In the vast realm of inorganic materials, the Cu2+-containing cuprates form one of the richest classes. Due to the combined effect of crystal-field, covalency and strong correlations, all undoped cuprates are magnetic insulators with well-localized spins S=1/2, whereas the charge and orbital degrees of freedom are frozen out. The combination of the spin-only nature of their magnetism with the unique structural diversity renders cuprates as excellent model systems. The experimental studies, boosted by the discovery of high-temperature superconductivity in doped La2CuO4, revealed a fascinating variety of magnetic behaviors observed in cuprates. A digest of prominent examples should include the spin-Peierls transition in CuGeO3, the Bose-Einstein condensation of magnons in BaCuSi2O6, and the quantum critical behavior of Li2ZrCuO4. The magnetism of cuprates originates from short-range (typically, well below 1 nm) exchange interactions between pairs of spins Si and Sj, localized on Cu atoms i and j. Especially in low-dimensional compounds, these interactions are strongly anisotropic: even for similar interatomic distances |Rij|, the respective magnetic couplings Jij can vary by several orders of magnitude. On the other hand, there is an empirical evidence for the isotropic nature of this interaction in the spin space: different components of Si are coupled equally strong. Thus, the magnetism of cuprates is mostly described by a Heisenberg model, comprised of Jij(Si*Sj) terms. Although the applicability of this approach to cuprates is settled, the model parameters Jij are specific to a certain material, or more precisely, to a particular arrangement of the constituent atoms, i.e. the crystal structure. Typically, among the infinite number of Jij terms, only several are physically relevant. These leading exchange couplings constitute the (minimal) microscopic magnetic model. Already at the early stages of real material studies, it became gradually evident that the assignment of model parameters is a highly nontrivial task. In general, the problem can be solved experimentally, using elaborate measurements, such as inelastic neutron scattering on large single crystals, yielding the magnetic excitation spectrum. The measured dispersion is fitted using theoretical models, and in this way, the model parameters are refined. Despite excellent accuracy of this method, the measurements require high-quality samples and can be carried out only at special large-scale facilities. Therefore, less demanding (especially, regarding the sample requirements), yet reliable and accurate procedures are desirable. An alternative way to conjecture a magnetic model is the empirical approach, which typically relies on the Goodenough-Kanamori rules. This approach links the magnetic exchange couplings to the relevant structural parameters, such as bond angles. Despite the unbeatable performance of this approach, it is not universally applicable. Moreover, in certain cases the resulting tentative models are erroneous. The recent developments of computational facilities and techniques, especially for strongly correlated systems, turned density-functional theory (DFT) band structure calculations into an appealing alternative, complementary to the experiment. At present, the state-of-the-art computational methods yield accurate numerical estimates for the leading microscopic exchange couplings Jij (error bars typically do not exceed 10-15%). Although this computational approach is often regarded as ab initio, the actual procedure is not parameter-free. Moreover, the numerical results are dependent on the parameterization of the exchange and correlation potential, the type of the double-counting correction, the Hubbard repulsion U etc., thus an accurate choice of these crucial parameters is a prerequisite. In this work, the optimal parameters for cuprates are carefully evaluated based on extensive band structure calculations and subsequent model simulations. Considering the diversity of crystal structures, and consequently, magnetic behaviors, the evaluation of a microscopic model should be carried out in a systematic way. To this end, a multi-step computational approach is developed. The starting point of this procedure is a consideration of the experimental structural data, used as an input for DFT calculations. Next, a minimal DFT-based microscopic magnetic model is evaluated. This part of the study comprises band structure calculations, the analysis of the relevant bands, supercell calculations, and finally, the evaluation of a microscopic magnetic model. The