Cette thèse comporte deux volets distincts. Le premier, qui fait l'objet du chapitre 2, porte sur les modèles mathématiques en calcul de structures électroniques, et consiste plus particulièrement en l'étude des modèles de type Kohn-Sham avec fonctionnelles d'échange-corrélation LDA et GGA. Nous prouvons, pour un système moléculaire neutre ou chargé positivement, que le modèle Kohn-Sham LDA étendu admet un minimiseur, et que le modèle Kohn-Sham GGA pour un système contenant deux électrons admet un minimiseur. Le second volet de la thèse traite de problématiques diverses en homogénéisation. Dans les chapitres 3 et 4, nous nous intéressons à un modèle de matériau aléatoire dans lequel un matériau périodique est perturbé de manière stochastique. Nous proposons plusieurs approches, certaines rigoureuses et d'autres heuristiques, pour calculer au second ordre en la perturbation le comportement homogénéisé de ce matériau de manière purement déterministe. Les tests numériques effectués montrent que ces approches sont plus efficaces que l'approche stochastique directe. Le chapitre 5 est consacré aux couches limites en homogénéisation périodique, et vise notamment, dans le cadre parabolique, à comprendre comment prendre en compte les conditions aux limites et initiale, et comment corriger en conséquence le développement à deux échelles sur lequel repose classiquement l'homogénéisation, pour obtenir des estimations d'erreur dans des espaces fonctionnels adéquats / This thesis is divided into two parts. The first part, that coincides with Chapter 2, deals with mathematical models in quantum chemistry, and specifically focuses on Kohn-Sham models with LDA and GGA exchange-correlation functionals. We prove, for a neutral or positively charged system, that the extended Kohn-Sham LDA model admits a minimizer, and that the Kohn-Sham GGA model for a two-electron system admits a minimizer. The second part is concerned with various issues in homogenization. In Chapters 3 and 4, we introduce and study a model in which the material of interest consists of a random perturbation of a periodic material. We propose different approaches, either rigorous or formal, to compute the homogenized behavior of this material up to the second order in the size of the perturbation, in an entirely deterministic way. Numerical experiments show the efficiency of these approaches as compared to the direct stochastic homogenization process. Chapter 5 is devoted to boundary layers in periodic homogenization, in particular in the parabolic setting. It aims at giving a better understanding of how to take into account boundary and initial conditions, and how to correct the two-scale expansion on which homogenization is classically grounded, to obtain fine error estimates

Orientadores: Yuri Dimitrov Bozhkov, Antonio Carlos Gilli Martins / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-09-24T19:39:04Z (GMT). No. of bitstreams: 1 Freire_IgorLeite_D.pdf: 977261 bytes, checksum: b8ba44493aeac3de0d37cdfff2fc581b (MD5) Previous issue date: 2008 / Resumo: Neste trabalho provamos um teorema que faz a classificacão completa dos grupos de simetrias de Lie da equação semilinear de Kohn - Laplace no grupo de Heisenberg tridimensional. Uma vez que tal equação possui estrutura variacional, determinamos quais são suas simetrias de Noether e a partir delas estabelecemos suas respectivas leis de conservação / Abstract: In this work, we carry out a complete group classification of Lie point symmetries of semilinear Kohn - Laplace equations on the three-dimensional Heisenberg group. Since this equation has variational structure, we determine which of its symmetries are Noether's symmetries. Then we establish their respectives conservation laws / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

