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Electron-lattice interactions are often considered not to play a major role in material's properties as they are assumed to be small, the second-order effects. However, this study shows the importance of taking these effects into account in the simulations. My results demonstrate the impact of the electron-lattice interaction on the physics of the material and our understanding from it. One way to study these effects is to add them as perturbations to the unperturbed Hamiltonians in numerical simulations. The main objective of this thesis is to study electron-lattice interactions in molecular and magnetic crystals. It is devoted to developing numerical techniques considering model Hamiltonians and first-principles calculations to include the effect of lattice vibrations in the simulations of the above mentioned classes of materials. In particular, I study the effect of adding the non-local electron-phonon coupling on top of the Holstein Hamiltonian to study the polaron stability and polaron dynamics in molecular crystals. The numerical calculations are based on the semi-empirical Holstein-Peierls model in which both intra (Holstein) and inter (Peierls) molecular electron-phonon interactions are taken into account. I study the effect of different parameters including intra and intermolecular electron-phonon coupling strengths and their vibrational frequencies, the transfer integral and the electric field on polaron stability. I found that in an ordered two dimensional molecular lattice the polaron is stable for only a limited range of parameter sets with the polaron formation energies lying in the range between 50 to 100 meV. Using the stable polaron solutions, I applied an electric field to the system and I observed that the polaron is dynamically stable and mobile for only a limited set of parameters. Adding disorder to the system will result in even more restricted parameter set space for which the polaron is stable and moves adiabatically with a constant velocity. In order to study the effect of temperature on polaron dynamics, I include a random force in Newtonian equations of motion in a one dimensional molecular lattice. I found that there is a critical temperature above which the polaron destabilizes and becomes delocalized. Moreover, I study the role of lattice vibrations coupled to magnetic degrees of freedom in finite temperature paramagnetic state of magnetic materials. Calculating the properties of paramagnetic materials at elevated temperatures is a cumbersome task. In this thesis, I present a new method which allows us to couple lattice vibrations and magnetic disorder above the magnetic transition temperature and treat them on the same footing. The method is based on the combination of disordered local moments model and ab initio molecular dynamics (DLM-MD). I employ the method to study different physical properties of some model systems such as CrN and NiO in which the interaction between the magnetic and lattice degrees of freedom is very strong making them very good candidates for such a study. I calculate the formation energies and study the effect of nitrogen defects on the electronic structure of paramagnetic CrN at high temperatures. Using this method I also study the temperature dependent elastic properties of paramagnetic CrN. The results highlight the importance of taking into account the magnetic excitations and lattice vibrations in the studies of magnetic materials at finite temperatures. A combination of DLM-MD with another numerical technique namely temperature dependent effective potential (TDEP) method is used to study the vibrational free energy and phase stability of CrN. We found that the combination of magnetic and vibrational contributions to the free energy shifts down the phase boundary between the cubic paramagnetic and orthorhombic antiferromagnetic phases of CrN towards the experimental value. I used the stress-strain relation to study the temperature-dependent elastic properties of paramagnetic materials within DLM-MD with CrN as my model system. The results from a combinimation of DLM-MD with another newly developed method, symmetry imposed force constants (SIFC) in conjunction with TDEP is also presented as comparison to DLM-MD results.I also apply DLM-MD method to study the electronic structure of NiO in its paramagnetic state at finite temperatures. I found that lattice vibrations have a prominent impact on the electronic structure of paramagnetic NiO at high temperatures and should be included for the proper description of the density of states. In summary, I believe that the proposed techniques give reliable results and allow us to include the effects from electron-lattice interaction in simulations of materials

Theoretical Description of the Electron-Lattice Interaction in Molecular and Magnetic Crystals