ground state and the magnetic excitation spectrum of the evaluated model are analyzed using various simulation techniques, such as quantum Monte Carlo, exact diagonalization and density-matrix renormalization groups, while the choice of a particular technique is governed by the dimensionality of the model, and the presence or absence of magnetic frustration. To illustrate the performance of the approach and tune the free parameters, the computational scheme is applied to cuprates featuring rather simple, yet diverse magnetic behaviors: spin chains in CuSe2O5, [NO]Cu(NO3)3, and CaCu2(SeO3)2Cl2; quasi-two-dimensional lattices with dimer-like couplings in alpha-Cu2P2O7 and CdCu2(BO3)2, as well as the 3D magnetic model with pronounced 1D correlations in Cu6Si6O18*6H2O. Finally, the approach is applied to spin liquid candidates --- intricate materials featuring kagome-lattice arrangement of the constituent spins. Based on the DFT calculations, microscopic magnetic models are evaluated for herbertsmithite Cu3(Zn0.85Cu0.15)(OH)6Cl2, kapellasite Cu3Zn(OH)6Cl2 and haydeeite Cu3Mg(OH)6Cl2, as well as for volborthite Cu3[V2O7](OH)2*2H2O. The results of the DFT calculations and model simulations are compared to and challenged with the available experimental data. The advantages of the developed approach should be briefly discussed. First, it allows to distinguish between different microscopic models that yield similar macroscopic behavior. One of the most remarkable example is volborthite Cu3[V2O7](OH)2*2H2O, initially described as an anisotropic kagome lattice. The DFT calculations reveal that this compound features strongly coupled frustrated spin chains, thus a completely different type of magnetic frustration is realized. Second, the developed approach is capable of providing accurate estimates for the leading magnetic couplings, and consequently, reliably parameterize the microscopic Hamiltonian. Dioptase Cu6Si6O18*6H2O is an instructive example showing that the microscopic theoretical approach eliminates possible ambiguity and reliably yields the correct parameterization. Third, DFT calculations yield even better accuracy for the ratios of magnetic exchange couplings. This holds also for small interchain or interplane couplings that can be substantially smaller than the leading exchange. Hence, band structure calculations provide a unique possibility to address the interchain or interplane coupling regime, essential for the magnetic ground state, but hardly perceptible in the experiment due to the different energy scales. Finally, an important advantage specific to magnetically frustrated systems should be mentioned. Numerous theoretical and numerical studies evidence that low-dimensionality and frustration effects are typically entwined, and their disentanglement in the experiment is at best challenging. In contrast, the computational procedure allows to distinguish between these two effects, as demonstrated by studying the long-range magnetic ordering transition in quasi-1D spin chain systems. The computational approach presented in the thesis is a powerful tool that can be directly applied to numerous S=1/2 Heisenberg materials. Moreover, with minor modifications, it can be largely extended to other metallates with higher value of spin. Besides the excellent performance of the computational approach, its relevance should be underscored: for all the systems investigated in this work, the DFT-based studies not only reproduced the experimental data, but instead delivered new valuable information on the magnetic properties for each particular compound. Beyond any doubt, further computational studies will yield new surprising results for known as well as for new, yet unexplored compounds. Such "surprising" outcomes can involve the ferromagnetic nature of the couplings that were previously considered antiferromagnetic, unexpected long-range couplings, or the subtle balance of antiferromagnetic and ferromagnetic contributions that "switches off" the respective magnetic exchange. In this way, dozens of potentially interesting systems can acquire quantitative microscopic magnetic models. The results of this work evidence that elaborate experimental methods and the DFT-based modeling are of comparable reliability and complement each other. In this way, the advantageous combination of theory and experiment can largely advance the research in the field of low-dimensional quantum magnetism. For practical applications, the excellent predictive power of the computational approach can largely alleviate designing materials with specific properties.

