Density functional theory (DFT) is a popular approach to solving the many-electron Schrödinger equation, in order to investigate the properties of matter from first principles. While DFT can give the exact ground state electronic density of a system, in practice, an approximation is required for the many-body effects contained in the exchange-correlation functional. The accuracy of calculations performed using DFT is strongly related to the choice of approximation. In this thesis we will investigate and build upon a fully non-local approach to modeling exchange-correlation in the form of the weighted density approximation (WDA). Central to the WDA is the model function chosen for the coupling-constant averaged pair-correlation function (PCF). We show that a model PCF can be selected from a set to give excellent bulk properties for a particular system. However, this model is not necessarily transferable to other systems and there is no method of selecting an appropriate model from this set a priori. We suggest that the model PCF can be improved systematically by satisfying known physical constraints. One such constraint is the Kimball cusp condition, which we include in our model and implement. We demonstrate that surfaces are systems that require a non-local treatment of exchange-correlation by applying the WDA to metal surfaces and investigate the dissociative adsorption of H2 on the Cu(100) surface. A new framework for a model PCF with spin resolution is developed, providing a route for more physical constraints to be satisfied within a weighted spin density approximation (WSDA). A simple model is suggested and implemented and comparisons are made to the coupling-constant averaged PCF in the homogeneous electron gas. We then apply a selection of our new models to a number of materials and show that our model for the WSDA gives improved band gaps over the local density approximation. Application of the WSDA to spin polarised materials reveals shortcomings in our simple model. We then suggest further refinements to our implementation of the WSDA. It is expected that the inclusion of additional physical constraints will systematically improve results given in a weighted-density based approximation to exchange-correlation.

<p>Approximating the exchange-correlation energy in density functional theory (DFT) is a crucial task. As the only missing element in the Kohn-Sham DFT, the search for better exchange-correlation functionals has been an active field of research for fifty years. Many models and approximations are known and they can be summarized in what is known as the Jacob’s ladder. All the functionals in that ladder are local in the sense that they rely on the information of only one electronic coordinate. That is, even though the exchange-correlation hole, the cornerstone in density functional theory, is a two-electron coordinate quantity, one of the coordinates is averaged over in “Jacob’s ladder functionals.” This makes the calculations considerably more efficient. On the other hand, some of the important constraints on the form of the exchange-correlation functional become inaccessible in the one-point forms. The violation of these constraints leads to functionals plagued by systematic errors, leading to qualitatively incorrect descriptions of some chemical and physical processes.</p> <p>In this thesis the idea of a weighted density approximation (WDA) is explored. More specifically, a symmetric and normalized two-point functional is proposed for the exchange-correlation energy functional. The functional is based entirely on the hole for the uniform electron gas. By construction, these functionals fulfill two of the most important constraints: the normalization of the exchange-correlation hole and the uniform electron gas limit. The findings suggest that we should pursue a whole new generation of “new Jacob’s ladder” functionals.</p> <p>A further step was considered. Given the relevance of the long-range behavior of the exchange-correlation hole, a study of the electronic direct correlation function was performed. The idea was to build up the long-range character of the hole as convoluted pieces of the simple and short-ranged direct correlation function. This direct correlation function provides better results, at least for the correlation energy in the spin-polarized uniform electron gas.</p> <p>The advantage of one-point functionals is their computational efficiency. We therefore attempted to develop new methods that mitigate the relative computational inefficiency of two-point functionals. This led to new methods for evaluating the six-dimensional integrals that are inherent to the exchange-correlation energy.</p> / Doctor of Philosophy (PhD)

Neste trabalho propomos a representação de uma densidade bivariada segundo a natureza geométrica das marginais. Uma densidade contínua pode ser aproximada pela exponencial de um polinômio num domínio limitado. Tal aproximação permite estudar a influência do comportamento marginal e condicional sobre a densidade conjunta. Esse estudo é realizado encontrando os conjuntos dos possíveis valores dos coeficientes do polinômio segundo a informação dada sobre as densidades marginais ou as densidades condicionais. Também analisamos os coeficientes segundo algumas medidas de dependência. Concluímos mostrando a influência do comportamento marginal e condicional sobre variáveis aleatórias discretas. / In this work we propose a representation of a bivariate density according to the geometrical nature of the marginals. A continuous density can be approximated by the exponencial of a polynomial in a finite domain. Such approximation let us study the influence of the marginal and conditional behavior on the joint density. This study is done by finding the possible values of the polynomial coefficients upon the given information about the marginal or the conditional densities. We also analyze the coefficients according some dependence measures. We conclude showing the influence of the marginal and conditional behavior on discrete random variables.

