Adaptation of the projector-augmented-wave formalism to the treatment of orbital-dependent exchange-correlation functionals

This paper presents the formulation and numerical implementation of a self-consistent treatment of orbital-dependent exchange-correlation functionals within the projector-augmented-wave method of Blöchl [Phys. Rev. B 50, 17953 (1994)] for electronic structure calculations. The methodology is illustrated with binding energy curves for C in the diamond structure and LiF in the rock salt structure, by comparing results from the Hartree-Fock (HF) formalism and the optimized effective potential formalism in the so-called KLI approximation [Krieger, Li, and Iafrate, Phys. Rev. A 45, 101 (1992)] with those of the local density approximation. While the work here uses pure Fock exchange only, the formalism can be extended to treat orbital-dependent functionals more generally.

Figure 1

Plot of total energy differences $\Delta E$ for spin-unpolarized spherical atoms in their ground-state configurations for $Z=1$ (H) through $Z=36$ (Kr), using the self-consistent Hartree-Fock energy as reference. Results for $\Delta E$ obtained using self-consistent OEP or Kreiger, Li, and Iafrate (KLI) calculations are compared with results obtained using self-consistent generalized gradient approximation (GGA) or local density approximation (LDA) wave functions to “postprocess” the Fock exchange energies.

Figure 2

Plot of energy differences $\Delta E$ for spin-unpolarized spherical atoms in their ground-state configurations using LDA or GGA exchange correlation functionals, each referenced to the corresponding energies obtained using the same wave functions to “postprocess” the Fock exchange contribution.

Figure 3

Comparison of OEP and KLI functions of $V_x\left(r\right)$ for Br, Cl, and F in their ground states.

Figure 4

Plot of ionization energies for spherically averaged $3d$ atoms, assuming transitions $3d^{x}4s^2\to 3d^{x}4s^1$, using the HF, KLI, and LDA exchange approximations. The results designated as HFV were calculated using the frozen-core potential approximation as described in Ref.28.

Figure 5

Comparison of KLI all-electron [$V_x\left(r\right)$] and pseudo [${\tilde{V}}_x\left(r\right)$] local exchange potentials for Br, Cl, and F in their ground states.

Figure 6

Plots of ${\tilde{V}}_{\mathrm{loc}}^a\left(r\right)$ for F, Cl, and Br, comparing KLI results with the corresponding LDA results.

Figure 7

Binding energy curve for C in the diamond structure.

Figure 8

Binding energy curve for LiF in the rock salt structure.

Figure 9

Shell contributions to the combined shift function for the C atom.

Figure 10

Orbital shift functions ${\mathrm{g}}_p\left(r\right)$ for the C atom.