This thesis contains four articles related to mathematical aspects of Density Functional Theory. In Paper A, the theoretical justification of density methods formulated with current densities is addressed. It is shown that the set of ground-states is determined by the ensemble-representable particle and paramagnetic current density. Furthermore, it is demonstrated that the Schrödinger equation with a magnetic field is not uniquely determined by its ground-state solution. Thus, a wavefunction may be the ground-state of two different Hamiltonians, where the Hamiltonians differ by more than a gauge transformation. This implies that the particle and paramagnetic current density do not determine the potentials of the system and, consequently, no Hohenberg-Kohn theorem exists for Current Density Functional Theory formulated with the paramagnetic current density. On the other hand, by instead using the particle density as data, we show that the scalar potential in the system's Hamiltonian is determined for a fixed magnetic field. This means that the Hohenberg-Kohn theorem continues to hold in the presence of a magnetic field, if the magnetic field has been fixed. Paper B deals with N-representable density functionals that also depend on the paramagnetic current density. Here the Levy-Lieb density functional is generalized to include the paramagnetic current density. It is shown that a wavefunction exists that minimizes the "free" Hamiltonian subject to the constraints that the particle and paramagnetic current density are held fixed. Furthermore, a convex and universal current density functional is introduced and shown to equal the convex envelope of the generalized Levy-Lieb density functional. Since this functional is convex, the problem of finding the particle and paramagnetic current density that minimize the energy is related to a set of Euler-Lagrange equations. In Paper C, an N-representable Kohn-Sham approach is developed that also include the paramagnetic current density. It is demonstrated that a wavefunction exists that minimizes the kinetic energy subject to the constraint that only determinant wavefunctions are considered, as well as that the particle and paramagnetic current density are held fixed. Using this result, it is then shown that the ground-state energy can be obtained by minimizing an energy functional over all determinant wavefunctions that have finite kinetic energy. Moreover, the minimum is achieved and this determinant wavefunction gives the ground-state particle and paramagnetic current density. Lastly, Paper D addresses the issue of a Hohenberg-Kohn variational principle for Current Density Functional Theory formulated with the total current density. Under the assumption that a Hohenberg-Kohn theorem exists formulated with the total current density, it is shown that the map from particle and total current density to the vector potential enters explicitly in the energy functional to be minimized. Thus, no variational principle as that of Hohenberg and Kohn exists for density methods formulated with the total current density. / <p>QC 20140523</p>

Matter on a length scale comparable to that of a chemical bond is governed by the theory of quantum mechanics, but quantum mechanics is a many body theory, hence for the sake of chemistry or solid state physics, finding solutions to the governing equation, Schrodinger's equation, is hopeless for all but the smallest of systems. As the number of electrons increases, the complexity of solving the equations grows rapidly without bound. One way to make progress is to treat the electrons in a system as independent particles and to attempt to capture the many-body effects in a functional of the electrons' density distribution. When this approximation is made, the resulting equation is called the Kohn-Sham equation, and instead of requiring solving for one function of many variables, it requires solving for many functions of the three spatial variables. This problem turns out to be easier than the many body problem, but it still scales cubically in the number of electrons. In this work we will explore ways of obtaining the solutions to the Kohn-Sham equation in the framework of real-space pseudopotential density functional theory. The Kohn-Sham equation itself is an eigenvalue problem, just as Schrodinger's equation. For each electron in the system, there is a corresponding eigenvector. So the task of solving the equation is to compute many eigenpairs of a large Hermitian matrix. In order to mitigate the problem of cubic scaling, we develop an algorithm to slice the spectrum into disjoint segments. This allows a smaller eigenproblem to be solved in each segment where a post-processing step combines the results from each segment and prevents double counting of the eigenpairs. The efficacy of this method depends on the use of high order polynomial filters that enhance only a segment of the spectrum. The order of the filter is the number of matrix-vector multiplication operations that must be done with the Hamiltonian. Therefore the performance of these operations is critical. We develop a scalable algorithm for computing these multiplications and introduce a new density functional theory code implementing the algorithm. / text

We studied some semilinear elliptic equations on the entire space R^N. Our approach was variational, and the major obstacle was the breakdown in compactness due to the unboundedness of the domain. First, we considered an asymptotically linear Scltrodinger equation under the presence of a steep potential well. Using Lusternik-Schnirelmann theory, we obtained multiple solutions depending on the interplay between the linear, and nonlinear parts. We also exploited the nodal structure of the solutions. For periodic potentials, we constructed infinitely many homoclinic-type multibump solutions. This recovers the analogues result for the superlinear case. Finally, we introduced weights on the linear and nonlinear parts, and studied how their interact ion affects the local and global compactness of the problem. Our approach is based on the Caffarelli-Kohn-Nirenberg inequalities.

