Les travaux présentés dans ce manuscrit de thèse peuvent être divisés en deux parties. Dans une première partie, nous nous sommes intéressé à une extension multiconfigurationnelle de la théorie de la fonctionnelle de la densité (DFT) par l'intermédiaire d'une séparation de portée permettant un traitement hybride entre DFT et fonction d'onde multiconfigurationnelle « state-averaged ». Ainsi, nous récupérons en même temps la corrélation dynamique et la corrélation statique. De plus, cette étude est réalisée en considérant la DFT pour les ensembles afin de considérer une alternative à la méthode usuelle utilisée (DFT dépendante du temps) pour la détermination des états excités d'une molécule, évitant ainsi certains problèmes théoriques rencontrés avec cette approche. En particulier, les intersections coniques entre états excités nous intéressent particulièrement car il s'agit de cas pour lesquels une approche multiconfigurationnelle est primordiale. Dans une seconde partie, le développement de nouvelles fonctionnelles est réalisé sur le dimère de Hubbard asymétrique afin de tester de nouvelles approximations et d'étudier plus en détail les processus auto-cohérents. De plus, des couplages non-adiabatiques sont calculés en utilisant des énergies déterminées dans le cadre de la DFT pour les ensembles ayant la particularité de ne pas être dépendant du temps. / This thesis manuscript can be divided in two parts. In the first one, we are interested in a multiconfigurational extension for the density functional theory (DFT) including a range separation to deal with a hybrid theory between DFT and state-averaged wave function theory. In this case, we recover, at the same time, the dynamical correlation and the static correlation. Moreover, this study is performed considering the ensemble DFT to use an alternative to the usual method (time-dependent DFT) to describe the excited states of a molecule, avoiding some theoretical problems known with this approach. Particularly, conical intersections between excited states are interesting because a multiconfigurational approach is necessary. In the second part, new functionals development are performed and applied on the non-symmetric Hubbard dimer in order to test new approximations and to study more in detail self-consistency processes. In addition, non-adiabatic couplings are calculated using energies from ensemble DFT framework without time-dependence.

Made available in DSpace on 2016-12-12T20:15:52Z (GMT). No. of bitstreams: 1 Jardel C Rosa.pdf: 4774739 bytes, checksum: 9ea21d5b6cb09221260d2f1f58e1333f (MD5) Previous issue date: 2016-09-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we propose a model to study the effect of electron correlation in the phase diagrams of itinerant magnetocaloric materials, where the conduction electrons are described by the Hubbard model, and that also includes the exchange interaction Jf d between localized and itinerant moments, an external magnetic field, and the exchange interaction Jf f between localized moments. We discuss the applicability of the model to intermetallic materials containing rare earth elements and transition metals, where the simultaneous presence of interacting localized moments and itinerant magnetism produces a peculiar behavior that is reflected in their magnetocaloric properties. As a first step, we treat the model within the mean-field approximation. From the computaion of the magnetization curves on a simple cubic lattice, we obtain the phase diagrams and investigate the magnetic phase transition and the magnetocaloric effect. In a second step, we use the alloy analogy approximation to treat Hubbard model, which provides a more detailed and realistic description for these materials. / Neste trabalho, propomos um modelo para estudar o efeito da correlação eletrônica nos diagramas de fases de materiais magnetocalóricos itinerantes, no qual os elétrons de condução d são descritos pelo modelo de Hubbard, e que inclui ainda uma interação de troca Jf d entre os momentos localizados e itinerantes, um campo magnético externo, e uma interação de troca Jf f entre os momentos localizados. Discutimos a aplicabilidade do modelo em materiais intermetálicos contendo elementos de terras raras e metais de transição, onde a presença simultânea de momentos localizados interagentes e do magnetismo itinerante produz um comportamento peculiar, que é refletido nas suas propriedades magnetocalóricas. Num primeiro estágio, tratamos o modelo dentro da aproximação de campo médio. A partir do cálculo das curvas de magnetização em uma rede cúbica simples, obtemos os diagramas de fases e investigamos as transições de fase magnéticas e o efeito magnetocalórico. Num segundo estágio, utilizamos a aproximação da analogia de liga para tratar o modelo de Hubbard, que fornece uma descrição mais detalhada e realista para esses materiais.