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

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

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

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.

Submitted by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2017-03-15T09:40:27Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertação - Adriane S. Reis.pdf: 2903401 bytes, checksum: 5cb1050cca283d0c9adcd3d1fe7a1899 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2017-03-15T09:40:43Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertação - Adriane S. Reis.pdf: 2903401 bytes, checksum: 5cb1050cca283d0c9adcd3d1fe7a1899 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2017-03-15T09:40:58Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertação - Adriane S. Reis.pdf: 2903401 bytes, checksum: 5cb1050cca283d0c9adcd3d1fe7a1899 (MD5) / Made available in DSpace on 2017-03-15T09:40:58Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertação - Adriane S. Reis.pdf: 2903401 bytes, checksum: 5cb1050cca283d0c9adcd3d1fe7a1899 (MD5) Previous issue date: 2016-11-18 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / For a long time it was believed that polymers were materials which didn't have any conductive properties associated to them. However, several studies has been made through the years to show the birth of a new class of polymeric materials, known as intrinsically conductive polymers. This class of polymers generally consist of chains having a conjugation structure, with alternating double and single bonds of carbon atoms. This combination allows electronic transport through the polyaniline structure. Since polyaniline represents a many-particle system, we present an introduction to the Schrodinger equation, the Thomas-Fermi theory, an to the Khon-Sham equations, which are important to understand the methods used, in order to calculate the electronic transport properties of doped polyaniline with different chloride contents. To calculate the electronic structure and electronic transport properties, we used the first principles, also known as textit ab-initio method , by the density functional theory technique (DFT). The electronic properties of polyaniline were studied through the bands diagram, density of states per atom and orbital, charge density, transmittance and electrical current for each state of oxidation/reduction of this conjugated polymer. The results obtained show a good qualitatively agreement with the experimental results. It was possible to observe the transition process of polyanilines, affording to obtain the semiconductor and insulating phases, as shown in the bands diagrams and electric current spectra. / Durante muito tempo, acreditou-se que os polímeros eram materiais que não pos-suíam quaisquer propriedades condutoras. Contudo, diversas pesquisas realizadas ao longo dos anos, resultaram no nascimento de uma nova classe de materiais poliméricos, conhe-cidos como polímeros intrinsecamente condutores. Esta classe de polímeros, em geral, são compostas por cadeias que possuem conjugação em sua estrutura, ou seja, possuem ligações simples e duplas alternadas dos átomos de carbono. Essa conjugação leva a polia-nilina a possuir propriedades de transporte eletrônico através da estrutura, que é o motivo do presente estudo. Como a polianilina representa um sistema de muitos corpos, apresentamos uma introdução à equação de Schrôdinger, à teoria de Thomas-Fermi e às equações de Khon-Sham, que são de grande utilidade na compreensão dos métodos que utilizamos nesta dis-sertação, visando calcular as propriedades de transporte eletrônico da polianilina dopada com diferentes teores de cloro. Para o cálculo da estrutura eletrônica e as propriedades de transporte eletrônico , utilizamos o método de primeiros princípios, também conhecido como método ab-initio, via teoria do funcional da densidade (DFT). As propriedades eletrônicas da polianilina foram estudadas por meio do diagrama de bandas, densidade de estados por átomo e por orbital, densidade de carga, transmitân-cia e corrente elétrica para cada estado de oxidação/redução desse polímero conjugado. Os resultados alcançados mostraram a concordância qualitativa com os resultados experimentais. Pôde-se observar o processo de transição das polianilinas, conseguindo-se obter as fases isolantes e semicondutoras, como mostrado nos diagramas de bandas e nos espectros de corrente elétrica.

As the foundation for a new computational implementation, we survey the calculation of the complex electrical conductivity tensor based on the Kubo-Greenwood (KG) formalism (Kubo, 1957; Greenwood, 1958), with emphasis on derivations and technical aspects pertinent to use of projector augmented wave datasets with plane wave basis sets (BIlichl, 1994). New analytical results and a full implementation of the KG approach in an open-source Fortran 90 post-processing code for use with Quantum Espresso (Giannozzi et al., 2009) are presented. Named KGEC ([K]ubo [G]reenwood [E]lectronic [C]onductivity), the code calculates the full complex conductivity tensor (not just the average trace). It supports use of either the original KG formula or the popular one approximated in terms of a Dirac delta function. It provides both Gaussian and Lorentzian representations of the Dirac delta function (though the Lorentzian is preferable on basic grounds). KGEC provides decomposition of the conductivity into intra- and inter band contributions as well as degenerate state contributions. It calculates the dc conductivity tensor directly. It is MPI parallelized over k-points, bands, and plane waves, with an option to recover the plane wave processes for their use in band parallelization as well. It is designed to provide rapid convergence with respect to k-point density. Examples of its use are given.

No description available.

No description available.

No description available.

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>