Ce travail porte sur un problème inverse de sources dipolaires et son application à l'identification des sources de l'activité cérébrale telle qu'elle peut être mesurée par l'Electro-Encéphalo-Graphie (EEG). Des résultats d'identifiabilité et de stabilité ont été établis. Par ailleurs, une étude du problème de Cauchy en 3D, motivée par l'application de la méthode d'identification dite "algébrique", a été faite à l'aide de la méthode itérative introduite par Kozlov, Maz'ya et Fomin et au moyen des équations intégrales de frontières. En outre, une autre méthode basée sur une fonctionnelle coût de type Kohn et Vogelius a été considérée pour l'identification des sources et dont les résultats numériques sont avérés plus performants que ceux donnés par la méthode des moindres carrés.

Cette thèse comporte deux volets distincts. Le premier, qui fait l'objet du chapitre 2, porte sur les modèles mathématiques en calcul de structures électroniques, et consiste plus particulièrement en l'étude des modèles de type Kohn-Sham avec fonctionnelles d'échange-corrélation LDA et GGA. Nous prouvons, pour un système moléculaire neutre ou chargé positivement, que le modèle Kohn-Sham LDA étendu admet un minimiseur, et que le modèle Kohn-Sham GGA pour un système contenant deux électrons admet un minimiseur. Le second volet de la thèse traite de problématiques diverses en homogénéisation. Dans les chapitres 3 et 4, nous nous intéressons à un modèle de matériau aléatoire dans lequel un matériau périodique est perturbé de manière stochastique. Nous proposons plusieurs approches, certaines rigoureuses et d'autres heuristiques, pour calculer au second ordre en la perturbation le comportement homogénéisé de ce matériau de manière purement déterministe. Les tests numériques effectués montrent que ces approches sont plus efficaces que l'approche stochastique directe. Le chapitre 5 est consacré aux couches limites en homogénéisation périodique, et vise notamment, dans le cadre parabolique, à comprendre comment prendre en compte les conditions aux limites et initiale, et comment corriger en conséquence le développement à deux échelles sur lequel repose classiquement l'homogénéisation, pour obtenir des estimations d'erreur dans des espaces fonctionnels adéquats

Made available in DSpace on 2014-06-11T19:23:29Z (GMT). No. of bitstreams: 0 Previous issue date: 2007-12-18Bitstream added on 2014-06-13T19:50:17Z : No. of bitstreams: 1 nacbar_dr_me_bauru.pdf: 1169690 bytes, checksum: c7a661675601c87b8e63ac301a1c144c (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / São calculadas e analisadas as funções de Wannier de localização máxima para elétrons em cristais unidimensionais. Essas funções formam uma base apropriada para descrever estados eletrônicos em materiais sólidos. Para cristais com simetria de inversão é utilizado o método desenvolvido por Bruno-Alfonso e Hai [J. Phys: Condensed Matter 15, 6701 (2003)]. Cada banda de energia é classificada segundo a simetria das funções de Bloch nos pontos 'gama' e 'qui' da zona de Brillouin. Para cada classe de banda a fase das funções de Bloch é escolhida para que as funções de Wannier tenham localização máxima. A simetria da últimas é determinda pelo tipo de banda. São apresentados resultados analíticos e numéricos para o modelo de Kronig-Penney obtidos através da técnica da matriz de transferência e do método tight binding. Posteriormente, apresenta-se um novo procedimento para calcular funções de Wannier de localização máxima em cristais sem simetria de inversão. Para isso são utilizadas técnicas do Cálculo Variacional. A teoria é aplicada para obter e analisar funções de Wannier de elétrons de condução em duas superredes de materiais semicondutores. Uma dessas estruturas tem simetria de inversão e a outra, não. O comportamento assintótico das funções de Wannier é predito analiticamente e verificado através dos cálculos numéricos. As funções de Wannier de localização máxima mostram um decaimento exponencial multiplicado por um decaimento em lei de potência, ambos isotrópicos. O mesmo acontece com parte das funções que não tem localização máxima. Porém, há outras que que apresentam decaimento exponecial reduzido e anisotropia em seu decaimento em lei de potência. Esses resultados novos são explicados levando em conta pontos de ramificação da continuação analítica das funções de Bloch sobre o plano de vetor de onda complexo. / The maximally localized Wannier functions of electrons in one-dimensional crystals are calculated and analyzed. Those functions form a suitable basis to describe localized states in solid materials. For crystals with inversion symmetry we use the procedure of Bruno-Alfonso and Hai [J. Phys: Condensed Matter 15, 6701 (2003)]. Each energy band is classified according to the symmetry of the Bloch functions at the points 'gama' e 'qui' of the Brillouin zone. For each band class, the phase of the Bloch functions in chosen to give the maximally localized Wannier functions. The symmmetry of those functions depends on the band class. Analytical and numerical results are presented for the Kronig-Penney model. Those result are obtained through the tight-binding method or a transfer-matrix technique. A new procedure to calculate the maximally localized Wannier functions in crystals without inversion symmetry is established. This involves techniques of the Variational Calculus. The theory is applied to obtain the Wannier functions of conduction electrons in superlattices of semiconductor materials. One of the superlattices presents inversion symmetry, but the other does not. The asymptotic behavior of the Wannier functions is predicted analytically and verified through numerical calculations. The maximally localized Wannier functions display an isotropic exponetial decal times an isotropic power-law decay. The same applies to a class of non-optimal Wannier functions. However, there is another class of non-optimal Wannier functions with reduced exponential decay and anisotropic power-law decay. Such new results are explained by taking into account branch points in the analytical continuation of the Bloch functions into the plane of complex wave vector.