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The study about magnetic properties of strongly correlated electrons has been an problem of interest in physics by the great technological importance of various materials with this characteristic. A well-known theoretical approach, known as the Hubbard model, has been used in an attempt to describe such phenomena. This model considers the interactions and electron mobility in a simplified way for describing phenomena like itinerant magnetism and the metal-insulator transition. In this study were analyzed the thermodynamic clusters properties of structures with fcc lattice symmetry containing 4, 5, 6, 7 and 8 sites. An exact numerical diagonalization method was used, where the subspaces of the problem were separatly analyzed and put together on, which is a procedure that reduces the processing data time. In order to improve the diagram of quantum states also carried out a ground state analysis of the clusters, known in the literature and obtained from systems with 4, 5 and 6 sites. On this subject a new diagram was developed showing the quantum states transition using the Lanczos method, applied to clusters with 4 to 8 sites in this case. Finally, a comparison was made between the thermodynamic properties and the results found in the new quantum state diagram. The agreement between them is good. / O estudo de propriedades magnéticas de elétrons fortemente correlacionados tem sido um problema de interesse da física pela grande importância tecnológica de vários materiais com essa característica. Uma famosa abordagem teórica, conhecida como modelo de Hubbard, vem sendo utilizada na tentativa de descrever esse tipo fenômeno. Esse modelo considera interações e a mobilidade eletrônica descrevendo de forma simplificada fenômenos como magnetismo itinerante e transição metal-isolante. Neste trabalho foram analisadas propriedades termodinâmicas de clusters de estruturas com simetria da rede fcc contendo 4, 5, 6, 7 e 8 sítios. Foi utilizado um método de diagonalização numérica exata, onde os subespaços do problema foram analisados separadamente e reunidos posteriormente, procedimento que reduz o tempo de processamento dos dados. Foi também realizada uma análise do estado fundamental dos clusters com o intuito de aprimorar o diagrama de estados quânticos, conhecido na literatura, obtido a partir de sistemas com 4, 5 e 6 sítios. Sobre esse assunto, foi desenvolvido um novo diagrama que indica a transição de estados quânticos a partir da utilização do método de Lanczos, neste caso, aplicado a clusters de 4 a 8 sítios. Por fim, foi feita uma análise comparativa entre as propriedades termodinâmicas e os resultados encontrados no novo diagrama de estados quânticos que mostraram boa coerência entre si.

Les propriétés d'électrons en deux dimensions (2D) soulèvent des questions fondamentales qui ont été largement explorées au moyen des techniques théoriques de la matière condensée. L'extension de modèles classiques tel le modèle de Hubbard en 2D, en incluant par exemple plusieurs bandes électroniques, ore la possibilité d'accéder à des phénomèmes plus complexes, comme l'interaction du transport électronique et du magnétisme observé dans les composés de fermions lourds. Ces modèles sont en lien direct avec la question de couches minces couplées, les hétérostructures, qui sont depuis peu l'objet d'intenses recherches et orent la possibilité d'intéressantes applications. Dans ce contexte, nous étudions numériquement diérents syst èmes au moyen de la méthode du Monte Carlo Quantique du Déterminant. Tout d'abord, l'eet de la corrélation électronique dans un isolant de bande est évaluée, montrant en particulier l'absence d'une phase métallique intermédiaire. Un deuxième système est composé de deux bandes électroniques couplées, dans lequel l'eet de la largeur de bande de la partie corrélée est exploré de façon systématique. Finalement, nous étudions une interface métal-isolant, qui présente une phase intermédiaire surprenante lorsque le couplage à l'interface est ajusté. / The properties of electrons in two dimensions (2D) raise fundamental questions that have been extensively explored by condensed matter theory. Extending standard frameworks such as the 2D Hubbard model by accounting for more than one electronic band oers the opportunity to access more complex phenomena, such as the interplay between transport and magnetism found in heavy-fermions materials. Such models are directly connected to the problem of coupled layers in complex materials known as heterostructures, which have been widely studied and synthesized in recent years, and are expected to lead to important applications. In that context, we study numerically several systems, by mean of the Determinant Quantum Monte Carlo Method (DQMC). We rst analyze the eect of electronic correlation in a band insulator, showing in particular the absence of an intermediate metallic phase. A second system consists of two coupled bands that modelize a heavy fermion model, in which the role of the bandwidth of the correlated band is systematically investigated. Finally, we consider the case of a metal-insulator interface, unveiling an intriguing intermediate phase as the interfacial coupling is tuned.

