Energy correction for isolated impurities under periodic boundary conditions

Electrostatic interactions in finite systems treated with periodic boundary conditions: application to linear-scaling density functional theory

We present a comparison of methods for treating the electrostatic interactions of finite, isolated systems within periodic boundary conditions (PBCs), within density functional theory (DFT), with particular emphasis on linear-scaling (LS) DFT. Often, PBCs are not physically realistic but are an unavoidable consequence of the choice of basis set and the efficacy of using Fourier transforms to compute the Hartree potential. In such cases the effects of PBCs on the calculations need to be avoided, so that the results obtained represent the open rather than the periodic boundary. The very large systems encountered in LS-DFT make the demands of the supercell approximation for isolated systems more difficult to manage, and we show cases where the open boundary (infinite cell) result cannot be obtained from extrapolation of calculations from periodic cells of increasing size. We discuss, implement, and test three very different approaches for overcoming or circumventing the effects of PBCs: truncation of the Coulomb interaction combined with padding of the simulation cell, approaches based on the minimum image convention, and the explicit use of open boundary conditions (OBCs). We have implemented these approaches in the ONETEP LS-DFT program and applied them to a range of systems, including a polar nanorod and a protein. We compare their accuracy, complexity, and rate of convergence with simulation cell size. We demonstrate that corrective approaches within PBCs can achieve the OBC result more efficiently and accurately than pure OBC approaches.

Charged Defect Formation Energies in TiO2 Using the Supercell Approximation

TiO2 has been intensively studied as a wide band-gap transition metal oxide partially due to the multi-valence nature of its cation. Here, density-functional theory calculations within the supercell approximation are carried out to determine the preferred charge state of point defects in rutile TiO2. The first component of this work is to investigate the dependence of the defect formation energies on supercell size and the electrostatic Makov-Payne correction. The results show that the Makov-Payne correction improves the convergence of defect formation energies as a function of supercell size for positively charged titanium interstitials and negatively charged titanium vacancies. However, in the case of positively charged oxygen vacancies, applying the Makov-Payne correction gives the wrong sign for the defect formation energy correction. This is attributed to the shallow nature of the transition levels for this defect in TiO2. Finally, we combine the calculated defect formation energies with thermodynamic data to evaluate the influence of temperature on the relative stabilities of point defects. The results indicate that when the Makov- Payne correction is applied, a stable charge transition occurs for titanium interstitials. In addition, as the temperature increases, the dominant point defect in TiO2 changes from oxygen vacancies to titanium interstitials.