Orientador: Alexys Bruno Alfonso / Banca: Guo-Qiang Hai / Banca: Aguinaldo Robinson de Souza / O Programa de Pós-Graduação em Ciência e Tecnologia de Materiais, PosMat, tem caráter institucional e integra as atividades de pesquisa em materiais de diversos campi da Unesp / Resumo: São calculadas e analisadas as funções de Wannier de localização máxima para elétrons em cristais unidimensionais. Essas funções formam uma base apropriada para descrever estados eletrônicos em materiais sólidos. Para cristais com simetria de inversão é utilizado o método desenvolvido por Bruno-Alfonso e Hai [J. Phys: Condensed Matter 15, 6701 (2003)]. Cada banda de energia é classificada segundo a simetria das funções de Bloch nos pontos 'gama' e 'qui' da zona de Brillouin. Para cada classe de banda a fase das funções de Bloch é escolhida para que as funções de Wannier tenham localização máxima. A simetria da últimas é determinda pelo tipo de banda. São apresentados resultados analíticos e numéricos para o modelo de Kronig-Penney obtidos através da técnica da matriz de transferência e do método tight binding. Posteriormente, apresenta-se um novo procedimento para calcular funções de Wannier de localização máxima em cristais sem simetria de inversão. Para isso são utilizadas técnicas do Cálculo Variacional. A teoria é aplicada para obter e analisar funções de Wannier de elétrons de condução em duas superredes de materiais semicondutores. Uma dessas estruturas tem simetria de inversão e a outra, não. O comportamento assintótico das funções de Wannier é predito analiticamente e verificado através dos cálculos numéricos. As funções de Wannier de localização máxima mostram um decaimento exponencial multiplicado por um decaimento em lei de potência, ambos isotrópicos. O mesmo acontece com parte das funções que não tem localização máxima. Porém, há outras que que apresentam decaimento exponecial reduzido e anisotropia em seu decaimento em lei de potência. Esses resultados novos são explicados levando em conta pontos de ramificação da continuação analítica das funções de Bloch sobre o plano de vetor de onda complexo. / Abstract: The maximally localized Wannier functions of electrons in one-dimensional crystals are calculated and analyzed. Those functions form a suitable basis to describe localized states in solid materials. For crystals with inversion symmetry we use the procedure of Bruno-Alfonso and Hai [J. Phys: Condensed Matter 15, 6701 (2003)]. Each energy band is classified according to the symmetry of the Bloch functions at the points 'gama' e 'qui' of the Brillouin zone. For each band class, the phase of the Bloch functions in chosen to give the maximally localized Wannier functions. The symmmetry of those functions depends on the band class. Analytical and numerical results are presented for the Kronig-Penney model. Those result are obtained through the tight-binding method or a transfer-matrix technique. A new procedure to calculate the maximally localized Wannier functions in crystals without inversion symmetry is established. This involves techniques of the Variational Calculus. The theory is applied to obtain the Wannier functions of conduction electrons in superlattices of semiconductor materials. One of the superlattices presents inversion symmetry, but the other does not. The asymptotic behavior of the Wannier functions is predicted analytically and verified through numerical calculations. The maximally localized Wannier functions display an isotropic exponetial decal times an isotropic power-law decay. The same applies to a class of non-optimal Wannier functions. However, there is another class of non-optimal Wannier functions with reduced exponential decay and anisotropic power-law decay. Such new results are explained by taking into account branch points in the analytical continuation of the Bloch functions into the plane of complex wave vector. / Mestre

