END_REF]:

n ind (r, t) = ˆdt ˆd3 r χ [n GS ] r, r , t U ext r , t .
As a remainder, χ denotes in its Lehmann representation in frequency space:

χ (r, r , ω) = k =0 ρ k (r)ρ * k (r ) ω+ k -iη - ρ * k (r)ρ k (r ) ω-k +iη , ρ k (r) = N, k| n (r) |N, 0 ,
where ρ k are charge uctuations and ( k = E N,k -E N,0 ) neutral excitation energies which one wants to calculate [START_REF] Strinati | Application of the Green's functions method to the study of the optical properties of semiconductors[END_REF][START_REF] Tiago | Optical excitations in organic molecules, clusters, and defects studied by rst-principles Green's function methods[END_REF]. Within the DF T framework, the exact reducible polarizability χ can be expressed in terms of a closed Dyson-like equation following [START_REF] Marques | Time-Dependent Density Functional Theory[END_REF]:

χ(r, r , ω) = χ 0 (r, r , ω) + ˆdr 1 r 2 χ 0 (r, r 1 , ω) [v (r 1 , r 2 ) + f xc (r 1 , r 2 , ω)] χ(r 2 , r , ω). (1.70)
This is the central equation of TDDFT linear response in frequency space, where all quantities are functionals of the ground state density. Whereas time-independent DF T -K S eigenvalues can not be mapped straightforwardly on physical quantities, i.e. quasiparticle excitation energies, TDDFT directly provides optical excitation energies and oscillator strengths from the poles of χ [START_REF] Kronik | Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals[END_REF]. One needed ingredient is χ 0 , the non-interacting Kohn-Sham polarizability dened as: χ 0 rt, r t ≡ ∂n ind (r, t) ∂V tot (r , t ) .

Here, the total potential V tot consists of the applied external potential, the induced Hartree and the induced exchange-correlation potential. The non-interacting Kohn-Sham polarizability χ 0 can be readily obtained from a standard time-independent DF T -K S calculation, where one uses the time-independent DF T -K S eigenstates φ v,c and eigenvalues ε v,c to construct it:

χ 0 r, r , ω = m,l φ l (r) φ * l (r ) φ m (r ) φ * m (r) ω -ε 0 l -ε 0 m + iη - φ m (r) φ * m (r ) φ l (r ) φ * l (r) ω + ε 0 l -ε 0 m -iη .
Here, l stands for unoccupied and m for occupied states. It thus corresponds to a special type of a non-interacting polarizability P IP , see equation (1.50). However, its physical interpretation is dicult due to the missing correspondence of Kohn-Sham eigenvalues and single-particle excitation energies. Expression (1.70) is very similar to the presented Dyson equation for χ within the Green's functions approach (see below), however, important differences have to be noted. First, the TDDFT framework allows to nd a closed Dyson-like equation for the exact χ. Within the Green's function approach, this is only achieved within the random-phase approximation for χ, see equation (1.51). Instead, for the exact χ, the irreducible polarizability P enters the Dyson equation:

χ(1, 2) = P (1, 2) + ˆd34 P (1, 3)v (3, 4) χ(4, 2).
The denition of P (1.34) is analogue to that of χ 0 , however, it does not include the induced exchange-correlation potential, but solely the external and the induced Hartree potential. It is a complex object itself, resulting in a highly non-trivial equation for χ, see equation (1.53).

On the contrary, by utilizing χ 0 , it is possible to shift the complexity of the problem into the time-dependent exchange correlation kernel f xc . As a result, one has a closed equation for the two-point polarizability χ and avoids the detour by the four-point polarizability L. However, the exact kernel, dened as

f xc rt, r t = ∂V xc [n (r, t)] ∂n (r , t ) n=n GS ,
is unknown and one has to nd appropriate approximations to it. The most common approach is the adiabatic local-density approximation (TDLDA), where f xc is approximated by the derivative of the static and local LDA exchange-correlation functional [START_REF] Onida | Electronic excitations: density-functional versus many-body Green's-function approaches[END_REF]:

f LDA xc r, r = δ r -r ∂V LDA xc [n (r)] ∂n (r)
.

This is a drastic approximation, but its simplicity is nevertheless tempting.

In the case of frequency-independent exchange-correlation kernels and systems with a discrete excitation energy spectrum, such as e.g. nite systems, it is convenient to reformulate the response equation (1.70) in terms of an eigenvalue problem. By rewriting the equations in terms of a transition basis ψ exc (r, r ) consisting of products of Kohn-Sham occupied φ m (r) and empty φ l (r ) states, one obtains, analogously to the BSE approach, an eective two-particle Hamiltonian H 2p . The latter is composed of (anti-)resonant H res and coupling K coupl blocks and is non-Hermitian. For real wave functions and in the case of TDDFT, Casida demonstrated that one can simplify the problem to the following eigenvalue equation [START_REF] Marques | Time-Dependent Density Functional Theory[END_REF][START_REF] Onida | Electronic excitations: density-functional versus many-body Green's-function approaches[END_REF][START_REF] Casida | Recent Advances in Density Functional Methods, Part I[END_REF][START_REF] Tiago | Optical excitations in organic molecules, clusters, and defects studied by rst-principles Green's function methods[END_REF]:

H diag 1 2 H diag + 4 H exch + [H xc ] H diag 1 2 [X] = ε 2 m [X] ,
for which ecient algorithms, such as Haydock's recursion method [START_REF] Haydock | Recursive Solution of the Schroedinger Equation[END_REF], can be applied [START_REF] Marques | Time-Dependent Density Functional Theory[END_REF][START_REF] Onida | Electronic excitations: density-functional versus many-body Green's-function approaches[END_REF]. The eigenvalues are the square of the excitation energies and the eigenvectors are related to oscillator strengths. H diag , H exch and [H xc ] represent matrix operators, whose elements read:

H diag ml,m l = δ mm δ ll ε 0 l -ε 0 m , H exch ml,m l = ´drdr φ m (r) φ * l (r) v (r, r ) φ * m (r ) φ l (r ) , H xc ml,m l = ´drdr φ m (r) φ * l (r) f xc (r, r ) φ * m (r ) φ l (r ) .
A direct comparison of the resonant part H res of the BSE and the TDDFT formalism,

H res = H diag + H exch + H scr/xc ,
reveals that in TDDFT one also starts from a single-particle excitation spectrum (the diagonal H diag term) and then includes electron-hole interactions through the two remaining contributions. The diagonal terms within BSE and TDDFT are similar, however, in one strengths. However, from a computational point of view, the four-point representation is only convenient if the diagonalization of the eective eigenvalue problem is less demanding than the inversion of the response equation for several frequencies, which automatically implies frequency-independent kernels [START_REF] Marques | Time-Dependent Density Functional Theory[END_REF]. Moreover, the quadratic basis of contributing occupied and unoccupied states must be smaller than the real-space (reciprocal-space) basis for the corresponding two-point problem. In TDDFT, this is only the case for small nite systems with well separated energy levels, where only a small part of the optical spectrum is of interest [START_REF] Marques | Time-Dependent Density Functional Theory[END_REF]. For these systems, Casida's TDDFT approach to calculate optical excitation spectra is most popular. Later in this work, a detailed comparison of the seemingly so similar, but nevertheless conceptually dierent TDDFT linear response and GW/BSE approach is presented for organic molecules in the gas phase. given in the following.

Ab initio calculations using a Gaussian basis

The equations derived in the last chapter, ranging from Hartree-Fock to TDDFT, provide mathematical formalisms to describe matter at a quantum level. Although approximations and simplications have already been carried out in their derivation, one ends up with computationally demanding integro-dierential equations, depending on a multitude of arguments.

Due to their complexity, it is not evident that an analytic solution exists.

The Linear Combination of Atomic orbitals (LCAO) approach

Numerically, dierent approaches to tackle these complicated equations have been developed.

One approach consists of expressing these problems in terms of basis sets. The advantage is that the resulting equations can be readily transformed into matrix equations. This reduces the original problem to a system of linear equations, a task for which computers are more than suited and for which ecient algorithms have been developed [START_REF] Press | Numerical Recipes in Fortran: The Art of Scientic Computing[END_REF]. This will be demonstrated for eigenvalue problems of the type [START_REF] Szabo | Modern Quantum Chemistry -Introduction to Advanced Electronic Structure Theory[END_REF]:

Ô |f = w |f . (2.1)
Here, |f is an eigenvector to the operator Ô with eigenvalue w. In the following, we are working in the real space representation, i.e. f (r) = r |f and Ô (r) = r| Ô |r . Several equations already encountered in preceding chapters represent eigenvalue problems, such as the Hartree-Fock, the Kohn-Sham or the quasiparticle equation.

The basis set is introduced by expanding the eigenfunctions f (r) in a complete set of basis functions {α µ } with weighting coecients c µ :

f (r) = ∞ µ=1 c µ α µ (r) . (2.2)
We assume the general case of non-orthogonal basis sets, for which the matrix elements of the overlap matrix [S] are given by:

S µν = ˆd3 r α * µ (r) α ν (r) .
Inserting (2.2) into equation (2.1) leads to:

∞ µ=1 Ô (r) c µ α µ (r) = ∞ µ=1 w c µ α µ (r) .
Multiplication by an arbitrary basis function α ν (r) from the left and integration over d 3 r results in: In principle, the {α µ } form a complete basis set, i.e. on the one hand any f (r) is reproduced exactly, but on the other hand Ō becomes an innite matrix. In practice, a certain cut-o N is chosen limiting the amount of available basis functions:

∞ µ=1 c µ ˆd3 r α ν (r) Ô (r) α µ (r) = ∞ µ=1 w c µ ˆd3 r α ν (r) α µ (r) =
f (r) ≈ N µ=1 c µ α µ (r) .
In that way, the eigenvalue problem is reduced to a N × N problem, where the calculated N eigenvalues approximate the true eigenvalues. This implies that the basis set has to be selected very carefully, since it has a strong impact on the quality of the results within the chosen level of theory (H F , DF T -K S, GW ). In the case of orthogonal basis sets, the inclusion of more and more basis functions always improves the completeness, i.e. these basis sets are systematic.

However, as it will be detailed later in this section, this is not the case for non-orthogonal basis sets, where therefore special care must be taken.

Within the Linear Combination of Atomic Orbitals (LC AO) approach, molecular wave functions are expanded in atom-centered basis functions, such as Gaussian functions. These atom-centered functions consist of single-electron functions localized at the nuclei of the specic atoms, which very eciently describe the critical region near the nuclei. It tends to produce, in particular for nite size systems, much more compact basis sets than plane wave or real-space basis sets, even though convergency is a more dicult issue. A detailed review of state-of-the-art atom-centered basis sets is provided by Ref. [START_REF] Hill | Gaussian basis sets for molecular applications[END_REF].

From a product to an auxiliary basis

Introducing a basis {α} consisting of N basis functions leads for matrix elements of non-local operators such as the screened Coulomb potential W , the irreducible susceptibility P and the Fock operator V F to four-center-two-electron integrals of the type:

´d3 r d 3 r α * µ (r) α * κ (r ) Ô (r, r ) α κ (r) α ν (r ) .
(2.4)

Evaluating these matrix elements is equivalent to working in a so-called product basis:

P µκ (r) = α * µ (r) α κ (r) , containing N 2
2 dierent elements. This implies rather large basis sets compared to the original {α (r)} basis and consequently high computational costs. Moreover, assuming a Gaussian function basis set for {α (r)}, their product is not further centered on single atoms, but on barycentric points P. By way of example, for two unnormalized primitive s-type Gaussian functions G 1 = e -γ 1 (r-A) 2 and G 2 = e -γ 2 (r-B) 2 centered at atoms A and B one nds (see Appendix A.10):

G 1 G 2 = C e -η(r-P) 2 , with (η = γ 1 + γ 2 ), P = γ 1 A+γ 2 B η and C = e - γ 1 γ 2 (A-B) γ 1 +γ 2 2
. The mentioned points suggest not to work in the full product basis, but in a smaller basis consisting of L elements which approximately reproduces the product basis. Such a basis is called an auxiliary basis {β (r)}:

P µκ (r) L λ c µκλ β λ (r) ,
where c µκλ are weighting coecients. The equality can become exact for L → ∞. Concerning the character of the auxiliary basis {β (r)}, no general form is specied. For {α (r)} consisting of Gaussian functions of the type e -µ(r-A) 2

, it seems convenient to choose an auxiliary basis with Gaussian functions centered on specic atoms. The number of considered auxiliary basis functions per atom and the best exponential coecients µ represent a crucial point which requires extensive accuracy tests. Even though representing a strong approximation to the full product basis, auxiliary basis sets strongly reduce computational costs and make the evaluation of matrix elements for large systems technically feasible.

Supposing for the moment that {β (r)} is complete, one can dene two dierent closure relations, namely:

λλ |β λ S -1 λλ β λ | = 1 (2.5)
and

λλ |vβ λ [v] -1 λλ β λ | = 1, (2.6
α µ (r) α ν (r)| Ô (r, r ) |α τ (r ) α σ (r ) = λλ c µνλ c * τ σλ Ô λλ ,
where the coecients are given by:

c µνλ = α µ (r) α ν (r)| β λ (r) c * τ σλ = β λ (r )| α τ (r ) α σ (r ) .
The matrix elements read:

Ô λλ = S -1 λλ Ô λλ S -1 λλ , Ô λλ = β λ (r)| Ô (r, r ) |β λ (r ) .
W = v + v [P ] W , (2.8)
as it can be easily veried by inserting (2.5). These equations are solved by inversion, following:

W = v ( 1 -v [P ]) -1 .
Within the resolution of the identity Coulomb metric technique (RI-V), matrix elements are expressed according to equation (2.6):

α µ (r) α ν (r)| Ô (r, r ) |α τ (r ) α σ (r ) = λλ α µ (r) α ν (r)| vβ λ (r) [v] -1 λλ β λ (r ) Ô α τ (r ) α σ (r ) .
To conclude, within the resolution of the identity framework, four-center-two-electron integrals are transformed to expressions containing only two-and three-center integrals. The latter can be eciently computed in a Gaussian basis, which will be demonstrated later in this ) with an even-tempered distribution of the localization coecients α ranging from α min = 0.10 Bohr -2 to α max = 3.2 Bohr -2 and N Gaussian functions per l-channel, with l max = 1 . . . 3. Clearly, the gap converges signicantly faster with respect to N and l max within the RI-V approach, allowing thus for the utilization of smaller basis sets within this method. Figure

reproduced by courtesy of P. Boulanger.

chapter. The RI-SVS method bears the advantage of producing sparse matrices, reducing signicantly computational costs. This is not the case for the RI-V approach, which is therefore more expensive (by way of example see Table 2.1). Reducing the number of auxiliary basis functions to L, an error on the matrix elements is introduced. Provided that the operator Ô in the above equations corresponds to the Coulomb operator v, i.e. for Coulomb integrals, it can be shown that the error cancels to rst order in the RI-V approach [START_REF] Ren | Resolution-of-identity approach to Hartree-Fock, hybrid density functionals, RPA, MP2 and GW with numeric atom-centered orbital basis functions[END_REF]. On the contrary, within RI-SVS non-zero rst order terms arise. Consequently, errors in the Coulomb integrals due to the incompleteness of the auxiliary basis are minimized using the RI-V rather than the RI-SVS technique (see Fig. 2.1).

Common atom-centered basis sets

Slater-type orbitals Slater introduced so-called Slater-type orbitals (ST O), which resemble the wave functions of the hydrogen atom. They consist of a radial and angular part, following:

with α ST O = α ST O n (|r|) Y lm (φ, ϕ) , α ST O n (|r|) = |r| n-1 e -γ|r| .
Y lm are spherical harmonics describing the angular part of the wave function and γ is a constant accounting for the screening of the nuclear charge by the surrounding electrons. The main quantum number is presented by n, the angular momentum quantum number by l and the magnetic quantum number by m. The radial part of the wave function is represented by α ST O nl (|r|). This basis is very suited for atomic calculations, since the cusp condition at the origin and the exponential decay known from the hydrogen atom are well satised. However, matrix elements become computationally very demanding, because simple analytic relations valid for e.g. Gaussian functions do not exist.

Gaussian-type functions To circumvent this problem, Boys introduced Gaussian-type orbitals (GT O) in 1950 [START_REF] Boys | Electronic Wave Functions. I. A General Method of Calculation for the Stationary States of Any Molecular System[END_REF][START_REF] Boys | Automatic Fundamental Calculations of Molecular Structure[END_REF]. The angular part is still described by spherical harmonics, however, for the radial part Gaussian functions are introduced:

α GT O (r) = e -γr 2 .
As it will be shown later in this chapter, matrix elements are tremendously simplied in this basis, counterbalancing their improper behavior both for r at the nucleus and at innity. Indeed, they show neither a cusp at the origin nor the right e -|r| decay for r → ∞.

Combining both numerical eciency and a reasonable physical behavior is achieved by introducing contracted Gaussian functions (C GF ). They are based on the fact that a Slatertype function can already be well reproduced using a linear combination of only a few dierent Gaussian-type functions [START_REF] Clementi | Electronic structure of large molecular systems[END_REF]:

α CGF = n d n α GT O n .
Typically, (n = 2-6) Gaussian-type functions are sucient to t a Slater-type orbital. The simplest basis set using these contracted Gaussian functions is the STO-nG basis set. By way of example, in the case of the STO-3G basis set, one attempts to t a Slater-type function by a linear combination of three dierent GT Os.

Minimal basis sets STO-nG belong to the so-called group of minimal basis sets. The latter include only one CGF per occupied orbital in the neutral atom. For the hydrogen atom, only a single 1s CGF , while for the carbon atom ve CGF for the 1s, 2s, and three 2p orbitals are considered. The CGF , in turn, are obtained by linear combinations of Gaussian-type functions. Calculations performed with these basis sets are very fast, however, they provide only qualitative results suited as a starting point for more precise calculations.

Split-valence basis sets An expansion to minimal basis sets are so-called split-valence basis sets. They are taking into account that valence electrons are much more involved in chemical processes compared to the nearly chemically inert core electrons. Therefore, instead of a single CGF , a linear combination of several CGF is used to describe a valence orbital. Corresponding to the number of CGF included, these basis sets are called split-valence (double-, triple-, quadruple-)zeta basis sets. The group of Pople [START_REF] Ditcheld | Self-consistent Molecular-Orbital Methods. IX. An Extended Gaussian-Type Basis for Molecular Orbital Studies of Organic Molecules[END_REF] introduced the following notation for split-valence basis sets:

X -Y Zg,
where X represents the number of Gaussian-type functions g comprised in the t of each core orbital. Y and Z denote that valence orbitals are a linear combination of two CGF , each of them composed of Y and Z Gaussian-type functions. Split-valence triple-or quadruple-zeta basis sets are denoted X -Y ZW g and X -Y ZW V g, respectively.

Only increasing the number of CGF per atomic orbital does not automatically improve the quality of the basis. In addition, one should also include CGF s whose character signicantly diers from the ones already considered. One possibility is to add polarization functions, i.e.

functions with higher angular channels. They increase the exibility of the atom to form chemical bonds in every direction and thus improve the calculated molecular structures. In the notation of Pople, an asterisk denotes additional polarization orbitals:

X -Y Zg * .

Two asterisks signify that polarization orbitals are also added to light atoms. For the hydrogen atom, this means adding a p-like function to the basis. Another possibility is to include diuse functions, i.e. very shallow Gaussian-type functions:

X -Y Z + g.

In Pople's notation, they are represented by one/two plus sign(s). They are used to describe more accurately the tail of molecular orbitals. For the sake of illustration, a 6-31+g* basis set contracts six GT O to one CGT for each core orbital, whereas the valence orbitals are described by linear combinations of two CGT one of them consisting of three GT O, the other one of a single GT O. Moreover, except for light atoms, polarization and diusive functions are added.

Gaussian function basis sets and analytic properties

Basis sets using Gaussian functions [START_REF] Boys | Electronic Wave Functions. I. A General Method of Calculation for the Stationary States of Any Molecular System[END_REF][START_REF] Hill | Gaussian basis sets for molecular applications[END_REF][START_REF] Boys | Automatic Fundamental Calculations of Molecular Structure[END_REF]

α GT O (r) = (r x -A x ) h (r y -A y ) i (r z -A z ) j e -γ(r-A) 2 ,
where (h, i, j) are natural numbers and γ the exponential coecient. Similar to the wave functions of the hydrogen atom, we can dene the angular momentum quantum number using the exponents like (l = h + i + j). For (l = 0), we obtain a s-orbital-like Gaussian function,

(α GT O s (r) = e -γr 2
), for (l = 1) p-like functions, (α GT O p (r) = r x e -γr 2 , r y e -γr 2 , r z e -γr 2

), et cetera. Besides Cartesian Gaussian functions, so-called Spherical Gaussian functions are widely used:

α GT O = |r -A| l e -γ(r-A) 2 Y lm (φ, ϕ) ,
where m is the magnetic quantum number and Y lm (φ, ϕ) spherical harmonics. Cartesian and Spherical Gaussian function basis sets are only equivalent up to (l = 1), so one has to be careful which type of Gaussian functions is actually implemented. Nevertheless, unitary transformations allow one to easily switch between one representation and the other.

Advantages and subtleties of Gaussian basis sets

Using Gaussian function basis sets results in important computational advantages. Demanding parts of the calculation, which are otherwise only solvable with much computational eort, can be treated analytically. However, some subtleties have to be considered in order to produce accurate results. Once these points are respected, the ratio between eciency and accuracy is impressive.

Calculating matrix elements in a Gaussian basis Calculating matrix elements denotes the main step in most ab initio calculations and, once this is done, ecient linear algebra routines exist to solve systems of linear equations. Basis sets with Gaussian functions bare the advantage that a lot of matrix elements can be transformed in analytic expressions using for example the Gaussian product theorem or the Gaussian integral properties (see Appendix A.10). In turn, these analytic expressions can be evaluated eciently on computers. For the sake of illustration, this is demonstrated for two examples. The rst one is the three-center integral

M µνλ = α µ α ν | β λ ,
which has been already encountered previously within the RI-SVS formalism and which is frequently needed in calculations dealing with non-local operators. For the sake of clarity, we choose α µ , α ν and β κ to be unnormalized primitive s-like Gaussian functions centered at atoms R µ , R ν and R λ , respectively. Following the Gaussian product theorem, the product of three Gaussians yields a Gaussian function, multiplied by a constant C and centered at the barycentric point P = µRµ+λR λ +νRν η with an exponential coecient (η = µ + λ + ν):

M µνλ = ´d3 r e -µ(r-Rµ) 2 e -λ(r-R λ ) 2 e -ν(r-Rν ) 2 = C µνλ ´d3 r e -η(r-P) 2 .
This Gaussian integral can be straightforwardly solved by separation and gives:

M µνλ = C µνλ ´d3 r x e -η(rx-Px) 2 ´d3 r y e -η(ry-Py) 2 ´d3 r z e -η(rz-Pz)

2 = C µνλ π η 3 2 ,
an analytic expression easily computed. Moreover, taking for illustration the derivative of M µνλ with respect to P x , leads to the product of a primitive s-like and p x -like Gaussian function, initiating by successive derivations ecient recursion relations.

We now come to the central Coulomb integrals, which, in an auxiliary basis {β} within the RI-SVS representation, read:

[V C ] λλ = ¨d3 rd 3 r β * λ (r) β λ (r ) |r-r | .
In a Gaussian function auxiliary basis, this demanding non-local integral reduces to a simple analytic expression. This is demonstrated in detail in Appendix A.10 for two unnormalized, primitive s-like Gaussian functions centered at A and B and results in:

[V C ] s,s = ´´d 3 r d 3 r e -κ|r-A| 2 e -λ|r -B| 2 |r-r | = π κ 3 2 π λ 3 2 1 R erf R 2 4γ ,
where we dened (R = A -B) and γ = 1 4κ + 1 4λ . We also introduced the error function erf (y) = 2 √ π ´y 0 du e -u 2 . The latter is directly related to the so-called Boys function F n of order n [START_REF] Boys | Electronic Wave Functions. I. A General Method of Calculation for the Stationary States of Any Molecular System[END_REF][START_REF] Boys | Automatic Fundamental Calculations of Molecular Structure[END_REF], obeying:

F n (x) = ´1 0 dt t 2n e -xt 2 F 0 (x) = √ π 2 √ x erf ( √ x) , x > 0.
The Boys function plays a key role in one-or two-electron Coulomb integral evaluation, since ecient evaluation methods based on upward/downward recursion exist [START_REF] Shavitt | Methods in Computational Physics[END_REF]:

F n (x) = 1 2x [(2n=1) F n=1 (x)=e =x ] , F n (x) = 1 2n+1 [2x F n+1 (x)=e =x ] .
Rewriting the Coulomb matrix elements for two s-like Gaussian functions in terms of the Boys function gives:

[V C ] s,s = 1 √ π √ γ π κ 3 2 π λ 3 2 F 0 R 2 4γ .
[V C ] s,s can not only be eciently calculated, moreover it serves as a starting point for the calculation of the Coulomb integrals between higher l-number orbitals, such as [V C ] s,px . A derivation from scratch is not necessary, but solutions can be obtained using the analytical expression obtained for [V C ] s,s . This is shown for the s-like and p x -like Gaussian interaction in Appendix A.10, resulting in:

[V C ] s,px = - 1 κ 1 √ π γ -3 2 π κ 3 2 π λ 3 2 R x F 1 R 2 4γ .
Following the same procedure and taking [V C ] s,px as a starting point, we can derive an analytical expression for a p x -like and p y -like Gaussian interaction:

[V C ] px,py = - 1 8κ 2 λ 1 √ π γ -5 2 π κ 3 2 π λ 3 2 R x R y F 2 R 2 4γ .
Step by step, expressions for the dierent Coulomb matrix elements can be derived, using the previous analytic solutions. As a result, we obtain analytical expressions for the twoelectron bare exchange interaction containing Boys functions of dierent order. As already mentioned, these can be eciently evaluated applying recursion. This clearly demonstrates the computational advantages of Gaussian basis sets.

Non-Orthogonality Despite the advantages a Gaussian basis oers, one has to bear in mind that these basis sets are not orthogonal. As a consequence, basis sets have to be chosen carefully in order to minimize linear dependencies and generalized eigenvalue problem routines have to be used, removing overlap matrix eigenvectors which yield too small eigenvalues for numerical stability (see below).

Another subtlety arising from non-orthogonality is the non-systematic extension of the basis set. For orthogonal basis sets, a complete basis set (CBS), i.e. the inclusion of N → ∞ basis functions, reproduces wave functions exactly. Since this is technically impossible to realize, introducing a nite basis set represents a major approximation. The calculated wave functions and observables are only solutions in the function space of the basis within the chosen level of theory. The introduced error is called basis set superposition error (BSSE), which can be diminished by including more basis functions. For non-orthogonal basis sets, however, the inclusion of more basis functions does not automatically improve the result, i.e. the problem does not converge systematically like it is the case for orthogonal basis sets. Adding a basis function does not necessarily add supplemental information, but the added function can be very similar to a function already contained in the basis set. That leads to an unphysical weighting of directions and consequently unpredictable changes in the overlap and other observables. Therefore, convergence tests are of crucial importance for non-orthogonal basis sets. In the case of Gaussian functions, special basis sets, such as the correlation-consistent bases of Dunning [START_REF] Dunning | Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen[END_REF], have been specically designed to smoothly converge to the complete basis set limit. The basic idea of these basis sets is to compare the incremental change in correlation energy calculated at the conguration interaction to second order (CISD) level upon addition of higher angular momentum functions. The important outcome is that added functions separate in groups dened by the amount of correlation energy they cover. Each added group after the rst one has less and less contributions to the correlation energy. GW calculations using the FIESTA package. The label 6G in the auxiliary basis specication stands for a basis containing 6 Gaussian functions per angular momentum channel with an even-tempered distribution of the exponential coecients between 0.1 and 3.2. Similar, 4G represents an auxiliary basis with 4

Gaussian functions per angular momentum channel with an even-tempered distribution of the exponential coecients between 0.2 and 3.2. The number of auxiliary basis functions corresponds to the number after the basis renement, where linear dependencies have been removed. The given wall time, i.e. the real elapsed time per processor, corresponds to a G 0 W 0 calculation, where the GW correction has been explicitly calculated for several bands around the gap (see chapter GW/BSE in practice).

been applied. By way of example, in the case of the T ZDP basis, the rst basis orbitals of the valence s, p-channels are taken to be the 2s and 2p eigenfunctions of isolated atoms in the corresponding pseudopotential approximation. This is analogue to strategies developed for post-Hartree-Fock correlated calculations, along the line of natural atomic orbitals (NAO).

The additional valence channels are taken to be two primitive Gaussian functions, which are optimized in order to minimize the total energy at the DF T -LDA level. For carbon, the resulting most diuse Gaussian functions present a decay coecient α =0. The used auxiliary basis is composed of atomic-like orbitals centered at the atoms A, with real spherical harmonics for the angular part and a radial dependence composed of Gaussian functions: and oscillator strength with respect to the used standard correlation-consistent Kohn-Sham basis set of Dunning [START_REF] Dunning | Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen[END_REF] for the cyanine CN 5 molecule (see Ref.

β (r, φ, ϕ) = |r -A| l e -α(r-A) 2 R lm (φ, ϕ) .
[141] and Fig. 2.3). It is important to note that the augmentation by diuse orbitals is very important in order to obtain highly converged excitation energies.

The oscillator strengths are less aected, except for the smallest cc-pvdz basis.

computationally more ecient [START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF]. The choice of the optimal decay coecients α for the radial part, i.e. the specic extent of the basis functions, is crucial for the quality of the result. For the presented calculations, even-tempered auxiliary basis sets have been employed [START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF][START_REF] Reeves | Use of Gaussian Functions in the Calculation of Wavefunctions for Small Molecules. II. The Ammonia Molecule[END_REF][START_REF] Cherkes | Spanning the Hilbert space with an even tempered Gaussian basis set[END_REF]. The latter come from the assumption that it is advantageous to generate a series of α coecients with (α i + 1) /α i = constant, rather than taking uniformly spread values between α min and α max . This goes back to the observation that the overlap of two Gaussian functions depends on the ratio of their α coecients. Imposing a constant overlap between joining Gaussian functions with α i and α i±1 allows to better span the corresponding Hilbert space [START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF][START_REF] Cherkes | Spanning the Hilbert space with an even tempered Gaussian basis set[END_REF]. Consequently, α min , α max and the number of Gaussian functions per l-channel given, the remaining decay coecients can be readily generated. An important point to mention are numerical diculties arising from the non-orthogonality and the possible over-completeness of Gaussian function basis sets. Calculating the overlap between auxiliary basis functions, one observes that it tends to be rather large for diuse auxiliary functions on neighboring atoms. While auxiliary functions belonging to the same atom can be easily orthogonalized using e.g. a Gram-Schmidt procedure, a dierent scheme is necessary to avoid singularities in the overlap matrix stemming from neighboring basis functions [START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF]. Referring to a method conceived for full product basis sets [START_REF] Aryasetiawan | Product-basis method for calculating dielectric matrices[END_REF][START_REF] Foerster | On the Kohn-Sham density response in a localized basis set[END_REF], the problem is transformed to the eigenvector space of the overlap matrix S, where eigenvectors yielding eigenvalues smaller than 10 =8 are typically removed [START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF]. This does not signicantly reduce the number of basis functions, however, prevents from possible numerical instabilities associated with the inversion of a quasi-singular S matrix and the amplication of errors due to the transformations

Ô = S -1 λλ Ô S -1 λλ .
Apart from computational advantages a Gaussian basis oers by providing analytic expressions for demanding matrix elements, another convenient factor is that these basis sets are widely applied in the quantum chemistry community. That way, one can resort to a large experience, which is crucial when converging non-orthogonal basis sets. Moreover, there is also the possibility of using eigenvalues and eigenstates generated by DF T quantum chemistry codes, where all-electron calculations and the use of hybrid functionals are standard. This

N CN 5 40 Σ Σ x Σ c Σ = iGW = Σ x + Σ c = iGv + iG(W -v). Σ x Σ c Σ c r, r , E = i 2π ∞ -∞ dω e iηω G r, r , E + ω W r, r , ω ,
where η → 0 + and W = W -v. The main diculty stems from the evaluation of the nonlocal and frequency dependent screened Coulomb potential W , whose calculation implies the construction of the irreducible polarizability P . Imagine the rst iteration of a GW cycle, P is based on single-particle wave functions and eigenvalues, following:

P IP (r 1 , r 2 ; ω) = m,l φ l (r 1 ) φ * l (r 2 ) φ m (r 2 ) φ * m (r 1 ) ω -ε 0 l -ε 0 m + iη - φ m (r 1 ) φ * m (r 2 ) φ l (r 2 ) φ * l (r 1 ) ω + ε 0 l -ε 0 m -iη .
Due to its more complex form as compared to the Green's function G involving the summation over four wave functions, the polarizability represents, even within the presented RP A expression, a computationally demanding object. The screened Coulomb potential is then obtained through a Dyson-like equation, which reads in matrix notation following equation (2.8):

W RP A = v + v [P IP ] W RP A .
Here, W is expressed in terms of the auxiliary basis within the RI-SV S representation. The above equation is solved by inversion following:

W RP A = ( 1 -v [P IP ]) -1 v .
Dealing with frequency dependent quantities, the above procedure has to be performed on a frequency grid in order to obtain the frequency dependence of W . Since G (ω) and W (ω)

represent strongly varying functions along the real frequency axis, the latter have to be evaluated on rather closely spaced grid points. The calculation of W presents one of the crucial steps in a GW calculation and improvements in terms of eciency have a large impact on the necessary computational requirements.

The plasmon pole approximation One way to facilitate the calculation of the correlation contribution to the self-energy Σ c is the so-called plasmon pole approximation (P P A).

Here, the frequency dependence of the screened Coulomb potential is not explicitly calculated, but a simpler and physically meaningful form of W is constructed, yielding an approximate correlation contribution to the self-energy. The screened Coulomb potential W is composed of the bare Coulomb potential v and the inverse of the dielectric function ε -1 , following: W (12) = ´d3 -1 (13) v [START_REF] Hohenberg | Inhomogeneous Electron Gas[END_REF]. Dierent approaches to model W (ω) exist [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF]143,144,[START_REF] Hedin | Solid State Physics[END_REF][START_REF] Von Der Linden | Precise quasiparticle energies and Hartree-Fock bands of semiconductors and insulators[END_REF][START_REF] Hamada | Self-energy correction for the energy bands of silicon by the full-potential linearized augmented-plane-wave method: Eect of the valence-band polarization[END_REF][START_REF] Engel | Generalized plasmon-pole model and plasmon band structures of crystals[END_REF]. They are based on the observation that the imaginary part of -1 (ω) is a function with peaks at the neutral excitation energies. It is supposed that one main peak is dominating the spectrum, originating from plasmon excitations. In the most straightforward approach, -1 (ω) is simply approximated by a single narrow Lorentzian peak. The resulting dielectric function -1 then also has a simple peaked form, where the peak position ωk and the peak strength A k have to be tted for the particular system and must obey certain limits or constraints. The advantage of the P P A is that the convolution of G and W simplies to an analytic expression, superseding the demanding integration in equation (2.9). However, one can not access the imaginary part of the self-energy, i.e. quasi-particle lifetimes, since ImΣ c is zero everywhere except at ωk . Further, molecular systems do usually not possess a simple peaked structure for -1 .