We use the first-principles molecular-dynamics¡@simulation method (MD), which is based on the density functional theory (DFT) with local-density approximation (LDA), to calculate the sticking property of a carbon atom on hydrogen covered C(100) surface. We focused on trajectories and kinetic energy transfer of the gas-phase C atom for four incident angles of =0, £k/8, £k/6 and £k/4. We find that the calculated trajectories and the kinetic energy transfer of the gas-phase atom, Cn, overall are not very sensitive to the change of the incident angle. The insensitivity of the sticking property on the incident angle may be due to a large chemisorption energy, which bends the trajectory of Cn toward the surface, so that Cn is confined to move within a small range.

Neste trabalho propomos a representação de uma densidade bivariada segundo a natureza geométrica das marginais. Uma densidade contínua pode ser aproximada pela exponencial de um polinômio num domínio limitado. Tal aproximação permite estudar a influência do comportamento marginal e condicional sobre a densidade conjunta. Esse estudo é realizado encontrando os conjuntos dos possíveis valores dos coeficientes do polinômio segundo a informação dada sobre as densidades marginais ou as densidades condicionais. Também analisamos os coeficientes segundo algumas medidas de dependência. Concluímos mostrando a influência do comportamento marginal e condicional sobre variáveis aleatórias discretas. / In this work we propose a representation of a bivariate density according to the geometrical nature of the marginals. A continuous density can be approximated by the exponencial of a polynomial in a finite domain. Such approximation let us study the influence of the marginal and conditional behavior on the joint density. This study is done by finding the possible values of the polynomial coefficients upon the given information about the marginal or the conditional densities. We also analyze the coefficients according some dependence measures. We conclude showing the influence of the marginal and conditional behavior on discrete random variables.

Title: Stochastic Catastrophe Model Cusp Author: Jan Voříšek Department: Department of Probability and Mathematical Statistics Supervisor: Prof. Ing. Miloslav Vošvrda, CSc., Czech Academy of Sciences, Institute of Information Theory and Automation Abstract: The goal of this thesis is to analyze the stochastic cusp model. This task is divided into two main topics. The first of them concentrates on the stationary density of the cusp model and statistical testing of its bimodality, where power and size of the proposed tests are simulated and compared with the dip test of unimodality. The second main topic deals with the transition density of the stochastic cusp model. Comparison of approximate maximum likelihood approach with traditional finite difference and numerical simulations indicates its advantage in terms of speed of estimation. An approximate Fisher information matrix of general stochastic process is derived. An application of the cusp model to the exchange rate with time-varying parameters is estimated, the extension of the cusp model into stochastic bimodality model is proposed, and the measure of probability of intrinsic crash of the cusp model is suggested. Keywords: stochastic cusp model, bimodality testing, transition density ap- proximation

In this thesis, we provide convergent numerical solutions to non-linear forward-BSDEs (Backward Stochastic Differential Equations). Applications in mathematical finance, financial economics and financial econometrics are discussed. Numerical examples show the effectiveness of our methods.

