This thesis focuses on the phase transitions, including ferro-magnetic, anti-ferromagnetic, metal to "Mott" insulator and structural transitions in perovskite and Ruddlesden-Popper ruthenates. The thesis is mainly composed of two parts. The first half presents Density Functional Theory (DFT)+Dynamical Mean Field Theory (DMFT) studies of the electronically driven phase transitions in various ruthenate materials. We study cubic perovskite BaRuO$_3$ via DFT add DMFT method using interaction parameters which were found in previous studies to be appropriate for the related materials, CaRuO$_3$ and SrRuO$_3$. The calculated variation in transition temperature between the Ba and Sr compounds is consistent with experiment, confirming the assignment of the compounds to the Hund's metal family of materials, the appropriateness of the single-site dynamical mean field approximation for these materials as well as confirming the appropriateness of the values for the interaction parameters. The results provide insights into the origin of magnetism and the role of the van Hove singularity in the physics of Hund's metals.

We also study the metal-insulator transition (MIT) and magnetic transitions in Ca$_2$RuO$_4$. The Ru-O bonds lengths are found to be the most important control parameters for the metal-insulator transitions and rotations are found to be less important. The calculation successfully captures the important features of the para-magnetic (PM) "Mott" insulating state, including the orbital occupancy disproportionation and the orbitally resolved electron spectral function. It shows the advantage of single set DFT+DMFT in dealing with strongly correlated multi-orbital systems without the assumption of spin symmetry breaking. In the second half, we present a Landau free energy model that incorporates the electronic energetics, the coupling of the electronic state to local distortions and the coupling of local distortions to long-wavelength strains. The model is used to elucidate important experimental features in thermal and current-induced MIT in Ca$_2$RuO$_4$ and Ca$_3$Ru$_{2-x}$Ti$_x$O$_7$ materials.

The investigation of lattice and electronic energetics and determination of parameters using DFT+DMFT methods is explained. The change in lattice energy across the metal-insulator transition is shown to be comparable to the change in electronic energy. Important consequences are a strongly first order transition, a sensitive dependence of the phase boundary on pressure and that the geometrical constraints on in-plane lattice parameter associated with epitaxial growth on a substrate typically change the lattice energetics enough to eliminate the metal-insulator transition entirely. The change in elasto-resistance across the MIT is determined. The DFT+U relaxation study shows the octahedron relaxation with respect to uniaxial strain on a and b axes are very different. This sensitive a and b axes dependence is observed in calculations on both Ca$_2$RuO$_4$ and Ca$_3$Ru$_2$O$_7$. The theory model is also generalized to investigate spatially non-homogeneous solutions. Important features of the stripe patterns at the domain boundaries of metallic and insulating phases are discussed and compared with experiments.