Real-space formulation of orbital-free density functional theory using finite-element discretization: The case for Al, Mg, and Al-Mg intermetallics

We propose a local real-space formulation for orbital-free density functional theory (DFT) with density dependent kinetic energy functionals and a unified variational framework for computing the configurational forces associated with geometry optimization of both internal atomic positions as well as the cell geometry. The proposed real-space formulation, which involves a reformulation of the extended interactions in electrostatic and kinetic energy functionals as local variational problems in auxiliary potential fields, also readily extends to all-electron orbital-free DFT calculations that are employed in warm dense matter calculations. We use the local real-space formulation in conjunction with higher-order finite-element discretization to demonstrate the accuracy of orbital-free DFT and the proposed formalism for the Al-Mg materials system, where we obtain good agreement with Kohn-Sham DFT calculations on a wide range of properties and benchmark calculations. Finally, we investigate the cell-size effects in the electronic structure of point defects, in particular, a monovacancy in Al. We unambiguously demonstrate that the cell-size effects observed from vacancy formation energies computed using periodic boundary conditions underestimate the extent of the electronic structure perturbations created by the defect. On the contrary, the bulk Dirichlet boundary conditions, accessible only through the proposed real-space formulation, which correspond to an isolated defect embedded in the bulk, show cell-size effects in the defect formation energy that are commensurate with the perturbations in the electronic structure. Our studies suggest that even for a simple defect like a vacancy in Al, we require cell sizes of $\ensuremath{\sim}{10}^{3}$ atoms for convergence in the electronic structure.