Neste trabalho estudamos as propriedades físicas das impurezas de elementos de terra rara (TR) nos cristais de GaN, nas estruturas cristalinas zincblenda e wurtzita, e de ZnO. Para tal, consideramos as impurezas de Eu, Gd, Tb, Dy, Ho, Er e Tm substitucionais no sítio do cátion (Ga ou Zn), pois esta posição é mais estável do que no sítio do ânion (N ou O). As investigações foram efetuadas através de simulações computacionais baseadas em métodos de primeiros princípios dentro do formalismo da teoria do funcional da densidade e utilizando o método FP-LAPW (Full Potential Linear Augmented Plane Waves), implementado no código computacional WIEN2k, dentro do esquema de supercélula, com relaxações atômicas tratadas de modo apropriado. No tratamento dos estados de valência 3d (Ga ou Zn) e 4f dos átomos de TR foi introduzida uma correção local de Hubbard, para levar em conta efeitos de alta correlação eletrônica. Inicialmente, estudamos as propriedades eletrônicas e estruturais dos elementos de TR em suas fases metálicas. Estes resultados serviram para validar a metodologia de obtenção dos valores dos parâmetros U de Hubbard de modo autoconsistente, pois para este grupo de materiais existem resultados experimentais. Comparando a localização e o desdobramento entre os estados ocupados e desocupados relacionados aos orbitais 4f, em relação ao nível de Fermi dos sistemas, obteve-se uma ótima concordância. Nossos resultados para os sistemas dopados, utilizando a correção do potencial U de Hubbard, mostram que a suas descrições estão adequadas. Somente com a introdução da correção é possível localizar corretamente, em relação ao topo da banda de valência, os estados ocupados e desocupados relacionados aos orbitais 4f. Estes resultados estão de acordo com esquemas propostos na literatura, mostrando que o estado de oxidação das impurezas de TR é trivalente e que, em geral, em ambos sistemas cristalinos, não há a introdução de níveis de energia na região do gap dos materiais. Nosso estudo da correção do potencial U de Hubbard, obtido autoconsistemente, na teoria do funcional da densidade, mostra que seus valores não são universais, dependendo do estado de carga e do ambiente em que o átomo está inserido. Mais ainda, mostra que o procedimento adotado é totalmente apropriado para descrever a correlação eletrônica dos elétrons 4f. / In this work we studied the physical properties of rare earth (RE) impurities in GaN, in the zincblend and the wurtzite crystal structures, and in ZnO. Eu, Gd, Tb, Dy, Ho, Er and Tm substitutional impurities in the cation site (Ga or Zn) were considered, since this position is more stable than anion site (N or O). The investigations were carried out by computational simulations using ab initio methods, based on the density functional theory and the FP-LAPW (Full Potential Linear Augmented Plane Waves) method, as implemented in the WIEN2k code, considering the supercell approach and atomic relaxations. The 3d-Ga or 3d-Zn and the 4f-RE valence states were treated with the introduction of on-site Hubbard correction, in order to correctly describe the strongly correlated electrons. First, the electronic and structural properties of RE metallic systems were investigated and the results were compared with available experimental data, showing a good agreement. Those results helped to identify the appropriate procedure to compute the Hubbard U potential. This procedure should also provide a reliable description about the electronic properties of RE elements as impurities in semiconductors. Then, we have computed the properties of substitutional RE impurities in the cation site using the same methodology and procedures. The Hubbard U potential was necessary to correctly describe the position of the 4f occupied and unoccupied states, related to the valence band top. These results are in agreement with proposed electronic properties of RE doped GaN and ZnO found in the literature. The RE impurities oxidation states are in general trivalent in both crystal systems and introduce no energy levels in the gap region. Our Hubbard U potential correction, obtained in a self-consistent way, depends on the considered element and the neighborhood, and it is not an universal parameter. Moreover, our investigation shows that the adopted procedure is totally appropriate to describe the electronic correlation of the 4f electrons.

