Development and applications of a real-space multigrid method for the first-principles electronic structure calculations

Abstract

We present an efficient real-space multigrid method for first-principles electronic structure calculations in the framework of the local-density-functional approximation and the generalized gradient approximation. The Laplacian and gradient operators are discretized on mesh in real space using a higher-order finite difference method. The Poisson and Kohn-Sham equations are solved by a multigrid method that uses different relaxations on several grid levels. Testing various systems, convergence rates are found to be nearly independent of the number of real-space grids. For materials with localized d orbitals such as GaN, the number of iterations is not much increased. For charged Si clusters, the real-space method with use of nonperiodic boundary conditions gives exact total energies, as compared to plane-wave supercell calculations. The errors caused by the use of supercells are enlarged as excess or deficit charges from neutral clusters increase. In bulk materials where periodic-boundary conditions are employed, the Slater-Janak transition-state theorem is used to calculate the total energies of charged defects. We investigate several defects in GaN, including the Ga 3d electrons in the valence states. The effect of the Ga 3d electrons on the energetics of defects is discussed. Including spin polarization effects, our real-space calculations is applied to oxygen-related defects such as oxygen vacancy, oxygen interstitial, and oxygen molecule in $SiO_2$. For the neutral charge state, the oxygen vacancy exhibits a dimer configuration with a Si-Si bond. However, for the doubly positively charged state, the oxygen vacancy undergoes a spontaneous structural transformation from the dimer configuration to the puckered configuration, accompanied with breaking a Si-Si bond due to hole capture. For the neutral and positive charge states, the peroxy linkage with a Si-O-O-Si...