Strongly correlated materials are a class of materials that cannot be properly described by the Density Functional Theory (DFT), which is a single-particle approximation to the original many-body electronic Hamiltonian. These systems contain d or f orbital electrons, i.e., transition metals, actinides, and lanthanides compounds, for which the electron-electron interaction (correlation) effects are too strong to be described by the single-particle approximation of DFT. Therefore, complementary many-body methods have been developed, at the model Hamiltonians level, to describe these strong correlation effects. Dynamical Mean Field Theory (DMFT) and Rotationally Invariant Slave-Boson (RISB) approaches are two successful methods that can capture the correlation effects for a broad interaction strength. However, these many-body methods, as applied to model Hamiltonians, treat the electronic structure of realistic materials in a phenomenological fashion, which only allow to describe their properties qualitatively. Consequently, the combination of DFT and many body methods, e.g., Local Density Approximation augmented by RISB and DMFT (LDA+RISB and LDA+DMFT), have been recently proposed to combine the advantages of both methods into a quantitative tool to analyze strongly correlated systems. In this dissertation, we studied the possible improvements of these approaches, and tested their accuracy on realistic materials. This dissertation is separated into two parts. In the first part, we studied the extension of DMFT and RISB in three directions. First, we extended DMFT framework to investigate the behavior of the domain wall structure in metal-Mott insulator coexistence regime by studying the unstable solution describing the domain wall. We found that this solution, differing qualitatively from both the metallic and the insulating solutions, displays an insulating-like behavior in resistivity while carrying a weak metallic character in its electronic structure. Second, we improved DMFT to describe a Mott insulator containing spin-propagating and chargeless fermionic excitations, spinons. We found the spinon Fermi-liquid, in the Mott insulating phase, is immiscible to the electron Fermi-liquid, in the metallic phase, due to the strong scattering between spinons in a metal. Third, we proposed a new approach within the slave-boson (Gutzwiller) framework that allows to describe both the low energy quasiparticle excitation and the high energy Hubbard excitation, which cannot be captured within the original slave-boson framework. In the second part, we applied LDA+RISB to realistic materials modeling. First, we tested the accuracy of LDA+RISB on predicting the structure of transition metal compounds, CrO, MnO, FeO, CoO, CoS, and CoSe. Our results display remarkable agreements with the experimental observations. Second, we applied LDA+RISB to analyze the nature of the Am-O chemical bonding in the CsAm(CrO_4)_2 crystal. Our results indicate the Am-O bonding has strongly covalent character, and they also address the importance of the correlation effects to describe the experimentally observed electronic structure. In summary, we proposed three extensions within DMFT and RISB framework, which allow to investigate the domain wall structure in metal-Mott insulator coexistence regime, the metal-to-Mott-insulator transition with spinons excitation in the Mott-insulating phase, and the Hubbard excitation within RISB approach. Furthermore, we demonstrated that LDA+RISB is a reliable approximation to the strongly correlated materials by applying it to the transition metal compounds and the Americian chromate compounds.