Doctor of Philosophy / Department of Chemistry / Christine Aikens / Thiolate-protected noble metal nanoparticles (~2 nm size) are efficient solar photon harvesters, as they favorably absorb within the visible region. Clear mechanistic insights regarding the photo-physics of the excited state dynamics in thiolate-protected noble metal nanoclusters are important for future photocatalytic, light harvesting and photoluminescence applications. Herein, the core and higher excited states lying in the visible range are investigated using the time-dependent density functional theory method for different thiolate-protected nanoclusters. Nonadiabatic molecular dynamics simulations are performed using the fewest switches surface hopping approach with a time-dependent Kohn-Sham (FSSH-TDKS) description of the electronic states with decoherence corrections to study the electronic relaxation dynamics. Calculations on the [Au₂₅ (SH)₁₈]⁻¹ nanocluster showed that relaxations between core excited states occur on a short time scale (2-18 ps). No semiring or other states were observed at an energy lower than the core-based S₁ state, which suggested that the experimentally observed picosecond time constants could be core-to-core transitions rather than core-to-semiring transitions. Electronic relaxation dynamics on [Au₂₅ (SH)₁₈]⁻¹ with different R ligands (R = CH₃, C₂H₅, C₃H₇, MPA) [MPA = mercaptopropanoic acid] showed that all ligand clusters including the simplest SH model follow a similar trend in decay within the core states. In the presence of higher excited states, R= H, CH₃, C₂H₅, C₃H₇ demonstrated similar relaxations trends, whereas R=MPA showed a different relaxation of core states due to a smaller LUMO+1-LUMO+2 gap. Overall, the S₁ state gave the slowest decay in all ligated clusters. An examination of separate electron and hole relaxations in the [Au₂₅ (SCH₃)₁₈]⁻¹ nanocluster showed how the independent electron and hole relaxations contribute to its overall relaxation dynamics. Relaxation dynamics in the Au₁₈(SH)₁₄ nanocluster revealed that the S₁ state has the slowest decay, which is a semiring to core charge transfer state. Hole relaxations are faster than electron relaxations in the Au₁₈(SH)₁₄ cluster due its closely packed HOMOs. The dynamics in the Au₃₈(SH)₂₄ nanocluster predicted that the slowest decay, the decay of S₁₁ or the combined S₁₁-S₁₂, S₁-S₂-S₄-S₇ and S₄-S₅-S₉-S₁₀ decay, involves intracore relaxations. The phonon spectral densities and vibrational frequencies suggested that the low frequency (25 cm⁻¹) coherent phonon emission reported experimentally could be the bending of the bi-icosahedral Au₂₃ core or the “fan blade twisting” mode of two icosahedral units. Relaxation dynamics of the silver nanoparticle [Ag₂₅ (SR)₁₈]⁻¹ showed that both [Ag₂₅(SH)₁₈]⁻¹ and [Au₂₅ (SH)₁₈]⁻¹ follow a common decay trend within the core states and the higher excited states.