Exact calculation of Σ c using contour deformation techniques The Fiesta package [START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF][START_REF] Faber | First-principles GW calculations for DNA and RNA nucleobases[END_REF][START_REF] Blase | Charge-transfer excitations in molecular donor-acceptor complexes within the many-body Bethe-Salpeter approach[END_REF] does not make use of plasmon pole models, instead, a reformulation of equation (2.9) using contour deformation techniques is applied. The latter opens the way to an exact and, in addition, ecient evaluation of the correlation energy integral as compared to the direct integration along the real frequency axis.

In order to nd a suited reformulation of equation (2.9), one tries to replace the integral along the real frequency axis by an equivalent, but computationally less demanding expression [START_REF] Godby | Self-energy operators and exchangecorrelation potentials in semiconductors[END_REF][START_REF] Schöne | Self-Consistent Calculations of Quasiparticle States in Metals and Semiconductors[END_REF]. For convenience and without loss of generality, the non-interacting Green's function G 0 and W in the Lehmann representation are used for the demonstration:

G 0 (r, r´, ω + E) = occ m φ * m (r) φ m (r ) E + ω -ε m -iη + unocc l φ * l (r ) φ l (r) E + ω -ε l + iη W (r, r´, ω) = k 2 ω k V k (r) V * k (r ) (ω + ω k -iδ) (ω -ω k + iδ) , with V k (r) = ´d3 r v (r,
¸dω f (ω) = ¸C dω f (ω) + ¸D dω f (ω) ¸C dω f (ω) = ´∞ 0 dω f (ω ) + ´π 2 0 dϕ iRe iρ f Re iρ -´∞ 0 dω f (ω ) ¸D dω f (ω) = ´0 -∞ dω f (ω ) + ´π 3π 2 dΘ iRe iΘ f Re iΘ -´0 -∞ dω f (ω ) ,
where f (ω) = i 2π e iη ω G 0 (r, r , ω + E) W (r, r , ω) with η → 0 + . One thus deals with integrals of the form:

˛dω f (ω) = ˛C dω g (ω) e iη ω (ω -ωi ) n , (2.10)
where n is the order of the poles ωi originating from the Green's function and the screened Coulomb potential.

A short overview of the theorems used in the following is provided in Appendix A.9. Corresponding to the residue theorem, a closed path integral over a function f (z) with z∈ C yields:

1 ω = ω + iω = R e iϕ with R = (ω ) 2 + (ω ) 2 and ϕ = arctan ω ω . The poles of the non-interacting single-particle Green's function G 0 (r, r , ω)

are in the upper-half plane for occupied states and in the lower-half complex plane for unoccupied states (blue crosses), while the poles of G 0 (r, r , ω + E) are shifted by E with respect to them (yellow dots). Consequently, part of these poles are within the closed contour C.

˛dz f (z) =    0,
if no poles are enclosed 2πi i Res (f, z i ) , if poles at z i are enclosed.

Res (f, z i ) denotes the so-called residue of the function f at the enclosed pole z i of order n. It is the coecient of the (z -z i ) -1 summand in a Laurent expansion and is evaluated following:

Res (f, z i ) = 1 (n -1)! d n-1 dz n-1 [(z -z i ) n f (z)] z=z i . (2.11)
The poles of the screened Coulomb potential are not enclosed by the chosen contour, thus they do not require further discussion. However, for the poles of the Green's function, a case-by-case study is necessary. As depicted in Fig. (2.4), the energy zero is chosen to be in the center of the energy gap between occupied and unoccupied states. The contour C thus contains poles, if (ω = ε i -E > 0). These poles stem from the poles of the Green's function for occupied states, i.e. ε i < 0 and consequently E < 0. In this case, the contour D encloses no poles and one thus deduces from the residue theorem:

˛dω f (ω) = ˛C dω f (ω) = 2πi Θ (-ε i ) Θ (ε i -E) i Res (f, ωi ) .
For the case (ω = ε i -E < 0), the contour integral along C vanishes. On the contrary, contour D now encloses poles of the Green's function for unoccupied states, i.e. ε i > 0 and thus E > 0:

˛dω f (ω) = ˛D dω f (ω) = 2πi Θ (ε i ) Θ (E -ε i ) i Res (f, ωi ) .
A detailed discussion of the two presented cases follows. In the rst one, the fact that contour D vanishes can be used to deduce an expression for the integral along the positive real frequency axis:

¸D dω f (ω) = ´0 -∞ dω f (ω ) + ´π 3π 2 dΘ iRe iΘ f Re iΘ -´0 -∞ dω f (ω ) = 0, ´0 -∞ dω f (ω ) = -´π 3π 2 dΘ iRe iΘ f Re iΘ + ´0 -∞ dω f (ω ) .
The integral in the lower half plane from Θ = 3π 2 . . . π can be expressed in terms of ϕ = 0 . . . π

,

following:

Θ = -(ϕ + π) dΘ = -dϕ Re iΘ = -R cos (ϕ + π) -iR sin (ϕ + π) = R cos ϕ + iR sin ϕ = Re iϕ ,
where we applied the addition theorem for sine and cosine. For the integral it follows with ω = Re iϕ : ˆπ 3π 2 dΘ iRe iΘ f Re iΘ = -ˆπ 0 dϕ iRe iϕ g Re iϕ (Re iϕ -ωi ) n e iηR cos ϕ e -ηR sin ϕ and for the limit R → ∞, respectively:

lim R→∞ ˆπ 0 dϕ . . . e -ηR sin ρ R n-1 .
Corresponding to Jordan's lemma, the circular integral vanishes and one thus obtains:

ˆ0 -∞ dω f ω = ˆ0 -∞ dω f ω ,
i.e. the integral along the negative real axis can be replaced by an integral along the negative imaginary axis. Concerning the positive real axis, the contour integral along C has to be evaluated. In this case, it directly follows from Jordan's lemma that the circular integral vanishes, resulting in:

ˆ∞ 0 dω f ω = ˆ∞ 0 dω f ω + 2πi Θ (-ε i ) Θ (ε i -E) i Res (f, ωi ) .
Following equation (2.11), the residues are calculated to:

2πi i Res (ω i ) = - i φ * i (r) φ i r W r, r , ε i -E ,
where η has been set to zero. Combining the obtained results, the correlation part of the self-energy Σ c along the entire real axis can be evaluated through:

Σ c (r, r , E) = ´∞ -∞ dω f (ω ) = ´∞ -∞ dω f (ω ) -Θ (-ε i ) Θ (ε i -E) i φ * i (r) φ i (r ) W (r, r , ε i -E) .
(2.12)

For the second case, i.e. (ω = ε i -E < 0), an analogue expression is obtained. Again, the circular integrals vanish, resulting in:

´∞ 0 dω f (ω ) = ´∞ 0 dω f (ω ) ´0 -∞ dω f (ω ) = ´0 -∞ dω f (ω ) + 2πi Θ (ε i ) Θ (E -ε i ) i Res (f, ωi ) ,
where the residues are:

2πi i Res (ω i ) = - i φ * i (r) φ i r W r, r , ε i -E .
Combining the two cases leads for Σ c to:

Σ c (r, r , E) = ´∞ -∞ dω f (ω ) -[Θ (-ε i ) Θ (ε i -E) + Θ (ε i ) Θ (E -ε i )] i φ * i (r) φ i (r ) W (r, r , ε i -E) .
The residues in equation (2.12) can be eciently calculated, since W has only to be evaluated at the poles ε i of the Green's function, which fall within contour C or D. Replacing the integration along the real axis by an integration along the imaginary axis has the advantage that one avoids the strong pole structure along the real axis, since G 0 and W are well behaved along the imaginary frequency axis [START_REF] Godby | Self-energy operators and exchangecorrelation potentials in semiconductors[END_REF][START_REF] Schöne | Self-Consistent Calculations of Quasiparticle States in Metals and Semiconductors[END_REF]. Consequently, the frequency grid on which the integral is numerically evaluated can be much rougher compared to a grid on the real axis.

As implemented in the FIESTA package, the smooth function W (ω ) can be fold onto the nite [0, 1] interval by a change of variable, which allows to perform a Gaussian quadrature procedure for the integral,

ˆdx f (x) ≈ N i=1 w i f (x i ),
with as little as N = 12 points [START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF]. The eciency and accuracy of the presented contour deformation techniques method has been recently successfully tested for molecular systems [START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF][START_REF] Faber | First-principles GW calculations for DNA and RNA nucleobases[END_REF][START_REF] Faber | Electron-phonon coupling in the C 60 fullerene within the many-body GW approach[END_REF][START_REF] Tiago | Optical excitations in organic molecules, clusters, and defects studied by rst-principles Green's function methods[END_REF][START_REF] Bruneval | Ionization energy of atoms obtained from GW self-energy or from random phase approximation total energies[END_REF]. Furthermore, the results presented later in this work clearly show that contour deformation techniques oer an accurate and fairly ecient way to evaluate the correlation contributions to the self-energy.

3 | The GW/BSE method in practice 3.1. Preliminary considerations and motivation

The band gap problem

In order to predict suited material combinations for organic solar cells and to study in detail the underlying fundamental processes from rst principles, one needs an universal ab initio formalism. By universal, we mean a parameter-free approach likely to be reliable for diverse systems, such as bulk systems, surfaces, molecules, metals, semiconductors or insulators. Since gap energies, band osets and optical excitation energies are the experimental quantities of interest and since usually the experimentally aspired accuracy is of the order of 0.1-0.2 eV, our computational results are only interesting, if they meet the same error range. In addition, since typical molecular building blocks of interest for the targeted photovoltaic applications may contain hundreds of atoms, such as functionalized polythiophene chains, the chosen computational approach must be tractable on available computers.

Due to its favorable scaling, DF T -K S is the method of choice concerning ab initio calculations on ground state properties. DF T within the Kohn-Sham formulation allows one to access ground state total energies and densities. By way of example, related quantities such as bond lengths, bulk moduli or phonon frequencies usually come in excellent agreement (< 1%) with experiments [START_REF] Pickett | Density Functional theory in Solids: II. Excited States[END_REF][START_REF] Louie | Electronic Structure, Dynamics, and Quantum Structural Properties of Condensed Matter[END_REF]. The available codes scale with N 3 or even linearly [136,[START_REF] Mohr | Daubechies wavelets for linear scaling density functional theory[END_REF][START_REF] Skylaris | Introducing ONETEP: Linear-scaling density functional simulations on parallel computers[END_REF]. However, serious discrepancies are observed, when the DF T -K S approach is used to predict excited state properties. In principle, the ground state density contains all information to set up the many-body Hamiltonian. However, up to now, no solution schemes where in addition approximate exchange-correlation functionals are applied. This is motivated by the apparent similarities between the Kohn-Sham and the Hartree-Fock equations.

For the latter, Koopmans theorem directly connects the HF eigenvalues to excitation energies within the frozen orbital approximation. Following this approach on the DF T level, in some cases the obtained band structures agree surprisingly well with experiment at least from a qualitative point of view [START_REF] Bachelet | Relativistic and core-relaxation eects on the energy bands of gallium arsenide and germanium[END_REF]. Concerning quantitative results, systematic studies on the band gap have been carried out [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF][START_REF] Aryasetiawan | The GW method[END_REF][START_REF] Pickett | Density Functional theory in Solids: II. Excited States[END_REF][START_REF] Bachelet | Relativistic and core-relaxation eects on the energy bands of gallium arsenide and germanium[END_REF]157,[START_REF] Pickett | Local-density approximation for dynamical correlation corrections to single-particle excitations in insulators[END_REF][START_REF] Bechstedt | Proc. 19th Internat. Conf. Phys. Semiconductors[END_REF]. In these studies, the true fundamental gap, dened as the dierence between the ionization energy IE and the electron

anity (EA = E N 0 -E N +1 lu ), E g = IE -EA = E N +1 lu + E N -1 ho -2E N , (3.1)
is compared to the standard DF T -K S gap:

E KS g = ε DF T lu -ε DF T ho ,
where ε DF T ho represents the eigenvalue of the highest occupied (ho) level and ε DF T lu the eigenvalue of the lowest unoccupied (lu) Kohn-Sham level in a N particle system. These studies demonstrated that standard DF T -K S calculations applying (semi)local exchange-correlation functionals, such as LDA, severely underestimate the band gap by up to several eV [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF][START_REF] Aryasetiawan | The GW method[END_REF][START_REF] Pickett | Density Functional theory in Solids: II. Excited States[END_REF][START_REF] Bachelet | Relativistic and core-relaxation eects on the energy bands of gallium arsenide and germanium[END_REF]157,[START_REF] Pickett | Local-density approximation for dynamical correlation corrections to single-particle excitations in insulators[END_REF][START_REF] Bechstedt | Proc. 19th Internat. Conf. Phys. Semiconductors[END_REF]. This is known as the band gap problem and is depicted in

Excited states within DFT-KS

The calculation and interpretation of excited state properties takes a prominent place within the DF T community. The reason is that the determination of excitation energies from Kohn-Sham eigenvalues is not straightforward and associated with certain subtleties. Since the quality of the obtained excitation energies can strongly depend both on the system itself and on the used approximate functional, much care is needed. This problem also aects our manybody perturbation theory GW/BSE results, since DF T -K S eigenvalues and eigenstates serve as a starting point. As it will be pointed out later, especially for the single-shot G 0 W 0 approach, a reliable DF T -K S starting point is indispensable.

In order to get to the heart of the band gap problem and to estimate the errors introduced both by the use of approximate exchange-correlation functionals and the association of DF T eigenvalues with single-particle excitation energies, two limiting cases are discussed in the following the homogeneous electron gas and the free atom. These are thought as a guideline to assess the reliability of DF T -K S calculations. For the electron gas, the exact exchangecorrelation functional V xc is by construction the LDA one. This oers the possibility to carry out an exact DF T calculation and to compare the obtained eigenvalues to many-body perturbation theory quasiparticle energies [START_REF] Hedin | Solid State Physics[END_REF]. Concerning the highest occupied eigenvalue, the two methods perfectly agree. This underlines the validity of the ionization potential theorem. However, for the remaining spectrum, discrepancies occur, which simply arise from the association of DF T -K S eigenvalues with excitation energies. Dierences are small for (un)occupied states close to the Fermi energy E F , however, increase for states farther from the SCF method [START_REF] Jones | The density functional formalism, its applications and prospects[END_REF][START_REF] Chelikowsky | Quantum Connement and Optical Gaps in Si Nanocrystals[END_REF][START_REF] Godby | Density-Relaxation Part of the Self-Energy[END_REF][START_REF] Tiago | Electronic and optical excitations in Ag n clusters (n = 1˘8): Comparison of density-functional and many-body theories[END_REF]. The procedure consists of performing three dierent DF T -K S ground state calculations, namely for the N -electron and for the corresponding (N ± 1) system. Subtracting the resulting ground state energies provides the ionization energy IE, the electron anity EA and consequently also the gap E g following equation (3.1).

Since ground state energies calculated within DF T are usually in very good agreement with experiment, one obtains reliable binding energies and gaps at reasonable computational costs.

It is important to note, however, that SCF only provides the ionization energy and the electron anity, whereas the remaining excitation energies can not be accessed. This is due to the fact that the charged system is relaxed in its ground state and consequently total energies other than E N ±1 s=0 are not accessible. Another drawback is that this method can only be applied for nite systems. Usually, periodic boundary conditions are used in solids, leading to a Coulomb explosion, i.e. an unphysical charging of the system by adding an electron (hole)

to each of the unit cells. Even though techniques exist, where a positive background charge is introduced as compensation, it is not clear how to straightforwardly deduce the fundamental gap from the obtained total energies [START_REF] Kraisler | Fundamental gaps with approximate density functionals: The derivative discontinuity revealed from ensemble considerations[END_REF][START_REF] Sharma | Reduced density matrix functional for many-electron systems[END_REF][START_REF] Chan | Ecient Band Gap Prediction for Solids[END_REF].

The DFT-KS derivative discontinuity As an alternative approach to obtain within DF T -K S the ionization energy, the electron anity and thus the band gap, the ionization potential theorem suggests to calculate the derivative of the total energy for a N 0 and (N 0 + 1) system with respect to the fractional occupation number ñi ∈ [0, 1]:

-IE (N 0 ) = ε DF T ho=N 0 = ∂E N 0 ∂ ñho ñho =1-η , -EA (N 0 ) = ε DF T ho=N 0 +1 = ∂E N 0 +1 ∂ ñho ñho =1-η , (3.2)
where η = 0 + . Since charged congurations are disadvantageous for innite systems, one makes use of the fact that the total energy is a piecewise function with respect to the fractional particle number N (see Fig. 1.4a). This allows to evaluate the electron anity by analyzing the right limit ε DF T ho (N 0 + δ) of the neutral system instead of working with the left limit of the charged system ε DF T ho (N 0 + 1 -δ) as suggested by equation (3.2):

E g = IE -EA = ε N 0 +δ ho -ε N 0 -δ ho . (3.3)
As shown in Fig. (1.4b), when innitesimally crossing the integer value N 0 from (N 0 -δ) to (N 0 + δ), the Kohn-Sham eigenvalue ε ho is subject to an abrupt change. In this context, it is important to mention that the Kohn-Sham eigenvalues, as calculated in a single DF T -K S calculation on N 0 particles, correspond to the left limit, i.e. (N 0 -δ) [START_REF] Kronik | Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals[END_REF]. Since the external and the Hartree potential are continuous with respect to the density, within the DF T -K S formalism, the discontinuity can stem either from the kinetic energy of the non-interacting electrons or the exchange-correlation energy contribution [START_REF] Kronik | Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals[END_REF]. First, the abrupt slope change in the kinetic energy is due to a change in the leading orbital, i.e. the fact that the state labeled ho at the left is not identical to the ho state on the right, since one already starts to in-nitesimally occupy the next available state. The ho state on the right thus corresponds to the lowest unoccupied state (lu) on the left. Second, the change in the exchange-correlation energy is due to a spatially constant discontinuity in the exchange-correlation potential, usually termed derivative discontinuity ∆ xc [START_REF] Kronik | Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals[END_REF]. The ho state on the left can therefore be expressed in terms of the lu state on the left and the derivative discontinuity [START_REF] Kronik | Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals[END_REF][START_REF] Kraisler | Fundamental gaps with approximate density functionals: The derivative discontinuity revealed from ensemble considerations[END_REF]:

ε N 0 +δ ho = ε N 0 -δ lu + ∆ xc .
Consequently, one obtains for the band gap:

E g = IE -EA = ε N 0 -δ lu -ε N 0 -δ ho + ∆ xc . (3.4)
This is an important outcome, since it shows that even for the exact exchange-correlation functional, the Kohn-Sham gap (E KS g

= ε N 0 -δ lu -ε N 0 -δ ho
) obtained from a single calculation on a N -body system does not exactly yield the fundamental gap E g . Instead, the derivative discontinuity has to be taken into account. Depending on the system, the magnitude of the latter can vary from negligible amounts, i.e. the jump is too a large amount described by the kinetic energy term, to values of the order of E KS g [START_REF] Kronik | Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals[END_REF][START_REF] Jones | The density functional formalism, its applications and prospects[END_REF]. For approximate exchangecorrelation functionals, the degree by which equation (3.4) is fullled diers [START_REF] Kraisler | Fundamental gaps with approximate density functionals: The derivative discontinuity revealed from ensemble considerations[END_REF]. In these cases, discrepancies between E KS g and the true gap can thus arise both from the discontinuity and from the approximate nature of the functional. Determining the discontinuity is a highly non-trivial task and the reader is referred to Ref. [START_REF] Kraisler | Fundamental gaps with approximate density functionals: The derivative discontinuity revealed from ensemble considerations[END_REF] for a comprehensive overview of current approaches.

An alternative to the standard Kohn-Sham scheme is the so-called generalized Kohn-Sham (GKS) approach, where non-local potentials are introduced. The idea is that discontinuity eects, which are not already covered by the kinetic energy, are not solely shouldered by the unknown local KS exchange-correlation potential. Instead, they are supposed to automatically come in through non-local potentials, such as e.g. the Fock term [START_REF] Kronik | Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals[END_REF]. The GKS equations take the following form:

Ô [{φ i }] + V ext (r) + V r [n] (r) φ i (r) = ε i φ i (r) ,
where Ô [{φ i }] is a non-local, orbital-dependent operator and V r the local remainder potential determined by the derivative of the remainder energy with respect to the density.

In the case where Ô [{φ i }] is chosen to be the single-particle kinetic operator, one retrieves the standard Kohn-Sham equations, where the remainder contains the Hartree and the usual exchange-correlation contribution. If instead Ô [{φ i }] is the kinetic energy operator and the Hartree-Fock operator, one obtains a Hartree-Fock-Kohn-Sham scheme, which is in principle exact and where the remainder accounts for correlation only. Unfortunately, only very little is known about the correct form of the remainder in this case, making this scheme dicult to apply in practice [START_REF] Kronik | Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals[END_REF]. Apart from these two limiting cases, the most common applications of the GKS are hybrid exchange-correlation functionals. The latter contain a certain amount α of Fock exchange VF , a complementary amount (1 -α) of the standard approximate (semi)local exchange V sl x and correlation V sl c contributions [82]:

- ∇ 2 2 + V ext (r) + V H [n] (r) + α VF + (1 -α) V sl x [n] (r) + V sl c [n] (r) φ i (r) = ε i φ i (r) .
These functionals are a special case of the GKS scheme, where Ô [{φ i }] consists of the kinetic energy operator and a fraction of the Hartree-Fock operator [START_REF] Kronik | Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals[END_REF]. As a consequence, one obtains the kinetic energy and a fraction of the Hartree and of the Fock contribution, whereas one chooses the remainder to contain the rest of the Hartree term and the (semi)local exchange-correlation contributions. By way of example, the LDA and P BE [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] functionals correspond to (α = 0), while Hartree-Fock is characterized by (α = 1) and V sl x = V sl c = 0 .

One of the most popular hybrid functionals, the already introduced B3LYP functional [START_REF] Becke | A new mixing of Hartree-Fock and local density-functional theories[END_REF],

is determined by (α = 0.2), the similar PBE0 functional [START_REF] Ernzerhof | Assessment of the Perdew-Burke-Ernzerhof exchange-correlation functional[END_REF] by (α = 0.25) [START_REF] Bruneval | Benchmarking the Starting Points of the GW Approximation for Molecules[END_REF]. The choice of α crucially inuences the accuracy of the methods. Usually, α is either obtained by semiempirical tting on a set of study case systems or formal considerations. However, there is no transferability from one system to another, so describing semi-conducting, metallic,nite or extended systems on the same footing is questionable [START_REF] Kronik | Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals[END_REF]. Nevertheless, as already discussed for the limiting case of a free atom and as depicted in part. This is achieved by a decomposition of the Coulomb potential, following e.g.:

1 |r -r | = 1 -erf (κ, |r -r |) |r -r | + erf (κ, |r -r |) |r -r | , (3.5)
where erf is the error function and κ a parameter characterizing the range-separation, i.e.

the spatial extent of the short-range interaction [START_REF] Gerber | Range separated hybrid density functional with long-range Hartree-Fock exchange applied to solids[END_REF]. At a characteristic distance of around ∼ 2/κ, short-range contributions become negligible. Bearing an important inuence on the accuracy of the obtained results, the range-separation parameter κ has to be carefully chosen, either through tting or physical considerations. The rst term in equation (3.5), i.e.

the short-range contribution V sr,κ x , is not evaluated directly, but replaced by a standard (semi)local DF T exchange functional, whereas the long-range is governed by a Hartree-Fock like operator V lr,κ F

[82]:

- ∇ 2 2 + V ext (r) + V H [n] (r) + V lr,κ F + V sr,κ x [n] (r) + V sl c [n] (r) φ i (r) = ε i φ i (r) .
This oers the possibility to include the full exact exchange in the long range regime, yielding

H 2 DF T KS G 0 W 0 @LDA G 0 W 0 @HF evGW @LDA DF T LDA DF T B3LY P HF GW HF DF T LDA evGW G 0 W 0 @LDA G 0 W 0 @HF Ô [{φ i }] xc [α + β (κ, |r -r |)] |r -r | LDA P BE 0.0 α = β = 0 1.0 V sl x = V sl c = 0 α = 1 β = 0 (-1/ |r -r |) 1 |r -r | = 1 -[α + β (κ, |r -r |)] |r -r | + α + β (κ, |r -r |) |r -r | . α β (0 ≤ α ≤ 1) (0 ≤ β ≤ 1) (0 ≤ α + β ≤ 1) DF T |r -r | = ∞ |r -r | = 0 DF T α |r -r | + β (κ, |r -r |) |r -r | , |r -r | = 0 DF T α |r -r | = ∞ [1 -(α + β)]
remaining contribution of (α + β). As a consequence, for (semi)local DF T exchange functionals which vanish in this limit, the exchange interaction is solely governed by the second term and thus reduces from the correct physical limit of 1 to (α + β). Later in this chapter, consequences of this reduction will be discussed in detail in the context of optical charge-transfer excitations. A popular representative of the CAM functionals, which will be important in the following, is the CAM-B3LYP functional [START_REF] Yanai | A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)[END_REF] with α = 0.19, β = 0.46 and κ = 0.33. Apart from CAM functionals, a variety of RSH functionals has been developed, yielding very satisfying results for gap energies (see Fig. In the following, the band gap problem is revisited from the viewpoint of many-body perturbation theory. Here, the calculated quasiparticle energies can be directly connected to excitation energies. Even though usually more expensive in terms of necessary computing power than DF T -K S, the universality and accuracy of the GW/BSE approach are striking.

The many-body perturbation theory GW formalism in practice

The rst full GW calculation of the self-energy goes back to the 1960s, where Hedin studied the electron gas and demonstrated a systematic way to expand the electron-electron self-energy in terms of the screened Coulomb potential [START_REF] Hedin | New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas Problem[END_REF]. However, due to computational diculties, GW calculations on real systems have not been carried out until the 1980s [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF][START_REF] Godby | Self-energy operators and exchangecorrelation potentials in semiconductors[END_REF][START_REF] Hybertsen | First-Principles Theory of Quasiparticles: Calculation of Band Gaps in Semiconductors and Insulators[END_REF][START_REF] Godby | Accurate Exchange-Correlation Potential for Silicon and Its Discontinuity on Addition of an Electron[END_REF]. The latter provided promising results, yielding signicantly improved excitation energies compared to DF T -K S. A compelling example is the case of lanthanum hydride LaH 3 , where DF T -LDA predicts a semi-metal (see Fig. 3.4), whereas experiments document a semiconducting behavior. The latter is well reproduced within GW [START_REF] Chang | Quasiparticle band structure of lanthanum hydride[END_REF]. It is interesting to note that for this system, DF T -LDA already gives good qualitative characteristics and that GW then corrects the quantitative values by an opening of the band gap. This is a typical outcome and therefore GW is often equated with a scissor operator, which cuts the band structure within the gap and shifts occupied and unoccupied states apart from each other. Up to now, systematic studies on a variety of systems demonstrated the reliability and the accuracy of the GW method for the qualitative and quantitative prediction of band structures [START_REF] Aryasetiawan | The GW method[END_REF]157], with the exception of strongly-correlated materials, such as e.g. Mott insulators. A compilation is

GW for nite systems

Originating from the solid state community, most of the popular GW implementations are conceived for periodic systems [START_REF] Gonze | ABINIT: First-principles approach to material and nanosystem properties[END_REF][START_REF] Deslippe | BerkeleyGW: A massively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures[END_REF][START_REF] Marini | Yambo: An ab initio tool for excited state calculations[END_REF]. Finite systems, such as molecules or clusters, can in principle be treated within the same framework by introducing an articial periodicity. This implies that one constructs periodically repeated supercells containing the nite electronic system under study. In order to avoid an unphysical interaction between the cells, GW studies on molecules in gas phase. After the pioneering work of Shirley in 1993 [START_REF] Shirley | GW quasiparticle calculations in atoms[END_REF],

only very recently, GW has been applied to atomic and molecular systems and implementations have been designed to specically account for the requirements which come along with localized systems [START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF][START_REF] Bruneval | Benchmarking the Starting Points of the GW Approximation for Molecules[END_REF][START_REF] Tiago | Optical excitations in organic molecules, clusters, and defects studied by rst-principles Green's function methods[END_REF][START_REF] Bruneval | Ionization energy of atoms obtained from GW self-energy or from random phase approximation total energies[END_REF][START_REF] Rohlng | Electron-hole excitations and optical spectra from rst principles[END_REF][START_REF] Grossman | High Accuracy Many-Body Calculational Approaches for Excitations in Molecules[END_REF][START_REF] Tiago | First-principles GW-BSE excitations in organic molecules[END_REF][START_REF] Dori | Valence electronic structure of gas-phase 3,4,9,10-perylene tetracarboxylic acid dianhydride: Experiment and theory[END_REF][START_REF] Stan | Fully self-consistent GW calculations for atoms and molecules[END_REF][START_REF] Morris | Vertex corrections in localized and extended systems[END_REF]191,[START_REF] Palummo | Ab initio electronic and optical spectra of free-base porphyrins: The role of electronic correlation[END_REF][START_REF] Rostgaard | Fully self-consistent GW calculations for molecules[END_REF][START_REF] Ke | All-electron GW methods implemented in molecular orbital space: Ionization energy and electron anity of conjugated molecules[END_REF][START_REF] Foerster | An O(N[sup 3]) implementation of Hedin's GW approximation for molecules[END_REF][START_REF] Marom | Electronic structure of copper phthalocyanine from G 0 W 0 calculations[END_REF][START_REF] Samsonidze | Simple Approximate Physical Orbitals for GW Quasiparticle Calculations[END_REF][START_REF] Marom | Benchmark of GW methods for azabenzenes[END_REF][START_REF] Sharifzadeh | Quantitative molecular orbital energies within a G0W0 approximation[END_REF][START_REF] Umari | Importance of semicore states in GW calculations for simulating accurately the photoemission spectra of metal phthalocyanine molecules[END_REF][START_REF] Körzdörfer | Strategy for nding a reliable starting point for G 0 W 0 demonstrated for molecules[END_REF][START_REF] Caruso | Unied description of ground and excited states of nite systems: The self-consistent GW approach[END_REF][START_REF] Van Setten | The GW-Method for Quantum Chemistry Applications: Theory and Implementation[END_REF][START_REF] Pham | GW calculations using the spectral decomposition of the dielectric matrix: Verication, validation, and comparison of methods[END_REF]. This meets the demand of upcoming elds, such as organic electronics or organic photovoltaics, which denitely seek for accurate single-particle excitation energies. Nevertheless, contrary to the case of periodic systems, systematic studies on the performance of the GW formalism for molecular systems remain scarce. Hence, the present work focuses on exploring the merits and limitations of GW for molecules.

From

G 0 W 0 to self-consistency
The G 0 W 0 approach As already detailed in preceding chapters, in order to obtain oneparticle excitation energies within the many-body Green's function framework, one has to solve the quasiparticle eigenvalue equation:

H 0 (r) ψ qp ν (r) + ˆdr Σ r, r , ε qp ψ qp ν r = ε qp ψ qp ν (r) , (3.7)
where H 0 contains contributions from the kinetic energy, the external potential and the Hartree potential. All interactions beyond Hartree are governed by the self-energy Σ, which is non-local, non-Hermitian and energy-dependent. As a consequence, the resulting quasiparticle wave functions ψ qp ν are not orthogonal and quasiparticle energies are complex.

In most GW applications, a perturbative approach is used to solve equation (3.7). Based on the similarity of the quasiparticle and the Kohn-Sham equations, one usually supposes that the Kohn-Sham wave functions are in close agreement with the quasiparticle wave functions and that the perturbation (Σ -V xc ) is small with respect to the complete Hamiltonian [START_REF] Hedin | Electron correlation: Keeping close to an orbital description[END_REF].

It has been demonstrated early, in the case of silicon, that the self-energy is indeed diagonal in the DF T -LDA basis [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF]. To rst order in the perturbation, one can then approximate the quasiparticle energy by:

ε qp,ν = ε KS ν + ψ KS ν Σ (ε qp,ν ) -V xc ψ KS ν . (3.8)
Linearizing the above equation in the vicinity of the Kohn-Sham eigenvalues ε KS ν removes the dependence of Σ on the solution ε qp,ν and leads to:

ε qp,ν ≈ ε KS ν + Z KS ν ψ KS ν Σ ε KS ν -V xc ψ KS ν , (3.9)
where the renormalization factor,

Z KS ν =   1 - ∂Σ νν (ω) ∂ω ω=ε KS ν   -1 with Σ νν (ω) = ψ KS ν |Σ (ω)| ψ KS ν ,
accounts for dynamical eects of the self-energy. In the standard procedure, quasiparticle energies are obtained within a single-shot GW approach from equation (3.9), usually referred to as G 0 W 0 method [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF]. Single-particle Kohn-Sham eigenvalues are used to construct the non-interacting (mean-eld) Green's function G 0 , from which in turn the non-interacting irreducible polarizability, the zero-order screened Coulomb potential and the single-shot selfenergy are obtained:

ε KS , ψ KS ⇓ G 0 = ν ψ * KS,ν (r)ψ KS,ν (r ) ω-ε KS ν ±iη =⇒ P 0 = -iG 0 G 0 =⇒ W 0 = v + vP 0 W 0 =⇒ Σ = iG 0 W 0 .
Here, the laborious calculation of self-consistent GW quantities is avoided, thus providing a computationally feasible and consequently a very popular GW scheme. However, since input single-particle energies are corrected within one cycle only, a dependence of the GW result on the underlying single-particle calculation is induced. This has been recently demonstrated for several molecular systems, where both Hartree-Fock and DF T calculations applying diverse (semi)local and hybrid functionals have been evaluated as a starting point [START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF][START_REF] Faber | First-principles GW calculations for DNA and RNA nucleobases[END_REF][START_REF] Bruneval | Benchmarking the Starting Points of the GW Approximation for Molecules[END_REF][START_REF] Ren | Resolution-of-identity approach to Hartree-Fock, hybrid density functionals, RPA, MP2 and GW with numeric atom-centered orbital basis functions[END_REF][START_REF] Marom | Electronic structure of copper phthalocyanine from G 0 W 0 calculations[END_REF][START_REF] Körzdörfer | Strategy for nding a reliable starting point for G 0 W 0 demonstrated for molecules[END_REF]. Discrepancies concerning the quasiparticle energies amount to 1 eV for very small molecules. Further, it was shown that standard G 0 W 0 calculations based on DF T -LDA obviously underestimate fundamental gaps (see Fig. 3.2), due to an overscreening when building the screened Coulomb potential with too small DF T -LDA gaps. DF T -K S with hybrid functionals seems to be a more suited starting point compared to (semi)local functionals such as LDA or P BE [START_REF] Bruneval | Benchmarking the Starting Points of the GW Approximation for Molecules[END_REF]. However, it is indicated that none of the approaches can be considered as generally reliable and that the amount of exact exchange needed to be included in the hybrid functional is not clear at rst sight. This points out the need to improve on the standard G 0 W 0 scheme. By way of example, there are approaches searching for a unique and consistent DF T -K S starting point by determining the appropriate amount of exact exchange in the hybrid functional for each system under study [START_REF] Körzdörfer | Strategy for nding a reliable starting point for G 0 W 0 demonstrated for molecules[END_REF]. In this work, we rather focus on self-consistency in the GW scheme as a mean to overcome the starting point dependence.

