Energy density in density functional theory: Application to crystalline defects and surfaces

We propose a method for decomposing the total energy of a supercell containing defects into contributions of individual atoms, using the energy density formalism within the density functional theory. The spatial energy density is unique up to a gauge transformation, and we show that unique atomic energies can be calculated by integrating over Bader and charge-neutral volumes for each atom. Numerically, we implement the energy density method in the framework of the Vienna ab initio simulation package (VASP) for both norm-conserving and ultrasoft pseudopotentials and the projector augmented-wave method, and we use a weighted integration algorithm to integrate the volumes. Surface energies and point defect energies can be calculated by integrating the energy density over the surface region and the defect region, respectively. We compute energies for several surfaces and defects: the (110) surface energy of GaAs, the monovacancy formation energies of Si, the (100) surface energy of Au, and the interstitial formation energy of O in a hexagonal close-packed Ti crystal. The surface and defect energies calculated using our method agree with size-converged calculations of the difference in the total energies of a system with versus a system without defects. Moreover, the convergence of the defect energies with size can be found from a single calculation.