Neste trabalho mostramos que uma grande variedade de modelos exatamente solúveis através do ansatz de Bethe coordenadas podem também ser resolvidos através de um ansatz do produto matricial. Estes modelos são descritos no caso unidimensional por cadeias quânticas, e por matrizes de transferência no caso de sistemas clássicos bi-dimensionais. Diferentemente do ansatz de Bethe, em que as auto-funções do modelo são escritas como uma combinação de ondas planas, no nosso ansatz do produto matricial elas são dadas por produtos de matrizes, onde as matrizes obedecem a uma álgebra associativa apropriada. Estas relações algébricas são obtidas impondo-se que as auto-funções escritas em termos do ansatz satisfaçam à equação de auto-valor do operador Hamiltoniano ou da matriz de transferência. A consistência das relações de comutatividade entre os elementos da álgebra implicam na exata integrabilidade do modelo. Além disso, o ansatz que propomos permite uma formulação simples e unificada para vários Hamiltonianos quânticos exatamente solúveis. Apresentamos nesta tese a formulação do nosso ansatz do produto matricial para uma grande família de redes quânticas, como os modelos anisotrópico de Heisenberg, Fateev-Zamolodchikov, Izergin-Korepin, Sutherland, t-J, Hubbard etc. Mais ainda, formulamos nosso ansatz para processos estocásticos de partículas com tamanhos e classes diferentes difundindo assimetricamente na rede. Por fim, com o objetivo de dar suporte a nossa conjectura de que todos os modelos exatamente solúveis através do ansatz de Bethe coordenadas, associados a Hamiltonianos quânticos unidimensionais ou matrizes de transferência bidimensionais, também podem ser resolvidos através de um ansatz do produto matricial, apresentamos a formulação do nosso ansatz para a matriz de transferência do modelo de seis-vértices com condição de contorno toroidal / In this work we show that a large family of exactly solved models through the coordinate Bethe ansatz can also be solved through a matrix product ansatz. The models are described in the one dimensional case by quantum Hamiltonians, and by transfer matrices in the case of two dimensional classical models. Differently from the Bethe ansatz, where the model\'s eigenfunctions are described by a plane wave combination, in our matrix product ansatz they are given by a matrix product, where the matrices obey a suitable associative algebra. Theses algebraic relations are obtained by imposing that the eigenfunctions described in terms of the ansatz satisfy the eigenvalue equation for the associated Hamiltonian or transfer matrix. The consistency of the commutativity relations among the elements of the algebra implies the exact integrability of the model. Furthermore, the matrix product ansatz we propose allows an unified and simple formulation for the solution of several exact integrable quantum Hamiltonians. We present on this thesis the formulation of our matrix product ansatz for a huge family of quantum chains such as the anisotropic Heisenberg model, Fateev-Zarnolodchikov model, Izergin-Korepin model, Sutherland model, t- J model, Hubbard model, etc. Moreover, we formulated our ansatz for stochastic process of particles with different sizes and classes diffusing asymmetrically on the lattice. Finally, in order to support our conjecture that all exactly solved models through the coordinate Bethe ansatz, associated to unidimensional quantum Hamiltonians or two-dimensional transfer matrices, can also be solved through a matrix product ansatz, we present the formulation of our ansatz, for the transfer matrix of the six-vertex model with toroidal boundary condition

Nesta dissertação estudaremos sistemas de bósons ultrafrios armadilhados em uma rede ótica quadrada bidimensional sem levar em consideração o confinamento harmônico. A dinâmica desses sistemas é bem descrita pelo modelo de Bose-Hubbard, que prevê uma transição de fase quântica de um superfluido para um isolante de Mott a temperaturas baixas, e pode ser induzida variando a profundidade do potencial da rede ótica. Apresentaremos o diagrama de fases dessa transição construído a partir de uma aproximação de campo médio e também com um cálculo numérico usando um algoritmo de Monte Carlo Quântico, denominado algoritmo Worm. Encontramos o ponto crítico para o primeiro lobo de Mott em ambos os casos, concordando com trabalhos anteriores. / This work study the two-dimensional ultracold bosonic atoms loaded in a square optical lattice, without harmonic confinement. The dynamics of this system is described by the Bose-Hubbard model, which predicts a quantum phase transition from a superfluid to a Mott-insulator at low temperatures that can be induced by varying the depth of the optical potential. We present here the phase diagram of this transition built from a mean field approach and from a numerical calculation using a Quantum Monte Carlo algorithm, namely the Worm algorithm. We found the critical transition point for the first Mott lobe in both cases, in agreement with the standard literature.