Self-consistent schemes In order to bypass the starting point dependency of the perturbative G 0 W 0 approach, it would be desirable to have a self-consistent scheme starting from input single-particle eigenstates and eigenvalues, which are iteratively updated and converged.

The most intuitive way is to start from a non-interacting Green's function G 0 , which is used to obtain the corresponding self-energy through Hedin's cycle. The latter in turn serves to calculate an updated Green's function through the inversion of Dyson's equation. One thus repeats the following cycle until the Green's function obtained from the Dyson equation equals the G used to calculate Σ:

ε KS , ψ KS ⇓ G 0 , G =⇒ P = -iGG ⇑ ⇓ Σ = iGW ⇐= W = v + vP W
Such a fully-self-consistent scheme guarantees that the obtained results are independent of the starting point. It allows to obtain self-consistent Green's functions, from which converged spectral functions or total energies, e.g. through the Galitskii-Migdal equation [START_REF] Galitskii | Application of quantum eld theory methods to the many body problem[END_REF], can be directly deduced. If one is interested in the excitation energy spectrum, a similar scheme involving the self-consistent solution of the quasiparticle equation can be applied. This is computationally very demanding, since, due to the non-Hermitian nature of the self-energy, left and right eigenvectors of the Hamiltonian (H 0 + Σ) have to be considered and its energydependence implies a matrix eigenvalue problem for each energy grid point. Due to the enormous workload fully self-consistent schemes imply, only few works have been carried out so far. For solids, the quality of fully self-consistent schemes is strongly debated [START_REF] Schöne | Self-Consistent Calculations of Quasiparticle States in Metals and Semiconductors[END_REF][START_REF] Holm | Fully self-consistent GW self-energy of the electron gas[END_REF][START_REF] Ku | Band-Gap Problem in Semiconductors Revisited: Eects of Core States and Many-Body Self-Consistency[END_REF][START_REF] Delaney | Comment on 'Band-Gap Problem in Semiconductors Revisited: Eects of Core States and Many-Body Self-Consistency[END_REF][START_REF] Faleev | All-Electron Self-Consistent GW Approximation: Application to Si, MnO, and NiO[END_REF]. By way of example, for the homogeneous electron gas, a worsening of the spectral function with a strong underestimation of the quasiparticle weight as compared to the incoherent part has been observed, whereas the total energy shows an excellent agreement with Monte Carlo calculations [START_REF] Holm | Fully self-consistent GW self-energy of the electron gas[END_REF]. On the contrary, in the case of Si and Ge an improvement of the band gaps compared to perturbative single-shot calculations has been demonstrated [START_REF] Ku | Band-Gap Problem in Semiconductors Revisited: Eects of Core States and Many-Body Self-Consistency[END_REF]. Concerning nite systems, such as atoms or molecules, full self-consistency seems to perform better than G 0 W 0 starting from (semi)local functionals, however, G 0 W 0 calculations based on hybrid functionals yield an equivalent quality. Again, owing to the complexity of the problem, only few studies are available [START_REF] Stan | Fully self-consistent GW calculations for atoms and molecules[END_REF][START_REF] Rostgaard | Fully self-consistent GW calculations for molecules[END_REF][START_REF] Caruso | Unied description of ground and excited states of nite systems: The self-consistent GW approach[END_REF].

As an alternative to fully self-consistent schemes, dierent less demanding self-consistent self-consistency on:

wave functions eigenvalues evCOHSEX

√ scCOHSEX √ √ G 0 W 0 evGW √ Table 3.1.
: Short overview of the dierent many-body perturbation theory GW and COH-SEX schemes available in the FIESTA code [START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF].

GW approaches have been recently introduced. Faleev and coworkers [START_REF] Faleev | All-Electron Self-Consistent GW Approximation: Application to Si, MnO, and NiO[END_REF] developed the so-called quasiparticle self-consistent GW scheme (QP scGW ), where the true self-energy is rst approximated by a Hermitian self-energy, following:

i| Σ QP scGW |j = 1 2 [ i| Σ (ε qp,j ) |j + j| Σ (ε qp,i ) |i ] .
This greatly simplies the fully self-consistent solution. After convergency is reached, equation (3.8) is evaluated at the obtained quasiparticle energies. This approach has been shown to yield accurate quasiparticle energies in solids [START_REF] Faleev | All-Electron Self-Consistent GW Approximation: Application to Si, MnO, and NiO[END_REF][START_REF] Bruneval | Eect of self-consistency on quasiparticles in solids[END_REF]. However, even though much less demanding than a standard fully self-consistent scheme, its computational costs are not negligible [START_REF] Bruneval | Ionization energy of atoms obtained from GW self-energy or from random phase approximation total energies[END_REF]. In this work, we want to access the reliability of two self-consistent schemes, which are less expensive and thus promising for calculations on large molecular systems. One is a simple partially self-consistent approach, where only the eigenvalues are updated in each cycle, whereas the DF T -K S input eigenstates are unchanged [START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF][START_REF] Faber | First-principles GW calculations for DNA and RNA nucleobases[END_REF][START_REF] Faber | Many-body Green's function GW and Bethe-Salpeter study of the optical excitations in a paradigmatic model dipeptide[END_REF][START_REF] Shishkin | Self-consistent GW calculations for semiconductors and insulators[END_REF]. This approach will be labeled evGW in the following. The other one consists of a self-consistent COHSEX calculation to improve the DF T -K S starting point, followed by a partially self-consistent GW calculation to obtain converged quasiparticle energies. Self-consistency at the COHSEX level can be partial, i.e. on the eigenvalues only (evGW @evCOHSEX), or both on the eigenvalues and eigenfunctions (evGW @scCOHSEX) [START_REF] Faber | Many-body Green's function GW and Bethe-Salpeter study of the optical excitations in a paradigmatic model dipeptide[END_REF][START_REF] Bruneval | Eect of self-consistency on quasiparticles in solids[END_REF]. See Table (3.1) for an overview of the dierent approaches. ) per l-channel, up to l = 2 orbitals for rst row elements. The localization coecients α possess an even-tempered distribution [START_REF] Cherkes | Spanning the Hilbert space with an even tempered Gaussian basis set[END_REF] ranging from α min = 0.10 Bohr -2 to α max = 3.2 Bohr -2 , except for hydrogen, where the coecient goes from α min = 0.10 Bohr -2 to α max = 1.5 Bohr -2 .

The GW formalism applied

Beyond the scissor operator: level crossings

Following the results of previous G 0 W 0 @LDA studies, showing that this approach leads to too small ionization energies, gaps and optical excitation energies [START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF][START_REF] Faber | First-principles GW calculations for DNA and RNA nucleobases[END_REF][START_REF] Blase | Charge-transfer excitations in molecular donor-acceptor complexes within the many-body Bethe-Salpeter approach[END_REF], we test the partial self-consistency evGW scheme for the dipeptide with an update of the eigenvalues [START_REF] Faber | Many-body Green's function GW and Bethe-Salpeter study of the optical excitations in a paradigmatic model dipeptide[END_REF].

It is important to note that we explicitly calculate the quasiparticle correction for several occupied and unoccupied energy levels around the gap, while the remaining eigenvalues are correspondingly shifted. 2 This is an extension to standard G 0 W 0 approaches, in the following denoted GW Scissor, where usually only a quasiparticle correction to the HOMO and LUMO is calculated. DF T -LDA and G 0 W 0 @LDA eigenvalues are switched, in order to match them to the correct evGW ordering. The presented GW results correspond to a quasiparticle correction explicitly calculated for 20 states around the gap. DF T -LDA eigenvalues strongly dier from evGW , whereas a self-consistent scCOHSEX already gives the right ordering and level spacing. Even though the occupied levels are shifted by about 1 eV to too low energies, the scCOHSEX approach is a valuable starting point for evGW and even G 0 W 0 calculations (see column evGW@scCOHSEX and G 0 W 0 @scCOHSEX). Concerning the G 0 W 0 @LDA approach, even though several states around the gap are corrected, the right ordering can not be restored within a single-shot G 0 W 0 calculation. Consequently, the reliability of the G 0 W 0 Scissor approach as commonly applied seems questionable in this case.

Hermitian within this approach. This opens the door to ecient fully self-consistent quasiparticle calculations, since the eigenvectors are orthogonal and solutions to the same static and Hermitian operator.

The approach studied in this work [START_REF] Faber | Many-body Green's function GW and Bethe-Salpeter study of the optical excitations in a paradigmatic model dipeptide[END_REF] consists of fully self-consistent scCOHSEX calculations with an update of both eigenvalues and eigenfunctions. These serve as an input for subsequent G 0 W 0 or evGW calculations [START_REF] Bruneval | Eect of self-consistency on quasiparticles in solids[END_REF]. The obtained results are compiled in Table (3.2). In the case of transparent conductive oxides and quaternary thin lms for photovoltaics [START_REF] Vidal | Eects of Electronic and Lattice Polarization on the Band Structure of Delafossite Transparent Conductive Oxides[END_REF][START_REF] Vidal | Strong Interplay between Structure and Electronic Properties in CuIn(S,Se) 2 : A First-Principles Study[END_REF] or for bulk gold [START_REF] Rangel | Band structure of gold from many-body perturbation theory[END_REF], the G 0 W 0 @scCOHSEX approach has been shown to yield excellent results in semiconductors combining extended and localized states [START_REF] Bruneval | Eect of self-consistency on quasiparticles in solids[END_REF][START_REF] Gatti | Understanding Correlations in Vanadium Dioxide from First Principles[END_REF]. Very recently, Korbel and coworkers conrmed this observation for small transition metal clusters using the FIESTA package [START_REF] Korbel | Benchmark Many-Body GW and Bethe-Salpeter calculations for small transition-metal molecules[END_REF].

In this section, we are interested in assessing the quality of the self-consistent scCOHSEX scheme as a starting point for evGW calculations on molecules. It is important to have in mind that in the present contour deformation implementation, calculating repeatedly the GW correction to several states away from the gap turns out to be expensive due to the poles contribution. In addition, we want to test one of the common approximations in the GW community, namely the assumption that the Kohn-Sham and quasiparticle eigenfunctions strongly overlap, even though the energy gap may dier signicantly. By way of example, in the case of simple bulk systems, an agreement of 99.9% has been claimed [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF], which justies the presented perturbative GW approach. However, a well-known example, where such an is a π 1 state, located (0.25, 0.74, 0.99) eV above the σ 1 , π 2 and σ 2 states, respectively. This can be compared to spacings of (0.26, 0.75, 0.97) eV within the evGW @scCOHSEX value.

Such an excellent agreement in level ordering and energy spacing, together with a better HOMO-LUMO gap, indicates that the scCOHSEX eigenvalue spectrum is certainly a better starting point for evGW calculations as compared to the DF T -LDA approach. In particular, a simple G 0 W 0 @scCOHSEX Scissor approach, correcting only the HOMO and LUMO, yields already very similar results as compared to a full evGW @scCOHSEX calculation correcting a large number of states around the fundamental gap.

Inferring a better quality of the scCOHSEX eigenfunctions from the strongly ameliorated energy spectrum, as compared to Kohn-Sham DF T -LDA calculations, remains a dicult issue. However, the analysis of the scCOHSEX σ 1 and π * where the charge is averaged within planes perpendicular to the molecular "axis". For the occupied σ 1 state, the DF T -LDA, scCOHSEX and Hartree-Fock wave functions (dotted lines) are nearly indistinguishable. However, for the π * 2 state (full lines), dierences start to appear in particular at the Hartree-Fock level. Clearly, the scCOHSEX wave function is closer to the DF T -LDA one, even though the scCOHSEX (and evGW ) quasiparticle spectrum is closer to the Hartree-Fock one.

Conclusion

In conclusion, the presented GW results on the model dipeptide aimed at studying the performance of two computationally ecient self-consistent GW schemes. We now summarize the main conclusions of the present study:

The G 0 W 0 @LDA Scissor approach, where only HOMO and LUMO are explicitly corrected, fails as the HOMO-LUMO gap is underestimated and, importantly, the ordering of occupied states close to the gap is wrong. This subsequently leads to optical spectra dicult to compare to experiment.

While DF T calculations with hybrid functionals may possibly be a better starting point, we explored inexpensive partially self-consistency GW schemes in order to come up with an approach that does not depend on the starting guess. Moreover, this approach is supposed to work equally well for extended solids of various kind (sp-or spd-systems) and molecular systems.

The evGW @LDA scheme, namely an approach where only the eigenstates are updated, has been shown to yield very accurate quasiparticle energies for the present π-conjugated molecules without transition metal atoms. This was demonstrated for example during our diploma thesis in the case of DNA/RNA nucleobases (for details see Ref. [START_REF] Faber | First-principles GW calculations for DNA and RNA nucleobases[END_REF]),

showing similar important level crossings from DF T -LDA to GW . However, the problem of level crossings requires that GW calculations are performed several times for a large number of corrected states, implying important computational costs related to the present contour deformation techniques (contribution from the poles).

The inexpensive self-consistent scCOHSEX scheme leads to a much better starting point for GW calculation. Not only the gap is closer to the quasiparticle value, but the ordering and spacing of the occupied state is found to be accurate. Consequently, an evGW correction calculated for the HOMO and LUMO only, or even a G 0 W 0 @scCOHSEX calculation, leads to results of the same quality than performing evGW on a large set of levels. Therefore, this is becoming our preferred option for large molecular systems.

For the model dipeptide, despite the presence of localized σ-states suering from significant self-interaction problems, the scCOHSEX wave functions were not found to dier very signicantly from the DF T -LDA ones, just as usually assumed in extended solids with delocalized (s,p) electrons.

We nally conclude this section by admitting that, contrary to the DNA/RNA nucleobases, our evGW calculations could not be compared to higher level theory or experimental data.

Since the "proof of the pudding is in the eating", we now turn to the absorption spectrum for which comparisons can be directly drawn.

3.3. The BSE formalism applied: charge-transfer excitations

Charge-transfer excitations

In the presented BSE calculations, we study the optical properties of two systems, one is the already introduced dipeptide, the other one is a family of organic dyes, namely coumarin molecules of interest for dye-sensitized solar cells. They have in common that their lowestlying optical excitations are so-called charge-transfer (C T ) excitations. The latter are a special class of electron-hole excitations, where the excited electron and the created hole are spatially separated from each other. These non-local excitations are very interesting both from a fundamental and an applied point of view. Apart from the fact that they are often the principal excitations in organic molecules composed of functional groups with dierent electro-negativity (push-pull systems), they are believed to represent the intermediate state between bound electron-hole pairs and dissociated free carriers in organic solar cells [START_REF] Sariciftci | Photoinduced Electron Transfer from a Conducting Polymer to Buckminsterfullerene[END_REF][START_REF] Schmidt-Mende | Self-Organized Discotic Liquid Crystals for High-Eciency Organic Photovoltaics[END_REF][START_REF] Hardin | The renaissance of dye-sensitized solar cells[END_REF]. However, the exact mechanisms leading to charge separation remain rather controversial [START_REF] Bakulin | The Role of Driving Energy and Delocalized States for Charge Separation in Organic Semiconductors[END_REF][START_REF] Caruso | Long-range exciton dissociation in organic solar cells[END_REF][START_REF] Yost | Electrostatic Eects at Organic Semiconductor Interfaces: A Mechanism for "Cold" Exciton Breakup[END_REF], urging for computational quantum mechanical studies which allow an accurate exploration of local and CT excitations at various energies. Moreover, since it is also believed that they allow an ecient charge separation, molecules characterized by low-lying CT excitations are promising candidates for organic photovoltaics [START_REF] Stein | Prediction of charge-transfer excitations in coumarin-based dyes using a range-separated functional tuned from rst principles[END_REF][START_REF] Hara | Molecular Design of Coumarin Dyes for Ecient Dye-Sensitized Solar Cells[END_REF][START_REF] Hara | Design of new coumarin dyes having thiophene moieties for highly ecient organic-dye-sensitized solar cells[END_REF][START_REF] Kurashige | Theoretical Investigation of the Excited States of Coumarin Dyes for Dye-Sensitized Solar Cells[END_REF][START_REF] Wong | Coumarin dyes for dye-sensitized solar cells: A longrange-corrected density functional study[END_REF][START_REF] Sanchez-De Armas | Coumarin derivatives for dye sensitized solar cells: a TD-DFT study[END_REF].

Recently, we already demonstrated that our GW/BSE approach works well for intermolecular CT excitations, where the excited electron and hole are located on two dierent molecules [START_REF] Blase | Charge-transfer excitations in molecular donor-acceptor complexes within the many-body Bethe-Salpeter approach[END_REF][START_REF] Duchemin | Short-Range to Long-Range Charge-Transfer Excitations in the Zincbacteriochlorin-Bacteriochlorin Complex: A Bethe-Salpeter Study[END_REF]. We now want to tackle a situation which often occurs in nature, namely the problem of intramolecular CT excitations, where donor and acceptor are situated on the same molecule.

Treating CT excitations theoretically using ab initio methods remains dicult. Wave function-based quantum chemistry methods such as complete active space second order perturbation theory (CASPT2) or multi-reference conguration interaction (MRCI) methods [START_REF] Leszczynski | Handbook of Computational Chemistry[END_REF] yield accurate results, however, they are computationally too demanding to treat systems with more than a few tens of atoms. At the density functional theory level, constrained DF T formalisms [START_REF] Wu | Constrained Density Functional Theory and Its Application in Long-Range Electron Transfer[END_REF][START_REF] Ghosh | Computational approaches to charge transfer excitations in a zinc tetraphenylporphyrin and C[sub 70] complex[END_REF] have proven to be extremely ecient in providing a good description of the lowest-lying CT excitations in rather large systems, but generalizing such techniques to higher excited states remains a dicult issue. Further, excited states wave functions, needed to calculate e.g. transfer rates, are not available. Another candidate is time-dependent density functional theory (TDDFT) [START_REF] Runge | Density-Functional Theory for Time-Dependent Systems[END_REF][START_REF] Marques | Time-Dependent Density Functional Theory[END_REF][START_REF] Casida | Time-dependent density-functional theory for molecules and molecular solids[END_REF], where one obtains optical excitations properties of systems signicantly larger than that amenable to e.g. CASPT2 or MRCI approaches. While describing very well local optical excitations, applied with standard (semi-)local functionals, it fails in reproducing CT excitations in most instances and can not be considered reliable.

The reason is that long-range interactions between electrons and holes which are distant to each other and which are (almost) not overlapping can not be treated properly with these local functionals [START_REF] Dreuw | Long-range charge-transfer excited states in time-dependent density functional theory require non-local exchange[END_REF][START_REF] Tozer | Relationship between long-range charge-transfer excitation energy error and integer discontinuity in Kohn-Sham theory[END_REF][START_REF] Dreuw | Failure of Time-Dependent Density Functional Theory for Long-Range Charge-Transfer Excited States: The Zincbacteriochlorin-Bacteriochlorin and Bacteriochlorophyll-Spheroidene Complexes[END_REF]. This can be made clear by considering the TDLDA case, where one nds within Casida's eective eigenvalue formulation [START_REF] Casida | Recent Advances in Density Functional Methods, Part I[END_REF] that the resonant matrix elements read (see preceding chapters):

H diag vc,v c = δ vv δ cc ε 0 c -ε 0 v , H exch vc,v c = ´drdr φ v (r) φ * c (r) v (r, r ) φ * v (r ) φ c (r ) , H LDA vc,v c = ´drφ v (r) φ * c (r) ∂V LDA xc ∂n(r) φ * v (r) φ c (r) .
Clearly, the product of φ v (r) and φ c (r), and consequently the two electron-hole interaction terms, go to zero for excitations, where the electron φ c and the hole φ v do not overlap, since the wave functions depend on the same space variable. As a consequence, one ends up with the diagonal part, i.e. with the time-independent Kohn-Sham eigenvalue dierences ε 0 c -ε 0 v , and electron-hole interactions are neglected. More specically, for long-range CT excitations, where an electron is promoted from a donor to a distant acceptor, [START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF] Mulliken's rule should be satised. The latter states that the lowest charge-transfer excitation energy E exc CT equals [START_REF] Kronik | Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals[END_REF][START_REF] Mulliken | Structures of Complexes Formed by Halogen Molecules with Aromatic and with Oxygenated Solvents[END_REF]:

E exc CT = IE D -EA A - 1 R ,
where IE D is the ionization energy of the donor, EA A the electron anity of the acceptor and R = | r e -r h | the donor-acceptor distance, given by the average position of the excited electron r e and hole r h . This implies that the charge-transfer excitation energy is determined by the fundamental energy gap of the entire complex (IE D -EA A ) and the 5 It is assumed that the chemical interaction between the donor and the acceptor is negligible.

electron-hole binding energy (1/R). Within TDLDA, this limit is not correctly reproduced and the 1/R term is completely neglected [START_REF] Stein | Prediction of charge-transfer excitations in coumarin-based dyes using a range-separated functional tuned from rst principles[END_REF]. Further, (IE D -EA A ) reduces to the too small DF T -LDA gap. Within the BSE approach, the exchange contribution H exch also vanishes for CT excitations. However, the screened part H scr does not necessarily go to zero, since the electron wave function φ c and the hole wave function φ v depend on dierent space coordinates. Instead, φ v (φ c ) and φ v (φ c ) depend on the same space coordinate, which yields non-zero densities for v = v and c = c weighted by the non-local screened Coulomb potential W :

H scr vc,v c = -ˆdrr φ v (r) φ * c r W stat r , r φ * c r φ v (r) .
Clearly, the correct 1/R Mulliken limit is automatically obtained within BSE, which can be

made clear setting v = v , c = c , |φ v (r)| 2 = δ (r -r h ) and |φ c (r)| 2 = δ (r -r e ).
This yields for the screened exchange term:

H scr LR = - 1 ε M R ,
which is the long-range limit of W stat . For donor-acceptor systems in vacuum, the macroscopic dielectric function equals ε M = 1. Since the diagonal part of the Bethe-Salpeter resonant Hamiltonian yields the quasiparticle energy dierences ε GW c -ε GW v and since the exchange term goes to zero for charge-transfer excitations, one directly arrives within the long-range limit of the screened exchange contribution H scr LR in vacuum at the Mulliken limit:

E exc CT = IE GW D -EA GW A - 1 R .
This has been recently shown for the case of intermolecular CT excitations in the zincbacteriochlorin/bacteriochlorin complex [START_REF] Duchemin | Short-Range to Long-Range Charge-Transfer Excitations in the Zincbacteriochlorin-Bacteriochlorin Complex: A Bethe-Salpeter Study[END_REF], as depicted in Fig. (3.9).

The encountered diculties within TDLDA paved the way for the success of non-local hybrid functionals [START_REF] Yanai | A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)[END_REF][START_REF] Kronik | Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals[END_REF][START_REF] Tawada | A long-rangecorrected time-dependent density functional theory[END_REF][START_REF] Iikura | A long-range correction scheme for generalized-gradient-approximation exchange functionals[END_REF][START_REF] Baer | Density Functional Theory with Correct Long-Range Asymptotic Behavior[END_REF]. Analogue to the time-independent case, one can also set up a generalized TDDFT scheme with Fock-like non-local potentials. In this case, an additional term appears in the H xc vc,v c contribution:

H xc vc,v c = ´drφ v (r) φ * c (r) f sl,γ xc (r) φ * v (r) φ c (r) + ´drdr φ v (r) φ * v (r) u κ (r, r ) φ * c (r ) φ c (r ) . (3.10)
The short-range exchange and correlation are governed by the (semi)local exchange-correlation kernel f sl,γ xc , whereas long-range is included through u κ (r, r ) = erf(κ|r,r ) |r-r | . This is the most general form, also applicable for range-separated hybrid functionals. Concerning limiting cases, provided that u κ is zero and f sl,γ xc the standard LDA exchange-correlation functional, one retrieves the TDLDA scheme. The case of TDDFT with hybrid functionals is obtained by setting u κ to (α/ |r -r |), i.e. by introducing a xed amount of exact exchange [START_REF] Kronik | Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals[END_REF].

Concerning charge-transfer optical excitations, from equation (3.10), it becomes immediately clear that similar to (semi)local functionals, also standard hybrid functionals do not correctly Ref. [START_REF] Duchemin | Short-Range to Long-Range Charge-Transfer Excitations in the Zincbacteriochlorin-Bacteriochlorin Complex: A Bethe-Salpeter Study[END_REF].

reproduce the long-range Mulliken limit. Since H exch and the local exchange-correlation term in H xc vanish for a small overlap of the excited electron and the hole, the lowest excitation energy is given by:

E exc CT = IE D -EA A -ˆdrdr φ v (r) φ * v (r) u κ r, r φ * c r φ c r .
As it is depicted in Fig. (3.3), only certain, specically designed range-separated hybrid functionals show the right 1/R dependence in the long-range and correctly describe chargetransfer excitations. However, for standard hybrid functionals, the lowest CT excitation energy reduces to:

E exc CT = IE D -EA A - α R ,
i.e. one nds a reduced electron-hole binding energy of e.g. 20% in the case of B3LYP.

Range-separated hybrids give precise results both for local and CT excitations [START_REF] Yanai | A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)[END_REF][START_REF] Kronik | Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals[END_REF][START_REF] Tawada | A long-rangecorrected time-dependent density functional theory[END_REF][START_REF] Iikura | A long-range correction scheme for generalized-gradient-approximation exchange functionals[END_REF][START_REF] Baer | Density Functional Theory with Correct Long-Range Asymptotic Behavior[END_REF]. However, they contain adjustable parameters, which are not known a priori. They have to be determined either semi-empirically by tting ground state properties of a given set of molecules or by taking into account physical criteria applying to the specic system [START_REF] Stein | Prediction of charge-transfer excitations in coumarin-based dyes using a range-separated functional tuned from rst principles[END_REF]. The transferability from one system to another remains dicult, since it has been shown that the optimal value for these parameters can be strongly system-dependent and even statedependent [START_REF] Wong | Coumarin dyes for dye-sensitized solar cells: A longrange-corrected density functional study[END_REF][START_REF] Lange | Charge-Transfer Excited States in a pi-Stacked Adenine Dimer, As Predicted Using Long-Range-Corrected Time-Dependent Density Functional Theory[END_REF][START_REF] Stein | Reliable Prediction of Charge Transfer Excitations in Molecular Complexes Using Time-Dependent Density Functional Theory[END_REF]. Within GW/BSE, on the contrary, the screening is determined by the system itself instead of being xed at the beginning. This makes this approach universal and systems ranging from metals with a very eective screening to molecules with poor screening properties should be equally accessible. The strength of the screening is automatically adjusted and thus both local and CT excitations are reproduced. In the following, this will be discussed rst for the model dipeptide and second for the coumarin family, both showing intramolecular CT excitations. High-quality quantum chemistry and TDDFT calculations serve as a valuable reference.

3.3.2. Charge-transfer excitations in the model dipeptide

Background and Notation

The studied model dipeptide was originally introduced in Ref. [START_REF] Serrano-Andres | Theoretical Study of the Electronic Spectroscopy of Peptides. III. Charge-Transfer Transitions in Polypeptides[END_REF] to rationalize the origin of the 7.3-7.5 eV absorption structure common to many polypeptides in gas phase. This work revealed the importance of intramolecular CT excitations between neighboring peptide units and subsequently, the dipeptide served as a test case for intramolecular CT excitations.

It is one of the rst examples, where large errors have been observed at the TDDFT level.

This triggered its study by a large variety of approaches, including CASPT2 [START_REF] Serrano-Andres | Theoretical Study of the Electronic Spectroscopy of Peptides. III. Charge-Transfer Transitions in Polypeptides[END_REF], TDDFT with various (semi)local, hybrid or range-separated hybrid functionals [START_REF] Yanai | A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)[END_REF][START_REF] Peach | Excitation energies in density functional theory: An evaluation and a diagnostic test[END_REF][START_REF] Akinaga | Range-separation by the Yukawa potential in long-range corrected density functional theory with Gaussian-type basis functions[END_REF] and also a Bethe-Salpeter study based on an empirical scissor approach [START_REF] Rocca | Ab initio calculations of optical absorption spectra: Solution of the BetheSalpeter equation within density matrix perturbation theory[END_REF]. Diculties were encountered to reproduce the CASPT2 results [START_REF] Serrano-Andres | Theoretical Study of the Electronic Spectroscopy of Peptides. III. Charge-Transfer Transitions in Polypeptides[END_REF] with unusual discrepancies between the mentioned state-of-the-art techniques. Moreover, a very large sensitivity of CT excitation energies on the chosen functional parameters within e.g. the same CAM-B3LYP TDDFT framework was observed [START_REF] Yanai | A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)[END_REF][START_REF] Peach | Excitation energies in density functional theory: An evaluation and a diagnostic test[END_REF].

In this work, we focus on two dierent kinds of optical excitations. On the one hand, we are interested in valence transitions between states localized on the same peptide unit.

These are labeled W 1 and W 2 and the electron is promoted from an occupied σ i -state to an unoccupied π * i -state. On the other hand, we study CT excitations between states localized on dierent peptide groups, namely the CT a exciton, a σ 1 → π * 2 transition, and the CT b exciton, a π 1 → π * Concerning the technical details of the BSE calculations presented in the following, we go beyond the already introduced Tamm-Danco approximation (TDA). This implies that the coupling between resonant (R) and anti-resonant (R * ) transitions is explicitly treated, i.e. The main outcome of the TDLDA or TDDFT-PBE calculations is that CT excitation energies are much too small. The CT excitations are located below the lowest intramonomer W 1 or W 2 transitions. This is in great contrast to the CASPT2 results, where the CT excitations are found to lie about 1.4 eV to 2.4 eV above the W 1 and W 2 excitations. Our TDLDA value (4.63 eV) for the CT a transition, which consists nearly entirely of a transition between the Kohn-Sham HOMO and LUMO, can be compared to the HOMO-LUMO Kohn-Sham gap of 4.62 eV. This conrms that within TDDFT using local exchange-correlation functionals, the electron-hole interaction term vanishes for spatially separated electron and hole states and one is left with the energy dierence between Kohn-Sham states. On the other hand, the local W 1 and W 2 transitions, with a strong overlap between nal and initial states, are much better described, even though showing a 0.2 -0.3 eV red shift for the W 1 transition as compared to CASPT2.

W i σ i → π i CT a σ 1 → π * 2 CT b π 1 → π * 2 C C * 0.3 (φ v φ c ) GW/BSE W 1 W 2 CT a CT b GW/BSE GW/BSE CT 0.02 W 1 W 2 0.
Introducing some amount of exact exchange to the TDDFT kernel allows even spatially separated electrons and holes to partially interact. Previous TDDFT-B3LYP calculations (see Table 3.3) indeed show some improvement as compared to TDLDA by locating the CT states above the W 1 and W 2 transitions. However, compared to CASPT2 calculations, the CT excitations energies are still about 1 eV to 1.8 eV too small, as a reminder that the B3LYP functional captures only 20% of exact exchange. This problem can be cured using range-separated functionals such as LC-BLYP or CAM-B3LYP, where the CT excitations come in much better agreement [START_REF] Yanai | A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)[END_REF][START_REF] Peach | Excitation energies in density functional theory: An evaluation and a diagnostic test[END_REF] with the quantum-chemistry reference as indicated in Table (3.3). Nevertheless, within the CAM-B3LYP method itself, one observes energy dierences of the order of 0.7 eV for the CT a exciton, leading to the standard question of the proper choice of the parameters (α + β = 0.65 or α + β = 0.8 in the present case).

Comparing our GW/BSE calculations (@LDA column in Table 3.3) to CASPT2 values, we nd an excellent agreement for the W 1, W 2 and the CT b excitation. The maximum discrepancy is 0.07 eV for the W 1 transition, while remarkably both the local W 2 and charge-transfer CT b excitation agree within 0.02 eV. 7 Clearly, tuning the (α, β) and range-separation param- eters may bring the CAM-B3LYP calculations in better agreement with CASPT2 values, but we emphasize that the present GW/BSE scheme does not contain any adjustable parameters.

Concerning the oscillator strengths of the respective transitions (see numbers in parenthesis in Table 3.3), the GW/BSE values are in reasonable agreement with the CASPT2 reference.

The LC-BLYP and CAM-B3LYP values also agree for the transitions with vanishing oscillator strength, whereas they signicantly underestimate the value of the oscillator strength for the CT b exciton, where the GW/BSE oscillator strength is closer to the CASPT2 value.

Obtaining an excellent agreement between the various formalisms proves more dicult for the oscillator strengths than for the corresponding excitation energies.

The largest discrepancy between the present GW/BSE@LDA and available CASPT2 calculations is of 0.3 eV for the CT a excitation. For such a transition, our GW/BSE value is 7 The use of the Tamm-Danco approximation at the GW/BSE level leads to increased excitation energies and a deteriorated spectrum as compared to CASPT2. In agreement with the 0.15 eV blue shift reported by Rocca and coworkers [START_REF] Rocca | Ab initio calculations of optical absorption spectra: Solution of the BetheSalpeter equation within density matrix perturbation theory[END_REF], the largest TDA induced shift concerns the CT b excitation energy, which is blue-shifted by 0.17 eV in our calculations. The TDA further induces a small blue-shift of 0.03 eV for the W 1 and W 2 transitions, in perfect agreement with Ref. [START_REF] Rocca | Ab initio calculations of optical absorption spectra: Solution of the BetheSalpeter equation within density matrix perturbation theory[END_REF]. The CT a charge-transfer state is marginally aected by a 0.01 eV blue-shift.

in nearly perfect agreement with the LC-BLYP prediction, lying in between the two CAM-B3LYP values. As evidenced in Table (3.3) and Fig. (3.11), observing the rather large ∼ 0.7

eV variation between the two CAM-B3LYP values, such a transition is clearly very sensitive to the details of the exchange and correlation potential. Before commenting on such a deviation, we will test the impact of using frozen Kohn-Sham LDA eigenstates in the present GW and Bethe-Salpeter approach here below.

