Real-space pseudopotential method for first principles calculations of general periodic and partially periodic systems

We present a real-space method for electronic-structure calculations of systems with general full or partial periodicity. The method is based on the self-consistent solution of the Kohn-Sham equations, using first principles pseudopotentials, on a uniform three-dimensional non-Cartesian grid. Its efficacy derives from the introduction of a new generalized high-order finite-difference method that avoids the numerical evaluation of mixed derivative terms and results in a simple yet accurate finite difference operator. Our method is further extended to systems where periodicity is enforced only along some directions (e.g., surfaces), by setting up the correct electrostatic boundary conditions and by properly accounting for the ion-electron and ion-ion interactions. Our method enjoys the main advantages of real-space grid techniques over traditional plane-wave representations for density functional calculations, namely, improved scaling and easier implementation on parallel computers, as well as inherent immunity to spurious interactions brought about by artificial periodicity. We demonstrate its capabilities on bulk GaAs and Na for the fully periodic case and on a monolayer of Si-adsorbed polar nitrobenzene molecules for the partially periodic case.

Figure 1

Schematic representation of the $\widehat{u}$, $\widehat{v}$, and $\widehat{u}-\widehat{v}$ directions. Here the angle between $\widehat{u}$ and $\widehat{v}$ is acute so that $\widehat{u}-\widehat{v}$ is the preferred direction for derivation—see text for details.

Figure 2

Real-space calculation within the local density approximation for the band structure of GaAs along selected high-symmetry directions in the Brillouin zone, using (a) scalar-relativistic and (b) fully relativistic (i.e., including spin-orbit coupling) pseudopotentials.

Figure 3

Real-space calculation within the local density approximation for the band structure of Na along selected high-symmetry directions in the Brillouin zone. The Fermi energy has been chosen as the zero energy.

Figure 4

(Color online) Electrostatic dipole per cell as a function of position, for a nitrobenzene monolayer adsorbed on the Si(111) surface, computed within a partially periodic PARSEC calculation and a fully periodic dipole-corrected VASP calculation. The unit cell structure is given above the dipole curves on the same scale.