In this thesis in the first part, using first-principle method we investigated charged Oxygen vacancies in rutile TiO 2 by exploiting gap correction approach such as GGA+U, GW many body correctio and hybrid functional . Although GGA+U improve the character of localized d-orbitals and also enlarge the band gap, but it would be unable to give the neutral and singly-ionized vacancy induced levels inside the gap and stabilize them. We apply GW on top of GGA+U which is better starting point and closer features to the quasiparticle picture than GGA . The obtained results shows that application of GW correction severely depends on and follows the behavior of the starting point and therefore cannot stabilize the neutral and singly-ionized vacancy in the gap. Moreover , In the whole variation of Fermi energy across the gap, singly ionized vacancy is never stable. Employing hybrid functional PBE0 and HSE , in addition to gap correction, would transfer the neutral and singly-ionized vacancy states into the gap in qualitative agreement with experiment. Also the results shows doubly –ionized vacancy have lower formation energy in Ferrni-energy variation . Hence oxygen vacancy is shallow donor with the doubly –ionized vacancy is the most stable charge. In the second part, we performed ab-inito methods to the investigation of Mn and As centers magnetic nanowires. Using two phenomenological models based on dangling bond energies and position-dependent atomic energies , show periodic behavior of stability which originate from edge energy and compositions. Additionally , using ab-initio thermodynamics in real temperature and pressure, all phase diagram show periodic stability in which the magnetic edges is undesirable.