Ce travail de thèse est une étude par des méthodes de simulation de structure électronique du phénomène de dégradation en milieu aqueux de la zircone yttriée. La zircone yttriée est notamment utilisée pour la fabrication de prothèses dont la durée de vie dépend du matériau et de son environnement. Pour ces applications, la zircone yttriée est de structure tétragone, mais en fonction du dopage en yttrium et de l'environnement, cette phase est en compétition avec des structures monoclinique et cubique. Cette compétition est cruciale dans ce travail car elle peut à la fois, augmenter la résistance, ou favoriser la détérioration de ce matériau. L'étude réalisée se concentre sur des taux de dopage inférieurs à 20% mol. en oxyde d'yttrium Y2O3.Dans un premier temps, les structures atomiques d'équilibre à basse température ont été déterminées dans les trois phases en utilisant une méthode de calcul basée sur la Théorie de la Fonctionnelle de la Densité, dans l'approximation de la densité locale. Les résultats incluent de nouvelles structures qui n'ont jusqu'alors pas été discutées dans la littérature et qui sont cohérentes avec le diagramme de phase de la zircone yttriée à basse température. Une analyse plus détaillée suggère que le mécanisme de stabilisation de la zircone yttriée tetragonale est un effet du dépistage ionocovalent particulièrement efficace dans cette phase et dont la signature dans la structure s'exprime par la présence d'ions zirconium en coordination 7. Ceci représente un point important permettant de lier stabilité et structure dans ces systèmes. Une deuxième partie de ce travail est consacrée à l’effet de l'eau dans la zircone yttriée en volume. Pour conclure, ce manuscrit représente une contribution originale à la compréhension de mécanismes à l'échelle atomique qui sont à l'origine des propriétés de la zircone yttriée et de ses applications.yttriée et de ses applications. / This PhD thesis is an electronic structure simulation work to study the stabilization mechanism of yttria stabilized zirconia (YSZ) and the phenomena of the degradation of YSZ due to the presence of an aqueous media. YSZ is used especially in dental and orthopedics application sbut its service depends on the environment. For these biomedical applications a crystallographic tetragonal YSZ (t-YSZ) structure is used, but depending on the concentration of yttria (Y2O3) and the environment, this tetragonal structure is thermodynamically in competition with a monoclinic and cubic phases. This competition is crucial in this work, because it has both effects : increase the resistance or promote the degradation of this material. The study is conducted for Y2O3 concentrations less than 20% mol. First, atomic structures of the three phases at low temperature were determined using density functional theory (DFT)with the local density approximations (LDA). The results include new structures that were not discussed in the literature and which are consistent with the phase diagram of YSZ at low temperature. A more detailed analysis suggests that the stabilization mechanism in YSZ is due to an effective ionocovalent screening : particularly in t-YSZ phase, its signature is expressed in Zr-ions with a coordination number of 7. This represents an important point for linking stability and structure in these systems. A second part of this work is devoted to the effect of water on YSZ bulk systems which leads to low temperature degradation (LTD) of YSZ based bioceramics materials. In conclusion, this PhD thesis represents an original contribution to the understanding of the mechanism and properties of YSZ and its applications at the atomic scale.

Im Hinblick auf das immense Potential von Diamant als Material für die Mikroelektronik wurden im Rahmen dieser Arbeit undotierte und dotierte Diamantfilme mittels chemischer Gasphasenabscheidung auf Silizium präpariert und anschließend auf ihre elektronischen Eigenschaften hin untersucht. Für Letzteres wurde hauptsächlich die Elektronen-Energieverlustspektroskopie in Transmission verwendet. In situ Gasphasendotierung oder Ionenimplantation wurde zur Dotierung der Filme mit Bor, Lithium oder Phosphor eingesetzt. Bei der Ionenimplantation wurde aufgrund der Erzeugung von Strahlenschäden generell eine Erhöhung des sp2-Anteils beobachtet: Letzterer konnte jedoch im Falle der Bordotierung durch eine, den Implantationsprozeß folgende, Hochtemperaturtemperung wieder deutlich vermindert werden. Für die in situ Dotierung mit Bor wurde eine Verringerung des sp2-Gehaltes mit steigender Dotierkonzentration gefunden. Für den Film mit der höchsten Borkonzentration konnte auch die B1s Absorptionskante untersucht werden. Sie gibt Hinweise auf den überwiegenden Einbau der Boratome in einer tetragonalen Orientierung. Das hiermit verbundene Vorhandensein von Akzeptoren führt zu elektronischen Anregungen im Energiebereich der Bandlücke, welche mittels Infrarotspektroskopie und EELS nachgewiesen werden konnten. Aus den EELS Messungen lassen sich Akzeptorkonzentrationen berechnen, welche wiederum den hohen Anteil an tetraedrisch eingebauten Boratomen bestätigen. Desweiteren untersucht wurden, als interessante Materialklasse mit weitreichendem technologischem Potential, undotierte und stickstoffdotierte, diamantartige amorphe Kohlenstoffilme und hierbei insbesondere die Abhängigkeit der elektronischen und optischen Eigenschaften von der Ionenenergie und dem Stickstoffpartialdruck während der Filmpräparation. Die Plasmonenergien, Massendichten, sp3-Anteile und die optischen Bandlücken der Filme wurden quantitativ bestimmt, wobei das jeweilige Maximum bei