In the vast realm of inorganic materials, the Cu2+-containing cuprates form one of the richest classes. Due to the combined effect of crystal-field, covalency and strong correlations, all undoped cuprates are magnetic insulators with well-localized spins S=1/2, whereas the charge and orbital degrees of freedom are frozen out. The combination of the spin-only nature of their magnetism with the unique structural diversity renders cuprates as excellent model systems. The experimental studies, boosted by the discovery of high-temperature superconductivity in doped La2CuO4, revealed a fascinating variety of magnetic behaviors observed in cuprates. A digest of prominent examples should include the spin-Peierls transition in CuGeO3, the Bose-Einstein condensation of magnons in BaCuSi2O6, and the quantum critical behavior of Li2ZrCuO4. The magnetism of cuprates originates from short-range (typically, well below 1 nm) exchange interactions between pairs of spins Si and Sj, localized on Cu atoms i and j. Especially in low-dimensional compounds, these interactions are strongly anisotropic: even for similar interatomic distances |Rij|, the respective magnetic couplings Jij can vary by several orders of magnitude. On the other hand, there is an empirical evidence for the isotropic nature of this interaction in the spin space: different components of Si are coupled equally strong. Thus, the magnetism of cuprates is mostly described by a Heisenberg model, comprised of Jij(Si*Sj) terms. Although the applicability of this approach to cuprates is settled, the model parameters Jij are specific to a certain material, or more precisely, to a particular arrangement of the constituent atoms, i.e. the crystal structure. Typically, among the infinite number of Jij terms, only several are physically relevant. These leading exchange couplings constitute the (minimal) microscopic magnetic model. Already at the early stages of real material studies, it became gradually evident that the assignment of model parameters is a highly nontrivial task. In general, the problem can be solved experimentally, using elaborate measurements, such as inelastic neutron scattering on large single crystals, yielding the magnetic excitation spectrum. The measured dispersion is fitted using theoretical models, and in this way, the model parameters are refined. Despite excellent accuracy of this method, the measurements require high-quality samples and can be carried out only at special large-scale facilities. Therefore, less demanding (especially, regarding the sample requirements), yet reliable and accurate procedures are desirable. An alternative way to conjecture a magnetic model is the empirical approach, which typically relies on the Goodenough-Kanamori rules. This approach links the magnetic exchange couplings to the relevant structural parameters, such as bond angles. Despite the unbeatable performance of this approach, it is not universally applicable. Moreover, in certain cases the resulting tentative models are erroneous. The recent developments of computational facilities and techniques, especially for strongly correlated systems, turned density-functional theory (DFT) band structure calculations into an appealing alternative, complementary to the experiment. At present, the state-of-the-art computational methods yield accurate numerical estimates for the leading microscopic exchange couplings Jij (error bars typically do not exceed 10-15%). Although this computational approach is often regarded as ab initio, the actual procedure is not parameter-free. Moreover, the numerical results are dependent on the parameterization of the exchange and correlation potential, the type of the double-counting correction, the Hubbard repulsion U etc., thus an accurate choice of these crucial parameters is a prerequisite. In this work, the optimal parameters for cuprates are carefully evaluated based on extensive band structure calculations and subsequent model simulations. Considering the diversity of crystal structures, and consequently, magnetic behaviors, the evaluation of a microscopic model should be carried out in a systematic way. To this end, a multi-step computational approach is developed. The starting point of this procedure is a consideration of the experimental structural data, used as an input for DFT calculations. Next, a minimal DFT-based microscopic magnetic model is evaluated. This part of the study comprises band structure calculations, the analysis of the relevant bands, supercell calculations, and finally, the evaluation of a microscopic magnetic model. The ground state and the magnetic excitation spectrum of the evaluated model are analyzed using various simulation techniques, such as quantum Monte Carlo, exact diagonalization and density-matrix renormalization groups, while the choice of a particular technique is governed by the dimensionality of the model, and the presence or absence of magnetic frustration. To illustrate the performance of the approach and tune the free parameters, the computational scheme is applied to cuprates featuring rather simple, yet diverse magnetic behaviors: spin chains in CuSe2O5, [NO]Cu(NO3)3, and CaCu2(SeO3)2Cl2; quasi-two-dimensional lattices with dimer-like couplings in alpha-Cu2P2O7 and CdCu2(BO3)2, as well as the 3D magnetic model with pronounced 1D correlations in Cu6Si6O18*6H2O. Finally, the approach is applied to spin liquid candidates --- intricate materials featuring kagome-lattice arrangement of the constituent spins. Based on the DFT calculations, microscopic magnetic models are evaluated for herbertsmithite Cu3(Zn0.85Cu0.15)(OH)6Cl2, kapellasite Cu3Zn(OH)6Cl2 and haydeeite Cu3Mg(OH)6Cl2, as well as for volborthite Cu3[V2O7](OH)2*2H2O. The results of the DFT calculations and model simulations are compared to and challenged with the available experimental data. The advantages of the developed approach should be briefly discussed. First, it allows to distinguish between different