Die vorliegende Arbeit beschäftigt sich mit der mikroskopischen Modellierung von strominduziertem Schalten der Magnetisierung in magnetischen Tunnelstrukturen. Die Tunnelstruktur besteht aus zwei durch einen nichtmagnetischen Isolator voneinander getrennten Ferromagneten und einem Paramagneten, der als Elektronenreservoir dient. Die Ferromagnete werden beide durch das Hubbard-Modell beschrieben. Durch Anlegen einer Spannung verschieben sich die chemischen Potentiale auf beiden Seiten des Isolators, wodurch ein endlicher Tunnelstrom entsteht. Dieser wird im Rahmen des Modells durch eine Hybridisierung zwischen benachbarten Schichten simuliert. Das Modell muss im Nichtgleichgewicht gelöst werden, da aufgrund der unterschiedlichen chemischen Potentiale thermodynamisches Gleichgewicht nicht angenommen werden darf. Daher wird zur analytischen Auswertung der Keldysh-Formalismus verwendet, der eine Erweiterung der Viel-Teilchen-Theorie ins Nichtgleichgewicht darstellt. Da es sich beim Hubbard-Modell um ein nicht exakt lösbares Viel-Teilchen-Modell handelt, wurde in der Arbeit eine approximative Lösung, der sogenannte Nichtgleichgewichtsspektraldichteansatz, entwickelt. Dieser beruht auf einer Hochenergieentwicklung der retardierten Greenfunktion mit Hilfe der exakt berechenbaren Spektralmomente. Die numerischen Resultate stimmen qualitativ mit dem Experiment überein. Insbesondere gelingt es, das Hystereseverhalten der Magnetisierung des freien Ferromagneten in Abhängigkeit der angelegten Spannung korrekt zu reproduzieren. Es kann somit allein durch Anlegen einer Spannung kontrolliert zwischen paralleler und antiparalleler Ausrichtung der Magnetisierungen geschaltet werden. Dieses Phänomen ist anhand der entsprechenden Quasiteilchenzustandsdichten erklärbar. Weiterhin wird das Verhalten der kritischen Spannung systematisch in Form von Phasendiagrammen dargestellt und diskutiert. / This thesis is concerned with the microscopic modelling of current-induced switching of magnetization in magnetic tunnel junctions. The tunnel junction consists of two ferromagnets which are divided by a nonmagnetic insulator and a paramagnet, which acts as an electron reservoir. The ferromagnets are both described by the Hubbard model. By applying a voltage the chemical potentials on both sides of the insulator are shifted which results in a finite tunneling current. Within the model the current is simulated by a hybridization between neighbouring regions. The model has to be solved in non-equilibrium since thermal equilibrium requires a constant chemical potential for the whole system, which is not the case due to the voltage. Thus the Keldysh formalism will be used for evaluating the model. Since the Hubbard model is not exactly solvable one needs approximations. In this work a non-equilbrium spectral density approach is developed. It is based on a high-energy expansion of the retarded Green''s function and takes interactions beyond the mean field level into account. The numerical results of the theory are in qualitative agreement with experiments. It will be shown that it is possible to correctly get the hysterisis behaviour of the magnetization of the free ferromagnet in dependence on the applied voltage. Thus the relative alignment of the two magnetizations can be switched just by applying an electric field. This can be explained with the corresponding quasiparticle densities of state. Furthermore the behaviour of the critical voltage will be discussed systematically by calculating phase diagrams of the tunnel junction.

