Anisotropic charge screening and supercell size convergence of defect formation energies

One of the main sources of error associated with the calculation of defect formation energies using plane-wave Density Functional Theory (DFT) is finite size error resulting from the use of relatively small simulation cells and periodic boundary conditions. Most widely-used methods for correcting this error, such as that of Makov and Payne, assume that the dielectric response of the material is isotropic and can be described using a scalar dielectric constant $\epsilon$. However, this is strictly only valid for cubic crystals, and cannot work in highly-anisotropic cases. Here we introduce a variation of the technique of extrapolation based on the Madelung potential, that allows the calculation of well converged dilute limit defect formation energies in non-cubic systems with highly anisotropic dielectric properties. As an example of the implementation of this technique we study a selection of defects in the ceramic oxide Li$_2$TiO$_3$ which is currently being considered as a lithium battery material and a breeder material for fusion reactors.