GW/BSE calculations starting from self-consistent COHSEX eigenstates

The results of our GW/BSE study starting from self-consistent COHSEX eigenstates and eigenvalues is presented in the column "@C OHSEX" of Table (3.3). As compared to GW/BSE calculations where the Kohn-Sham eigenstates are kept frozen ("@LDA" column), the W 1 and W 2 excitation energies hardly change by a maximum of 0.03 eV for the W 1 transition. The largest variation is again related to the CT a transition, with an increase of 0.22

eV. This worsens the agreement with the CASPT2 value, but brings our GW/BSE calculations in excellent agreement with the CAM-B3LYP (α + β = 0.8) results. Such an evolution can be traced back to a ∼ 0.2 eV blue-shift of the π * 2 energy level within GW @COHSEX as compared to GW @LDA. The oscillator strength associated with this transition is also seen to adopt a smaller value, worsening the agreement with the CASPT2 value, but improving the agreement with the CAM-B3LYP result.

It is interesting to observe that what we may consider to be our most accurate values, namely our evGW/BSE calculations based on scCOHSEX eigenstates, come in excellent agreement with the CAM-B3LYP value with enhanced long-range exchange, namely setting (α + β) to 0.8 instead of the original 0.65 value. In the case of CT excitations, the correct long-range Mulliken limit can only be reproduced with a long-range (α + β = 1) parametrization of the CAM-B3LYP functional (see Fig. 3.3). As such, the (α + β = 0.8) functional provides in principle a better description of the long-range CT electron-hole interaction. Very consistently, the LC-BLYP functional, with a proper asymptotic scaling, locates the CT a transition at 8.38 eV [START_REF] Yanai | A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)[END_REF], in much better agreement with our GW/BSE values than the CASPT2 prediction.

However, the analysis of the contributing wave functions in Fig. (3.10) shows that the CT a transition in the dipeptide is far from the ideal case of the long-range well-separated electronhole CT limit. Overall, our GW/BSE@COHSEX results show a mean absolute error of 0.1 eV and 0.08 eV as compared to CAM-B3LYP (α + β = 0.8) and LC-BLYP, respectively.

Regarding previous studies on CT excitations within the present GW/BSE formalism, with typical errors of the order of 0.1 eV as compared to experiment, TDDFT with optimized range-separated functionals or CASPT2 calculations [START_REF] Blase | Charge-transfer excitations in molecular donor-acceptor complexes within the many-body Bethe-Salpeter approach[END_REF][START_REF] Duchemin | Short-Range to Long-Range Charge-Transfer Excitations in the Zincbacteriochlorin-Bacteriochlorin Complex: A Bethe-Salpeter Study[END_REF][START_REF] Baumeier | Excited States of Dicyanovinyl-Substituted Oligothiophenes from Many-Body Green's Functions Theory[END_REF], the present 0.3 eV to 0.5 eV discrepancies for the CT a transition are somehow unusual, even though dramatically smaller than the typical errors induced by TDDFT calculation with (semi)local kernels or even B3LYP. The 0.7 eV dierence obtained between CAM-B3LYP calculations with various parametrizations [START_REF] Yanai | A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)[END_REF] indicates that such variations cannot be explained by dierences in running parameters (basis sizes and type, pseudopotential, etc.), but really hinge on the sensitivity of this transition onto the balance between short-and long-range exchange and correlation.

While we cannot comment on the accuracy and limitations of the available CASPT2 calculations, we certainly can emphasize in particular the lack of double-excitations in the present GW/BSE formalism and in TDDFT calculations, a possible explanation that would require more sophisticated treatments such as the inclusion of dynamical eects in the screened Coulomb potential matrix elements at the BSE level [START_REF] Sangalli | Double excitations in correlated systems: A many-body approach[END_REF]. While this is certainly beyond the scope of the present work, we can conclude that as it stands, the present parameter-free GW/BSE approach oers an accuracy comparable to TDDFT calculations performed with the best available parametrized range-separated functionals.

Conclusion

We studied within the many-body Green's function GW/BSE approach the excitation energies of a paradigmatic dipeptide. The latter served as a benchmark for describing intramolecular CT excitations in organic systems within various theoretical frameworks, including TDDFT with local, classical hybrid and range-separated hybrid functionals, CASPT2 calculations and a previous Bethe-Salpeter study based on an empirical GW Scissor approach. Based on evGW calculations, our calculated optical excitation energies are found to agree with CASPT2 calculations within 0.07 eV for the local W 1, W 2 and the charge-transfer CT b excitation and a maximum discrepancy of 0.3 eV for the CT a transition. The eect of further updating self-consistently the quasiparticle wave functions within the static COHSEX approximation leads to rather marginal variations for the W 1, W 2 and CT b excitations, but shifts the discrepancy to 0.5 eV as compared to CASPT2 for the ubiquitous CT a transition.

Our BSE calculations based on the GW @COHSEX eigenvalues and eigenfunctions agree very well with both CAM-B3LYP calculations with enhanced long-range exchange (α + β = 0.8) and the original LC-BLYP formulation, with a maximum mean absolute error of 0.1 eV. The present results conrm the reliability of the parameter-free GW/BSE formalism in describing local and charge-transfer excitations in organic systems of interest e.g. for photovoltaics, photosynthesis or photocatalysis. The most prominent modern DSSCs, also known as Grätzel cells, consist of porous layers of titanium dioxide (TiO 2 ) nanoparticles covered by a monolayer of an organic dye absorbing the sun light. The working principle is schematically illustrated in Fig. (3.12). While the most ecient sensitizers are composed of ruthenium dye complexes, intense research is conducted so as to nd molecular alternatives which are cheaper, easier to synthesize and free from the resource limitations related to the noble metal ruthenium. As a promising direction, Hara and coworkers demonstrated that coumarin-based dyes, such as the so-called NKX-2xxx fam-ily (see Fig. 3.13), could lead to conversion eciencies approaching that of ruthenium-based DSSCs [START_REF] Hara | Molecular Design of Coumarin Dyes for Ecient Dye-Sensitized Solar Cells[END_REF][START_REF] Hara | Design of new coumarin dyes having thiophene moieties for highly ecient organic-dye-sensitized solar cells[END_REF][START_REF] Hara | Oligothiophene-Containing Coumarin Dyes for Ecient Dye-Sensitized Solar Cells[END_REF][START_REF] Wang | A High-Light-Harvesting-Eciency Coumarin Dye for Stable Dye-Sensitized Solar Cells[END_REF].

Coumarins show very good photoelectric properties [START_REF] Hara | Molecular Design of Coumarin Dyes for Ecient Dye-Sensitized Solar Cells[END_REF][START_REF] Hara | Oligothiophene-Containing Coumarin Dyes for Ecient Dye-Sensitized Solar Cells[END_REF] and they are an impressive example of chemical molecular design. Starting from the originally tested C343 coumarin [START_REF] Rehm | Femtosecond Electron-Transfer Dynamics at a Sensitizing Dye-Semiconductor (TiO2) Interface[END_REF], the introduction of (-C=C-) methine fragments between the coumarin unit and the terminal (-COOH) carboxyl group (see Fig. 3.14) induces a red shift of the absorption spectrum [START_REF] Hara | Molecular Design of Coumarin Dyes for Ecient Dye-Sensitized Solar Cells[END_REF].

This improves the light harvesting in the visible range, since the absorption edge is shifted to lower energies (see Fig. group enhances the acceptor character of the combined (-COOH) and (-C=N) cyanoacrylic acid group, increasing the charge-transfer character of the internal excitations. This is believed to favor the injection of the photoelectron into the TiO 2 conduction band through the anchoring (-COOH) carboxylic unit [START_REF] Hara | Molecular Design of Coumarin Dyes for Ecient Dye-Sensitized Solar Cells[END_REF]. Finally, the replacement of the methine spacer by thiophene chains reduces the adverse aggregation of dyes onto the TiO 2 surface, leading to the NKX-2677 dye with a solar-energy-to-electricity conversion eciency of 7.7% [START_REF] Hara | Design of new coumarin dyes having thiophene moieties for highly ecient organic-dye-sensitized solar cells[END_REF].

Due to its internal donor-acceptor structure, this family of molecules became also a bench-Figure 3.12.: Functional principle of an exemplary DSSC with the NKX-2677 coumarin dye: light is absorbed in the monolayer of the NKX-2677 dye, creating as lowestlying excitation a CT state (averaged electron/ hole distribution in green/ orange). The excited electron then goes from the conduction band of the dye into the conduction band of TiO 2 (lower in energy) and nally in the anode (not shown). The circuit is closed, as the redox system I -/I - 3 transports electrons from the cathode (not shown) to the dye, which in turn recovers charge neutrality. mark for theoretical studies aiming at solving the already mentioned problem of describing CT excitations within time-dependent density functional theory (TDDFT) [START_REF] Dreuw | Long-range charge-transfer excited states in time-dependent density functional theory require non-local exchange[END_REF][START_REF] Dreuw | Failure of Time-Dependent Density Functional Theory for Long-Range Charge-Transfer Excited States: The Zincbacteriochlorin-Bacteriochlorin and Bacteriochlorophyll-Spheroidene Complexes[END_REF]. In particular, TDDFT calculations with (semi)local [START_REF] Sanchez-De Armas | Coumarin derivatives for dye sensitized solar cells: a TD-DFT study[END_REF] and hybrid [START_REF] Stein | Prediction of charge-transfer excitations in coumarin-based dyes using a range-separated functional tuned from rst principles[END_REF][START_REF] Kurashige | Theoretical Investigation of the Excited States of Coumarin Dyes for Dye-Sensitized Solar Cells[END_REF][START_REF] Wong | Coumarin dyes for dye-sensitized solar cells: A longrange-corrected density functional study[END_REF] kernels were conducted and compared to reference quantum chemistry coupled-cluster CC2 calculations [START_REF] Kurashige | Theoretical Investigation of the Excited States of Coumarin Dyes for Dye-Sensitized Solar Cells[END_REF] in order to assess the accuracy of the various approaches. This oers us a broad spectrum of ab initio reference data. In the present work, we analyze the (singlet) excitation energies of various coumarin-based molecules such as the parent C343 dye and the related NKX-2388, NKX-2311, NKX-2586 and NKX-2677 structures.

Technical details

The used molecular structures have been relaxed at the all-electron DFT-B3LYP 6-311G(d,p) level using the Gaussian09 package [START_REF] Frisch | Gaussian 09 Revision B.01[END_REF]. As a single-particle starting point for the GW/BSE calculations, DF T -LDA eigenstates and eigenvalues as provided by the SIESTA package [136] with a large triple-zeta plus double polarization basis (TZDP) are used.

The GW correction is explicitly calculated for the 10 highest occupied and lowest unoccupied levels, whereas the remaining levels are rigidly shifted. We performed iterative GW calculations with a simple self-consistency on the eigenvalues as presented in the preceding section. Our auxiliary basis contains six e -αr 2 Gaussian functions for the radial part of each (s, p, d) channel, with an even-tempered distribution of the localization coecients α ranging from 0.1 to 3.2, except for hydrogen, where the range is set to 0.1 to 1.5. As such, our auxiliary basis contains typically 54 orbitals per atom. Using these running parameters, the results of Refs. [START_REF] Kurashige | Theoretical Investigation of the Excited States of Coumarin Dyes for Dye-Sensitized Solar Cells[END_REF] and [START_REF] Wong | Coumarin dyes for dye-sensitized solar cells: A longrange-corrected density functional study[END_REF] (lled black circles) and the TD-LC-BLYP data from Ref. [START_REF] Wong | Coumarin dyes for dye-sensitized solar cells: A longrange-corrected density functional study[END_REF] (blue triangles up). The coumarins are indicated by their number (removing the NKX prex) with (c) standing for (cis) and (t)

for (trans). The axes' physical length is scaled according to their respective energy range.

0.48 eV for the NKX-2677 structure. This is certainly the signature that in the long-range charge-separation limit, the restricted amount of exact exchange in the B3LYP functional [START_REF] Becke | A new mixing of Hartree-Fock and local density-functional theories[END_REF] is not enough to account for the correct electron-hole interaction. To illustrate that point, we plot in Fig. We now come to the central results of the present study, namely the many-body pertur- bation theory data. In contrast to the TDLDA, TD-PBE or even TD-B3LYP results, our GW/BSE values (empty red squares in Fig. 3.15) are in much better agreement with the CC2 data points, with a mean absolute error of 0.06 eV. Such an agreement is remarkable accounting for the fact that the present GW/BSE approach does not contain any adjustable parameter. Concerning the longest NKX-2677 dye, which shows the largest discrepancy with CC2 calculations, we observe that our result falls within the values provided by the rangeseparated hybrid BNL functional study of Ref. [START_REF] Stein | Prediction of charge-transfer excitations in coumarin-based dyes using a range-separated functional tuned from rst principles[END_REF], where two dierent strategies to optimize non-empirically the range-separation parameter have been tested.

8 As compared to the RSH- BNL study, our GW/BSE results dier by a MAE ranging from 0.04 to 0.07 eV, which is well within the target maximum error of 0.1 eV. As emphasized in Ref. [START_REF] Stein | Prediction of charge-transfer excitations in coumarin-based dyes using a range-separated functional tuned from rst principles[END_REF], the CC2 approach is also not free from approximations and dierences of the order of 0.1 eV as compared to more accurate e.g. CASPT2 calculations are certainly to be expected. 9

8 The range separation parameter is obtained ab initio by minimizing the MAE between the Kohn-Sham HOMO and/or LUMO eigenvalues and the corresponding quantities obtained with a much more accurate ∆SCF approach for the neutral and charged systems. Depending on the chosen criteria, i.e optimization of the HOMO only of the neutral and charged systems ( J1 scheme) or of the HOMO and LUMO of the neutral system ( J2 scheme), slightly dierent values can be obtained. See Ref. [START_REF] Stein | Prediction of charge-transfer excitations in coumarin-based dyes using a range-separated functional tuned from rst principles[END_REF]. TD-LC-BLYP, TD-BNL and CC2 calculations. We provide the TD-B3LYP results from Ref. [START_REF] Kurashige | Theoretical Investigation of the Excited States of Coumarin Dyes for Dye-Sensitized Solar Cells[END_REF], which are in excellent agreement with the results of Ref. [START_REF] Wong | Coumarin dyes for dye-sensitized solar cells: A longrange-corrected density functional study[END_REF]. The numbers in parentheses indicate the associated oscillator strengths.

LDA/PBE a B3LYP b LC-BLYP c BNL d J 1 /J 2 CC2 b GW -BSE C343
The coumarins are indicated by their number (without the NKX prex), where (c) stands for (cis) and (t) for (trans).

a Ref. [START_REF] Sanchez-De Armas | Coumarin derivatives for dye sensitized solar cells: a TD-DFT study[END_REF], b Ref. [START_REF] Kurashige | Theoretical Investigation of the Excited States of Coumarin Dyes for Dye-Sensitized Solar Cells[END_REF], c Ref. [START_REF] Wong | Coumarin dyes for dye-sensitized solar cells: A longrange-corrected density functional study[END_REF], d Ref. [START_REF] Stein | Prediction of charge-transfer excitations in coumarin-based dyes using a range-separated functional tuned from rst principles[END_REF] Clearly, as compiled in Table (3.4), TDDFT calculations with the LC-BLYP functional [START_REF] Iikura | A long-range correction scheme for generalized-gradient-approximation exchange functionals[END_REF] also provide excellent results [START_REF] Wong | Coumarin dyes for dye-sensitized solar cells: A longrange-corrected density functional study[END_REF], with a MAE of 0.03 eV as compared to CC2, smaller than our GW/BSE MAE of 0.06 eV. However, as emphasized in Ref. [START_REF] Wong | Coumarin dyes for dye-sensitized solar cells: A longrange-corrected density functional study[END_REF], the range-separation parameter κ in the LC-BLYP study has been precisely adjusted to minimize the root mean square error compared to CC2 calculations. The best-t κ value for these systems (κ = 0.17) is found to be much smaller than the original value (κ = 0.33) advocated by Iikura and coworkers [START_REF] Iikura | A long-range correction scheme for generalized-gradient-approximation exchange functionals[END_REF]. The strong dependence of the optical excitation energies as a function of κ indicates that the choice of the originally recommended κ = 0.33 value would lead to a signicant overestimation of the transition energies (by as much as 0.30.4 eV; see Fig. 4 of

Ref. [START_REF] Wong | Coumarin dyes for dye-sensitized solar cells: A longrange-corrected density functional study[END_REF]). This leads to the standard question of the choice and transferability of the rangeseparation parameter. One observes, however, that with the best-t κ value, the correlation between LC-BLYP and CC2 results is very remarkable, showing that this class of systems can be described by a unique parameter.

Bearing important consequences on the use of such dyes in DSSCs, our GW/BSE calculations show as expected that the onset of absorption is signicantly red shifted with increasing size length (see Fig. 3.17). This evolution is in clear contrast with the behavior of CT excitations in well-separated gas phase donor-acceptor dyads where the exciton binding energy scales as the inverse distance between the two molecules, leading to an increase of the absorption energy onset. However, contrary to well separated donor or acceptor systems, the quasiparticle HOMO-LUMO gap in donor or acceptor dyads connected by a conducting π-conjugated bridge does not remain constant with varying bridge length. This is clearly exemplied in Table (3.5), where the GW gap is found to quickly decrease from the C343 molecule to the longest NKX-2677 dye. Except for the large variation of the electronic anity (EA) from the C343 parent to the NKX-2388 system, analysis of the GW HOMO and LUMO quasiparticle energies indicates that this gap reduction stems both from a destabilization of the HOMO been performed with a limited SV(P) basis. See Ref. [START_REF] Kurashige | Theoretical Investigation of the Excited States of Coumarin Dyes for Dye-Sensitized Solar Cells[END_REF]. explaining that it leads to one of the largest conversion eciencies in coumarin based DSSCs.

GW

Conclusion

In conclusion, we have studied within the many-body Green's function GW/BSE approach the excitation energies of a family of coumarin dyes recently shown to be very promising candidates for replacing ruthenium-based chromophores in dye-sensitized solar cells (DSSCs).

In such donor-bridge-acceptor molecules, the lowest singlet excitations are characterized by a charge-transfer character that varies with the length of the π-conjugated bridge. As a result, TD-B3LYP calculations can lead to an error as large as 0.5 eV as compared to reference quantum chemistry coupled-cluster CC2 calculations, despite the 20% of exact exchange contained in its functional form. We demonstrate that the GW/BSE approach leads to an excellent agreement with CC2 data with a mean absolute error of the order of 0.06 eV for the excitation energies. Such an accuracy is comparable to the best results provided by TDDFT calculations with optimized long-range corrected range-separated hybrids, but with a parameter-free approach that performs equally well for extended insulating or metallic systems and gas phase organic molecules. Such an excellent agreement is also demonstrated for the related oscillator strengths. The ability of the GW/BSE approach to describe both localized and chargetransfer excitations in nite size molecular systems or extended semiconductors originates in particular from the use of the screened Coulomb potential W that automatically adjusts the strength and range of the Coulomb interactions. This exibility may prove as a signicant advantage in the study of DSSCs, where both the organic dye and the extended TiO2 semiconductor must be treated with sucient accuracy.

After having carefully tested the validity range and accuracy of the presented GW/BSE approach with respect to ab initio quantum chemistry methods, DFT and TDDFT, we go on by studying its quality concerning a property directly related to the quasiparticle energies, namely the electron-phonon coupling (EPC). As discussed in detail in the subsequent section, electron-phonon coupling in molecular systems is at the heart of several important physical phenomena, such as the mobility of charge carriers in organic electronic devices [START_REF] Gosar | Linear-Response Theory of the Electron Mobility in Molecular Crystals[END_REF][START_REF] Fratini | Bandlike Motion and Mobility Saturation in Organic Molecular Semiconductors[END_REF][START_REF] Ortmann | Theory of charge transport in organic crystals: Beyond Holstein's small-polaron model[END_REF][START_REF] Ciuchi | Molecular Fingerprints in the Electronic Properties of Crystalline Organic Semiconductors: From Experiment to Theory[END_REF]. In the following, we study the electron-phonon coupling in two dierent systems: the molecular Buckminster fullerene C 60 and the most popular semi-metal graphene.

Ecient ab initio calculation of electron-phonon coupling matrix elements

Electron-phonon coupling (EPC), i.e. the interplay between electrons and vibrational eigenmodes, takes an important place in diverse elds of research. By way of example, it is believed to play a prominent role for the transport properties of organic semiconductors [START_REF] Gosar | Linear-Response Theory of the Electron Mobility in Molecular Crystals[END_REF][START_REF] Fratini | Bandlike Motion and Mobility Saturation in Organic Molecular Semiconductors[END_REF][START_REF] Ortmann | Theory of charge transport in organic crystals: Beyond Holstein's small-polaron model[END_REF][START_REF] Ciuchi | Molecular Fingerprints in the Electronic Properties of Crystalline Organic Semiconductors: From Experiment to Theory[END_REF], for the exciton dissociation at the donor-acceptor interface in organic photovoltaics [START_REF] Tamura | Phonon-Driven Ultrafast Exciton Dissociation at Donor-Acceptor Polymer Heterojunctions[END_REF], for the life-time of hot electrons in semiconductors [START_REF] Lautenschlager | Phonon-induced lifetime broadenings of electronic states and critical points in Si and Ge[END_REF][START_REF] Gopalan | Temperature dependence of the shifts and broadenings of the critical points in GaAs[END_REF] and it is also at the heart of the BCS phonon-mediated theory of superconductivity [START_REF] Hebard | Superconductivity at 18K in potassium-doped C60[END_REF][START_REF] Gunnarsson | Superconductivity in fullerides[END_REF][START_REF] Ganin | Bulk superconductivity at 38K in a molecular system[END_REF][START_REF] Mitsuhashi | Superconductivity in alkali-metal-doped picene[END_REF]. Concerning organic systems, we recently demonstrated that the inclusion of electron-phonon coupling eects is crucial to obtain accurate band structures in the case of crystalline pentacene [START_REF] Ciuchi | Molecular Fingerprints in the Electronic Properties of Crystalline Organic Semiconductors: From Experiment to Theory[END_REF][START_REF] Faber | Electrons go green: Exploring organic semiconductors[END_REF]. Organic semiconductor crystals are, contrary to standard inorganic semiconductors with strongly covalently bound atoms, composed of molecules which are weakly connected through van der Waals interactions. As such, electronic bands dispersion and electron-phonon coupling strengths may be of the same magnitude. In the case of crystalline pentacene, which is a typical study case for organic semiconductors, accurate GW band structure calculations [START_REF] Tiago | Ab initio[END_REF] revealed important discrepancies with experimental ARPES results. Since the GW approach is among the best available ab initio band structure theories, the errors could not be explained on the electronic level. Instead, we demonstrated that it is indispensable to include the interaction of the electrons with the internal vibrations of the molecules. In a [247].

non-perturbative dynamical mean-eld theory (DMFT) approach (S. Ciuchi, S. Fratini), the intramolecular electron-phonon coupling strength (Holstein approximation) calculated for a single pentacene molecule at the GW level has been combined with an ab initio band structure of the pentacene crystal [START_REF] Yoshida | Crystallographic and electronic structures of three dierent polymorphs of pentacene[END_REF] in order to describe simultaneously the molecular (vibrational)

and crystalline nature (band structure) of these materials. The obtained band structure is in extremely good agreement with the experimental measurements, which is shown in Fig.

( 3.19) for the two highest occupied bands, important for hole carrier transport. Our combined experimental and theoretical study on pentacene has demonstrated unambiguously that ngerprints of the molecular constituents are clearly visible in the electronic energy spectrum.

Since our contribution was limited to providing on-site electron-phonon coupling matrix elements within the many-body GW formalism, a subject that we describe here below in detail for the Buckminster fullerene case, we do not discuss further this specic example, but directly start analyzing how accurately EPC energies can be calculated.

Concerning ab initio calculations of EPC matrix elements, up to now, mainly DF T -K S and its perturbative linear response extension (DF P T ) [START_REF] Gonze | Dynamical matrices, Born eective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory[END_REF][START_REF] Baroni | Phonons and related crystal properties from density-functional perturbation theory[END_REF] have been applied, providing remarkable information at the microscopic level. However, several recent studies questioned the accuracy of the EPC matrix elements calculated within the standard DF T approaches using (semi)local exchange-correlation functionals, such as LDA. The average electron-phonon coupling of specic phonon modes in graphene [START_REF] Lazzeri | Impact of the electron-electron correlation on phonon dispersion: Failure of LDA and GGA DFT functionals in graphene and graphite[END_REF], the value of the electron-phonon average potential in the electron-doped Buckminster fullerene C 60 [12, 260], superconducting bismuthates and transition-metal chloronitrides [START_REF] Yin | Correlation-Enhanced Electron-Phonon Coupling: Applications of GW and Screened Hybrid Functional to Bismuthates, Chloronitrides, and Other High-T c Superconductors[END_REF], or the renormalization of the photoemission band structure of pentacene [START_REF] Ciuchi | Molecular Fingerprints in the Electronic Properties of Crystalline Organic Semiconductors: From Experiment to Theory[END_REF] and diamond crystals [START_REF] Antonius | Many-Body Eects on the Zero-Point Renormalization of the Band Structure[END_REF], were all shown to be strongly aected by a signicant underestimation of the strength of the EPC matrix elements as calculated within DF T -K S and (semi)local exchange-correlation functionals. In the following we comment in detail on three of the mentioned cases, namely C 60 , graphene and diamond.

The magnitude of the EPC in C 60 has been subject to numerous theoretical and experi- mental studies since the early 90ies, oering a variety of reference data. Such a wealth of data can be explained by the observed superconducting transition of alkali-doped (n-doped) fullerides, which has been subject of intense research [START_REF] Gunnarsson | Superconductivity in fullerides[END_REF]. On the ab initio level, (semi)local

DF T -LDA [START_REF] Faber | Electron-phonon coupling in the C 60 fullerene within the many-body GW approach[END_REF] and DF T -PBE [START_REF] Laamme-Janssen | Electron-phonon coupling in C 60 using hybrid functionals[END_REF] meV [START_REF] Laamme-Janssen | Electron-phonon coupling in C 60 using hybrid functionals[END_REF], resulting in an excellent agreement with the lower and most recent experimental limit of 107 meV [START_REF] Hands | Vibronic interactions in the visible and near-infrared spectra of C60 anions[END_REF][START_REF] Iwahara | Vibronic coupling in C60 anion revisited: Derivations from photoelectron spectra and DFT calculations[END_REF]. This demonstrates the sensitivity of the EPC strength on the used amount of exact exchange. Certainly, it would be possible to build a hybrid functional, which yields a perfect match with experiment. However, without comparison to experimental reference data, an estimation of the appropriate amount of exact exchange is dicult and the predictive power of this method thus limited. Moreover, the needed amount of exact exchange may vary from one system to another. Such diculties do not arise within the GW formalism, where the screening is, as already discussed in preceding sections, intrinsically adjusted by the system itself and not articially xed at the beginning. That gives this method a large exibility and systems of dierent kinds should be accessible on the same footing. Concerning the C 60 molecule, our recent evGW study, as described here below, shows an EPC strength of 101 meV, which is in excellent agreement with the two most recent experimental results and 43% larger than the reference DF T -LDA value. This clearly points out the necessity to go beyond the DF T -LDA approach for the calculation of the EPC in C 60 .

A similar result has been recently obtained in the case of graphene [START_REF] Lazzeri | Impact of the electron-electron correlation on phonon dispersion: Failure of LDA and GGA DFT functionals in graphene and graphite[END_REF]. Here, the EPC matrix elements for the coupling of the Γ-E 2g and K-A 1 phonon modes with the electronic states at the Fermi level were studied within a non-self-consistent G 0 W 0 approach starting from DF T -LDA. Within G 0 W 0 , the square of the corresponding EPC matrix elements (see denition below) were shown to increase by 41% and 114%, respectively, as compared to DF T -LDA. This is consistent with the case of C 60 . Very importantly, this study on graphene also demonstrated that DF T -B3LYP yields, contrary to the presented results on C 60 , signicantly too large coupling constants as compared to experiment. This certainly points out the diculty in obtaining hybrid functionals, which are accurate both for nite and extended systems.

Concerning the case of diamond, very recently, the band-gap renormalization by zero-point motion has been studied by means of DF T -LDA, G 0 W 0 and GW [START_REF] Antonius | Many-Body Eects on the Zero-Point Renormalization of the Band Structure[END_REF]. In this context, the corresponding EPC matrix elements for band edges have been evaluated, since they can be directly related to the temperature dependent renormalization of the electronic bands. In Ref.

[264], it was already shown that standard DF T and DF P T methods strongly underestimate the zero-point renormalization of the direct gap of diamond compared to experiment. By increasing the zero-point renormalization by more than 40% compared to DF T -LDA, the recent G 0 W 0 /GW calculations from Ref. [START_REF] Antonius | Many-Body Eects on the Zero-Point Renormalization of the Band Structure[END_REF] remedied the discrepancies and restored a good agreement between theory and experiment.

To conclude, the hitherto results suggest that the EPC strength is signicantly aected by the GW correction in both nite and extended systems. It is therefore important to go beyond the DF T -K S approach. However, the computational costs of such techniques represent strong limitations to their application. Unfortunately, the kind of theories that are available within DF T , and in particular the ecient density functional perturbation theory, are not yet available within the framework of many-body perturbation theory. As detailed later in this section, for GW calculations, one has to pass by demanding frozenphonon techniques. These are only feasible for zone-center or zone-boundary phonon modes, since otherwise large unit cells are needed implying high computational costs. In this work, we explore less demanding many-body perturbation theory approaches to calculate the EPC.

Namely, we study the accuracy of the COHSEX approximation, both in its single-shot and its self-consistent version. Further, we also examine the accuracy of a constant screening approximation, namely we neglect the variations of the screened Coulomb potential W upon small changes of the atomic positions along the vibrational eigenmodes. These approximations are evaluated by means of studies on the molecule C 60 and the popular semi-metal graphene. Electronic properties As already mentioned before, electron-doped fullerenes, so-called fullerides, have attracted much attention, since they show phonon-mediated superconducting transitions [START_REF] Hebard | Superconductivity at 18K in potassium-doped C60[END_REF][START_REF] Gunnarsson | Superconductivity in fullerides[END_REF][START_REF] Ganin | Bulk superconductivity at 38K in a molecular system[END_REF]. Fullerenes are cage-like molecules, where one of the most prominent examples of the fullerene family is the Buckminster fullerene C 60 (see Fig. 3.20). Under doping, e.g. with alkali atoms such as sodium or potassium, the electronic structure of the neutral C 60 fullerene is not fundamentally changed. The s-type levels of the alkali atoms appear close to the second lowest-lying unoccupied C 60 level (denoted LUMO-1 in the follow- ing). Moreover, the C 60 LUMO becomes half-lled by the additional electrons of the dopant, shifting the Fermi level up from the gap center into the LUMO (see Fig. 3.21).

Following second order perturbation theory for electron-phonon scattering in extended solids, one obtains the eective phonon-mediated attractive potential V ep , i.e. a measure of the attractive interaction between electrons in a Cooper pair, via [START_REF] Antropov | Phonons, electron-phonon, and electron-plasmon coupling in C 60 compounds[END_REF]:

V ep = 1 M 1 [N (E F )] 2 i,j,k ν,q g ν,q ω 2 ν,q ψ i,k | ∂V ∂u ν,q |ψ j,k+q 2 δ (ε i,k ) δ (ε j,k+q -ε i,k -ω ν,q ) , (3.11)
where M is the atomic mass, N (E F ) the number of states at the Fermi level, ω ν,q the eigen-frequency of the phonon mode and g ν,q its degeneracy. Here, an electron in state ψ i with wave vector k is scattered into an unoccupied state ψ j with wave vector (k + q) through a phonon of frequency ω ν . The phonon gradient represents the variation of the self-consistent potential V felt by the electrons with respect to the distortion along the vibrational eigenmode u υ,q .

Energy conservation is ensured by the δ-functions. Since phonon energies are typically rather small, namely only of the order of several meV, only electrons near the Fermi level eectively scatter, where occupied and unoccupied states are close in energy. For molecules, the above equation valid for solids needs to be adapted to discrete energy levels and becomes in the so-called molecular limit [START_REF] Schlüter | Electron-phonon coupling and superconductivity in alkali-intercalated C 60 solid[END_REF][START_REF] Antropov | Phonons, electron-phonon, and electron-plasmon coupling in C 60 compounds[END_REF]:

V ep = 1 9M ν g ν ω 2 ν 3 i,j=1 ψ i | ∂V ∂u ν |ψ j 2 , (3.12)
where i and j run over states near the Fermi level, i.e. in the case of the electron-doped C 60 over the three degenerate LUMO levels.

Phonon properties Within the harmonic approximation, phonon frequencies ω ν and displacement patterns u Iα for an atom I along Cartesian components α can be obtained by solving the following secular equation:

J,β [D] αβ IJ -δ IJ δ αβ M I ω 2 u I,β = 0,
where [D] the so-called dynamical or inter-atomic force constant (IFC) matrix:

[D] IJ ≡ ∂ 2 E (R) ∂R α I ∂R β J .
Here, E (R) is the Born-Oppenheimer energy surface, i.e. the total energy of the electronic system depending parametrically on the ion core positions R. As detailed in Ref. [START_REF] Baroni | Phonons and related crystal properties from density-functional perturbation theory[END_REF],

within the framework of DF T -K S, density functional perturbation theory (DFPT) has been conceived to eciently access the second order derivative of the Born-Oppenheimer energy surface with respect to the atomic positions. Going into details of this formalism is clearly beyond the scope of this work and the reader is referred to Refs. [START_REF] Gonze | Dynamical matrices, Born eective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory[END_REF][START_REF] Baroni | Phonons and related crystal properties from density-functional perturbation theory[END_REF] for a comprehensive overview. The phonon eigenmodes and eigenfrequencies used in the presented study on the C 60 molecule are identical to those of Refs. [12, 260] and have been calculated at a DF T -B3LYP level with a 6-311G* basis. This was shown to yield excellent phonon eigenfrequencies as compared to Raman experiments [START_REF] Laamme-Janssen | Electron-phonon coupling in C 60 using hybrid functionals[END_REF].

Evaluation of the electron-phonon coupling We now make the connection between EPC matrix elements and GW calculations. It can be shown that within the present frozenphonon approach, the explicit deformation of the molecule diagonalizes the eigenstates with respect to the perturbation, leaving only the intraband transitions [START_REF] Faber | Electron-phonon coupling in the C 60 fullerene within the many-body GW approach[END_REF][START_REF] Laamme-Janssen | Electron-phonon coupling in C 60 using hybrid functionals[END_REF][START_REF] Antropov | Phonons, electron-phonon, and electron-plasmon coupling in C 60 compounds[END_REF]. Further, using the Hellman-Feynman theorem [START_REF] Hellmann | Einführung in die Quantenchemie[END_REF][START_REF] Feynman | Forces in Molecules[END_REF], the expectation value of (∂V /∂u ν ) on the |ψ i eigenstate can be expressed as the gradient (∂ε i /∂u ν ) of the corresponding energy level, namely:

V ep = 1 9M ν gν ω 2 ν 3 i=1 ψ i | ∂V ∂uν |ψ i 2 = 1 9M ν gν ω 2 ν 3 i=1 ∂ε i ∂uν 2 , (3.13)
where ε i are the LUMO energy eigenvalues as calculated within DF T -LDA, GW or COHSEX.