The localized/delocalized duality of 5f electrons plays an important role in understanding the complex physics of actinides. Band-structure calculations based on the ad hoc assumption that 5f electrons are simultaneously localized and delocalized explained the observed dHvA experiments very well. This ad hoc assumption also gives the correct equilibrium volume for delta-Pu. Experimentally, the duality of 5f electrons is observed by inelastic neutron scattering experiments, or by soft X-ray angle-resolved photoelectron spectroscopy. It is worth recalling that the origin of partial localization in the 3d and 5f systems is quite different. In compounds with 3d electrons, the large crystalline electric field set up by the surrounding environment of transition metal ions plays a major role. On the other hand, in 5f systems, the Hund's rule correlations play the key role whilst the crystalline electric field is less important. In this thesis we have studied the effect of intra-atomic correlations on anisotropies in hopping matrix elements of different 5f orbitals. For that purpose, we used the effective model that includes on-site interactions that are responsible for Hund's rules and effective hopping terms that result from the hybridization of different 5f orbitals with the environment. Two different approximations, namely, rotationally invariant slave-boson mean-field (RISBMF) and infinite time-evolving block decimation (iTEBD), have been used to investigate the ground-state properties of the Hamiltonian. We have demonstrated that Hund's rule correlations enhance strongly anisotropies in hopping matrix elements. For a certain range of 5f bandwidth parameters this effect may result in a complete suppression of hopping processes for some of 5f orbitals, i.e., the system is in a partially localized phase. Within the RISBMF method, we calculated the ground-state properties and the phase diagram of the system. The suppression of hopping processes in some of 5f orbitals due to Hund's rule correlations can be seen through orbital-dependent quasiparticle weights. In a mean-field theory, a quasiparticle weight of zero for an orbital means a complete suppression of hopping processes in this orbital. Thus, quasiparticle weights and occupation numbers were used to classify partially localized phases. In the calculated phase diagram we obtain four partially localized phases that can be separated into two different sets. In the first set electrons in two orbitals are localized. In the second, electrons in one orbital are localized. The difference between the two sets is not simply the number of localized orbitals but the mechanism for the partial localization. For the first set, the Hund's rule mechanism applies: only those 5f electrons that enable the remaining ones to form a Hund's rule state will delocalize. This mechanism requires to have at least two localized orbitals, therefore it is definitely not applicable to those phases with only one localized orbital. For the second set, a situation similar to a single-band Mott-Hubbard transition applies. The direct on-site Coulomb interaction between jz and -jz electrons plays the key role for understanding the partial localization transition. In order to assess the validity of the RISBMF results we have used the iTEBD method to calculate the ground-state properties of a 1D system. Qualitatively, the two approaches agree with each other. However, we found an area where the RISBMF yields an artificial ground-state. Note that the mean-field method is worst for a 1D system. Therefore one shoud not judge from it the quality of the RISBMF method for the more general case.

Made available in DSpace on 2016-12-12T20:15:51Z (GMT). No. of bitstreams: 1 Marsal Eduardo Bento.pdf: 1108751 bytes, checksum: c5c36b13c56d8d4fadcee9ddc5a9098e (MD5) Previous issue date: 2014-12-16 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The development of Density Functional Theory (DFT) has been focused primarily on two main pillars: (1) the pursuit of more accurate exchange-correlation (XC) density functionals; (2) the feasibility of computational implementation when dealing with many-body systems. In this context, this work is aimed on using one-dimensional quantum systems as theoretical laboratories to investigate the implementation of orbital functionals (OFs) of density. By definition, OFs are those which depend only implicitly on the density, via an explicit formulation in terms of Kohn-Sham orbitals. Typical examples are the XC functionals arising from the Perdew-Zunger self-interaction correction (PZSIC). Formally, via Kohn-Sham equations, the implementation of OFs must be performed by means of the optimized effective potential method (OEP), which is known by requiring an excessive computational effort even when dealing with few electrons systems (N ̴ 10). Here, we proceed a systematical investigation aiming to simplify or avoid the OEP procedure, taking as reference the implementation of the PZSIC correction applied to one-dimensional Hubbard chains. / O desenvolvimento da Teoria do Funcional da Densidade (DFT) tem se concentrado, sobretudo, em dois pilares fundamentais: (1) a busca por funcionais de troca e correlação (XC) mais precisos; (2) a viabilidade de implementação computacional diante de sistemas com muitos elétrons. Nesse contexto, o objetivo principal deste trabalho consiste em utilizar sistemas quânticos unidimensionais, mais simples de serem tratados numericamente, como laboratórios teóricos para o desenvolvimento de alternativas de implementação numérica de funcionais orbitais (OFs) da densidade. Por definição, OFs são todos aqueles que dependem apenas implicitamente da densidade, via formulação explícita em termos dos orbitais Kohn-Sham. Exemplos típicos são os funcionais XC advindos da correção de auto-interação de Perdew e Zunger (PZSIC). Formalmente, via equações de Kohn-Sham, a implementação de OFs deve ser procedida por meio do método do potencial efetivo otimizado (OEP) que, no contexto computacional, é conhecido por se tornar demasiadamente custoso, inclusive para sistemas com poucos elétrons (N ̴ 10). Sendo assim, investigamos, de forma sistemática, alternativas de simplificar ou evitar o procedimento OEP, tomando como referência a implementação da correção PZSIC aplicada a cadeias de Hubbard unidimensionais.