Simple Approximate Physical Orbitals for $GW$ Quasiparticle Calculations

Generating unoccupied orbitals within density functional theory (DFT) for use in $GW$ calculations of quasiparticle energies becomes prohibitive for large systems. We show that, without any loss of accuracy, the unoccupied orbitals may be replaced by a set of simple approximate physical orbitals made from appropriately prepared plane waves and localized basis DFT orbitals that represent the continuum and resonant states of the system, respectively. This approach allows for accurate quasiparticle calculations using only a very small number of unoccupied DFT orbitals, resulting in an order of magnitude gain in speed.

Figure 1

(a) Energy eigenvalues and (b) plane-wave components of the eigenvectors arranged in ascending order by kinetic energy for the electronic orbitals of benzene (${\mathrm{C}}_6{\mathrm{H}}_6$). Orbitals 1–87 are obtained from plane-wave DFT calculation. Orbitals 88–200 are the SAPOs $\{E_{n\mathbf{k}},{\Psi }_{n\mathbf{k}}(\mathbf{r})\}$ ($\mathrm{C}+\mathrm{R}/\mathrm{E}$ using the notation of Fig. 2). The zero of the energy scale in (a) is aligned with the vacuum level.

Figure 2

Self energy (a) in the static COHSEX approximation and (b) within the GPP model for the HOMO of benzene (${\mathrm{C}}_6{\mathrm{H}}_6$). The sums over orbitals in (a) irreducible polarizability and (b) self energy are computed using 17, 87, 481, and 2699 exact DFT orbitals complemented with the SAPOs made of the symmetrized plane waves and localized basis DFT orbitals that represent the continuum (C) and resonant (R) orbitals of benzene, respectively, up to the energy cutoff of 6 Ry above the vacuum level. The C and R orbitals are orthonormalized according to Eq. (2) with either the energy (E) correction of Eq. (4) applied or the DFT Hamiltonian (H) diagonalization of Eq. (5) carried out. The horizontal lines denote the values computed with 2699 exact DFT orbitals.