Clearly, assuming that the u ν vibrational modes are well described within DF T , the quality of the V ep is directly related to the that of the ε i eigenvalues.

For evaluating the matrix element of the electron-phonon coupling, the question arises, which phonon modes can couple to the LUMO states, i.e. for which modes the matrix element ψ i | ∂V ∂uν |ψ i is non-zero. Group theory analysis show that the Kronecker product t 1u ⊗ t 1u of the LUMO states character only projects on the non-degenerate A g modes and on the vefold degenerate H g modes (see Fig. 3.22). This signicantly reduces the number of matrix elements to be calculated. In total, ten modes can contribute to the coupling, two of A g and eight of H g symmetry. The electron-phonon coupling involving the A g modes does not lift the degeneracy, i.e. the three LUMO states change their energy by the same amount under distortion. On the contrary, concerning the coupling to the H g modes, the LUMO levels split A g [START_REF] Strinati | Application of the Green's functions method to the study of the optical properties of semiconductors[END_REF] H g [START_REF] Faber | First-principles GW calculations for DNA and RNA nucleobases[END_REF] 60

A g (2)
H g (7) Electronic properties For a subsequent discussion of the electron-phonon coupling, we rst want to briey comment on the electronic properties. Graphene consists of a honeycomb lattice of sp 2 -hybridized carbon atoms, i.e. it is characterized by a system of delocalized π- Electron-phonon coupling Graphene is a fundamentally important material, since the EPC strength for certain phonon modes can be directly obtained from experiment. This makes graphene a valuable reference for our ab initio EPC calculations. As elaborated in Ref. [START_REF] Piscanec | Kohn Anomalies and Electron-Phonon Interactions in Graphite[END_REF] and depicted in Fig. (3.25), the phonon dispersion of graphene possesses two discontinuities in the frequency derivative for the highest optical-phonon branch (HOB), namely at the zone center (q = Γ) and at the symmetry (q = K) point. In general, kinks in the phonon dispersion are called Kohn anomalies [START_REF] Kohn | Image of the Fermi Surface in the Vibration Spectrum of a Metal[END_REF]. They can be observed in metals, where the screening of the atomic vibrations by the conduction electrons can rapidly change for phonons related to certain q points. They are determined by the shape of the electronic Fermi surface and occur when the scattering electronic states |ψ i,k and |ψ j,k+q are both at the Fermi level.

H g α -0.1 ≥ α ≤ 0.1 α = 0.05 evGW 60 41 2 i=0 ∂ε lumo+i ∂u ν = 0, H g H g DF T GW DF T LDA 3 a 1 a 2 δ i i = 1, 2, 3 K K e -αr 2 l l = 3 α α min = 0.20 -2 α max = 3.2 -2 G 0 W 0 @LDA 7.
For graphene, this consequently implies kinks at (q = Γ) and (q = K). We thus consider processes, where electrons ψ i,K at (k = K) with Fermi energy are either vertically scattered in unoccupied states ψ i,K+q through phonons with wave vector q = Γ = 0, or horizontally in unoccupied states of the neighboring k = K point through phonons with wave vector q = K.

In Ref. [START_REF] Piscanec | Kohn Anomalies and Electron-Phonon Interactions in Graphite[END_REF], it has been demonstrated that the slope of the kink is proportional to the square of the EPC matrix element of the respective phonon mode. The phonon modes showing a Kohn anomaly are the so-called E 2g and A 1 phonon modes, corresponding to the HOB at Γ and at K, respectively. As a consequence, one can directly deduce the EPC strength of these modes from experimental phonon dispersions [START_REF] Lazzeri | Impact of the electron-electron correlation on phonon dispersion: Failure of LDA and GGA DFT functionals in graphene and graphite[END_REF][START_REF] Piscanec | Kohn Anomalies and Electron-Phonon Interactions in Graphite[END_REF]. The contribution to the EPC of the other phonon modes at Γ and K has been shown to be negligible, consistent with their well-behaved continuous dispersion [START_REF] Piscanec | Kohn Anomalies and Electron-Phonon Interactions in Graphite[END_REF].

In the following, it will be discussed how the EPC matrix elements D are accessed within a frozen-phonon approach. The latter originate from the following expression [START_REF] Lazzeri | Impact of the electron-electron correlation on phonon dispersion: Failure of LDA and GGA DFT functionals in graphene and graphite[END_REF]:

D = ψ i,k | ∂V ∂u ν,q |ψ j,k+q ,
where ∂V /∂u is the derivation of the eective potential with respect to the phonon modes.

In order to determine the EPC associated with the E 2g phonon, the basis atoms are stepwise displaced corresponding to the phonon pattern in Fig. (3.26a). Following Ref. [START_REF] Lazzeri | Impact of the electron-electron correlation on phonon dispersion: Failure of LDA and GGA DFT functionals in graphene and graphite[END_REF], the square of the corresponding EPC matrix element becomes in this frozen-phonon scheme for graphene:

D 2 Γ F = lim d→0 1 16 ∆E Γ d 2 .
Here, each atom is displaced by d and . . . F represents an average over the Fermi surface. ∆E Γ is the splitting between the doubly degenerate π-states with Fermi energy at (k = K), namely the opening of the gap induced by the coupling of these π-states with the Γ-E 2g phonon mode. In a similar way, the square of the EPC matrix element associated with the

Γ E 2g K A 1 √ 3 × √ 3 K A 1 D 2 K F = lim d→0 1 8 ∆E K d 2 , A 1 ∆E K (k = K) π K A 1 60 DF T LDA G 0 W 0 G 0 W 0 DF T LDA DF T LDA GW |a| = 2.46 20
18 × 18 × 1 k DF T 36 × 36 × 1 k GW k q = K 60 45 G W W GW
COHSEX The already introduced static COHSEX approach represents a drastic approximation to the GW self-energy and was shown to yield too large gaps in the case of semiconductors [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF]. By way of example, in the present case of C 60 , the COHSEX gap is found to be 5.3 eV (B3LYP geometry). This can be compared to the ∼ 4.9 eV experimental gap. Even though yielding a too large gap, we note that it is much better than the starting 1.6 eV DF T -LDA Kohn-Sham value.

While it cannot be claimed that the static COHSEX approach is a good approximation to absolute quasiparticle energies, we emphasize that we are interested in quasiparticle energy dierences upon small (innitesimal) atomic lattice displacements. The main assumption on which we rely to calculate the electron-phonon coupling within the COHSEX approximation is that the variations of the dynamical contribution to the self-energy can be neglected. This can be certainly rationalized by emphasizing that dynamical interactions are driven by the plasmons dynamics, collective excitations much less sensitive to small atomic displacements than single-particle excitation energies and wave functions. In this work, we both test singleshot COHSEX and partially self-consistent COHSEX calculations with an update of the eigenvalues, in the following labeled COHSEX and evCOHSEX, respectively. The obtained data is compared to G 0 W 0 and evGW reference values.

Constant screening In addition to the static COHSEX approximation, we want to assess the quality of the so-called constant screening approximation. Namely, the screened Coulomb potential W is calculated once for the undistorted structure, while it is read in for the distorted frozen phonon congurations. Namely, we assume that (∂W/∂u ν ) is zero. Since the calculation of W denotes one of the most expensive parts, this dramatically reduces computational costs.

Many-body calculations on the C 60 molecule are carried out using the FIESTA package.

The constant screening approximation is tested both within a single-shot COHSEX and an evCOHSEX approach with self-consistency on the eigenvalues. These approaches are labeled COHSEX(W ) and evCOHSEX(W ), respectively, in the following. We restrict our considerations on the (ev)COHSEX level, since the constant screening approach within evGW is computationally not straightforward. This is due to the evaluation of the correlation part of the self-energy using contour deformation techniques with an explicit calculation of the residues. Further, since calculations are performed within a non-orthogonal Gaussian basis, special care must be taken implementing such a constant-screening approximation. As already discussed in preceding chapters, non-local operators such as the bare and screened Coulomb potential are expressed in terms of an atom-centered auxiliary basis {β} within the RI-SVS technique, following:

[W ] β,β = ´´drdr β(r) W (r, r ) β (r ) W (r, r ) = β,β β(r) S -1 [W ]S -1 β,β β (r ),
where S is the overlap matrix in the auxiliary basis. Using now the notation: W and β for the screened Coulomb potential and the auxiliary basis for the slightly distorted system, the assumption: W (r, r ) W (r, r ) leads straightforwardly to the condition:

[W ] β,β S ββ [W ] β,β S β β ,
where S ββ =< β|β > is an overlap matrix between the auxiliary bases for the perturbed and unperturbed systems, respectively.

Calculations on graphene are performed using the Yambo code [START_REF] Marini | Yambo: An ab initio tool for excited state calculations[END_REF] within the plasmon pole approximation. We scrutinize the constant-screening approximation within a single-shot COHSEX approach, as in the case of C 60 , but also at the G 0 W 0 level (labeled G 0 W 0 (W ) in the following). This is possible due to the use of a xed plasmon pole frequency, independent of structural deformations. Further, starting from a correct band gap at the DF T -LDA level, the eect of self-consistency is not as important as in the case of C 60 and was shown to have only minor eects.

In order to rationalize the constant screening approach, we briey discuss similarities to the BSE formalism. As discussed in detail in preceding chapters, the latter is based on two main assumptions, which have been shown to be remarkably accurate [START_REF] Albrecht | Ab Initio Calculation of Excitonic Eects in the Optical Spectra of Semiconductors[END_REF][START_REF] Benedict | Optical Absorption of Insulators and the Electron-Hole Interaction: An Ab Initio Calculation[END_REF][START_REF] Rohlng | Excitonic Eects and the Optical Absorption Spectrum of Hydrogenated Si Clusters[END_REF]. The rst one is the replacement of the dynamically screened Coulomb potential by its static analogue, which is similar to the static COHSEX approximation. The second one is the neglect of the variation (∂W/∂G), assumed to be negligible. In the present constant screening approach, we also neglect variations of W . However, the perturbation is not the single-particle Green's function G, but the vibrational distortion of the system in this case.

Results and discussion

The COHSEX approximation Our results are compiled in Table (3.6) for C 60 and in Table (3.7) for graphene, respectively. For the sake of illustration, the evolution of the lowest unoccupied 3-fold electronic energy level with respect to the phonon displacement along the strongest coupling C 60 eigenmode H g [START_REF] Faber | First-principles GW calculations for DNA and RNA nucleobases[END_REF] is represented in Fig. (3.27).

For C 60 , the total evGW coupling potential is within 8% of that found by Ref. [START_REF] Faber | Electron-phonon coupling in the C 60 fullerene within the many-body GW approach[END_REF], as a result of the larger basis set we used. 27 eV value in Ref. [START_REF] Lazzeri | Impact of the electron-electron correlation on phonon dispersion: Failure of LDA and GGA DFT functionals in graphene and graphite[END_REF] to 7 eV and thus closer to the π-plasmon resonance in graphene.

Such dierences are negligible with respect to the more than 100% increase as compared to the DF T -LDA value. Further, we emphasize that we are interested in the dierences between the GW values and the presented COHSEX and constant screening approaches, where all running parameters are the same.

In the case of C 60 , comparing evGW and evCOHSEX calculations, a global raise of about 7% can be noted, which has to be compared to the 30% increase from DF T -LDA to evGW .

Besides the H g (4) and A g (1) modes, showing very small couplings, the evCOHSEX approx- of C 60 . This situation can be tentatively ascribed to the very peculiar nature of the Fermi surface and the presence of low-energy plasmons [START_REF] Trevisanutto | Ab Initio GW Many-Body Eects in Graphene[END_REF], which decreases the validity of the static approximation.

LDA GW H g (1) H g (2) H g (3) H g (4) H g (5) H g (6) H g (7) H g (8) A g (1) A g (2) H g a b A g a V ep ν 60 V ep DF T LDA H g A g a b GW DF T LDA GW W DF T LDA 60 Γ E 2g 18 G 0 W 0 K 1 29 31 54 DF T LDA G 0 W 0
The constant screening approach The results of the constant screening approximation are compiled in Table (3.6) for C 60 and in Table (3.7) for graphene. For C 60 , where we test the constant screening approximation at the COHSEX level, an excellent agreement is obtained within evCOHSEX(W ) compared to the corresponding evCOHSEX calculations. In total, evCOHSEX(W ) agrees within 1.5% compared to evCOHSEX.

In the case of graphene, we remind that diculties have been encountered at the COH-SEX level. Concerning the constant screening approach, the coupling with the zone-center optical mode Γ-E 2g provides the best results, with a 5% error when comparing G 0 W 0 (W ) to G 0 W 0 and COHSEX(W ) to COHSEX. For the coupling with the zone-corner K-A 1 mode, discrepancies are not larger than 6% and 7%, applying the constant screening approximation to G 0 W 0 and COHSEX, respectively. This is much smaller than the error induced by the standard DF T -LDA approach.

Clearly, among the two approximations tested above, the constant screening approach stands as a much better approximation than the static COHSEX one in the present case of electron-phonon coupling, in particular in the somehow pathological case of graphene.

Mode Coming back to the analogue approximations implemented in standard GW/BSE calculations, namely a static screened Coulomb potential W and (∂W/∂G = 0), it is commonly assumed that the GW/BSE approach is much more resistant to approximations on the screened Coulomb potential W as compared to the GW approach for (charged) excitations. This is due to cancellations of errors between the electron-electron and electron-hole interactions. Namely, any error introduced in W is expected to aect excitonic interactions and quasiparticle gaps in opposite ways. Clearly, the present GW study of the variations of a given quasiparticle energy with respect to ionic positions cannot benet from such cancellation of errors. Nevertheless, the constant screening approximation turns out to be a reliable approach, which allows to tremendously save on computational costs.

LDA G 0 W 0 G 0 W 0 (W ) COHSEX COHSEX(W ) Graphene Γ-E 2g 45
An important consequence of the present ndings is that once the screened Coulomb potential W (r, r ; ω) is built for the equilibrium geometry, the calculation of the variations of the quasiparticle energies with respect to the perturbation (λ) only requires the evaluation of the variations of the Green's function G with respect to the perturbation. This can be straightforwardly performed within standard DFPT techniques, at least in the case of non-self-consistent G 0 W 0 calculations, where the Green's function is directly constructed from input DFT eigenstates. This may invite, for molecular systems such as C 60 , to rely on G 0 W 0 calculations starting from DFT eigenstates obtained with hybrid functionals, which have been shown to be a better starting point for organic systems [START_REF] Bruneval | Benchmarking the Starting Points of the GW Approximation for Molecules[END_REF][START_REF] Marom | Benchmark of GW methods for azabenzenes[END_REF], instead of carrying out self-consistent GW calculations.

Conclusion

Electron-phonon coupling plays a prominent role in organic semiconductors. As shown here above for several important systems, its accurate description from rst principles necessitates to go beyond DF T /DF P T with standard (semi)local functionals. GW clearly demonstrated to be a promising alternative, however, up to now, this approach is limited to small systems and zone-center phonons due to tremendous computational costs within the frozen-phonon technique. This points out the need for methods which are accurate and computationally reasonable at the same time. Inspired from the GW/BSE formalism, we tested two approaches, namely the static COHSEX approximation and the constant screening approach. Since the calculation of the W denotes the most expensive part of a GW calculation, this greatly reduces the computational eort.

Concerning the accuracy of the tested approaches, our ndings suggest that the COHSEX approximation may not be reliable, in particular in the case of graphene. However, the constant screening hypothesis, namely assuming that W remains to rst order constant with respect to small ionic displacements, seems to be valid. In the case of C 60 , it introduces a small error of about 2%. Preliminary results on the extended semiconductor diamond conrm this excellent agreement. Concerning the paradigmatic case of graphene, where the phonon perturbation dramatically aects the Dirac cone and the semi-metallic character, a maximum discrepancy of 7% is found. The present results oer promising perspectives to carry on such many-body evaluations of the electron-phonon coupling gradients with much reduced computer costs on realistic systems. However, further studies are required both on a larger set of systems and physical observables in order to better assess the validity of the presented approach.

| Conclusion & Perspectives

Organic photovoltaics oers unique perspectives for a sustainable electricity generation from sun light. The assortment of appropriate organic semiconductors is vast and can be thereto largely increased by molecular engineering. However, the design of ecient organic solar cells is a highly non-trivial optimization problem, where material combinations with well matching electronic levels and optical gaps have to be found.

The present project aimed to assess the ability of the many-body perturbation theory GW/BSE formalism to describe electronic and optical properties of organic systems, in order to contribute to the understanding of fundamental microscopic processes and to actively steer experimental materials research. In this context, we studied the electronic and optical properties for molecules in the gas phase in order to assess the quality of the GW/BSE formalism for nite systems. On the electronic structure level, we chose a model dipeptide, a paradigmatic molecule which raised our interest due to signicant discrepancies in the optical spectra within reference TDDFT and quantum chemistry methods [START_REF] Faber | Many-body Green's function GW and Bethe-Salpeter study of the optical excitations in a paradigmatic model dipeptide[END_REF][START_REF] Serrano-Andres | Theoretical Study of the Electronic Spectroscopy of Peptides. III. Charge-Transfer Transitions in Polypeptides[END_REF]. Already its electronic structure is an interesting study case, where large discrepancies between the starting DF T -LDA and the GW electronic structure are observed. The many-body GW correction not only opens the gap and shifts the occupied levels to lower energies, but important level crossings and changes in the level spacings can be noted. This is similar to the previously studied case of the DNA/RNA nucleobases [START_REF] Faber | First-principles GW calculations for DNA and RNA nucleobases[END_REF] and demonstrates that a simple GW Scissor approach with an explicit correction of the HOMO and the LUMO only is not enough. Instead, the GW correction should be calculated self-consistently with an update of the eigenvalues (evGW approach) for several states around the gap. Since this approach quickly becomes expensive for large systems, we tested the quality of the inexpensive static COHSEX approximation. We showed that self-consistent COHSEX calculations, even though overestimating fundamental gaps, yield results in much better agreement with the nal evGW electronic structure than DF T -LDA and thus oer an ideal starting point for inexpensive non-self-consistent G 0 W 0 or GW Scissor operator calculations.

Concerning optical absorption properties, the model dipeptide is characterized by low-lying intramolecular charge-transfer excitations. As expected, TDDFT with (semi)local exchangecorrelation functionals fails to reproduce the latter, drastically underestimating the electronhole Coulomb interaction. TDDFT with range-separated hybrid functionals cures the problem and brings the CT excitations energies in good agreement with the higher-level correlated quantum chemistry reference. However, parameters are introduced which have to be adjusted for the respective problem. On the contrary, our combined GW/BSE approach comes in close agreement with the quantum chemistry data, while being parameter-free and systemindependent. As a second case study showing intramolecular charge-transfer excitations, we focused on the optical spectra of a family of coumarin dyes [START_REF] Faber | Many-body Green's function study of coumarins for dye-sensitized solar cells[END_REF]. These molecules recently attracted much attention, namely as all-organic, transition metal free absorbers in Grätzel cells with very promising power-conversion eciencies. Moreover, they are an impressive example of molecular design, where the gap and structural properties have been systematically optimized [START_REF] Hara | Molecular Design of Coumarin Dyes for Ecient Dye-Sensitized Solar Cells[END_REF][START_REF] Hara | Design of new coumarin dyes having thiophene moieties for highly ecient organic-dye-sensitized solar cells[END_REF]. Again, TDDFT with (semi)local exchange-correlation functionals has problems in reproducing the charge-transfer excitation energies from the quantum chemistry reference and range-separated hybrid functionals are needed. The tested GW/BSE approach, however, intrinsically adjusts the electron-hole Coulomb interaction and consequently gives charge-transfer excitation energies in good agreement with quantum chemistry calculations.

As a further important issue, we could also assess the quality of GW/BSE oscillator strengths, in the interesting case of molecular systems, where not only the onset of absorption, but also the oscillator strength has been chemically engineered in order to optimize the matching with the solar spectrum.

After having carefully tested the accuracy of the presented GW/BSE approach for the electronic and optical properties of the model dipeptide and the coumarin dyes, we also studied its quality concerning the electron-phonon coupling (EPC). The latter takes a prominent place in organic semiconductors and an accurate calculation from rst principles is indispensable for a realistic modeling. Even though DF T , and especially density functional perturbation theory (DF P T ), provides a most ecient way to access the electron-phonon coupling, recent studies showed a clear underestimation of the EPC of up to 50% as compared to experiment when using standard (semi-)local exchange-correlation functionals. The problem is cured on the many-body GW level, where EPC strengths come in close agreement with experiment [START_REF] Faber | Electron-phonon coupling in the C 60 fullerene within the many-body GW approach[END_REF][START_REF] Lazzeri | Impact of the electron-electron correlation on phonon dispersion: Failure of LDA and GGA DFT functionals in graphene and graphite[END_REF][START_REF] Antonius | Many-Body Eects on the Zero-Point Renormalization of the Band Structure[END_REF]. However, within GW , ecient techniques as in the case of DF P T are not available and an expensive frozen-phonon approach with step-wise atomic displacements along the phonon modes has to be carried out. In this work, we tested two alternative many-body approaches for the calculation of the EPC, namely the static COHSEX and the constant screening approximation, which are supposed to yield much less demanding frozen-phonon calculations. We studied their validity range by means of the fullerene C 60 and the popular two-dimensional semi-metal graphene. Concerning the static COHSEX approach, we demonstrated that it results in signicant discrepancies as compared to the GW reference, especially in the case of graphene. The constant screening approach, where we neglect the variations of the screened Coulomb potential with respect to small deformations around the equilibrium structure, gives, however, results in excellent agreement with the corresponding GW reference. Even though this approach has still to be validated for a larger variety of systems, the obtained results are promising and open the door to an inexpensive and reliable many-body treatment of the electron-phonon coupling.

Embedding techniques

δ (1, 2) = δ (r 1 -r 2 ) δ (t 1 -t 2 )
and integrals are carried out like

ˆd1 = ˆd3 r 1 ˆ+∞ -∞ dt 1 . Further, v (1, 2) = v (r 1 -r 2 ) δ (t 1 -t 2 ) = 1 |r 1 -r 2 | δ (t 1 -t 2 )
is the bare Coulomb potential and the operator h 0 represents the single-particle Hamiltonian,

h 0 = - 1 2 ∇ 2 + V ext (r) .
Functional derivatives For the following derivations, some mathematical background is needed, namely the theory of functional derivative. Usually, one deals with functions, which map one number x onto another number y: f (x) : x → y.

In many-body theory, functionals one often encounters functionals, i.e. functions whose arguments themselves are functions:

F [f (x)] : f (x) → y.
To dierentiate between a function and a functional, square brackets are used. One prominent example for a functional is the ground state energy of a many-body system depending on the ground state density which is in turn a function of r: E 0 [n 0 (r)]. Derivatives can be carried out following:

dF [f (x)] df (x) = ˆdx ∂F [f (x)] ∂f (

x) .

A short overview of very useful relations follows [START_REF] Strinati | Application of the Green's functions method to the study of the optical properties of semiconductors[END_REF]:

1.

∂F

∂F (2) = δ (1 -2) (1)
2. The product rule denotes:

∂ ∂Ψ (2) (F [Ψ (1)] G [Ψ (1)]) = ∂F [Ψ (1)] ∂Ψ (2) G [Ψ (1)] + ∂G [Ψ (1)] ∂Ψ (2) F [Ψ (1)] . (A.1) 3. ∂ ∂Ψ(2) F [Ψ(1)] G[Ψ(1)] = 1 G[Ψ(1)] 2 ∂F [Ψ(1)] ∂Ψ(2) G [Ψ (1)] -∂G[Ψ(1)] ∂Ψ(2) F [Ψ (1)] 4. For F [G [Ψ (1) ; 2]] a chain rule exists: ∂F ∂Ψ (1) = ˆd2 ∂F ∂G (2) ∂G (2) ∂Ψ (1)
. (A.

3)

The derivation starts with F (12) = ˜d3d4F (13) G (34) F (42) which is a reformulation of denition [START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF]. We proceed as follows: Field operators Field operators create ( Φ † (r, t)) or annihilate ( Φ(r , t )) a particle at (r, t)

∂F (12) ∂Ψ(3) = = ∂( ˜d4d5 F (14) G(45) F (52)) ∂Ψ(3) = ˜d4d5 ∂F (14) ∂Ψ(3) G (
and (r , t ), respectively [START_REF] Bruus | Many-Body Quantum Theory in Condensed Matter Physics -An Introduction[END_REF]. Their commutation relations are given by: Φ(r , t ), Φ † (r, t)

+ = δ(r -r )δ t -t Φ( †) (r, t), Φ( †) (r , t ) + = 0.
These relations hold for fermions, for bosons the eld operators commutate.

The Hamiltonian for a system of N interacting electrons,

Ĥ = N i=1 - 1 2 ∇ 2 i + V ext (r i ) + 1 2 i =j v (r i -r j ) ,
can be expressed in eld operator notation following the transformation rules for operators.

Sums over 1-particle operators Ô (r) are transformed by:

N i=1 Ô (r i ) = ˆd3 r Φ † (r, t) Ô (r) Φ(r, t).
Analogue, for 2-particle operators like he Coulomb potential, it holds:

N ij Ô (r i , r j ) = ˆd3 rd 3 r Φ † (r , t) Φ † (r, t) Ô r, r Φ(r, t) Φ(r , t).
For the Hamiltonian we thus obtain:

Ĥ = ˆd3 r Φ † (r) h 0 (r) Φ(r) + 1 2 ¨d3 rd 3 r Φ † (r, t) Φ † (r , t) v r -r Φ(r , t) Φ(r, t), with h 0 (r) = -1 2 ∇ 2 + V ext (r).
i ∂ Φ(r, t) ∂t = h 0 (r) Φ(r, t) + ˆd3 r v r -r Φ † (r , t) Φ(r , t) Φ(r, t). (A.4) i ∂G 1 (1,2) ∂Uext(3) = = Ψ N 0 |T ∂Sev ∂U ext (3) Φ(1) Φ † (2) |Ψ N 0 Ψ N 0 |T [Sev]|Ψ N 0 - Ψ N 0 |T [Sev Φ(1) Φ † (2)]|Ψ N 0 Ψ N 0 |T [Sev]|Ψ N 0 2 Ψ N 0 T [∂Sev] ∂Uext(3) Ψ N 0 = Ψ N 0 |T ∂Sev ∂U ext (3) Φ(1) Φ † (2) |Ψ N 0 Ψ N 0 |T [Sev]|Ψ N 0 - Ψ N 0 |T [Sev Φ(1) Φ † (2)]|Ψ N 0 Ψ N 0 |T [Sev]|Ψ N 0 Ψ N 0 | T [∂Sev ] ∂U ext (3) |Ψ N 0 Ψ N 0 |T [Sev]|Ψ N 0 = Ψ N 0 |T ∂Sev ∂U ext(3) Φ(1) Φ † (2) |Ψ N 0 Ψ N 0 |T [Sev]|Ψ N 0 -i G 1 (1, 2) Ψ N 0 | T [∂Sev ] ∂U ext(3) |Ψ N 0 Ψ N 0 |T [Sev]|Ψ N 0 .
The derivation of time-evolution operator with respect to the external eld obeys:

T [∂S ev ] ∂U ext (3) = 1 i T S ev Φ † (3) Φ(3) .
Consequently, we obtain:

i ∂G 1 (1,2) ∂Uext(3) = = 1 i Ψ N 0 |T [Sev Φ † (3) Φ(3) Φ(1) Φ † (2)]|Ψ N 0 Ψ N 0 |T [Sev]|Ψ N 0 -i G 1 (1, 2) 1 i Ψ N 0 |T [Sev Φ † (3) Φ(3)]|Ψ N 0 Ψ N 0 |T [Sev]|Ψ N 0 = -1 i Ψ N 0 |T [Sev Φ(1) Φ(3) Φ † (3) Φ † (2)]|Ψ N 0 Ψ N 0 |T [Sev]|Ψ N 0 + i G 1 (1, 2) 1 i Ψ N 0 |T [Sev Φ(3) Φ † (3)]|Ψ N 0 Ψ N 0 |T [Sev]|Ψ N 0 = -i G 2 (1, 3, 2, 3 + ) + i G 1 (1, 2) G 1 (3, 3 + ) → ∂G 1 (1,2) ∂Uext(3) = -G 2 (1, 3, 2, 3 + ) + G 1 (1, 2) G 1 (3, 3 + ) ,
where we introduced 3 + = (r 3 , t 3 + η) with η → 0 to ensure the right time order.

A.2. Lehmann representation of the one-particle Green's function

In the so-called Lehmann representation [START_REF] Bruus | Many-Body Quantum Theory in Condensed Matter Physics -An Introduction[END_REF], the eigenfunctions Ψ N n of the many-body Hamiltonian Ĥ, solutions to Ĥ Ψ N n = E N n Ψ N n , are used as a basis set, in which the Green's function is expressed. This set contains for example the ground state wave function Ψ N 0 .

Since the Hamiltonian is a Hermitian operator, its eigenvalues E N n are real. In eld operator notation, where Φ † (r, t) creates and Φ(r, t) annihilates a particle at (r, t), Ĥ is given by:

Ĥ = ˆd3 r Φ+ (r) h 0 (r) Φ(r) + 1 2 ¨d3 rd 3 r Φ † (r, t) Φ † (r , t) v r -r Φ(r , t) Φ(r, t),
where the one-particle potentials are collected in h 0 (r) = -1 2 ∇ 2 +V ext (r) and where v (r -r ) = 1 |r-r | is the Coulomb potential.

In the following, we derive an expression for the time-ordered single-particle Green's function in this basis. The single-electron Green's function at non-zero temperature is dened as the following thermal average:

G e (rt, r t ) = -i Φ(r, t) Φ † (r , t ) Θ (t -t ) = -i 1 Z n Ψ N n e -β Ĥ Φ(r, t) Φ † (r , t ) Ψ N n Θ (t -t ) ,
where Z is dened in the grand canonical ensemble as Z = n e βE N n , with β = 1/k B T . For the sake of clarity, we already used a shorthand notation, where we dened Ĥ ≡ Ĥ -µ N

e Ĥ Ψ N n = k H k k! Ψ N n = k E k n k! Ψ N n = e E N n Ψ N n ,
in order to obtain:

G e (rt, r t ) = -i 1 Z n Ψ N n e -βE N n Φ(r, t) Φ † (r , t ) Ψ N n Θ (t -t ) = -i 1 Z n e -βE N n Ψ N n Φ(r, t) Φ † (r , t ) Ψ N n Θ (t -t ) .
In the Heisenberg picture the time-dependence of an operator Ô is governed by the relation:

Ô(t) = e i Ĥt Ô(t = 0)e -i Ĥt ,
yielding for the single-electron Green's function:

G e (rt, r t ) = -i 1 Z n e -βE N n Ψ N n e i Ĥt Φ(r)e -i Ĥt e i Ĥt Φ † (r )e -i Ĥt Ψ N n Θ (t -t ) = -i 1 Z n e -βE N n Ψ N n e iE N n t Φ(r)e -i Ĥt e i Ĥt Φ † (r )e -iE N n t Ψ N n Θ (t -t ) .
Extracting the exponentials from the expectation value and inserting the completeness relation, m |Ψ m Ψ m | = 1, in between the eld operators results in:

G e (rt, r t ) = = -i 1 Z n,m e -βE N n e iE N n t e -iE N n t Ψ N n Φ(r)e -i Ĥt |Ψ m Ψ m | e i Ĥt Φ † (r ) Ψ N n Θ (t -t ) = -i 1 Z n,m e -βE N n e iE N n t e -iE N n t Ψ N n Φ(r) e -iEmt |Ψ m Ψ m | e iEmt Φ † (r ) Ψ N n Θ (t -t ) = -i 1 Z n,m e -βE N n e i(E N n -Em)t e i(Em-E N n )t Ψ N n Φ(r) |Ψ m Ψ m | Φ † (r ) Ψ N n Θ (t -t ) = -i 1 Z n,m e -βE N n e -i(Em-E N n )τ Ψ N n Φ(r) |Ψ m Ψ m | Φ † (r ) Ψ N n Θ (τ ) ,
where we introduced τ = t -t and where we assumed that {|Ψ m } are also eigenfunctions of the Hamiltonian with corresponding eigenvalues E m . From the last equality, we draw the conclusion that the {|Ψ m } have to be a basis of a (N + 1) Hilbert space. Bases of dierent Hilbert spaces, i.e. describing systems of varying particle number, form the Fock space and are orthogonal, Ψ N Ψ P =N = 0. This would cause for example Ψ m | Φ † (r ) Ψ N n = Ψ m | Ψ N +1 to be zero. That means, the index m denotes both the number of particles in the system: (N + 1) and the conguration, i.e. representing e.g. the ground state Ψ N +1 0 or the rst excited state of the N + 1 particle system. For the sake of clarity, we use the notation |Ψ m ≡ Ψ N +1 m here and in the following. Likewise, E N +1 m is the total energy of a N + 1 particle system in conguration m. Both Ψ N n and Ψ N +1 m can be solutions to the eigenvalue problem of H, since the latter is expressed in eld operator notation, where the number of particles is not xed. In exactly the same manner, the single-hole Green's function G h (r t , rt) can be rewritten, giving:

G h (r , r, τ ) = - i 1 Z n,l e -βE N n e -i(E N n -E N -1 l )τ Ψ N n Φ † (r ) |Ψ l Ψ l | Φ(r) Ψ N n Θ (-τ ) ,
where l stands for a state of a N -1 particle system.

Denitions In order to facilitate the notation, the following quantities are introduced:

the Lehmann amplitude of the (N + 1) system:

f m (r) = Ψ N +1 m Φ † (r) Ψ N n and f * m (r) = Ψ N n Φ(r) Ψ N +1 m the Lehmann amplitude of the (N -1) system: g l (r) = Ψ N -1 l Φ(r) Ψ N n and g * l (r) = Ψ N n Φ † (r) Ψ N -1
: ε l = E N 0 -E N -1 l
Using these denitions, the Green's functions G e and G h can be written in a more compact form:

G e (r, r , τ ) = - i 1 Z n,m e -βE N n e -i(E N +1 m -E N n )τ f * m (r)f m (r ) Θ (τ ) G h (r , r, τ ) = - i 1 Z n,l e -βE N n e -i(E N n -E N -1 l )τ g * l (r )g l (r) Θ (-τ ) .
The Green's function at zero temperature T Since we are interested in the low temperature regime, where k B T |E i -E 0 | holds, the thermal energy is not sucient to excite the initial system and consequently we assume the system to be in its ground state. Thus, the sum over excited states n of the N particle system becomes redundant and only the sums over m and l remain, because the latter describe excitations caused by photons or other higher energy sources. For instance, the prefactor Z becomes: Z = n e βE N n T 0

-→ e βE N 0 . This simplies the above equations to:

G e (r, r , τ ) = - i m e -i(E N +1 m -E N 0 )τ f * m (r)f m (r ) Θ (τ ) G h (r , r, τ ) = - i l e -i(E N 0 -E N -1 l )τ g * l (r )g l (r) Θ (-τ ) .
and

G e (r, r , τ ) = - i m e -iεmτ f * m (r)f m (r ) Θ (τ ) (A.8) G h (r , r, τ ) = - i l e -iε l τ g * l (r )g l (r) Θ (-τ ) , (A.9)
respectively.

