A comparison of orthogonal and non-orthogonal localized wavefunctions for Si in the diamond structure is carried out. We have used a real-space grid formulation of density functional theory in combination with the local density approximation for exchange and correlation to describe the energetics. Maximally localized wavefunctions, obtained from the extended Kohn-Sham states with and without an orthogonality constraint, are calculated and it is found that the wavefunctions calculated without any orthogonality constraint are the most localized. When solving directly for localized states, by applying a localization constraint to each electronic state, we find that there is a large difference between orthogonal and non-orthogonal states: when the localization region is a sphere with a radius of 3.0 Å, we get an error in the total energy due to the localization constraint of 0.2 and 2.7 eV/atom for non-orthogonal and orthogonal wavefunctions respectively.