einer Ionenenergie von 100 eV gefunden wurde. Alle eben genannten Größen verringern sich kontinuierlich mit zunehmendem Stickstoffanteil. Eine Kramers-Kronig Analyse der Verlustspektren gibt Zugriff auf den Real- und Imaginärteil der dielektrischen Funktion und damit auf das Spektrum der Einteilchenanregungen. Die Hybridisierung der Kohlenstoff- und der Stickstoffatome wurde detailliert aus den jeweiligen 1s Absorptionskanten bestimmt. Weiterhin wurde Diamant als Modellsystem eines Festkörpers mit rein kovalenten Bindungen untersucht, insbesondere die Verlustfunktion von Diamant entlang mehrerer Hochsymmetriekristallrichtungen über einen großen Energie- und Impulsbereich. Aus den EELS Messungen erschließt sich unmittelbar die stark anisotrope Plasmonendispersion von Diamant. Aus dem Vergleich der experimentellen Spektren mit ab initio LDA Rechnungen, die sowohl Kristallokalfeldeffekte als auch Austausch- und Korrelationseffekte beinhalten, lassen sich direkt Rückschlüsse auf den Einfluß der verschiedenen Effekte ziehen. Schon im optischen Limit, aber umso mehr mit steigendem Impulsübertrag q, wird eine Überlagerung der kollektiven Plasmonanregung mit Einteilchenanregungen im Energiebereich des Plasmons beobachtet, woraus eine Kopplung zwischen beiden Arten von Anregungen resultiert. Abgesehen vom deutlichen Einfluß der Bandstruktur auf die Plasmonendispersion läßt die überaus inhomogene Elektronenverteilung von Diamant auf nicht zuvernachlässigende Kristallokalfeldeffekte schließen. Der Vergleich zwischen experimentellen und berechneten Spektren zeigt deutlich, wie die Kristallokalfeldeffekte in der Tat mit steigendem Impulsübertrag an Gewicht zunehmen und die Struktur der Verlustfunktion mitbestimmen. / In the context of the immense potential of diamond as a material for use in the microelectronics industry, in this thesis pristine and doped diamond films have been deposited on silicon using chemical vapour deposition. Subsequently their electronic properties have been investigated using mainly electron energy-loss spectroscopy. Doping of the films with boron, lithium or phosphorous was carried out either via in-situ gas phase doping during film growth or using ion implantation. Upon ion implantation an increase of the carbon content with sp2 hybridisation has generally been found due to ion beam induced damage. In the case of boron doping it was possible to significantly reduce this sp2-contribution using a high temperature anneal. For the in-situ doping with boron, upon increasing doping concentration a decrease of the sp2-contribution was found. For the sample with the highest boron content the boron 1s absorption edge could also be investigated, providing evidence for the preferential incorporation of the boron atoms into tetrahedrally co-ordinated sites. This boron incorporation leads to the existence of electronic excitations in the energy range of the band gap, which could be observed using both infrared and electron energy-loss spectroscopy. From the electron energy-loss measurements it was possible to calculate acceptor concentrations which were consistent with the large amount of tetrahedrally co-ordinated boron atoms. A second theme in this thesis involved the study of pristine and nitrogen doped diamond-like amorphous carbon films, which are an interesting material class with far-reaching technological potential. Here the focus of the research concerned the dependency of the electronic and optical properties of the films upon the ion energy and the nitrogen partial pressure applied during the film preparation. The plasmon energies, mass densities, sp3 contribution and the optical band gaps of the samples were determined quantitatively, whereby the maximum in all these quantities was found to occur for ion energies of 100 eV. Furthermore, all of these characteristics were found to decrease continually with increasing nitrogen content. A Kramers-Kronig analysis of the loss spectra enabled the derivation of the real and imaginary parts of the dielectric function and with this of the complete spectrum of single particle excitations. The hybridization between the carbon and nitrogen atoms was also studied in detail from the analysis of the respective 1s absorption edges. Furthermore this thesis deals with the investigation of diamond as a model system for solids with pure covalent bonds. In particular, the loss function of diamond was measured along different high symmetry directions over a wide range of energy and momentum. Firstly, the EELS measurements showed directly the strongly anisotropic nature of the plasmon dispersion in diamond. Secondly, by the comparison of the experimental spectra with ab initio LDA-based calculations that include crystal local field effects as well as exchange and correlation contributions, conclusions can be drawn as to the influence of these quantities. In the optical limit, but even more so with increasing momentum transfer q, a superposition of the collective plasmon excitation and the single particle excitations in the energy range of the plasmon is observed. This energetic proximity results in a coupling between both types of excitations. Apart from the distinct influence of the bandstructure on the plasmon dispersion, the considerably inhomogeneous electron distribution of diamond would lead one to expect significant crystal local field effects in this system. The comparison between the experimental and the calculated spectra shows explicitly that the crystal local field effects increase with increasing momentum transfer and play an important role in defining the structure of the loss function.

