Implementation and performance of the frequency-dependent $GW$ method within the PAW framework

Algorithmic details and results of fully frequency-dependent $G_0{W}_0$ calculations are presented. The implementation relies on the spectral representation of the involved matrices and their Hilbert or Kramers-Kronig transforms to obtain the polarizability and self-energy matrices at each frequency. Using this approach, the computational time for the calculation of polarizability matrices and quasiparticle energies is twice as that for a single frequency, plus Hilbert transforms. In addition, the implementation relies on the PAW method, which allows to treat $d$-states with relatively modest effort and permits the reevaluation of the core-valence interaction on the level of the Hartree-Fock approximation. Tests performed on an $sp$ material (Si) and materials with $d$ electrons (GaAs and CdS) yield quasiparticle energies that are very close to previous all-electron pseudopotential and all-electron full-potential linear muffin-tin-orbital calculations.

Figure 1

Illustration of additive augmentation in the PAW method. Pseudized quantities are defined in the entire space on a regular (plane wave) grid (a). To obtain AE energies, the pseudo-wave-functions are reconstructed inside spheres and the corresponding one-center energy terms are subtracted (b), and finally the AE wave functions are reconstructed as well and the AE one-center energies are added (c).

Figure 2

Atomic scattering properties of the Si TM (left) and PAW (right) potentials used in the present work for QP calculations (Tables ). Shown are the logarithmic derivatives of the radial wave functions for different angular momenta for a spherical Si atom, evaluated at a distance of $r=1.3\mathrm{\AA }$ from the nucleus. Solid lines correspond to the all-electron full-potential, and dotted lines to the TM pseudopotential or PAW potential. The energy zero corresponds to the vacuum level. Circles indicate linearization energies for projectors.