Made available in DSpace on 2015-05-14T13:21:41Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 10515395 bytes, checksum: 24068367353a0ab8bc6c91ecaae5abf5 (MD5) Previous issue date: 2014-09-02 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Titanium dioxide (TiO2) in anatase phase has been the most studied metal oxide in the last two decade due to its potential technological aplication in many areas, such as dyes, (photo)catalysts and solar cells. Computational and experimental methods has been employed to understand the mechanical, electronical and catalytic properties of anatase. In this context, the introduction of dopants and induction of point deffects formation (mainly oxygen vacancies) are responsible for tunning those properties in many situatuions, e.g. increasing the (photo)catalytic efficience. In the present study, we investigated the effect of Ce-doping on the formation of oxygen vacancy and reduced Ti3+ and Ce3+ centers in anatase TiO2 (bulk and (001) surfaces) by means of computational quantum chemical calculations. All calculations were performed at the density functional theory (DFT) level taking into account the periodicity of the systems. The bulk phase was initially studied without dopant, and the Grimme DFT-D2 dispersion potential was reparameterized for octahedral [TiO6] interaction in anatase polymorph within the B3LYP (GTO) approach (hereafter named B3LYP-D*). Several properties were calculated from bulk with B3LYP-D* in good agreement with experimental values from anatase single crystals, such as: Wulff construction of single crystal in thermodynamic equilibrium, elastic constants, infrared and Raman vibrational frequencies, and electronic structure calculations (band structure and density of states). Two DFT approachs were used for mitigate the self-interaction error (SIE) in both reduced and/or Ce-doped systems: the on-site Dudarev DFT+U correction and the hybrid B3LYP (20% HF) functional with plane-wave or Gaussian-type basis set. Only a small local perturbation was associated with the Ce-dopant introduction in the octahedral site (bulk) and Ce(5c) (surface). The Ce-doped systems presented lower oxygen vacancy formation energy than pristine TiO2. The most stable VO configuration in the bulk were in the next neighbors from the reduced [CeO6]/ center, instead the low-coordinated [CeO5] centers. Similarly, the dopant on the subsurface distoted octahedral [CeO6]d site of (001) surface boosted the remotion of O(2c) in the outmost layer of the surface. This behavior was not observed with the dopant on the low-coordinated [CeO5] site. The Ti3+ [3d 1] and Ce3+ [4f 1] midgap states were found up to ~ 1.0 eV bellow de conduction band. These founds are in agreement with experimental evidences of the enhanced facilitation of VO formation in Ce-doped anatase, and superior (photo)catalytic activity when compared with undoped TiO2. In the general way, the vacancy formation energy decreased significantly in the following situations: (i) oxide reduction in the presence of Ce as dopant; (ii) VO in the nearest neighbor sites of the reduced [CeO6]/ octahedra (iii) introduction of two Ce dopants around VO; (iv) VO in the outmost layers plus [CeO6]/ at the subsurface. / Dióxido de titânio (TiO2) na fase anatase tem sido o óxido mais estudado nas últimas décadas devido à sua potencial aplicação tecnológica em diversas áreas, dentre as quais como pigmento, (foto)catalisador e em células solares. Não obstante, métodos experimentais e computacionais têm sido empregados na compreensão de suas propriedades mecânicas, eletrônicas e catalíticas da anatase. Nesse contexto, a introdução de dopantes com formação de defeitos pontuais do tipo vacâncias de oxigênio tem sido responsável por aumentar a eficiência desse material como catalisador sobre muitas reações. No presente estudo, investigamos, por meio de cálculos computacionais, o efeito da dopagem do TiO2 anatase com o lantanídeo Ce sobre a formação de vacâncias de oxigênio e exploramos suas consequências na formação de centros reduzidos Ti3+ e Ce3+, no bulk e na superfície (001). Todos os cálculos foram realizados ao nível da teoria do funcional da densidade (DFT) utilizando códigos computacionais que levam em consideração a periodicidade dos sistemas cristalinos (2D para superfícies ou 3D para bulk). Para uma maior compreensão da anatase sem o dopante, exploramos inicialmente o efeito das interações de dispersão entre os poliedros [TiO6] e reparametrizamos o potencial de Grimme DFT-D2 com funções de base GTO (o qual chamamos B3LYP-D*), a partir do qual várias propriedades estruturais foram obtidas, como construção de Wulff do hábito cristalino sob equilíbrio termodinâmico, propriedades elásticas, vibracionais e eletrônicas. Na presença do dopante e com os sistemas reduzidos, duas principais metodologias foram adotadas na tentativa de reduzir o erro de autointeração da DFT: o uso do funcional híbrido B3LYP e a correção de Dudarev DFT+U. Orbitais do tipo Gaussiana e ondas planas foram utilizadas como funções de base. Apenas uma pequena deformação local ocorreu com a introdução do dopante no sítio octaédrico