microscopic models that yield similar macroscopic behavior. One of the most remarkable example is volborthite Cu3[V2O7](OH)2*2H2O, initially described as an anisotropic kagome lattice. The DFT calculations reveal that this compound features strongly coupled frustrated spin chains, thus a completely different type of magnetic frustration is realized. Second, the developed approach is capable of providing accurate estimates for the leading magnetic couplings, and consequently, reliably parameterize the microscopic Hamiltonian. Dioptase Cu6Si6O18*6H2O is an instructive example showing that the microscopic theoretical approach eliminates possible ambiguity and reliably yields the correct parameterization. Third, DFT calculations yield even better accuracy for the ratios of magnetic exchange couplings. This holds also for small interchain or interplane couplings that can be substantially smaller than the leading exchange. Hence, band structure calculations provide a unique possibility to address the interchain or interplane coupling regime, essential for the magnetic ground state, but hardly perceptible in the experiment due to the different energy scales. Finally, an important advantage specific to magnetically frustrated systems should be mentioned. Numerous theoretical and numerical studies evidence that low-dimensionality and frustration effects are typically entwined, and their disentanglement in the experiment is at best challenging. In contrast, the computational procedure allows to distinguish between these two effects, as demonstrated by studying the long-range magnetic ordering transition in quasi-1D spin chain systems. The computational approach presented in the thesis is a powerful tool that can be directly applied to numerous S=1/2 Heisenberg materials. Moreover, with minor modifications, it can be largely extended to other metallates with higher value of spin. Besides the excellent performance of the computational approach, its relevance should be underscored: for all the systems investigated in this work, the DFT-based studies not only reproduced the experimental data, but instead delivered new valuable information on the magnetic properties for each particular compound. Beyond any doubt, further computational studies will yield new surprising results for known as well as for new, yet unexplored compounds. Such "surprising" outcomes can involve the ferromagnetic nature of the couplings that were previously considered antiferromagnetic, unexpected long-range couplings, or the subtle balance of antiferromagnetic and ferromagnetic contributions that "switches off" the respective magnetic exchange. In this way, dozens of potentially interesting systems can acquire quantitative microscopic magnetic models. The results of this work evidence that elaborate experimental methods and the DFT-based modeling are of comparable reliability and complement each other. In this way, the advantageous combination of theory and experiment can largely advance the research in the field of low-dimensional quantum magnetism. For practical applications, the excellent predictive power of the computational approach can largely alleviate designing materials with specific properties.:List of Figures List of Tables List of Abbreviations 1. Introduction 2. Magnetism of cuprates 3. Experimental methods 4. DFT-based microscopic modeling 5. Simulations of a magnetic model 6. Model spin systems: challenging the computational approach 7. Kagome lattice compounds 8. Summary and outlook Appendix Bibliography List of publications Acknowledgments

Le présent mémoire traite de la description du LaOFeAs, le premier matériau découvert de la famille des pnictures de fer, par la théorie de la fonctionnelle de la densité (DFT). Plus particulièrement, nous allons exposer l’état actuel de la recherche concernant ce matériau avant d’introduire rapidement la DFT. Ensuite, nous allons regarder comment se comparent les paramètres structuraux que nous allons calculer sous différentes phases par rapport aux résultats expérimentaux et avec les autres calculs DFT dans la littérature. Nous allons aussi étudier en détails la structure électronique du matériau sous ses différentes phases magnétiques et structurales. Nous emploierons donc les outils normalement utilisés pour mieux comprendre la structure électronique : structures de bandes, densités d’états, surfaces de Fermi, nesting au niveau de Fermi. Nous tirerons profit de la théorie des groupes afin de trouver les modes phononiques permis par la symétrie de notre cristal. De plus, nous étudierons le couplage électrons-phonons pour quelques modes. Enfin, nous regarderons l’effet de différentes fonctionnelles sur nos résultats pour voir à quel point ceux-ci sont sensibles à ce choix. Ainsi, nous utiliserons la LDA et la PBE, mais aussi la LDA+U et la PBE+U. / We present DFT calculations of the electronic structure of LaOFeAs, the parent compound of the new family of superconductors, the iron pnictides, in this master thesis. Specifically, we are going to take a look at the present state of the research done on this material before giving a quick introduction to DFT. Then, we will compare the optimized structural parameters as calculated by our DFT code with the experimental data as well as results obtained by other groups. We studied the electronic structure of LaOFeAs using the standard set of tools : band structure, density of state (DOS), fermi surface and fermi surface nesting. We used theoretical methods to determine the allowed phonon modes in this crystal structure. This, in turn, enabled us to explore the electron-phonon coupling in our material for the most important modes. We’ll also discuss the influence different functionals may have for calculating the electronic structure. This will allow us to validate our results. In detail, we will compare results obtained with the following functionals: LDA and PBE, as well as LDA+U and PBE+U.

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