Microhartree precision in density functional theory calculations

To address ultimate precision in density functional theory calculations we employ the full-potential linearized augmented plane-wave + local-orbital (LAPW + lo) method and justify its usage as a benchmark method. LAPW + lo and two completely unrelated numerical approaches, the multiresolution analysis (MRA) and the linear combination of atomic orbitals, yield total energies of atoms with mean deviations of 0.9 and $0.2\mu \mathrm{Ha}$, respectively. Spectacular agreement with the MRA is reached also for total and atomization energies of the G2-1 set consisting of 55 molecules. With the example of $\alpha$ iron we demonstrate the capability of LAPW + lo to reach $\mu \mathrm{Ha}/\mathrm{atom}$ precision also for periodic systems, which allows also for the distinction between the numerical precision and the accuracy of a given functional.

Figure 1

The error in the total LSDA energy of an oxygen atom when using local orbitals with angular momenta up to ${\ell }_{max}$ (left) and as a function of a plane-wave cutoff $R_{\mathrm{MT}}{G}_{max}$ (right). The limit of the total energy is estimated by using ${\ell }_{max}=6$ and by extrapolating its dependence on $R_{\mathrm{MT}}{G}_{max}$.

Figure 2

LSDA energies of atoms obtained by madness (MRA) and nwchem (LCAO) with exciting (LAPW + lo) results taken as a reference. Note the microhartree precision throughout.

Figure 3

Bulk moduli and lattice constants of $\alpha$ iron. The yellow triangles correspond LAPW + lo calculations, and the red circles correspond to results by other methods taken from Ref. [3]. The open and blue triangles represent older LAPW(+ lo) calculations [33]. The lines indicate results obtained in this Rapid Communication. The diamonds correspond to experimental data [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45].