do Ti no bulk e no sítio Ce(5c) da superfície. Em ambos bulk e superfície (001) a introdução do dopante levou à diminuição da energia de formação de VO. As configurações mais estáveis de VO no bulk ocorreram nos vizinhos próximos ao octaedro reduzido [CeO6]/ em vez daquelas que envolviam a diminuição da coordenação do dopante para [CeO5]. Analogamente, a presença do dopante na subsuperfície da (001) formando o poliedro distorcido [CeO6]d impulcionou a remoção de um O(2c) na camada mais externa da superfície, o que não foi observado quando a dopagem acontecia na última camada de cátions com coordenação [CeO-5]. Assim como no bulk, os estados intermediários no band gap associados com a vacância mais estável na superfície (001) foram referentes ao Ti3+ [3d 1] e Ce3+ [4f 1] localizados em torno de 0.5-1.0 eV abaixo da banda de condução, de acordo com achados experimentais e evidências no aumento da atividade (foto)catalítica dado pela criação destes canais de transferência eletrônica. De modo geral, a energia de formação das vacâncias de oxigênio diminuiu consideravelmente nas seguintes situações: (i) redução do óxido na presença do dopante Ce, (ii) desde que os sítios de criação das VO localizem-se nos primeiros vizinhos ao octaedro [CeO6]/ distorcido por expansão polarônica, (iii) introdução de dois dopantes, em direções opostas, em torno de VO, e (iv) criação de VO nas camadas mais externas da superfície, desde que o dopante esteja na subsuperfície e com coordenação [CeO6].

Dans la modélisation statistique, nous sommes le plus souvent amené à supposer que le phénomène étudié est généré par une structure pouvant s’ajuster aux données observées. Cette structure fait apparaître une partie principale qui représente le mieux possible le phénomène étudié et qui devrait expliquer les données et une partie supposée négligeable appelée erreur ou innovation. Cette structure complexe est communément appelée un modèle, dont la forme peut être plus ou moins complexe. Afin de simplifier la structure, il est souvent supposé qu’elle repose sur un nombre fini de valeurs, appelées paramètres. Basé sur les données, ces paramètres sont estimés avec ce que l’on appelle des estimateurs. La qualité du modèle pour les données à notre disposition est également fonction des estimateurs et de leurs propriétés, par exemple, est-ce que les estimateurs sont raisonnablement proches des valeurs idéales, c’est-à-dire les vraies valeurs. Des questions d’importance portent sur la qualité de l’ajustement d’un modèle aux données, ce qui se fait par l’étude des propriétés probabilistes et statistiques du terme d’erreur. Aussi, l’étude des relations ou l’absence de ces dernières entre les phénomènes sous des hypothèses complexes sont aussi d’intérêt. Des approches possibles pour cerner ce genre de questions consistent dans l’utilisation des tests portemanteaux, dits également tests de diagnostic. La thèse est présentée sous forme de trois projets. Le premier projet est rédigé en langue anglaise. Il s’agit en fait d’un article actuellement soumis dans une revue avec comité de lecture. Dans ce projet, nous étudions le modèle vectoriel à erreurs multiplicatives (vMEM) pour lequel nous utilisons les propriétés des estimateurs des paramètres du modèle selon la méthode des moments généralisés (GMM) afin d’établir la distribution asymptotique des autocovariances résiduelles. Ceci nous permet de proposer des nouveaux tests diagnostiques pour ce type de modèle. Sous l’hypothèse nulle d’adéquation du modèle, nous montrons que la statistique usuelle de Hosking-Ljung-Box converge vers une somme pondérée de lois de khi-carré indépendantes à un degré de liberté. Un test généralisé de Hosking-Ljung-Box est aussi obtenu en comparant la densité spectrale des résidus de l’estimation et celle présumée sous l’hypothèse nulle. Un avantage des tests spectraux est qu’ils nécessitent des estimateurs qui convergent à la vitesse n−1/2 où n est la taille de l’échantillon, et leur utilisation n’est pas restreinte à une technique particulière, comme par exemple la méthode des moments généralisés. Dans le deuxième projet, nous établissons la distribution asymptotique sous l’hypothèse de faible dépendance des covariances croisées de deux processus stationnaires en covariance. La faible dépendance ici est définie en terme de l’effet limité d’une observation donnée sur les observations futures. Nous utilisons la notion de stabilité et le concept de contraction géométrique des moments. Ces conditions sont plus générales que celles de l’invariance des moments conditionnels d’ordre un à quatre utilisée jusque là par plusieurs auteurs. Un test statistique basé sur les covariances croisées et la matrice des variances et covariances de leur distribution asymptotique est alors proposé et sa distribution asymptotique établie. Dans l’implémentation du test, la matrice des variances et covariances des covariances croisées est estimée à l’aide d’une procédure autorégressive vectorielle robuste à l’autocorrélation et à l’hétéroscédasticité. Des simulations sont ensuite effectuées pour étudier les propriétés du test proposé. Dans le troisième projet, nous