Transformation into frequency space Having in mind photoemission experiments, where the energy is usually expressed in terms of ω, it is suitable to transform equation (A.8) and (A.9) into frequency space using the Fourier transformation:

G(r, r , ω) = ´∞ -∞ dτ e iωτ G(r, r , τ ) G(r, r , τ ) = 1 2π ´∞ -∞ dω e -iωτ G(r, r , ω).
For the single-electron Green's function this gives:

G e (r, r , ω) = -i ´∞ -∞ dτ e iωτ Θ (τ ) m e -iεmτ f * m (r)f m (r ) = -i m f * m (r)f m (r ) ´∞ 0 dτ e i(ω-εm)τ = -i m f * m (r)f m (r ) 1 i(ω-εm) e i(ω-εm)τ ∞ τ =0
.

At this point, a case-by-case analysis is necessary. For τ = 0, we obviously get:

G e (r, r , ω) = - 1 m f * m (r)f m (r ) 1 ω -ε m .
On the contrary, the problem does not converge for τ → ∞. Therefore, we have to introduce a factor e i(iδ)τ with δ → 0:

G(r, r , ω) = ˆ∞ -∞ dτ e i(ω+iδ)τ G(r, r , τ ).
Like this, we get:

G e (r, r , ω) = -i m f * m (r)f m (r ) ´∞ 0 dτ e i(ω-εm+iδ)τ = -i m f * m (r)f m (r ) ´∞ 0 dτ e i(ω-εm)τ e -δτ . ⇒ G e (r, r , ω) = 1 m f * m (r)f m (r ) 1 ω -ε m + iδ .
An analogue calculation for the single-hole Green's function results in:

⇒ G h (r, r , ω) = - 1 l g * l (r )g l (r) 1 ω -ε l -iδ .
Combining the last two equations yields the time-ordered one-particle Green's function in the Lehmann representation:

G T (r, r , ω) = 1 unocc m f * m (r)f m (r ) ω -ε m + iδ + 1 occ l g * l (r )g l (r) ω -ε l -iδ .
We have to bear in mind that m stands for excitations, where an electron is inserted, and l for excitations, where an electron is removed from the N -particle system. Analogue, the retarded and advanced Green's functions are:

G R (r, r , ω) = 1 unocc m f * m (r)f m (r ) ω -ε m + iδ + 1 occ l g * l (r )g l (r) ω -ε l + iδ G A (r, r , ω) = 1 unocc m f * m (r)f m (r ) ω -ε m -iδ + 1 occ l g * l (r )g l (r) ω -ε l -iδ ,
G(rt, r t ) = 1 i Ψ N 0 T Φ(r, t) Φ † (r , t ) Ψ N 0 = G e (rt, r t ) -G h (r t , rt), = 1 i Ψ N 0 Φ(r, t) Φ † (r , t ) Ψ N 0 θ (t -t ) -1 i Ψ N 0 Φ † (r , t ) Φ(r, t) Ψ N 0 θ (t -t)
and using the product rule, one obtains: where we used the fact that the derivative of a step function is a δ-function:

∂θ(t-t ) ∂t = -∂θ(t -t) ∂t = δ(t -t ).

Further, using that Φ(r) Φ † (r )

+ = δ r -r
for equal times t and t [START_REF] Bruus | Many-Body Quantum Theory in Condensed Matter Physics -An Introduction[END_REF], the rst two terms of the above equation simplify to:

δ t, t Ψ N 0 Φ(r, t) Φ † (r , t ) + Φ † (r , t ) Φ(r, t) Ψ N 0 = δ t, t δ rr .

Concerning the remaining two terms, they can be combined to:

Ψ N 0 T ∂ Φ(r, t) ∂t Φ † (r , t ) Ψ N 0 .
Further, inserting expression (A.4) for the equation of motion of the annihilation operator results in:

i ∂G(rt,r t ) ∂t = δ (r -r ) δ (t, t ) + Ψ N 0 T ∂ Φ(r,t) ∂t Φ † (r , t ) Ψ N 0 = δ (r -r ) δ (t, t ) +

1 i
Ψ N 0 T h 0 (r) Φ(r, t) + ´d3 r v (r -r ) Φ † (r , t) Φ(r , t) Φ(r, t) Φ † (r , t ) Ψ N 0 = δ (r -r ) δ (t, t ) +

1 i
Ψ N 0 T h 0 (r) Φ(r, t) Φ † (r , t ) Ψ N 0 + 1 i

Ψ N 0 T ´d3 r v (r -r ) Φ † (r , t) Φ(r , t) Φ(r, t) Φ † (r , t ) Ψ N 0 .

Since h 0 (r) and v (r -r ) are just functions and no operators in second quantization, they can be put in front of the corresponding matrix elements:

i ∂G(rt,r t ) ∂t = δ (r -r ) δ (t, t ) + h 0 (r)

1 i Ψ N 0 T Φ(r, t) Φ † (r , t ) Ψ N 0 + ´d3 r v (r -r ) 1 i Ψ N
0 T Φ † (r , t) Φ(r , t) Φ(r, t) Φ † (r , t ) Ψ N 0 = δ (r -r ) δ (t, t ) + h 0 (r) G(rt, r t )+ ´d3 r v (r -r ) 1 i Ψ N 0 T Φ † (r , t) Φ(r , t) Φ(r, t) Φ † (r , t ) Ψ N 0 .

The time-ordered product containing four eld operators governs the full electron-electron interaction. We can rewrite this term respecting that every exchange of eld operators causes a minus sign:

i ∂G(rt,r t ) ∂t = δ (r -r ) δ (t, t ) + h 0 (r) G(rt, r t ) -´d3 r v (r -r ) 1 i Ψ N 0 T Φ(r, t) Φ(r , t) Φ † (r , t) Φ † (r , t ) Ψ N 0 Σ (12) = -i ˇd3d4d5d6 v (r 1 -r 3 ) G (15) ∂G -1 (56) ∂Uext(3) G (64) G -1 (42) = -i ˜d3d5 v (r 1 -r 3 ) G (15) ∂G -1 (52) ∂Uext(3) = -i ˝d3d4d5 v (r 1 -r 3 ) G (15) ∂G -1 (52) ∂Vtot( 4) ∂Vtot( 4) ∂Uext [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF] ⇒ Σ (12) = i ˜d4d5 G (15) Γ (52; 4) W ( 14) , (A. [START_REF] Benanti | Organic solar cells: An overview focusing on active layer morphology[END_REF] where we introduced a total potential V tot = U ext + V ind , the vertex function Γ(52; 4) = -∂G -1 (52) ∂Vtot( 4) and the screened Coulomb potential W (14) = ´d3 v (r 1 -r 3 ) ∂Vtot (4) ∂Uext( 3) .

Incorporating the Hartree potential in the one-particle operator (H 0 = h 0 + V H ), we can write the equation of motion (A.11) as follows :

i ∂ ∂t 1 -H 0 (r 1 ) G( 11) -ˆd3 Σ (13) G 31 = δ 1, 1 .

(A.14)

For a non-interacting system (Σ = 0), the latter simplies to:

+ ∂ReΣ ii (ω) ∂ω = ω -ε H,i -ReΣ ii (ε qp,i ) -∂ReΣ ii (ω) ∂ω (ω -ε H,i -ReΣ ii (ε qp,i ))
= ω -ε H,i -ReΣ ii (ε qp,i ) -∂ReΣ ii (ω) ∂ω (ω -ε qp,i ) .

A.4.2. The quasiparticle equation

In order to facilitate the derivation, we generalize all quantities from real frequency space ω to the complex plane z ∈ C. Consequently, the biorthonormal representation of the singleparticle Green's function is represented by [START_REF] Layzer | Properties of the One-Particle Green's Function for Nonuniform Many-Fermion Systems[END_REF]:

G T (r, r , z) = λ ψ λ (r, z) ψ λ (r , z) z -E λ (z) . The fourth integrand g 4 , + lim ω→ω l k,l e iη ω φ * l (r )φ l (r)(E+ω-ε l -iη)

g 4 = k,l e iη ω φ * l (r ) φ l (r) E + ω -ε l -iη 2 ε k V k (r) V * k (r ) (ω + ε k -iδ) (ω -ε k + iδ) ,
E+ω-ε l -iη

2 ε k V k (r)V * k (r ) (ω+ε k -iδ)(ω-ε k +iδ) = k,l e iη ωk φ * l (r )φ l (r) E+ω k -ε l -iη 2ε k V k (r)V * k (r ) -2ε k +i2δ
+ k,l e iη ωl φ * l (r ) φ l (r)

Having evaluated the four residues, one nds for the self-energy:

Σ(r, r , E) =m,k e iη ωk φ * m (r)φm(r ) E+ω k -εm+iη

2ε k V k (r)V * k (r ) -2ε k +i2δ
l e iη ωl φ * l (r ) φ l (r) v (r, r )

k,l e iη ωk φ * l (r )φ l (r) E+ω k -ε l -iη

2ε k V k (r)V * k (r ) -2ε k +i2δ
k,l e iη ωl φ * l (r ) φ l (r)

2 ε k V k (r)V * k (r )
(ω l +ε k -iδ)(ω l -ε k +iδ) .

Reordering yields:

Σ(r, r , E) = -occ k,l e iη ωl φ * l (r ) φ l (r) v (r, r ) + However, they are slightly shifted into the complex plane by the innitesimal value ±η. The poles of the screened Coulomb potential are found at the neutral excitation energies ε k (orange squares), which are shifted by the innitesimal value ±δ in the complex plane. Only the poles of the Green's function for occupied states at ωl = ε l + iη and the poles of W at ωk = -ε k + iδ are enclosed by the contour, whereas the remaining poles are not concerned.

2 ε k V k (r)V * k (r ) (ω l +ε k -iδ)(ω l -ε k +iδ)
Working in frequency space, the following integral has to be solved: where η is an innitesimal parameter, the pole structure of Σ x can be evaluated. The latter has poles in the complex plane both at ωl = ε l + iη and ωm = ε m -iη, however, only poles ωl for occupied states are enclosed by the contour. Using the same contour deformation rules the theorems used in the following can be found in Appendix A.9.

Expanding I (ω) results in: That implies that for each of the four summands, a closed integral has to be solved:

I (ω) =
˛C dω I (ω) = 4 i=1
˛C dω I i (ω) .

The residue theorem states that a closed integral gives either zero, if no poles of the function are enclosed, or yields 2πi j Res (I, ωj ) in the case when poles ωj of order n are contained.

The closed contour C is chosen to consist of the entire real axis and a semi-circle in the upper half plane (see Fig. (A.2)) or a semi-circle in the lower half plane. Corresponding to Jordan's lemma, the integral along the semi-circle vanishes for R → ∞ for higher order poles n > 1, both for contours in the upper and in the lower half plane. Since the above summands I i only show higher order poles, one can freely choose either an integration in the upper or in lower half plane and one obtains:

˛C dω I 2,3 (ω) = ˆ∞ -∞ dω I 2,3 ω .

The choice of the contour depends on the position of the poles and one tries to avoid including the latter into the contour. Since I 1 and I 4 only have poles either in the upper or in the lower half plane, one can choose the path correspondingly, resulting in vanishing integrals. However, GW approximation for Σ. For the kernel we thus get:

K T DSHF (3, 5; 4, 6) = ∂Σ GW (3,4) ∂G 1 (5,6) -iδ (3, 4) δ (5, 6) v (3, 5) = ∂iG(3,4)W (3,4) ∂G 1 [START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF][START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF] -iδ (3, 4) δ (5, 6) v [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF][START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF] .

Applying the product rule (A.1), yields: (3, 5; 4, 6) = i ∂G 1 (3,4) ∂G 1 (5,6) W (3, 4) + iG 1 (3, 4) ∂W (3,4) ∂G 1 (5,6) -iδ = G 1 (1, 2 ) G 1 (2, 1 ) + +i ´d56 G 1 (1, 5) G 1 (6, 1 ) W (5, 6) L(6, 2; 5, 2 ) -i ´d46 G 1 (1, 4) G 1 (4, 1 ) v (4, 6) L(6, 2; 6, 2 ) = L QIP (121 2 ) + +i ´d56 L QIP (161 5) W (5, 6) L(6, 2; 5, 2 ) -i ´d56 L QIP (151 5) v (5, 6) L(6, 2; 6, 2 ), (A.27) it is clear that we deal in this case with a time-dependent screened Hartree-Fock approach.

K T DSHF
The rst summand accounts for the dynamical screened exchange, while the second term represents the Hartree contribution.

Fourier transformation to frequency space and the static approximation Since L is a four-point quantity, in principle it depends on four dierent time arguments. However, the Coulomb potential v is instantaneous, v (1, 2) = v (x 1 x 2 ) δ (t 1 -t 2 ) , and also the one-particle Green's functions in equation (A.27) only depend on time dierences. This suggests that the time arguments can be contracted to a single one in order to permit a straightforward Fourier transformation to frequency space. However, this is not possible without further ado and approximations are implied. First, one assumes translational time invariance and an isochronous propagation of the electron and the hole. 2 In addition, the dynamically screened interaction W is approximated within its static limit:

W (1, 2) ≈ W stat (x 1 x 2 ) δ (t 1 -t 2 ) .

2 For L = L (1234), i.e. L = L (t1 -t4; t2 -t3), we set t1 = t3; t2 = t4. where τ = t 1 -t 2 and τ = t 4 -t 1 . For the sake of clarity we used a simplied notation, where we dropped the space coordinates. We consequently simplied the time-dependent equation for L so far that we are now able to perform a straightforward Fourier transformation using the convolution theorem:

L (ω) = L QIP (ω) + iL QIP (ω) L (ω) W stat (x 3 x 4 ) -iL QIP (ω) L (ω) v (x 3 x 4 ) .

We now reintroduce the space coordinates, yielding:

L x 1 x 2 ; x 1 x 2 ; ω = L QIP x 1 x 2 ; x 1 x 2 ; ω +i ´dx 3 x 4 L QIP x 1 x 4 ; x 1 x 3 ; ω W stat (x 3 x 4 ) L x 4 x 2 ; x 3 x 2 ; ω -i ´dx 3 x 4 L QIP x 1 x 3 ; x 1 x 3 ; ω v (x 3 x 4 ) L x 4 x 2 ; x 4 x 2 ; ω = L QIP x 1 x 2 ; x 1 x 2 ; ω + ´dx 3 x 4 x 5 x 6 L QIP x 1 x 4 ; x 1 x 3 ; ω L x 6 x 2 ; x 5 x 2 ; ω × × [iδ (x 5 , x 3 ) δ (x 6 , x 4 ) W stat (x 3 x 4 ) -iδ (x 3 , x 4 ) δ (x 5 , x 6 ) v (x 3 x 5 )] ,

unoccupied) are not important. This can be depicted in a (n 1 n 2 , n 3 n 4 ) matrix: where zeros on the diagonal appear when f n 2 = f n 1 . From now on, we only work in the physical meaningful (ml, lm) subspace, where [L QIP ] has no non-zero diagonal elements and is thus invertible. Moreover, we introduce an occupation matrix [F ],

F n 1 n 2 n 3 n 4 = (f n 2 -f n 1 ) δ (n 1 , n 3 ) δ (n 2 , n 4 ) , which also has only non-zero elements in the chosen subspace. It follows: Many mathematical simplications follow from the fact that -i LQIP is a diagonal matrix.

For example, its inverse is readily calculated to:

-i LQIP (ω)

-1 = ∆ε n 2 n 1 -ω 0 0 ∆ε n 2 n 1 -ω .
It is convenient to single out the frequency dependence:

-i LQIP (ω)

-1 = ∆ε n 2 n 1 0 0 -∆ε n 2 n 1 -ω 1 0 0 1 ,
enabling us to dene a frequency-independent two-particle eective Hamiltonian H 2p :

H 2p = ∆ε n 2 n 1 0 0 ∆ε n 2 n 1 -[F ] [iK] = ∆ε n 2 n 1 0 0 ∆ε n 2 n 1 + i K ml,m l K ml,l m
-K lm,m l -K lm,l m .

In general, the two-particle eective Hamiltonian is non-hermitian and it denotes:

H 2p = H res K coupl -K coupl * -(H res ) * .
H res is called resonant part, i.e. transitions between (occupied → unoccupied) states are treated. Contrary, -(H res ) * is an anti-resonant part, where transitions between (unoccupied → occupied) states and thus negative frequency transitions are considered. The K coupl and -K coupl * blocks couple the resonant to the anti-resonant part and include both (occupied → unoccupied) and (unoccupied → occupied) transitions. The matrix element H res ml,m l = ∆ε lm δ mm δ ll + iK ml,m l denotes in detail: H res ml,m l = ∆ε lm δ mm δ ll +i ´dx 1 x 1 x 2 x 2 φ m (x 1 ) φ * l x 1 φ * m (x 2 ) φ l x 2 K x 1 x 2 ; x 1 x 2 = ∆ε lm δ mm δ ll +i ´dx 1 x 1 x 2 x 2 φ m (x 1 ) φ * l x 1 φ * m (x 2 ) φ l x 2 × iδ (x 1 , x 2 ) δ x 1 , x 2 W stat x 1 x 1 -iδ x 1 , x 1 δ x 2 , x 2 v (x 1 x 2 ) = ∆ε lm δ mm δ ll + ´dx 1 x 1 x 2 x 2 φ m (x 1 ) φ * l x 1 φ * m (x 2 ) φ l x 2 δ x 1 , x 1 δ x 2 , x 2 v (x 1 x 2 ) -´dx 1 x 1 x 2 x 2 φ m (x 1 ) φ * l x 1 φ * m (x 2 ) φ l x 2 δ (x 1 , x 2 ) δ x 1 , x 2 W stat x 1 x 1 = H diag ml,ml + H exch ml,m l + H scr ml,m l , where we split the resonant part into a diagonal, an electron-hole exchange and a screened electron-hole interaction part following:

H diag ml,ml = ∆ε lm δ mm δ ll , H exch ml,m l = ´dx 1 x 2 φ m (x 1 ) φ * l (x 1 ) v (x 2 x 1 ) φ * m (x 2 ) φ l (x 2 ) , H scr ml,m l = -´dx 1 x 2 φ m (x 1 ) φ * l (x 2 ) W stat (x 1 x 2 ) φ * m (x 1 ) φ l (x 2 ) .

A.9. Contour deformation techniques

In the following, a short overview of contour deformation techniques is provided. For a detailed derivation, the reader is referred to Ref. [START_REF] Sormann | Elektronentheorie des Festkörpers[END_REF]. Assuming the following common type of an closed integral:

˛C dz I (z) = ˛C dz f (z) e itz (z -z i ) n , (A. [START_REF] Hara | Oligothiophene-Containing Coumarin Dyes for Ecient Dye-Sensitized Solar Cells[END_REF] where n is the order of the pole and z = (x, y) ∈ C. The closed contour C can be freely chosen.

The residue theorem According to the residue theorem, a closed path integral over a function f (z) yields:

˛dz f (z) =    0,
if no poles are enclosed 2πi i Res (f, z i ) , if poles at z i are enclosed.

Res (f, z i ) denotes the so-called residue of the function f at the enclosed pole z i and it is the coecient of the (z -z i ) -1 summand in a Laurent expansion. It can be calculated following:

Res (f, z i ) = 1 (n -1)! d n-1 dz n-1 [(z -z i ) n f (z)] z=z i . Jordan's lemma Analyzing the integral over the semi-circle for functions of the type A.30, one nds for the upper half plane with z = Re iϕ = R cos ϕ + iR sin ϕ and dz = iRe iϕ dϕ: ˆπ 0 dϕ iRe iϕ I Re iϕ = ˆπ 0 dϕ iRe iϕ f Re iϕ (Re iϕ -z i ) n e itR cos ϕ e -tR sin ϕ and for the limit, respectively: lim R→∞ ˆπ 0 dϕ . . . e -tR sin ρ R n-1 .

The integral in the lower half plane from Θ = π . . . 2π can be expressed in terms of ϕ = 0 . . . π, Jordan's lemma now states that for poles of higher order (n > 1), the integral vanishes for R → ∞. For poles of the order n = 1, a case-by-case analysis is necessary. For t > 0, the integral only vanishes for contours in the upper half plane, however, for contours in the lower half plane it diverges. Contrary, for t < 0, the integral vanishes for contours in the lower half plane and diverges for contours in the upper half plane. It is therefore crucial, to wisely choose the contour for a given problem.

Example: Step function The above theorems are applied to the step function Θ. In its integral form, the latter reads:

Θ (t) = - 1 2πi ˆ∞ -∞ dω e -iω t ω + iη .
The complex integral of interest is thus:

˛C dω e -iωt ω + iη ,

where ω = ω + iω and η → 0. One observes a pole of rst order at ω0 = -iη, i.e. in the lower half complex plane. Consequently, Jordan's Lemma has to be taken into account and one has to do a case-by-case study. For t > 0, the contour is chosen to consist of the entire

R → ∞

˛C dω e -iωt ω + iη = ˆ∞ -∞ dω e -iω t ω + iη = -2πiRes (ω 0 ) .

Res (ω 0 ) = lim . The same is valid for two Gaussian functions with arbitrary angular momentum, but the multiplication constant becomes more sophisticated. Concerning products of several Gaussian functions, it is evident that the solution is again a Gaussian function, since the product can be evaluated two after two Gaussian functions resulting in a Gaussian function centered at the common balance point.

Gaussian integrals Integrals over Gaussian functions often reduce to simple analytic expressions and can be found in mathematical tables. They are of great use, since they can be solved straightforwardly. Some of these helpful relations, which are also used for derivations in this section, are the following:

´∞ -∞ dx e -x 2 = √ π, ´∞ -∞ dx e -c 1 x 2 = π c 1 , ´∞ -∞ dx e -c 1 (x+c 2 ) 2 = π c 1 .
Normalization Since it is convenient to work in a normalized basis, one imposes to the so-called self-overlap SO: SO = ˆd3 r α * i (r) α i (r) = 1

and introduces a normalization constant N for a Cartesian Gaussian function centered at the origin:

α i (r) = N r h x r i y r j z e -γr 2 .

Thus, one obtains: SO = N 2 ˆd3 r r 2h x r 2i y r 2j z e -2γr 2 = 1.

This integral can be easily solved by separation and integration by parts: SO = N 2 ˆdr x r 2h x e -2γr 2 x ˆdr y r 2i y e -2γr 2 y ˆdr z r 2j z e -2γr 2 z = 1.

Consequently, one nds for the normalization constant of a primitive Gaussian function: ˆd3 q e iq(r-r ) q2 . This is one of the key steps in treating integrals containing the Coulomb potential, allowing to separate the integral:

N = 2 π
V s,s C = 4π (2π) 3 ˆˆˆd 3 q d 3 r d 3 r e iq(r-r ) q 2 e -κ|r-A| 2 e -λ|r -B| 2 V s,s C = 4π (2π) 3 ˆd3 q 1 q 2 ˆd3 r e iqr e -κ|r-A| 2 ˆd3 r e -iqr e -λ|r -B| 2 .

The integrations over r and r can be carried out independently from each other applying the same solution scheme. One starts by introducing the vector µ = r -A with e iqA e -q 2 4κ .

We take advantage of the similarity of I r and I r = ´d3 r e -iqr e -λ|r -B| 2 and proceed in the same way by paying attention to the dierent signs. As a result, one obtains:

I r = π λ V s,s C = 4π (2π) 3 π κ 3 2 π λ 3 2 ´d3 q 1
q 2 e iqA e -q 2 4κ e -iqB e -q 2 4λ = 4π (2π) 3 π κ q 2 e -γq 2 +iqR , where we dened R = A -B and γ = 1 4κ + 1 4λ . To solve the remaining integral over d 3 q, it is rst dierentiated and then reintegrated with respect to a variable τ . This allows to simplify the expression and to separate it: I q (τ ) = ˆd3 q 1 q 2 e -τ q 2 +iqR .

Here, τ corresponds to γ and we just renamed it in order to avoid confusion along the following procedure. Dierentiation of I q with respect to τ yields: dIq dτ = -´d3 q e -τ q 2 +iqR = -I qx q I qy q I qz q , I qx q = ´dq x e -τ q 2

x +iqxRx = e -R 2

x 4τ ´dq x e -τ (qx-iRx 2τ )

2 = e -R 2

x 4τ π τ .

Proceeding in the same way for I qy q and I qz q and combining all results, gives:

dI q dτ = -e -R 2 4τ π τ 3 2 .
Integrating over this quantity holds an expression for I q (τ ), since choosing wisely the integration limits, we get: ∞ γ dI q (τ ) dτ dτ = I q (∞) -I q (γ) = -I q (γ) ,

→ I q (γ) = - ∞ γ dI q (τ ) dτ dτ = ∞ γ dτ e -R 2 4x π τ 3 2 .
This integral can be solved by substitution, where:

u 2 = R 2 4τ , x = R 2 4u 2 , dτ du = -R 2 2u 3 .
Since τ goes from γ to ∞, u goes from R 2 4γ to lim τ →∞ R 2 4τ = 0. It follows:

I q (γ) = - 0 ˆ R 2 4γ du R 2 2u 3 e -u 2 4u 3 R 3 π 3 2 , I q (γ) = 4π 3 2 1 R R 2 4γ
0 du e -u 2 .

The integration over all space (A.32) has been replaced by a one-dimensional integration over a nite interval, which can be found in mathematical tables as error function erf(y):

ˆy 0 du e -u 2 = √ π 2 erf (y) .
This results in:

I q (γ) = 2π 2 1 R erf R 2 4γ .
Consequently, the Coulomb integral between two unnormalized s-like Gaussian functions can be calculated as:

V s,s C = π κ 3 2 π λ 3 2 1 R erf R 2 4γ . (A.33)
The introduced error function is strongly related to the so-called Boys function F n of order n [START_REF] Boys | Electronic Wave Functions. I. A General Method of Calculation for the Stationary States of Any Molecular System[END_REF][START_REF] Boys | Automatic Fundamental Calculations of Molecular Structure[END_REF], obeying:

´1 0 dt t 2 2γ (A x -B x ) e -R 2 t 2 4γ = -1 κ 1 √ π γ -3 2 π κ 3 2 π λ 3 2 R x ´1 0 dt t 2 e -R 2 t 2 4γ , V s,px C = - 1 κ 1 √ π γ -3 2 π κ 3 2 π λ 3 2 R x F 1 R 2 4γ .
Following the same procedure and taking V s,px π λ

3 2 R x R y F 2 R 2 4γ .
Step by step, expressions for the dierent bare exchange interactions can be derived using the previous solutions following the demonstrated scheme. As a result, we get analytical expressions for the Coulomb integral containing Boys functions of dierent order. As already mentioned above, these can be eciently evaluated applying recursion.

Figure 0

 0 Figure 0.1.: Recent compilation of the best research power conversion eciencies of diverse solar cell types, reaching from standard single-junction silicon cells to emerging quantum dot cells. Record eciencies of up to 45% are achieved with multi-junction cells, while single-junction silicon cells approach 28%. From 2001 to 2014, an impressive increase by a factor of around 4 can be noted for organic solar cells, reaching a laboratory eciency of 11%.Figure pro-

  Figure provided by the American National Renewable Energy Laboratory (NREL) at http://www.nrel.gov/ncpv/.

  Fig. (0.3a). The rst bilayer cell, as realized by Tang in 1986 [25], was based on copper phthalocyanine and a perylene tetracarboxylic derivative as donor and acceptor materials, respectively, resulting in a power conversion eciency of 1%. The respective molecular structures are depicted in Fig. (0.5b).

ContentsFigure 0 . 5 .

 05 Figure 0.5.: Atomic representation of molecules commonly used in exciton solar cells: a) Ruthenium dye b) Copper phthalocyanine (left) and the perylene tetracarboxylic derivative PTC (right), as applied as donor and acceptor materials in the rst organic solar cell with a donor/acceptor interface. c) In bulk heterojunction cells, typically PC 61 BM, a derivative of the Buckminster fullerene C 60 ,

1 | Methodology 1 . 1 .

 111 Photoemission spectroscopy has evolved to become one of the most established experimental techniques, playing a central role in probing the electronic structure of materials[START_REF] Hüfner | Photoelectron Spectroscopy -Principles and Applications[END_REF][START_REF] Reinert | Photoemission spectroscopy -from early days to recent applications[END_REF]. Various quantities related to the electronic structure are accessible, e.g. the chemical composition of materials, the Fermi surface or the electron-phonon coupling strength, leading to major progresses in the understanding of fundamental processes in solid state physics, chemistry and material sciences.The physical origin of photoemission spectroscopy (PES) is the photoelectric eect[START_REF] Hertz | Über einen Einuÿ des ultravioletten Lichtes auf die electrische Entladung[END_REF][START_REF] Hallwachs | Über den Einuÿ des Lichtes auf electrostatisch geladene Körper[END_REF].Discovered in 1887 by Hertz and Hallwachs, it describes the ejection of electrons from a sample due to irradiation with light. Today's measurement setups are still very similar to the pioneering experiments: monochromatic and polarized photons from a light source most commonly UV, X-Ray or synchrotron radiation hit the sample under a certain angle and cause electrons to be ejected. The kinetic energy of these photoelectrons is detected via an electrostatic analyzer. Due to the relation between the momentum p of a photoelectron and the wave vector k of the corresponding Bloch state in the crystal, an angular resolved measurement of the kinetic energy gives access to the electronic dispersion relation, i.e. the band structure (ARPES: angular resolved photoelectron spectroscopy).

Figure 1 . 1 .

 11 Figure 1.1.: Experimental valence band PES spectra of V 2 O 3 , measured at dierent incident photon energies. The spectrum signicantly changes going from small photon energies to the high energy limit. This is due to the energy dependence of the electron-photon cross section. Moreover, extrinsic processes, such as inelastic scattering with secondary electrons, are more eective for low kinetic energy photoelectrons. Figure taken from Ref. [56].

( 1 .

 1 [START_REF] Hull | Properties of crystalline silicon[END_REF] to frequency space via a Fourier transformation. Using the fact that the Fourier transformation of a derivative ∂ ∂t becomes (-iω) and that δ-functions δ (τ ) give unity, one arrives at:

Figure 1 . 6 .

 16 Figure 1.6.: The quasiparticle picture: the many-body system is approximated by replacing the bare Coulomb interaction v between electrons by quasiparticles that interact through the weaker screened Coulomb potential W . The quasiparticles represent bare particles (purple) surrounded by an inversely charged cloud (grayish),

  as occurring in the Lehmann representation of the Green's function. The connection between quasiparticle and excitation energies is depicted in Fig. (1.7).

  that HF or DF T -K S wave functions are almost identical to the unknown quasiparticle wave functions ψ qp ν ≈ ψ HF ν ≈ ψ KS ν and one diagonalizes equation (1.27) for example in a Kohn-Sham basis:

ν

  and ε qp ν are assumed to be of the same order. As a result, equation(1.28) no longer depends on the quasiparticle energy, but on the known DF T -K S energies. One has to keep in mind that the demonstrated approach is only valid if the DF T -K S wave functions are close to the true quasiparticle wave functions. The validity of this assumption will be discussed in detail in subsequent chapters. Finally, the last missing ingredient in equation(1.29) is the self-energy Σ ε KS ν

  ) is evaluated in terms of contour deformation techniques. The derivation is tedious and can be found in the Appendix A.5. It can be obtained upon inserting the time-ordered singleparticle Green's function G in its Lehmann representation and the screened Coulomb potential in terms of uctuation potentials V m in equation (1.39):

  1.11). Equation(1.44) provides a rst intuitive picture of the absorption process. However, comparing the model to experimental absorption spectra reveals large quantitative and qualitative discrepancies (see Fig.1.10).

(1. 45 ) 1 = 2 =

 4512 Due to the creation of bound excitonic states, sharp absorption peaks occur at energies (E exc n = E g -Ry * /n 2 ) below the gap. The rst peak in the spectrum is at (E exc E g -Ry * ), the second at (E exc E g -Ry * 4 ), etc. , i.e. for (n → ∞) peaks are getting closer and closer to each other. Since the function value is diminishing at the same time with (1/n 3 ), they sum up to a nite value close to ( ω = E g ), see Fig.(1.11).

( 1 .

 1 [START_REF] Born | Theoretical investigations on the relation between crystal dynamics and x-ray scattering[END_REF] i.e. a Hartree term K H = -iδ (3, 4) δ (5, 6) v[START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF][START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF] and the remainder K R = ∂Σ(3,4) ∂G 1 (5,6) . The remaining part K R including all interactions beyond Hartree denes the irreducible four-point polarizability L through:

69 )H

 69 res is called resonant part, i.e. transitions between (occupied → unoccupied) states are treated. On the contrary, -(H res ) * is an anti-resonant part, where transitions between (unoccupied → occupied) states and thus negative frequency transitions are considered. The K coupl and -K coupl * blocks couple the resonant to the anti-resonant part and include both (occupied → unoccupied) and (unoccupied → occupied) transitions. Within the the socalled Tamm-Danko approximation (T DA), these o-diagonal coupling terms in H 2p are neglected. As a result, H 2p becomes a block-diagonal, Hermitian matrix, whose eigenvalues are obtained by diagonalizing the Hermitian block matrix [H res ], while the eigenvalues of [-H res ] are just opposite in sign. The qualities and limitations of the T DA are discussed later in this work.

  case Kohn-Sham eigenvalues are taken as a starting point and in the other case quasiparticle excitation energies. The exchange terms are totally equal, whereas the screening is included through the static and non-local exchange-correlation kernel f xc (r, r ) within TDDFT and through the static and non-local screened Coulomb potential W stat (r, r ) within BSE, respectively. In the case of TDLDA, the exchange-correlation kernel becomes the static and local LDA functional, i.e. the screened exchange contribution simplies to:H xc,LDA ml,m l = ˆdr φ m (r) φ * l (r) ∂V LDA xc [n] ∂n (r) φ * m (r) φ l (r) .Important dierences between TDDFT with (semi)local kernels and BSE in the electron and hole space integration variables will be highlighted later in the course of the discussion of charge-transfer optical excitations.By expressing the response equation in transition space, one has to solve, similar to the BSE approach, a four-point problem. Whereas this is inevitable in BSE, one can choose between a representation in the transition space basis consisting of pairs of occupied and unoccupied states and a two-point problem in TDDFT, since χ is governed by a closed Dysonlike two-point response equation. The advantage of the transition space representation is that a direct comparison with BSE is possible. Moreover, one can directly access the character of a transition, i.e. the mainly contributing occupied and unoccupied orbitals, associated with a specic excitation energy. This information is not provided by the solution of the twopoint equation, where one only obtains excitation energies and the corresponding oscillator

2 |

 2 Details on the ImplementationMany-body perturbation theory GW/BSE calculations, as presented later in this work, have been carried out using the FIESTA package[START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF][START_REF] Blase | Charge-transfer excitations in molecular donor-acceptor complexes within the many-body Bethe-Salpeter approach[END_REF][START_REF] Faber | First-principles GW calculations for DNA and RNA nucleobases[END_REF]. The latter is a recent Gaussian-basis implementation of the GW/BSE formalisms, suited for calculations on nite systems. Computationally demanding non-local quantities, such as the irreducible polarizability or the screened Coulomb potential, are expressed within an auxiliary basis instead of a full product basis. Both the resolution of the identity technique (RI-SVS) and the resolution of the identity Coulomb metric technique (RI-V) are available to express the auxiliary basis. The self-energy is obtained by explicitly evaluating the frequency integral over the one-particle Green's function and the screened Coulomb potential using contour deformation techniques, i.e. one goes beyond any plasmon pole approximation. Details on the mentioned aspects are

  µ S µν , i.e. the introduction of a basis {α µ } transforms equation (2.1) into a generalized eigenvalue matrix equation: [O] [c] = w [c] [S] , (2.3) where the vector [c] contains the weighting coecients [c] µ = c µ and where the matrix elements of [O] denote O µν = ´dr α * µ (r) Ô (r) α ν (r). Once the matrix elements are calculated, ecient linear algebra routines exist to solve equation (2.3). In the case of orthogonal basis sets, the overlap matrix [S] simply reduces to the unitary matrix.