Im Hinblick auf das immense Potential von Diamant als Material für die Mikroelektronik wurden im Rahmen dieser Arbeit undotierte und dotierte Diamantfilme mittels chemischer Gasphasenabscheidung auf Silizium präpariert und anschließend auf ihre elektronischen Eigenschaften hin untersucht. Für Letzteres wurde hauptsächlich die Elektronen-Energieverlustspektroskopie in Transmission verwendet. In situ Gasphasendotierung oder Ionenimplantation wurde zur Dotierung der Filme mit Bor, Lithium oder Phosphor eingesetzt. Bei der Ionenimplantation wurde aufgrund der Erzeugung von Strahlenschäden generell eine Erhöhung des sp2-Anteils beobachtet: Letzterer konnte jedoch im Falle der Bordotierung durch eine, den Implantationsprozeß folgende, Hochtemperaturtemperung wieder deutlich vermindert werden. Für die in situ Dotierung mit Bor wurde eine Verringerung des sp2-Gehaltes mit steigender Dotierkonzentration gefunden. Für den Film mit der höchsten Borkonzentration konnte auch die B1s Absorptionskante untersucht werden. Sie gibt Hinweise auf den überwiegenden Einbau der Boratome in einer tetragonalen Orientierung. Das hiermit verbundene Vorhandensein von Akzeptoren führt zu elektronischen Anregungen im Energiebereich der Bandlücke, welche mittels Infrarotspektroskopie und EELS nachgewiesen werden konnten. Aus den EELS Messungen lassen sich Akzeptorkonzentrationen berechnen, welche wiederum den hohen Anteil an tetraedrisch eingebauten Boratomen bestätigen. Desweiteren untersucht wurden, als interessante Materialklasse mit weitreichendem technologischem Potential, undotierte und stickstoffdotierte, diamantartige amorphe Kohlenstoffilme und hierbei insbesondere die Abhängigkeit der elektronischen und optischen Eigenschaften von der Ionenenergie und dem Stickstoffpartialdruck während der Filmpräparation. Die Plasmonenergien, Massendichten, sp3-Anteile und die optischen Bandlücken der Filme wurden quantitativ bestimmt, wobei das jeweilige Maximum bei einer Ionenenergie von 100 eV gefunden wurde. Alle eben genannten Größen verringern sich kontinuierlich mit zunehmendem Stickstoffanteil. Eine Kramers-Kronig Analyse der Verlustspektren gibt Zugriff auf den Real- und Imaginärteil der dielektrischen Funktion und damit auf das Spektrum der Einteilchenanregungen. Die Hybridisierung der Kohlenstoff- und der Stickstoffatome wurde detailliert aus den jeweiligen 1s Absorptionskanten bestimmt. Weiterhin wurde Diamant als Modellsystem eines Festkörpers mit rein kovalenten Bindungen untersucht, insbesondere die Verlustfunktion von Diamant entlang mehrerer Hochsymmetriekristallrichtungen über einen großen Energie- und Impulsbereich. Aus den EELS Messungen erschließt sich unmittelbar die stark anisotrope Plasmonendispersion von Diamant. Aus dem Vergleich der experimentellen Spektren mit ab initio LDA Rechnungen, die sowohl Kristallokalfeldeffekte als auch Austausch- und Korrelationseffekte beinhalten, lassen sich direkt Rückschlüsse auf den Einfluß der verschiedenen Effekte ziehen. Schon im optischen Limit, aber umso mehr mit steigendem Impulsübertrag q, wird eine Überlagerung der kollektiven Plasmonanregung mit Einteilchenanregungen im Energiebereich des Plasmons beobachtet, woraus eine Kopplung zwischen beiden Arten von Anregungen resultiert. Abgesehen vom deutlichen Einfluß der Bandstruktur auf die Plasmonendispersion läßt die überaus inhomogene Elektronenverteilung von Diamant auf nicht zuvernachlässigende Kristallokalfeldeffekte schließen. Der Vergleich zwischen experimentellen und berechneten Spektren zeigt deutlich, wie die Kristallokalfeldeffekte in der Tat mit steigendem Impulsübertrag an Gewicht zunehmen und die Struktur der Verlustfunktion mitbestimmen. / In the context of the immense potential of diamond as a material for use in the microelectronics industry, in