The goal of computational research in the fields of engineering, physics, chemistry or as a matter of fact in any field, is to study the properties of systems from the various principles available. In computational engineering, particularly in nano-scale simulations involving low-energy physics or chemistry, the goal is to model such structures and understand their properties from first principles or better known as \textit{Ab Initio} calculations. Geometry optimization is the basic component used in modeling molecules. The calculations involved are used to find the coordinates or the positions of the atoms of the molecule where it has the minimum energy and is hence stable. Efficient calculation of the forces acting on the atoms is the most important factor to be able to study the stable geometry of a molecule. In this thesis, the approach used begins with efficient electronic structure calculations using all electron calculations which paves the way for efficient force calculations. Kohn-Sham equations Density functional theory (DFT) are used to find the electron wave functions as accurately as possible using a finite element basis that introduces minimum errors in calculations. FEAST, a highly efficient density matrix based eigenvalue solver, is used to obtain accurate eigenvalues. Derivation of forces is done using the Hellmann-Feynman theorem. To find the minimum energy configuration of the system, Newton's iterative method is used that converges to the desired coordinates where the energy at the global minimum is found. The theory behind energy minimization and the calculations involved will be elaborated in this thesis and a method to move the atom in the existing framework will be discussed.