  ) with[v] λλ = β λ (r)| v (r, r ) |β λ (r ) and S λλ = β λ | β λ . Based on these closure relations, matrix elements for non-local operators can be formulated in two ways. The resolution of the identity technique (RI-SVS) is based on equation (2.5) and yields:

(2. 7 )

 7 Consequently, only three ingredients are necessary in the RI-SVS representation: the inverse overlap between auxiliary basis functions S -1 , the overlap α µ α ν | β λ between the original basis {α} and the auxiliary basis {β}, and the matrix elements β λ | Ô |β λ , whose number is signicantly reduced due to the limited number of used auxiliary basis functions. For Dysonlike equations, such as W = v + vP W , one thus obtains within the RI-SVS equations of the type:

Figure 2 . 1 .

 21 Figure 2.1.: Convergence plot of the GW gap energy of the benzene molecule with respect to the number N of Gaussian functions per angular channel of the auxiliary basis within the RI-SVS and the RI-V approach, respectively. The calculations have been carried out using a TZ2P Kohn-Sham basis and a Gaussian function auxiliary basis (e -αr 2

Fig. ( 2

 2 .3) provides a typical plot of convergence concerning the number of used primitive Gaussian functions per angular momentum channel.

  r ) N, k| n (r ) |N, 0 . The indices m and l label occupied and unoccupied states, respectively. φ m,l and ε m,l are single-particle wave functions and energies. Neutral excitation energies are represented by ω k . Applying analytic continuation, equation (2.9) is transferred from real space to the complex plane (ω → ω). 1 The resulting complex integral is evaluated along the closed contour depicted in Fig. (2.4) using contour deformation techniques:

Figure 2 . 4 .

 24 Figure 2.4.: The chosen closed contour consists of contours C and D as illustrated. The poles of W at the neutral excitation energies are represented by orange squares.The poles of the non-interacting single-particle Green's function G 0 (r, r , ω)

  are available to extract information about excited states. Diculties are especially encountered, when DF T -K S eigenvalues are directly compared to one-particle excitation energies measured in photoemission experiments. In principle, as already discussed in detail in preceding chapters, DF T -K S eigenvalues are pure mathematical tools entering the formalism as Lagrange multipliers in order to ensure orthogonality for the wave functions. Referring to the ionization potential theorem, only the eigenvalue of the highest occupied level can be associated with the negative value of the ionization energy, IE ≡ E N -1 ho -E N 0 , provided that the exact exchange-correlation functional is used. Nevertheless, it is common practice to associate the complete Kohn-Sham eigenvalue spectrum with experimental band structures,

Fig. ( 3

 3 .1) for selected semiconductors and insulators.

Fig. ( 3 . 2 )

 32 for three organic molecules, hybrid functionals usually signicantly improve over (semi)local functionals such as LDA or GGA. However, since self-interaction errors are only partly removed, non-negligible discrepancies with correlated theories and experiment and thus the band gap problem still persist.A promising extension of hybrid functionals are so-called range-separated hybrid (RSH) functionals. Based on the work of Savin and coworkers[START_REF] Savin | On Degeneracy, Near-degeneracy and Density Functional Theory[END_REF][START_REF] Leininger | Combining long-range conguration interaction with short-range density functionals[END_REF][START_REF] Toulouse | Long-range/short-range separation of the electron-electron interaction in density-functional theory[END_REF], range-separated hybrid (RSH) functionals have been specically designed to tackle the band gap problem. The idea is to separate the exchange energy functional into a short-range (SR) and a long-range (LR)

  3.2). Important representatives are among others HSE06[START_REF] Krukau | Inuence of the exchange screening parameter on the performance of screened hybrid functionals[END_REF], long-range corrected (LRC or LC) RSH functionals reproducing the physical long-range limit, such as LC-BLYP[START_REF] Tawada | A long-rangecorrected time-dependent density functional theory[END_REF][START_REF] Iikura | A long-range correction scheme for generalized-gradient-approximation exchange functionals[END_REF], or the Baer, Neuhauser and Livshits (BNL) RSH functional[START_REF] Baer | Density Functional Theory with Correct Long-Range Asymptotic Behavior[END_REF][START_REF] Livshits | A well-tempered density functional theory of electrons in molecules[END_REF]. The exact exchange contribution [α + β erf (κ, |r -r |)] with respect to the inter-electron distance |r -r | is provided by Fig. (3.3) for diverse functionals.

  provided by Fig. (3.1).

Figure 3 . 4 .

 34 Figure 3.4.: From semi-metallic to semi-conducting: in the case of lanthanum hydride LaH 3 , DF T -LDA band structure calculations predict a semi-metal (left), whereas experiment yields a semi-conducting system. Within GW , the band gap is opened and the correct semi-conducting behavior is reproduced (right). Figure taken from Ref. [180].

3. 2 . 1 .Figure 3 . 6 :

 2136 Figure 3.6: Isocontour representation of the Kohn-Sham wave functions around the gap classied by their σor π-character. The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are singled out in a green and orange box, respectively. The asterisk denotes unoccupied orbitals, the subscripts (1, 2) the peptide unit on which the orbital is localized. The ordering corresponds to the DF T -LDA ordering. Carbon atoms are represented in gray, oxygen in red, nitrogen in orange and hydrogen in white, respectively.

2 states indicates that they project within 99 .

 99 8% and 98.9%, respectively, onto the corresponding DF T -LDA eigenstates. This demonstrates that the Kohn-Sham and scCOHSEX eigenstates do not dier signicantly, in contrast to the very large discrepancy for the eigenvalue spectra. For the sake of illustration, in Fig. (3.8), we plot the DF T -LDA, scCOHSEX and Hartree-Fock σ 1 and π * 2 wave functions,

Figure 3 .

 3 Figure 3.9.: GW/BSE optical excitation energies as a function of the zinc-bacteriochlorin (ZnBC) to bacteriochlorin (BC) distance R. The small box on the right indicates the represented GW/BSE intramolecular (Q-band and higher lying Soret B transitions) or charge-transfer transitions between the two molecules. The lowest lying ZnBC→BC charge-transfer excitation energy is compared to the (E GW g -e 2 /D) Mulliken limit (thick gray line), where E GW g is the GW quasiparticle gap of the dimer in the large D limit, where D is the distance between the two molecule centers. For the sake of comparison, the TDLDA values for the two lowest lying CT excitations are indicated by the two non-dispersive dotted lines below 1.5 eV. The two vertical arrows indicate the dierence between the TDLDA and GW/BSE values. Figure and caption (both modied) taken from

2

  transition. As a reminder, the corresponding Kohn-Sham wave functions are depicted in Fig. (3.6). Anticipating on our BSE results, Fig. (3.10) illustrates the studied excitations by providing an isocontour representation of the hole-averaged electron distribution (transparent green) as obtained from the expectation value of the electron density operator δ(r -r e ) on the corresponding two-body ψ(r e , r h ) BSE eigenstate. Similarly, the electron-averaged hole distribution is represented (gray wireframe). The clear CT character of the CT a transition and the partial CT character of the CT b excitation can be readily veried.

Figure 3 .

 3 Figure 3.11.: Excitation energies as provided within several theoretical frameworks plotted against the CASPT2 results (rst diagonal in red). The TDDFT values with the B3LYP (blue up triangles) and the CAM-B3LYP (open circles) functionals, and the present GW/BSE@LDA calculations (green squares) are depicted. Several CAM-B3LYP values are found for each excitation, showing in particular the spread of values as a function of the (α + β) parameter. Energies are given in eV.

3. 3 . 3 .

 33 Charge-transfer excitations in the coumarin family 3.3.3.1. The studied system: the coumarin molecules Promising to become a low-cost alternative to standard silicon-based photovoltaics, dyesensitized solar cells (DSSCs) have been intensively studied over the past 20 years [27, 242].

  3.16). The resulting molecular structures of the NKX-2xxx family are represented in Fig.(3.13) showing the so-called (cis) conformations, with a (trans) structural isomer represented in Fig.(3.14) for one of them. Further, inclusion of the cyano (-C=N)

Figure 3 .

 3 Figure 3.13.: Symbolic representation of the studied coumarins: (a) parent C343, (b) NKX-2388 (cis), (c) NKX-2311 (cis), (d) NKX-2586 (cis) and (e) NKX-2677. The dierence with the corresponding (trans) structures is represented in Fig. (3.14) for the NKX-2311 case. Black, white, red, blue and yellow atoms represent carbon, hydrogen, oxygen, nitrogen and sulfur, respectively.

Figure 3 .

 3 Figure 3.15.: Calculated lowest optical excitation energies (in eV) as a function of the coupled-cluster CC2 reference values. Results in perfect agreement with the CC2 calculations should fall on the diagonal (black line). The present GW/BSE calculations (empty red squares) are compared to the TD-B3LYP

  (3.16) the Kohn-Sham HOMO (a, d) and LUMO (b, e) eigenstates associated with the C343 and NKX-2677 structures. While the HOMO states are found to be rather delocalized, the LUMO in the NKX-2677 dye is clearly much more localized close to the electron-acceptor cyanoacrylic group. This results in an enhanced CT character as compared to the C343 parent molecule.The nature of the transitions can be better quantied by studying the electron and hole spatial localization in the excited states. This can be achieved by taking the expectation value of the electron position operator δ(r=r e ) over the two-body ψ(r e , r h ) BSE excitonic wave function, leading to an electron probability of presence averaged over the hole position.A similar quantity can be dened for the hole spatial distribution. The resulting densities are provided in Fig.(3.16) with an isocontour representation for the C343 and NKX-2677 molecules. These densities allow to obtain the mean electron or hole positions (see red arrows in Fig.3.16) and the related average electron-hole separation distance which amounts to 3.2 Å in C343. This clear CT character is certainly at the origin of the diculties met by TDLDA or TD-PBE to describe such an excitation. In the NKX-2677 case, this average distance increases to 4.6 Å as a signature of the enhanced CT character, explaining that the B3LYP results signicantly worsen from C343 to NKX-2677.

Figure 3 .

 3 Figure 3.16.: Isocontour representation of the C343 HOMO (a) and LUMO (b) Kohn-Sham eigenstates. Dierent colors indicate dierent signs of the wave function. In panels (d) and (e), similar plots for NKX-2677. In panels (c) and (f ), isocontour representation of the electron (yellow) and hole (light blue) probability distribution for the lowest C343 and NKX-2677 optical excited states, respectively, as obtained within BSE. The red arrows indicate the average hole (left arrow) and electron (right arrow) positions. The C343 and NKX-2677 molecules are not represented on the same scale.

Figure 3 .

 3 Figure 3.18.: Theoretical oscillator strength as a function of the GW/BSE excitation energies (eV). The GW/BSE values (red empty squares) are compared to the CC2 (black diamond), LC-BLYP (blue triangle up) and the BNL J 1 /J 2 (empty green circles) results. Results for the NKX-2677 and NKX-2586 (cis) molecule in the dashed box are reproduced in the upper-right inset.

Figure 3 .

 3 Figure 3.19.: (a) Color density plot of experimental ARPES data for the two highest occupied bands H1 and H2. The (green) dots are the peak positions as derived from Gaussian ts of the individual spectra. The gray line is the ab initio DF T band dispersion from Ref. [257], in close agreement with GW calculations within this energy window [256]. (b) Calculated spectrum in the presence of EPC interactions and disorder (see text). The dots are the experimental peak positions from panel (a), but now the color density plot originates from the theoretical spectra including electronic lifetimes and thus a broadening through electron-phonon scattering. Figure and caption (modied) taken from Ref.

3. 4 . 1 .

 41 The studied systems 3.4.1.1. Electron-phonon coupling in C 60

Figure 3 . 20 :

 320 Figure 3.20: Molecular structure of the Buckminster fullerene C 60 : this fullerene is often compared to a soccer ball, consisting of 12 pentagons and 20 hexagons. Every carbon atom is covalently bound to three neighboring atoms. The valence electrons form three σ-bonds per carbon atom at the molecule's surface, while the fourth electron is part of a delocalized π-system perpendicular to the surface. Due to the curvature of the molecular surface, πbonds have both a perpendicular and an in-plane component, i.e. C 60 is partly sp 2 -and partly sp 3hybridized.

Figure 3 . 21 :

 321 Figure 3.21: Schematic representation of the electronic structure of the alkali-doped C 60 fulleride: the completely lled HOMO level (blue) is ve-fold degenerate and has h u symmetry. The three-fold degenerate LUMO (orange) has t 1u symmetry and is half-lled by the additional electrons of the alkali dopants. Their sorbitals (black), which are nearly degenerate with the LUMO+1 state (black), are therefore empty. As a guide for the eye, the molecular levels are articially broadened.

Figure 3 .

 3 Figure 3.24.: Electronic dispersion of the valence and conduction bands in graphene as calculated within a tight-binding model. The zoom in shows the energy bands close to one of the Dirac points. Figure taken from Ref. [277].

[ 277 ]

 277 conjugated electrons. As it is depicted in Fig.(3.23), its honeycomb structure can be expressed as a triangular lattice with a basis of two atoms per unit cell. The latter is determined by two lattice vectors a 1 and a 2 , with length|a 1 | = |a 2 | = √ 3a ≈ 2.46 Å, where a ≈ 1.42 Å is the lattice constant. Its Brillouin zone (BZ) is spanned by two lattice vectors in reciprocal space, b 1 and b 2 [277]. Of particular importance for the physics of graphene are two points of the BZ, namely the so-called Dirac points K = 2π 3a , . The band structure in the vicinity of these points as obtained in a tight-binding approach is depicted in Fig. (3.24). The Dirac points are the only points in the band structure, where the valence and the conduction band touch each other, i.e. there is no gap between occupied and unoccupied states. Very importantly, in the vicinity of the Dirac points, the energy shows a linear dispersion with respect to k, just as for relativistic particles. Together with the large group velocity, which is the rst derivatives of the band energy with respect to k, these properties are amongst others often used to rationalize the excellent conductance of graphene.

Figure 3 .

 3 Figure 3.25.: Phonon dispersion of graphene (GE) as calculated (lines) at the experimental and equilibrium lattice spacings (a exp and a th ). The points represent experimental data [280]. The red straight lines at Γ and at K t the slopes. Figure and caption taken from Ref. [278].

  For graphene, our DF T -LDA and G 0 W 0 values are close to those of Ref. [51], namely 197 eV/Å within G 0 W 0 for the largest matrix element of the K-A 1 phonon in the present study, to be compared to 193 eV/Å in the previous one. The dierence can be explained by an increased ve point nite-dierence formula, instead of only two points, and by shifting the Godby-Needs plasmon model input nite frequency from the (default)

Figure 3 .

 3 Figure 3.27.: Evolution of the 3-fold degenerate lowest-unoccupied molecular orbital in C 60 with respect to a deformation α of -0.1 Å ≥ α ≤ 0.1 Å along the HG07 eigenmode, the strongest coupling mode. DF T -LDA (red circles) yields slopes and hence an electron-phonon potential which are much smaller than the corresponding evGW (blue downward triangles) calculations. COHSEX (green diamonds) and evCOHSEX (orange squares) are in very close agreement with evGW results. The evCOHSEX(W ) approach (black stars) excellently reproduces the evCOHSEX results.

(A. 2 ) 5 .

 25 For F [Ψ (1) ; 2, 3] being the inverse of G [Ψ (1) ; 2, 3] it holds: ˆd4 F [Ψ (1) ; 2, 4] G [Ψ (1) ; 4, 3] = δ (2 -3) .6. Denition (5) can be used to derive a very useful relation: ∂F [Ψ (1) ; 2, 3] ∂Ψ (6) = -ˆd4d5 F [Ψ (1) ; 2, 4] ∂G [Ψ (1) ; 4, 5] ∂Ψ (6) F [Ψ (1) ; 5, 3] .

∂Ψ( 3 )∂Ψ( 3 )=

 33 [START_REF] Hüfner | Photoelectron Spectroscopy -Principles and Applications[END_REF]) F (52) + ˜d4d5F (14) ∂G(45) ∂Ψ(3) F (52) + ˜d4d5F (14) G (45) ∂F(52) = ´d4 ∂F(14) ∂Ψ(3) δ (4 -2) + ˜d4d5 F (14) ∂G(45) ∂Ψ(3) F (52) + ´d5 δ (1 -5) ∂F (52) ∂F (12) ∂Ψ(3) + ˜d4d5 F (14) ∂G(45) ∂Ψ(3) F (52) + ∂F(12) ∂Ψ(3)→ ∂F(12) ∂Ψ(3) = -˜d4d5 F (14) ∂G(45) ∂Ψ(3) F (52) .

  We work in the Heisenberg representation, assuming that the Hamiltonian has no explicit time-dependence. The time-dependent eld operators fulll Heisenberg's equation of motion, describes the time evolution of a Heisenberg operator in the same way that the time-dependent Schrödinger equation describes the time evolution of a wave function. Inserting Ĥ and using the commutation relations for eld operators leads to an explicit expression for the equation of motion of an annihilation eld operator:

  including the chemical potential µ and the number operator N . Moreover, Ψ N n and E N n are eigenfunctions and eigenvalues to Ĥ -µ N , the Hamiltonian of a system exchanging energy and particles with a reservoir, i.e. a grand canonical ensemble. Since Ψ N n are eigenfunctions of Ĥ, we can use the property,

l

  the excitation energy needed to insert an electron to the N particle system resulting ina state Ψ N +1 m : ε m = E N +1 m -E N0 the excitation energy needed to remove an electron resulting in a state Ψ N -1 l

  with poles only in the upper (for G A ) or lower half plane (for G R ). Moreover, G A and G R are complex conjugates of each other: G R = G A * .A.3. Hedin's equationsEquation of motion of the single-particle Green's function Dierentiating the singleparticle Green's function with respect to one of its two time variables reveals important properties of the latter[START_REF] Strinati | Application of the Green's functions method to the study of the optical properties of semiconductors[END_REF][START_REF] Bruus | Many-Body Quantum Theory in Condensed Matter Physics -An Introduction[END_REF]. Starting from the time-ordered one-particle Green's function at zero temperature:

iθ+ Ψ N 0 ∂

 0 ∂G(rt,r t ) ∂t = Ψ N 0 Φ(r, t) Φ † (r , t ) Ψ (t -t) i ∂G(rt,r t ) ∂t = δ (t, t ) Ψ N 0 Φ(r, t) Φ † (r , t ) Ψ N 0 + Ψ N 0 Φ † (r , t ) Φ(r, t) Ψ N 0 Φ(r,t) ∂t Φ † (r , t ) Ψ N 0 θ (t -t ) -Ψ N 0 Φ † (r , t ) ∂ Φ(r,t) ∂t Ψ N 0 θ (t -t) ,

i ∂ ∂t 1 - 1 , 16 ) 1 - 31 = 11 G( 21 ) 21 G 21 =

 111613111212121 H 0 (r 1 ) G 0 11 = δ 1, Multiplying equation (A.14) by G 0 (21) from the left and integrating over d1 results in the Dyson equation: H 0 (r 1 ) G(11) -¨d3d1 G 0 (21) Σ (13) G ˆd1 G 0 (21) δ -¨d3d1 G 0 (21) Σ (13) G 31 = G 0 G 0 21 + ¨d3d1 G 0 (21) Σ (13) G 31 , (A.17)which can be reformulated using δ (12) = ´d3 G -1 (13) G (32) to:G -1 (54) = G -1 0 (54) -Σ (54) .Using the fact that the Fourier transformation of a derivative ∂ ∂t becomes -iω and that δfunctions δ (τ ) give unity, we can readily transform the above equation to frequency space, yielding:G -1 (r 5 r 4 , ω) = G -1 0 (r 5 r 4 , ω) -Σ (r 5 r 4 , ω) = δ (r 5 r 4 ) [ ω -H 0 (r 5 )] -Σ (r 5 r 4 , ω) .

1 = 1 -

 11 H,i -ReΣ ii (ε qp,i ) -∂ReΣ ii (ω) ∂ω ω=ε qp,i (ω -ε H,i -ReΣ ii (ε qp,i )) , G -1 coh (ω) = ω -ε H,i -ReΣ ii (ε qp,i ) -ω ∂ReΣ ii (ω) ∂ω + ε H,i ∂ReΣ ii (ω) ∂ω

ψ 2 ∇ 2 +

 22 λ and ψ λ are left and right eigenvectors of the non-Hermitian operator H, H (z) = Ĥ0 + Σ r, r , z = -1 V ext (r) + V H (r) + Σ r, r , z , and E λ (z) is the corresponding eigenvalue. Ĥ0 regroups the kinetic energy, the external potential V ext arising from the ion cores and the Hartree potential V H . Inserting this representation of the Green's function in the equation of motion (EOM) in frequency space (A.18)yields:[z -H 0 (r)] G (r, r , z) -´dr Σ (r, r , z) G (r , r , z) = δ (r -r ) λ [z -E λ (z)] -1 { [z -H 0 (r)] ψ λ (r, z) ψ λ (r , z) -ψ λ (r , z) ´dr Σ (r, r , z) ψ λ (r , z) } = δ (r -r ) λ [z -E λ (z)] -1 { [z -H 0 (r)] ψ λ (r, z) ´dr ψ ν (r , z) ψ λ (r , z) -´dr ψ ν (r , z) ψ λ (r , z) ´dr Σ (r, r , z) ψ λ (r , z) } = ´dr ψ ν (r , z) δ (r -r ) λ [z -E λ (z)] -1 { [z -H 0 (r)] ψ λ (r, z) δ νλ -δ νλ ´dr Σ (r, r , z) ψ λ (r , z) } = ψ ν (r, z) [z -H 0 (r)] ψ ν (r, z) -´dr Σ (r, r , z) ψ ν (r , z) = [z -E ν (z)] ψ ν (r, z)H 0 (r) ψ ν (r, z) + ´dr Σ (r, r , z) ψ ν (r , z) = E ν (z) ψ ν (r, z) .

poles of G 0

 0 and poles of W , respectively. The function g 2 ,g 2 = m,k e iη ω φ * m (r) φ m (r ) E + ω -ε m + iη 2 ε k V k (r) V * k (r ) (ω + ε k -iδ) (ω -ε k + iδ) ,has single poles in the upper half plane at ωk = -ε k + iδ which solely stem from the W contribution. According to equation (A.23), the residue can be calculated as follows:k Res 2 (g 2 , ωk ) = = lim ω→ω k m,k e iη ω φ * m (r)φm(r ) E+ω-εm+iη 2 ε k V k (r)V * k (r )(ω+ε k -iδ) (ω+ε k -iδ)(ω-ε k +iδ) = m,k e iη ωk φ * m (r)φm(r ) E+ω k -εm+iη 2ε k V k (r)V * k (r ) -2ε k +i2δ.For the third integrand g 3 ,g 3 = l e iη ω φ * l (r ) φ l (r) E + ω -ε l -iη v r,r , one identies single poles at ωl = ε l -E +iη originating from the Green's function for occupied states. The residue is: l Res 3 (g 3 , ωl ) = = lim ω→ω l l e iη ω φ * l (r )φ l (r)(E+ω-ε l -iη) E+ω-ε l -iη v (r, r ) = l e iη ωl φ * l (r ) φ l (r) v (r, r ) .

  includes poles in the upper half plane both at ωk = -ε k + iδ from W and at ωl = ε l -E + iη from the Green's function:kl Res 4 (g 4 , ωk,l ) = = i Res (g 4 , ωk ) + i Res (g 4 , ωl ) = lim ω→ω k k,l e iη ω φ * l (r )φ l (r) E+ω-ε l -iη 2 ε k V k (r)V * k (r )(ω+ε k -iδ) (ω+ε k -iδ)(ω-ε k +iδ)

-A. 6 .

 6 k,l,m e iη ωk 2ε k V k (r)V * k (r ) -2ε k +i2δ φ * m (r)φm(r ) E+ω k -εm+iη + φ * l (r )φ l (r) E+ω k -ε l -iη .Setting η and δ to zero, this equals:Σ(r, r , E) = -occ l φ * l (r ) φ l (r) W (r, r , E -ε l ) + k,o V k (r) V * k (r ) φ * o (r)φo(r ) E+ε k -εo = Σ SEX + Σ COH ,where we introduced the summation index o including occupied and unoccupied states and thus combining l and m. Calculation of the bare exchange using contour deformation techniquesIn the following, we demonstrate that the bare exchange contribution of the self-energy, Σ x r, r , τ = iG 0 r, r , τ v rr , is equivalent to the Fock exchange in Hartree-Fock theory, φ k,0 | VF |φ k,0 = -occ n ˆdrdr φ k,0 (r) φ n,0 (r ) φ * k,0 (r ) φ * n,0 (r) |r -r | .

Figure A. 1 .

 1 Figure A.1.: The complex integration along the contour C consists of a contour C 1 along the real axis and a semi-circle C 2 . The poles of the non-interacting Green's function are at the real single-particle energies ε l and ε m (blue crosses), respectively.

Σ

  x (r, r ) = i 2π v(r, r ) +∞ -∞ dω e iηω G 0 (r, r , ω).According to analytic continuation, we change to the complex plane (ω → ω = ω + iω ) and evaluate the complex integralΣ x (r, r ) = i 2π v r, r˛C dω e iη ωG 0 (r, r , ω)using contour deformation techniques. The contour C is chosen like depicted in gure (A.2),where the radius of the semi-circle R goes to innity. Inserting the non-interacting Green'sfunction G 0 , G 0 (r, r , ω) = occ l φ * l (r ) φ l (r) ω -ε l -iη + unocc m φ * m (r) φ m (r ) ω -ε m + iη ,

Figure A. 2 .

 2 Figure A.2.: The complex integral is carried out along the contour C, which decomposes into an integration along C 1 and C 2 . The poles of the Green's function are at the real excitation energies ε m (blue crosses), however, they are slightly shifted into the complex plane by the innitesimal value ±η. Only the poles of the Green's function for occupied states at ωl = ε l +iη are enclosed by the contour, whereas the poles for unoccupied states at ωm = ε m -iη are outside.

Following

  equation (A.[START_REF]Measuring internal quantum eciency to demonstrate hot exciton dissociation[END_REF], the residues of G 0 are calculated to:ωl Res (G 0 , ωl ) = l φ * l (r) φ l (r ) ,where η → 0. Finally, taking the expectation value of Σ x with single-particle states φ k,0 , one obtains:φ k,0 | Σ x |φ k,0 = -occ l ˆdrdr φ k,0 (r) φ l,0 (r ) φ * k,0 (r ) φ * l,0 (r) |r -r | ,where v (r, r ) = 1 |r-r | has been inserted. This is nothing else than the bare Fock exchange from Hartree-Fock theory.

Figure A. 3 .

 3 Figure A.3.: The complex integral is carried out along the contour C, which decomposes into an integration along C 1 and C 2 . The poles of one Green's function occur at ω = ε m,l ± iη (blue crosses), while the poles of the other one (yellow dots) are at ω = ε m,l -E ± iη. Only the poles for occupied states at ω1 = ε m1 + iη and ω2 = ε m2 -E + iη are enclosed by the contour, whereas the poles for unoccupied states are outlying.

fm(r 1 )f * m (r 2 )

 2 E+ω-ε m1 -iη fm(r 2 )f * m (r 1 ) ω-ε m2 -iη + f l (r 1 )f * l (r 2 ) E+ω-ε l1 +iη fm(r 2 )f * m (r 1 ) ω-ε m2 -iη + fm(r 1 )f * m (r 2 ) E+ω-ε m1 -iη f l (r 2 )f * l (r 1 ) ω-ε l2 +iη + f l (r 1 )f * l (r 2 ) E+ω-ε l1 +iη f l (r 2 )f * l (r 1 )ω-ε l2 +iη = I 1 + I 2 + I 3 + I 4 .

( 3 , 4 )

 34 δ (5, 6) v (3, 5) ≈ iδ (3, 5) δ (4, 6) W (3, 4) -iδ (3, 4) δ (5, 6) v (3, 5) , where -in addition to the GW approximation for the self-energy -we further neglected the dependence of the screened Coulomb potential on the propagation of the particle. Inserted into the Bethe-Salpeter equation,L T DSHF (1, 2; 1 , 2 ) = = G 1 (1, 2 ) G 1 (2, 1 ) + + ´d3456 G 1 (1, 3) G 1 (4,1 ) [iδ (3, 5) δ (4, 6) W (3, 4) -iδ (3, 4) δ (5, 6) v (3, 5)] L(6, 2; 5, 2 )

  ) = L QIP (τ ; -τ ) +i ´dτ L QIP (-τ ; τ ) W stat (x 3 x 4 ) L (τ + τ ; -τ -τ ) -i ´dτ L QIP (-τ ; τ ) v (x 3 x 4 ) L(τ + τ, -τ -τ ) L (τ ; -τ ) = L QIP (τ ; -τ ) +i ´dτ L QIP (-τ ; τ ) W stat (x 3 x 4 ) L (τ + τ ; -(τ + τ )) -i ´dτ L QIP (-τ ; τ ) v (x 3 x 4 ) L (τ + τ ; -(τ + τ )) L (τ ) = L QIP (τ ) +i ´dτ L QIP (τ ) L (τ -τ ) W stat (x 3 x 4 ) -i ´dτ L QIP (τ ) L (τ -τ ) v (x 3 x 4 ) ,

[(n 1 n 2 ) (n 3 n 4 )

 24 -iL QIP (ω)] = 1 -ω where ∆ε n 2 n 1 = ε n 2 -ε n 1 and ∆ε n 1 n 2 = -∆ε n 2 n 1 . It is convenient to rewrite the matrix element L n 1 n 2 n 3 n 4 QIP in a short notation using the occupation factors f i (f m = 1, f l = 0): -iL n 1 n 2 n 3 n 4 QIP (ω) = (f n 2 -f n 1 ) δ (n 1 , n 3 ) δ (n 2 , n 4 ) ∆ε n 2 n 1 -ω ,

[

  -iL QIP (ω)] = -i LQIP (ω) [F ] with -i LQIP (ω) [F ] = can rewrite the Bethe-Salpeter matrix equation as follows:[-iL (ω)] = -i LQIP (ω) [F ] + -i LQIP (ω) [F ] [iK] [-iL (ω)] = 1 --i LQIP (ω) [F ] [iK]

(A. 31 )

 31 In this specic case, the contour C is chosen to consist of a counter-clockwise path along the entire real and a semi-circle with radius R = x 2 + y 2 in the upper half or in the lower half plane (see Fig.(A.4)). In the former case, one deduces from the residue theorem:˛dz I (z) = ˆ∞ -∞ dx I (x) + lim R→∞ ˆπ 0 dϕ iRe iϕ I Re iϕ = 2πi i Res (I, z i )and in the latter:˛dz I (z) = -ˆ∞ -∞ dx I (x) + limR→∞ ˆ2π π dΘ iRe iΘ I Re iΘ = 2πi i Res (I, z i ) .

  iΘ = R cos (ϕ + π) + iR sin (ϕ + π) = R cos ϕ cos π + R sin ϕ sin π + iR sin ϕ cos π + iR cos ϕ sin π = -R cos ϕ -iR sin ϕ = -Re iϕ ,where we applied the addition theorem for sine and cosine. For the integral it follows with z = -Re iϕ :ˆ2π π dΘ iRe iΘ I Re iΘ = ˆπ 0 dϕ iRe iϕ f -Re iϕ (-Re iϕ -z i) n e -itR cos ϕ e +tR sin ϕ and for the limit, respectively: lim R→∞ ˆπ 0 dϕ . . . e tR sin ρ R n-1 .

G 1 2 G 3 P η CG 3 =-

 123η3 = e -γ 1 (r-A) 2 G 2 = e -γ 2 (r-B) C e -η(r-P) 2 , with η = γ 1 + γ 2 , P = γ 1 A+γ 2 B η and C = e

3 4 2 l γ l/ 2 [

 342 (2h -1)!! (2i -1)!! (2j -1)!!] 1/2 , where l = h + i + j is the angular momentum quantum number. For contracted Gaussian functions the normalization constant can be calculated in a similar way.A.10.2. Calculation of Coulomb integrals in a Gaussian basisAssume the interaction between two unnormalized s-like Gaussians centered at A and B with orbitals exponents κ and λ, the Coulomb integral V C = -occ n ´d3 rd 3 r α i (r)αn(r)α * n (r )α * j (r ) |r-r | reads in an auxiliary Gaussian basis {β i }: V s,s C = ˆˆd 3 r d 3 r e -κ|r-A| 2 e -λ|r -B| 2 |r -r | .