this thesis pristine and doped diamond films have been deposited on silicon using chemical vapour deposition. Subsequently their electronic properties have been investigated using mainly electron energy-loss spectroscopy. Doping of the films with boron, lithium or phosphorous was carried out either via in-situ gas phase doping during film growth or using ion implantation. Upon ion implantation an increase of the carbon content with sp2 hybridisation has generally been found due to ion beam induced damage. In the case of boron doping it was possible to significantly reduce this sp2-contribution using a high temperature anneal. For the in-situ doping with boron, upon increasing doping concentration a decrease of the sp2-contribution was found. For the sample with the highest boron content the boron 1s absorption edge could also be investigated, providing evidence for the preferential incorporation of the boron atoms into tetrahedrally co-ordinated sites. This boron incorporation leads to the existence of electronic excitations in the energy range of the band gap, which could be observed using both infrared and electron energy-loss spectroscopy. From the electron energy-loss measurements it was possible to calculate acceptor concentrations which were consistent with the large amount of tetrahedrally co-ordinated boron atoms. A second theme in this thesis involved the study of pristine and nitrogen doped diamond-like amorphous carbon films, which are an interesting material class with far-reaching technological potential. Here the focus of the research concerned the dependency of the electronic and optical properties of the films upon the ion energy and the nitrogen partial pressure applied during the film preparation. The plasmon energies, mass densities, sp3 contribution and the optical band gaps of the samples were determined quantitatively, whereby the maximum in all these quantities was found to occur for ion energies of 100 eV. Furthermore, all of these characteristics were found to decrease continually with increasing nitrogen content. A Kramers-Kronig analysis of the loss spectra enabled the derivation of the real and imaginary parts of the dielectric function and with this of the complete spectrum of single particle excitations. The hybridization between the carbon and nitrogen atoms was also studied in detail from the analysis of the respective 1s absorption edges. Furthermore this thesis deals with the investigation of diamond as a model system for solids with pure covalent bonds. In particular, the loss function of diamond was measured along different high symmetry directions over a wide range of energy and momentum. Firstly, the EELS measurements showed directly the strongly anisotropic nature of the plasmon dispersion in diamond. Secondly, by the comparison of the experimental spectra with ab initio LDA-based calculations that include crystal local field effects as well as exchange and correlation contributions, conclusions can be drawn as to the influence of these quantities. In the optical limit, but even more so with increasing momentum transfer q, a superposition of the collective plasmon excitation and the single particle excitations in the energy range of the plasmon is observed. This energetic proximity results in a coupling between both types of excitations. Apart from the distinct influence of the bandstructure on the plasmon dispersion, the considerably inhomogeneous electron distribution of diamond would lead one to expect significant crystal local field effects in this system. The comparison between the experimental and the calculated spectra shows explicitly that the crystal local field effects increase with increasing momentum transfer and play an important role in defining the structure of the loss function.