considérons un modèle périodique multivarié et cointégré. La présence de cointégration entraîne l’existence de combinaisons linéaires périodiquement stationnaires des composantes du processus étudié. Le nombre de ces combinaisons linéaires linéairement indépendantes est appelé rang de cointégration. Une méthode d’estimation en deux étapes est considérée. La première méthode est appelée estimation de plein rang. Dans cette approche, le rang de cointégration est ignoré. La seconde méthode est appelée estimation de rang réduit. Elle tient compte du rang de cointégration. Cette dernière est une approche non linéaire basée sur des itérations dont la valeur initiale est l’estimateur de plein rang. Les propriétés asymptotiques de ces estimateurs sont aussi établies. Afin de vérifier l’adéquation du modèle, des statistiques de test de type portemanteau sont considérées et leurs distributions asymptotiques sont étudiées. Des simulations sont par la suite présentées afin d’illustrer le comportement du test proposé. / In statistical modeling, we assume that the phenomenon of interest is generated by a model that can be fitted to the observed data. The part of the phenomenon not explained by the model is called error or innovation. There are two parts in the model. The main part is supposed to explain the observed data, while the unexplained part which is supposed to be negligible is also called error or innovation. In order to simplify the structures, the model are often assumed to rely on a finite set of parameters. The quality of a model depends also on the parameter estimators and their properties. For example, are the estimators relatively close to the true parameters ? Some questions also address the goodness-of-fit of the model to the observed data. This question is answered by studying the statistical and probabilistic properties of the innovations. On the other hand, it is also of interest to evaluate the presence or the absence of relationships between the observed data. Portmanteau or diagnostic type tests are useful to address such issue. The thesis is presented in the form of three projects. The first project is written in English as a scientific paper. It was recently submitted for publication. In that project, we study the class of vector multiplicative error models (vMEM). We use the properties of the Generalized Method of Moments to derive the asymptotic distribution of sample autocovariance function. This allows us to propose a new test statistic. Under the null hypothesis of adequacy, the asymptotic distributions of the popular Hosking-Ljung-Box (HLB) test statistics are found to converge in distribution to weighted sums of independent chi-squared random variables. A generalized HLB test statistic is motivated by comparing a vector spectral density estimator of the residuals with the spectral density calculated under the null hypothesis. In the second project, we derive the asymptotic distribution under weak dependence of cross covariances of covariance stationary processes. The weak dependence is defined in term of the limited effect of a given information on future observations. This recalls the notion of stability and geometric moment contraction. These conditions of weak dependence defined here are more general than the invariance of conditional moments used by many authors. A test statistic based on cross covariances is proposed and its asymptotic distribution is established. In the elaboration of the test statistics, the covariance matrix of the cross covariances is obtained from a vector autoregressive procedure robust to autocorrelation and heteroskedasticity. Simulations are also carried on to study the properties of the proposed test and also to compare it to existing tests. In the third project, we consider a cointegrated periodic model. Periodic models are present in the domain of meteorology, hydrology and economics. When modelling many processes, it can happen that the processes are just driven by a common trend. This situation leads to spurious regressions when the series are integrated but have some linear combinations that are stationary. This is called cointegration. The number of stationary linear combinations that are linearly independent is called cointegration rank. So, to model the real relationship between the processes, it is necessary to take into account the cointegration rank. In the presence of periodic time series, it is called periodic cointegration. It occurs when time series are periodically integrated but have some linear combinations that are periodically stationary. A two step estimation method is considered. The first step is the full rank estimation method that ignores the cointegration rank. It provides initial estimators to the second step estimation which is the reduced rank estimation. It is non linear and iterative. Asymptotic properties of the estimators are also established. In order to check for model adequacy, portmanteau type tests and their asymptotic distributions are also derived and their asymptotic distribution are studied. Simulation results are also presented to show the behaviour of the proposed test.