(A. 32 )

 32 Due to the operator 1/|r=r |, the above integral does not factorize into products of x-, yand z-components. To circumvent this problem, one uses the Fourier transform of the Coulomb potential instead of its representation in r-space:

2 = e iqA ´d3 µ e iqµ e -κµ 2 = 2 x= e -q 2 x 2 = e -q 2 x 4κ π κ , where we completed the square, -κ µ 2 x -2 iqxµx 2κ = -κ µ x -iqx 2κ 2 -iqx 2κ 2 ,

 2222222κ22 d 3 µ = d 3 r = dµ x dµ y dµ z : I r = ´d3 r e iqr e -κ|r-A| e iqA ˝dµ x dµ y dµ z e i(qxµx+qyµy+qzµz) e -κ(µ 2 x +µ 2 y +µ 2 z ) = e iqA I qx r I qy r I qz r , in order to separate the integration over dµ into three independent integrals which are identical and which can be solved in the same manner. Exemplary, we give here the single steps for the calculation of I qx r : I qx r = ´dµ x e iqxµx e -κµ 4κ ´dµ x e -κ(µx-iqx 2κ ) and where we used essential properties of Gaussian integrals as presented in the preceding section. The same procedure is used to evaluate I qy r and I qz r , respectively, and in total it follows for I r : I r = e iqA e -

F n (x) = ˆ1 0 dt t 2n e -xt 2 F 2 ) 2 |r

 222 This function plays a key role in one-or two-electron integrals evaluation, since ecient evaluation methods based on upward/ downward recursion exist [?, 134]:F n (x) = 1 2x (2n=1) F n=1 (x)=e =x , F n (x) = 1 2n+1 2x F n+1 (x)=e =x .Rewriting the Coulomb integral for two s-like Gaussian functions (A.33) in terms of the Boys function gives: not only be solved very eciently, moreover it serves as a starting point for the calculation of the bare exchange interaction between the remaining orbitals. It can be demonstrated that it is not necessary to carry out the derivation from scratch, but solutions can be obtained based on V s,s C . This is exemplary shown for the s-like and p x -like Gaussian ((r x -C x ) e -σ|r-C| Coulomb integral in the following: d dAx V s,s C = 2κ ´´d 3 r d 3 r (rx-Ax)e -κ|r-A| 2 e -λ|r -B|

C= 4κλ ´´d 3 r d 3 r 3 2-R 2 t 2 4γ 3 2-

 3323 as a starting point, we can derive an analytical expression for a p x -like and p y -like Gaussian (r y -C y ) e -σ|r-C| 2 (rx-Ax)e -κ|r-A| 2 (ry-By)e -λ|r -B| 2 |rR x ´1 0 dt t 4 2γ (A y -B y ) e R x R y ´1 0 dt t 4 e

  2.2. Calculation of the dynamical screened exchange using contour deformation techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The GW/BSE method in practice 3.1. Preliminary considerations and motivation . . . . . . . . . . . . . . . . . . . . 3.1.1. The band gap problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Excited states within DFT-KS . . . . . . . . . . . . . . . . . . . . . . 3.1.3. The many-body perturbation theory GW formalism in practice . . . . 3.1.3.1. GW for nite systems . . . . . . . . . . . . . . . . . . . . . . 3.1.3.2. From G 0 W 0 to self-consistency . . . . . . . . . . . . . . . . . 3.2. The GW formalism applied . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. The studied system: the model dipeptide . . . . . . . . . . . . . . . . .4.1.2. Electron-phonon coupling in graphene . . . . . . . . . . . . . 3.4.2. Tested approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3. Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.2. Technical details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Beyond the scissor operator: level crossings . . . . . . . . . . . . . . . 3.2.4. Improving the starting point: scCOHSEX . . . . . . . . . . . . . . . . 3.2.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. The BSE formalism applied: charge-transfer excitations . . . . . . . . . . . .

3.3.1. Charge-transfer excitations . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Charge-transfer excitations in the model dipeptide . . . . . . . . . . . 3.3.2.1. Background and Notation . . . . . . . . . . . . . . . . . . . . 3.3.2.2. Comparison to multi-reference quantum chemistry perturbation theory and TDDFT methods . . . . . . . . . . . . . . . 3.3.2.3. GW/BSE calculations starting from self-consistent COHSEX eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3. Charge-transfer excitations in the coumarin family . . . . . . . . . . . 3.3.3.1. The studied system: the coumarin molecules . . . . . . . . . 3.3.3.2. Technical details . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3.3. Comparison to quantum chemistry CC2 and TDDFT reference data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Ecient ab initio calculation of electron-phonon coupling matrix elements . . 3.4.1. The studied systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1.1. Electron-phonon coupling in C 60 . . . . . . . . . . . . . . . .

3

  Figure 1.3.: Comparison between a one-electron and a many-body picture of the PES process: The photoelectric eect creates a system reduced by one electron with respect to the initial one. Consequently, PES accesses nal state energies (after the ejection of the photoelectron), where the relaxation of the system due to the created hole is taken into account. These so-called binding energies are depicted

on the right-hand side, whereas on the left hand side single-electron ground state orbital energies are represented. The relation between initial ground state orbital energies and nal state energies is highly nontrivial, being a ngerprint of many-body interactions in the system. The energy levels are not only shifted in energy with respect to each other, but also the level spacings change. For the sake of clarity, broadening eects and satellites are not depicted. The only measurable quantities are binding energies, whereas orbital energies represent purely theoretical tools.

  1.2.1. The Hartree-Fock ApproachThe variational principle Solving the electronic Schrödinger equation (1.1) allows to predict the properties of any electronic system. Due to its complexity, an exact solution of the problem is, however, out of reach. Nevertheless, one can formulate strategies, where not a full solution is obtained, but where at least the ground state wave function Ψ 0 and the ground state energy E 0 are accessible. In this context, the total energy E trial is calculated using an arbitrary trial

many-body wave function Ψ trial . The variational principle then states that E trial is always larger than the ground state energy, except the case, where Ψ trial equals the ground state wave function:

  6 . Besides CI, determine the properties of the N -electron system. The Hamiltonian is completely determined by the number of electrons N , and by the charge Z i and the position of the ions {R 0 }, which enter in the external potential. These quantities are readily available through the ground state density. Integrating the density over the whole volume gives the number of electrons N . Moreover, due to Coulomb attraction, the core positions {R 0 } can be identied through nite maxima of the ground state density.

	dierent kinds of post-Hartree-Fock ab initio methods exist, where the most popular ones are e.g. MøllerPlesset perturbation theory and Coupled Cluster (CC) approaches. These 3 Later in this work, we adopt the common notation to restrict the word correlation to correlation eects their detailed discussion is beyond the scope of this thesis and the reader is referred to Refs. [34, 67, 68, 69]. 1.2.2. Density Functional Theory Density Functional Theory (DF T ) represents a powerful alternative to Hartree-Fock and post-Hartree-Fock approaches [59, 60, 70]. The already introduced electron density n (r) constitutes the central quantity in this method, bearing the advantage of being much easier to handle than the complicated many-body wave function Ψ (x 1 . . . x N ). Even though representing a much simpler object depending only on 3 instead of 3N spatial variables, very intuitive arguments demonstrate that it provides all necessary information to set up the many-body Hamiltonian beyond exchange interactions present in HF . methods are widely used in quantum chemistry electronic structure calculations, however, and consequently to

  The introduced Hohenberg-Kohn functional F HK [n 0 ] is universal and accounts for the kinetic energy T [n 0 ] and the electron-electron interaction E ee [n 0 ], whereas the specications of the system under study are completely governed by the external potential term. The explicit form of F HK [n 0 ] is not known, however, it is possible to divide E ee into the electrostatic Hartree energy E H and a non-classical energy E ncl , accounting for exchange and correlation:

  r , t, t .

	1.3.3. The Dyson equation and the self-energy
	The equation of motion technique As depicted in the previous section, the Green's func-
	tion enters in many observables. Studying its time-dependence represents a powerful tool to
	explore these quantities. Dierent approaches exist, where one of which is the equation of
	motion technique. Here, the quantity of interest, e.g. the single-particle Green's function, is
	dierentiated several times, creating a series of coupled dierential equations. Provided that
	this series constitutes a closed set of equations, the problem is in principle exactly solvable.
	However, if it is not a closed, physical arguments have to be carefully selected in order to
	truncate the series in a physically meaningful manner.
	As elaborated in Appendix A.3, the derivation of the equation of motion (EOM) of the
	time-ordered Green's function starts by dierentiating G with respect to one of its two time
	arguments, yielding:

  are commonly used in quantum chemistry calculations on molecules. First, physical meaningful ST O can be reproduced by contracting only a few GT O, i.e. the size of the basis is not signicantly augmented compared to a ST O basis. Moreover, Gaussian functions provide mathematical properties which transform

	complicated matrix elements in simple analytic expressions making calculations extremely
	ecient. These mathematical properties are presented in detail in this subsection.
	2.1.4.1. General denitions
	Before presenting in detail mathematical advantages, the common nomenclature is shortly

reviewed in order to avoid confusion. Gaussian functions are called primitive, if they consist of a single Gaussian function. They have to be distinguished from contracted Gaussian functions which are a linear combination of several primitive Gaussian functions located at the same center. Moreover, one distinguishes between Cartesian and spherical Gaussian functions. An unnormalized primitive Cartesian Gaussian function centered at A reads:

Table 2 .

 2 1.: Compilation of technical specications for three molecular systems, namely a model dipeptide, the Buckminster fullerene C60 and the coumarin molecule NKX-2677, in order to give a rough estimate of the computational costs for

	3d

By way of example, for the oxygen atom, adding a 3d-function has the largest eect in terms of correlation energy, this function is part of the rst group. Contrary,

1

  Bohr -2 , very close to the values optimized by Dunning at the cc-pVQZ level[START_REF] Dunning | Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen[END_REF]. Following Ref.[136], the rst d-channel orbital is taken to be the polarization orbital of the atomic p-orbital, namely the d-component of the perturbation induced by a uniform electric eld, complemented by a primitive Gaussian with decay constant α =0.3 Bohr -2 for carbon. Following Dunning, we nally add a single primitive Gaussian for the f -channel with e.g. α =0.76 Bohr -2 for carbon.

	See Table (2.2) for a typical convergency test with respect to the used correlation-consistent
	Kohn-Sham basis.

Table 2 .

 2 2.: Convergency test of the rst Bethe-Salpeter singlet excitation energy (in eV)

		Excitation energy	Oscillator strength
		cc-pVXZ	aug-cc-pVXZ	cc-pVXZ	aug-cc-pVXZ
	X=D	5.17	4.92	0.53	0.49
	X=T	4.96	4.82	0.49	0.48
	X=Q	4.87	4.81	0.49	0.47

Real spherical harmonics are used instead of complex spherical harmonics Y lm , since this is

Table 3 .

 3 2.: Compilation of the four highest occupied energy level eigenvalues as obtained within the dierent approaches discussed in the text. Notice that the given

Table 3 .

 3 3.: Singlet excitation energies for the model dipeptide as obtained within various TDDFT, many-body perturbation theory and CASPT2 approaches. Energies are in eV. For the CAM-B3LYP columns, the (0.65) and (0.8) numbers indicate the (α + β) parameter that controls in particular the percentage of long-range exchange. The @LDA and @COHSEX columns indicate evGW calculations with either DF T -LDA or selfconsistent COHSEX eigenstates as a starting point. Numbers in parenthesis are the oscillator strengths. Oscillator strengths in the (α + β = 0.65) CAM-B3LYP column are taken from Ref.[START_REF] Yanai | A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)[END_REF].

a

Refs.

[START_REF] Yanai | A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)[END_REF][START_REF] Peach | Excitation energies in density functional theory: An evaluation and a diagnostic test[END_REF]

b Ref.

[START_REF] Yanai | A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)[END_REF]

c Ref.

[START_REF] Serrano-Andres | Theoretical Study of the Electronic Spectroscopy of Peptides. III. Charge-Transfer Transitions in Polypeptides[END_REF]

(Table

II

, structure 1a)

Table 3 .

 3 4.: Calculated lowest transition singlet energies (eV). The GW/BSE results calculated in the present study are compared to the TDLDA, TD-PBE, TD-B3LYP,

Table 3 .

 3 [START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF].: Calculated GW ionization energy (IE), electronic anity (EA) and HOMO-LUMO gap E g . The given electron-hole binding energy E e-h

	/BSE
		CT
	3.44	2.57	2.56
		GW	GW/BSE
		GW/BSE
	GW/BSE
		22	GW/BSE
	GW/BSE		4

  calculations yield a total EPC strength of 73 meV and 76 meV, respectively, for the coupling of all modes to the LUMO level (see below). This is drastically smaller than available experimental values extracted from PES experiments on C 60 in the gas phase, which range from 107 meV to 158 meV. A signicant increase on the DF T level could be achieved by employing the hybrid functional B3LYP, yielding an EPC strength of 94 meV[START_REF] Laamme-Janssen | Electron-phonon coupling in C 60 using hybrid functionals[END_REF]. Moreover, by varying the amount of exact exchange in the hybrid functional from the original 20% value to 30%, the EPC strength could be augmented to up to 111

Table 3 .

 3 7.: Calculated EPC for graphene, which corresponds to the splitting of the degenerate HOMO and LUMO π-levels at the Dirac point. Values given in (eV /Å) 2 . The lower panel provides a visualization of the EPC as calculated within the dierent approaches, where the blue background represents the DF T -LDA value.

			65.5 (62.8 a )	69	77	81
	K-A 1	90	197 (193 a )	209	255	237

a Ref.

[START_REF] Lazzeri | Impact of the electron-electron correlation on phonon dispersion: Failure of LDA and GGA DFT functionals in graphene and graphite[END_REF]

;

  This signicantly simplies equation (A.27):L t 1 -t 2 ; t 2 -t 1 = = L QIP t 1 -t 2 ; t 2 -t 1 +i ´dt 3 t 4 L QIP t 1 -t 3 ; t 4 -t 1 W stat (x 3 x 4 ) δ (t 3 -t 4 ) L t 4 -t 2 ; t 2 -t 3 -i ´dt 3 t 4 L QIP t 1 -t 3 , t 3 -t 1 v (x 3 x 4 ) δ (t 3 -t 4 ) L(t 4 -t 2 , t 2 -t 4 ) = L QIP t 1 -t 2 ; t 2 -t 1 +i ´dt 4 L QIP t 1 -t 4 ; t 4 -t 1 W stat (x 3 x 4 ) L t 4 -t 2 ; t 2 -t 4 -i ´dt 4 L QIP t 1 -t 4 , t 4 -t 1 v (x 3 x 4 ) L(t 4 -t 2 , t 2 -t 4 ) L t 1 -t 2 ; t 2 -t 1 = = L QIP t 1 -t 2 ; t 2 -t 1 +i ´dt 4 L QIP (t 1 -t 4 ; t 4 -t 1 ) W stat (x 3 x 4 ) L t 4 -t 2 ; t 2 -t 4 -i ´dt 4 L QIP (t 1 -t 4 , t 4 -t 1 ) v (x 3 x 4 ) L(t 4 -t 2 , t 2 -t 4

The factor N/ (N -1) accounts for the fact that the particles are identical and indistinguishable.

Here, the factor N/ (N -1) disappears, since the two considered electrons are distinguished by their spin.

Subsequently, the evolution operator S(t, t ) = e -i Ĥ(t-t ) propagates the system from time t

P (1, 2) = -i ´d34 G(1, 3) Γ(3, 4; 2) G(4, 1) W (1, 2) = v(1, 2) + ´d34 v(1, 3) P (3, 4) W (4, 2).

Based on equation(1.43), ε2 can be evaluated in the vicinity of critical points of the band structure, the so-called van Hove singularities. Its spectral characteristics depend on the type and the dimension of the singularity. Equation (1.44) has been derived for a 3D-M0 singularity in a periodic semiconductor with parabolic bands. See Refs.[START_REF] Yu | Fundamentals of Semiconductors -Physics and Materials Properties[END_REF][START_REF] Schley | Optische Eigenschaften von InN und InN-basierten Halbleitern[END_REF][START_REF] Winzer | Optische und elektronische Eigenschaften von AlGaN/GaN-Heterostrukturen[END_REF] for more details.

We performed a convergency test with a GW correction on the 10/15/20 highest occupied and lowest unoccupied states. The resulting excitation energies for the W 1, W 2, CT a and the CT b exciton converge within 0.05 eV with respect to the number of self-consistently corrected (un)occupied states.

It is to be observed that due to computational costs at the time of publication, the CC2 calculations have

ε k V k (r)V * k (r ) (ω l +ε k -iδ)(ω l -ε k +iδ) .

e -iqB e -q 2 4λ .

The two-body Green's function G 2 An important conceptual quantity which will be needed in the following is the so-called two-particle Green´s function G 2 [START_REF] Strinati | Application of the Green's functions method to the study of the optical properties of semiconductors[END_REF][START_REF] Bruus | Many-Body Quantum Theory in Condensed Matter Physics -An Introduction[END_REF]. Analogue to the denition of the one-electron Green's function G e 1 , (i ) 2 G e 2 (1234) = Ψ N 0 Φ(r 1 , t 1 ) Φ(r 2 , t 2 ) Φ † (r 4 , t 4 ) Φ † (r 3 , t 3 ) Ψ N 0 θ t -t , describes the probability of nding an electron at (r 1 , t 1 ) and a second at (r 2 , t 2 ) after the injection of two electrons at (r 3 , t 3 ) and (r 4 , t 4 ), respectively. Similar, a Green's function for the propagation o two holes, G h 2 , can be dened, which in turn is combined with G e 2 to the time-ordered two-particle Green's function G 2 :

G 2 (1234) = 1 (i ) 2 Ψ N 0 T Φ(r 1 , t 1 ) Φ(r 2 , t 2 ) Φ † (r 4 , t 4 ) Φ † (r 3 , t 3 ) Ψ N 0 .

(A.5)

As a reminder, we are working in second quantization, so that Ψ N 0 . . . Ψ N 0 does not denote an integral ´d3 r, but an averaging over Fock states. G 2 is an important quantity giving direct access to many non-equilibrium properties like the electric conductivity or the magnetic susceptibility. It can be written in terms of the oneparticle Green's function G 1 and a non-local external potential U ext which is introduced in addition to the external core ion potential V ext included in h 0 . It is an articially introduced external potential which will be set to zero in the end:

G 2 1, r t; 2, rt + = G 1 (1, 2) G 1 r t, rt + -∂G 1 (1, 2) ∂U ext (r, r , t) .

(A.6)

For local potentials we nd:

The one-particle Green's function thus generates higher-order Green's function under the inuence of an external potential. The derivation of the above formulas starts by transforming G 1 from the Heisenberg representation into the interaction picture:

where the time evolution operator S ev is given by: T [S ev (∞, -∞)] = T e -i ´∞ -∞ dt ´drdr Φ † (r,t + )Uext(r,r ,t) Φ(r ,t) .

The time-dependence of the eld operators is still governed by the equilibrium Hamiltonian, while the time-dependence due to the additional potential is completely contained in S ev .

Dierentiating G 1 in the interaction representation with respect to a local external eld U ext and introduce the two-particle Green's function G 2 (A.5):

Finally one obtains for the equation of motion of the single-particle Green's function:

Concerning the shorthand notation, 1 represents (r, t), 1 ≡ (r , t ), 2 ≡ (r , t) and 2 + ≡ (r , t + η), where η with η → 0 is introduced to ensure the right time order.

The Dyson equation Following Schwinger's functional derivative approach for the derivation of Hedin's equations, we introduce a time-dependent external potential U ext , which is set to zero at the end of the derivation. This way, the two-particle Green's function in equation (A.10) can be substituted by equation (A.7):

This results in:

i ∂G( 11)

⇒ i ∂G( 11)

For the last identity we introduced the Hartree potential V H (r 1 ):

where we used that -i G(22 + ) equals the electron density n(2), which is the expectation value of the density operator n(2) = Φ † (2) Φ(2) as dened in Chapter I of the Appendix.

Moreover, we dened the self-energy Σ as:

where we used that δ (12) = ´d3 G -1 (13) G [START_REF] Hohenberg | Inhomogeneous Electron Gas[END_REF] . The expression for Σ can be further modied considering equation (A.3) and (A.2), leading to:

Consequently, for the EOM expressed in terms of the self-energy, we obtain:

Schwinger's functional derivative approach for the Hedin's equations The Dyson equation derived within Schwinger's functional derivative approach can be directly used to determine the irreducible vertex function Γ:

,

where ∂V tot = ∂U ext + ∂V H . It is important to keep in mind that G 0 originates from a system governed by the Hamiltonian H 0 = -1 2 ∇ 2 + V ext + V H , where V ext accounts for the core ion potential. Inserting expression (A.16) for G 0 , applying the chain rule (A.2) and using relation (A.3) leads to:

+ ˜d4d5 ∂Σ (12) ∂G( 45)

∂V H (3) + ˜d4d5 ∂Σ (12) ∂G( 45)

= δ (1, 2) δ (1, 3) + ˜d4d5 ∂Σ (12) ∂G( 45)

= δ (1, 2) δ (1, 3) -˜d4567 ∂Σ (12) ∂G( 45) G (46) ∂G -1 (67) ∂Vtot(3) G (75) .

This nally results in the Hedin equation for the irreducible vertex function Γ:

⇒ Γ(1, 2; 3) = δ(12)δ (13) + ˘d4567 ∂Σ (12) ∂G (45) G (46) Γ (6, 7; 3) G (75) .

(A. [START_REF] Shockley | Detailed balance limit of eciency of p-n junction solar cells[END_REF] Analogously, by using identity (A.3) and the denition of the vertex function Γ, the irreducible polarizability P (1, 2) ≡ ∂n ind (1) ∂Vtot(2) can be derived:

= ˆd3 -1 (13) v [START_REF] Hohenberg | Inhomogeneous Electron Gas[END_REF] .

Using the chain rule (A.2) and the denition of the Hartree potential (A.12), the inverse dielectric function can be expressed via:

= δ (12) + ´d34 v (13) P (34) -1 [START_REF] Onida | Electronic excitations: density-functional versus many-body Green's-function approaches[END_REF] .

From this, it follows for the screened Coulomb potential W (12) = ´d3 -1 (13) v(32): The single-particle Green's function can be written in a diagonal single-particle Hartree basis as follows:

.

Expanding the real part of the self-energy ReΣ (ω) around the quasiparticle poles ε qp in a

Taylor series and assuming ImΣ (ω) → 0,

yields for the Green's function a decomposition into a coherent and incoherent part:

The introduced quasiparticle weight Z qp,i is dened as

,

The obtained equation is still general and in order to arrive at the quasiparticle equation we only consider z at the complex quasiparticle energy ε qp , i.e. the pole of

Dening the quasiparticle wave functions as:

where we neglect the frequency dependency of the wave functions, and further imposing z = ε qp , we obtain the quasiparticle equation:

A.4.3. Approximation of the quasiparticle energy in a state-independent framework

The dicult part in calculating the quasiparticle energy following

is the dependence of the self-energy Σ on the quasiparticle energy ε qp,ν itself. An approximation is to expand Σ up to the linear term in a Taylor series around the known DFT Kohn-Sham energy ε KS ν :

Inserting the above expansion in equation (A.22) yields:

which can be transformed as follows:

to the nal equation:

A.5. The COHSEX formulation

We want to derive an expression for the self-energy Σ, where the latter is separated into two terms, namely the Coulomb hole Σ COH and screened exchange Σ SEX contribution:

For the derivation, we use the non-interacting Green's function G 0 ,

and the screened Coulomb potential W in its Lehmann representation:

where δ is an innitesimal parameter and where V k (r) represents uctuation potentials,

The neutral excitation energies are given by ε k = E N,k -E N,0 , where E N denote total energies of the N -particle system.

The self-energy integral is transformed into a complex integral following ω → ω = ω +iω = R e iϕ with R 2 = (ω ) 2 + (ω ) 2 and ϕ = arctan ω ω :

. This is advantageous, since complex contour integration techniques can be used. A detailed overview of the theorems used in the following can be found in Appendix A.9. The residue theorem states that a closed path integral over a function g(z) with z∈ C yields:

if no poles are enclosed if poles at z i are enclosed.

The residue Res (g, z i ) can be calculated following:

where n is the order of the pole. Expanding g (ω) results in:

That means we can split the problem in four subproblems, following:

˛C dω g (ω) = ˛dω g 1 (ω) + ˛dω g 2 (ω) + ˛dω g 3 (ω) + ˛dω g 4 (ω) .

To arrive at the COHSEX representation, the closed contour is chosen to consist of the entire real axis and a semi-circle with innite radius R in the upper half complex plane (see Fig.

A.1):

˛C dω g (ω) = ˛C1 dω g (ω) + ˛C2 dω g (ω) .

Since g 1 has only poles in the third and fourth quadrant, the contour encloses no poles and consequently the integral vanishes. Concerning the remaining integrals g 2-4 , the residues have to be calculated. Corresponding to Jordan's lemma, the integral along contour C 2 vanishes for R → ∞ in the case of higher order poles n > 1. This holds both for contours in the upper and in the lower half plane and concerns g 2 and g 4 . The integrand g 3 has poles of rst order, however, corresponding to Jordan's Lemma, for η > 0 the factor e iη ω ensures that the integral along a semi-circle in the upper half plane vanishes. 1 Consequently, one nds:

In order to arrive at the COHSEX formulation, we proceed in that way that we calculate the individual residues which are then grouped together in contributions originating from 1 Jordan's lemma states that for a factor e iωt and t > 0, the semi-circle has to be in the upper half plane.

For a semi-circle in the lower half plane, the integral diverges. For the same reasons, a semi-circle in the lower half plane has to be chosen for t < 0.

A.7. Lehmann representation of the irreducible polarizability

In the following, the Lehmann representation for the irreducible polarizability P (12) within GW -RP A is derived. In principle, P is a two-point quantity, however, we calculate with its four-point extension in order to draw direct conclusions for the quasi-independent-particle (QIP) polarizability:

The latter describes the independent propagation of an electron and its hole via the timeordered one-particle Green's functions G. The notation quasi-independent refers to the fact that the two particles indeed propagate without interacting with each other, however, both are quasi-particles and interact with the surrounding medium. The independent-particle quantity L IP can be obtained by inserting G 0 1 instead of G into the above equation:

To come back to the actual problem, P GW reads:

Assuming that the electron and the hole propagate simultaneously, we set t 1 = t 1 and t 2 = t 2 .

This yields:

P GW r 1 , r 2 ; r 1 , r 2 ; τ ≡ -iG r 1 , r 2 , τ G r 2 , r 1 , -τ , where τ = t 1 -t 2 . To derive the Lehmann representation, i.e. in order to express it in the basis spanned by the eigenfunctions of the Hamiltonian operator, we start by a Fourier transformation to frequency space. Further, we make use of the convolution theorem,

which states that the Fourier transform of a product of two functions is given by the convolution in Fourier space. Applied to P GW we obtain:

Inserting the one-particle Green's function in its Lehmann representation gives:

The integral can be solved by going to the complex plane according to analytic continuation (ω → ω = ω + iω , R = (ω ) 2 + (ω ) 2 ). As a consequence, a closed integral in the complex plane has to be evaluated using complex contour deformation theory. A detailed overview of since I 2 and I 3 have poles both in the upper and in the lower half plane, the residues have to be calculated:

The latter can be obtained by solving:

Choosing a semi-circle in the upper half plane, poles at ωm2 = ε m2 + iη are enclosed for I 2 . This yields for the residue:

.

For I 3 , poles at ωm1 = -E + ε m1 + iη are enclosed by a contour in the upper half plane, leading to:

.

Combining the obtained results leads to:

where we set ε l1 = ε l2 and ε m1 = ε m2 , because the discrimination had only been introduced

for the sake of clarity in the derivation.

From this result, the two-point irreducible polarizability within GW -RP A can be contracted:

Moreover, the Lehmann representation of P within the RP A can be readily deduced by replacing the Lehmann amplitudes f m,l and the excitation energies ε m,l by single-particle wave functions φ m,l and energies ε 0 m,l :

Furthermore, from equation (A.24), the quasi-independent-particle polarizability L QIP as needed for the Bethe-Salpeter equation can be readily obtained in its Lehmann representation by:

L QIP (1, 2; 1 , 2 ; τ ) = G (1, 2 ; τ ) G (2, 1 ; -τ ) = iP GW (1, 2; 1 , 2 ; τ )

L QIP (r 1 , r 2 ; r 1 , r 2 ; E) = iP GW (r 1 , r 2 ; r 1 , r 2 ; E)

E+(ε l -εm)-iη .

A.8. The Bethe-Salpeter equations Derivation of the Bethe-Salpeter equation for L In order to account for excitonic eects in optical absorption, we introduce the two-particle correlation function L:

which contains both the independent propagation of two particles through the one-particle Green's functions G 1 and their coupled motion through the two-particle Green's function G 2 .

From equation (A.6), we directly nd:

.

Using relation (A.3) yields:

From the Dyson equation:

where M is the mass operator containing all possible interactions and where G 0 1 is the one- particle Green's function for a system of non-interacting particles, we obtain:

Inserting this in the equation for L gives:

where we applied the chain rule (A.2) for the derivation of M with respect to the external potential and where

Simplifying the obtained equation even further, we nally arrive at a Dyson-like equation for

where the two-particle propagation is described by both the propagation of two quasi-independent particles and their motion coupled by an eective two-particle interaction, the so-called kernel

The Dyson-like equation for L has to be valid for arbitrary external time variables t and t , resulting in a compact notation:

which is known as the Bethe-Salpeter equation for L [START_REF] Strinati | Application of the Green's functions method to the study of the optical properties of semiconductors[END_REF][START_REF] Onida | Electronic excitations: density-functional versus many-body Green's-function approaches[END_REF][START_REF] Bussi | Eects of the Electron-Hole Interaction on the Optical Properties of Materials: the Bethe-Salpeter Equation[END_REF][START_REF] Sottile | Response functions of semiconductors and insulators: from the Bethe-Salpeter equation to time-dependent density functional theory[END_REF][START_REF] Bruneval | Exchange and Correlation in the Electronic Structure of Solids, from Silicon to Cuprous Oxide: GW Approximation and beyond[END_REF][START_REF] Rebolini | Electronic excitation energies of molecular systems from the Bethe-Salpeter equation: Example of the H2 molecule[END_REF]. Once the Bethe-Salpeter equation is solved, L is hence contracted to χ following χ (1, 2) = -iL (1, 2; 1, 2). This yields:

-iL(1, (3, 5; 4, 6) [-iL(6, 2; 6, 2)]

Reducible and irreducible quantities In order to single out the Coulomb term from the eective two-particle interaction K, we split the mass operator into the Hartree term and the self-energy Σ:

) . Inserting this into the kernel K gives: K = ∂Σ(3,4) ∂G 1 (5,6) + ∂V H (3,4) ∂G 1 (5,6)

= ∂Σ(3,4) ∂G 1 (5,6) -iδ (3, 4) ´d7 v (3, 7)

∂G 1( 7,7 + ) ∂G 1 [START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF][START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF] = ∂Σ(3,4) ∂G 1 (5,6) -iδ (3, 4) δ (5, 6) v [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF][START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF] .

That means we can separate the kernel K into the Hartree term K H = -iδ (3, 4) δ (5, 6) v [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF][START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF] and the remainder K R = ∂Σ (3,4) ∂G 1 [START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF][START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF] . This splitting can be used to dene the irreducible quantity L, which obeys: (3, 5; 4, 6) L(6, 2; 5, 2 ).

Having solved the integral equation for L, the reducible L can be obtained via:

L(1, 2; 1 , 2 ) = L(1, 2; 1 , 2 ) + ˆd3456 L(1, 4; 1 , 3) K H (3, 5; 4, 6) L(6, 2; 5, 2 ).

The choice of the kernel So far, the kernel K accounts for all correlations and is hence a very complicated object. A possible simplication consists of approximating the self-energy Σ by its bare exchange part Σ x :

For the kernel, we thus have:

K T DHF (3, 5; 4, 6) = ∂Σ x (3,4) ∂G 1 (5,6) -iδ (3, 4) δ (5, 6) v (3, 5) = i ∂G (3,4) ∂G 1 (5,6) v (3, 4) -iδ (3, 4) δ (5, 6) v (3, 5) = i δ(3, 5)δ (4, 6) v (3, 4) -iδ (3, 4) δ (5, 6) v [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF][START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF] .

Inserting this into the Bethe-Salpeter equation yields:

+i ´d56 G 1 (1, 5) G 1 (6, 1 ) v (5, 6) L(6, 2; 5, 2 ) -i ´d46 G 1 (1, 4) G 1 (4, 1 ) v (4, 6) L(6, 2; 6, 2 ), which represents a time-dependent Hartree-Fock approach. The rst summand accounts for the bare or Hartree-Fock exchange and the second for the classical Hartree contribution.

Correlation is not taken into account.

Another approach, accounting also for screening eects, is to use the already introduced with L QIP x 1 x 2 ; x 1 x 2 ; ω = 1 2π ˆdω G 1 x 1 x 2 ; ω + ω G 1 x 2 x 1 ; ω being the Fourier transform of the already introduced quasi-independent two-particle correlation function L QIP , where we also made use of the convolution theorem. In a short notation, we nally arrive at:

L x 1 x 2 ; x 1 x 2 ; ω = L QIP x 1 x 2 ; x 1 x 2 ; ω + ´dx 3 x 4 x 5 x 6 L QIP x 1 x 4 ; x 1 x 3 ; ω K (x 3 x 4 ; x 5 x 6 ) L x 6 x 2 ; x 5 x 2 ; ω ,

where

is the many-body perturbation theory kernel. We have to keep in mind that the Fourier transformation in the presented way is only possible due to the use of a static, i.e. strongly approximated, screened Coulomb potential W .

The eective two-particle problem Using the Lehmann representation of the Green's function in Fourier space, we can nd an explicit expression for the quasi-independent two-particle correlation function L QIP (see Appendix A.7):

-iL QIP x 1 x 2 ; x 1 x 2 ; ω ≈ m,l

Here, L QIP is approximated by using single-particle wave functions φ m,l instead of Lehmann amplitudes f m,l . The energies ε m,l originate from a DF T -KS calculation (i.e. L QIP ≈ L IP )

or can be quasiparticle energies obtained within a GW calculation. The index m denotes occupied, l unoccupied states. The form of L QIP suggests to work in transition space, i.e. to dene a two-particle excitonic basis {ψ exc },

based on the same single-particle basis functions φ m,l as G 0 and L RP A QIP . The transformation of any four-point quantity F (x 1 x 2 x 3 x 4 ) to this basis follows:

For the Bethe-Salpeter equation (A.28), we thus obtain: At this point it is useful to multiply the Bethe-Salpeter equation with a factor (-i) following equation (A.26), since in the end of the calculation one is interested in contracting L to χ:

χ (1, 2) = -iL(1, 2; 1 + , 2 + ).

One thus nds for the matrix equation: -iL n 1 n 2 n 3 n 4 QIP (ω) = -i ´dx 1 x 1 x 2 x 2 φ n 1 x 1 φ * n 2 (x 1 ) L QIP x 1 x 2 ; x 1 x 2 ; ω φ * , where m stands for occupied and l for unoccupied states and where we assumed an orthogonal and complete basis:

´drφ * i (r) φ j (r) = δ (i, j) i φ * i (r) φ i (r ) = δ (r -r ) .

We can draw an important conclusion from this result, namely that [L QIP ] is diagonal in the transition basis (n 1 = n 3 , n 2 = n 4 ).