Optimized norm-conserving Vanderbilt pseudopotentials

Fully nonlocal two-projector norm-conserving pseudopotentials are shown to be compatible with a systematic approach to the optimization of convergence with the size of the plane-wave basis. A reformulation of the optimization is developed, including the ability to apply it to positive-energy atomic scattering states and to enforce greater continuity in the pseudopotential. The generalization of norm conservation to multiple projectors is reviewed and recast for the present purposes. Comparisons among the results of all-electron and one- and two-projector norm-conserving pseudopotential calculations of lattice constants and bulk moduli are made for a group of solids chosen to represent a variety of types of bonding and a sampling of the periodic table.

Figure 1

All-electron $\left(\psi \right)$ and pseudo- $\left(\phi \right)$ wave functions for the Ge $d$ scattering state at 0.25 Ha, illustrating the use of a soft barrier to induce a bound-state-like tail, which allows residual energy optimization.

Figure 2

Ge pseudopotentials illustrating the smooth behavior characteristic of the residual energy optimization approach introduced here.

Figure 3

Illustration of the pseudopotential continuity results from requiring $M$ $=$ 3, 4, or 5 value plus derivative continuity constraints on the pseudo-wave-function, for the weakly-bound unoccupied Ca $d$ state. $r_c$ $=$ 2.5 a${}_{\mathrm{B}}$ is indicated by the vertical bar.

Figure 4

Convergence of the total energy with plane-wave cutoff for Si and Cu solids, showing the minor effect of the second projector of the OV method compared to KB. The points are residual energies from the first-projector atomic calculations.

Figure 5

K logarithmic derivatives vs energy plotted as $\mathrm{atan}[r_c(d\psi /dr)/\psi ]/\pi$ at $r_c$. All-electron, OV, and KB are compared. Curves are offset for clarity.

Figure 6

K 4$s$ wave functions computed with the all-electron potential and KB and OV pseudopotentials.