Maximally localized Wannier functions for GW quasiparticles

We review the formalisms of the self-consistent GW approximation to many-body perturbation theory and of the generation of optimally localized Wannier functions from groups of energy bands. We show that the quasiparticle Bloch wave functions from such GW calculations can be used within this Wannier framework. These Wannier functions can be used to interpolate the many-body band structure from the coarse mesh of Brillouin-zone points on which it is known from the initial calculation to the usual symmetry lines, and we demonstrate that this procedure is accurate and efficient for the self-consistent GW band structure. The resemblance of these Wannier functions to the bond orbitals discussed in the chemical community led us to expect differences between density-functional and many-body functions that could be qualitatively interpreted. However, the differences proved to be minimal in the cases studied. Detailed results are presented for ${\text{SrTiO}}_3$ and solid argon.

Figure 1

(Color online) Isosurface plots of ${\text{SrTiO}}_3$ valence-band maximally localized Wannier functions for GW quasiparticles, at isosurface values $\pm 1/\sqrt{V}$, where $V$ is the unit-cell volume, positive values red/light gray, and negative blue/dark gray. (a) is an O-centered $p_z$-like function showing sigma bonding with the $\text{Ti}{d}_{z^2}$ orbital, and (b) an $\text{O}{p}_x$-like function showing pi bonding with $\text{Ti}{d}_{xz}$.

Figure 2

(Color online) Isosurface plots for ${\text{SrTiO}}_3$ conduction-band MLWFs at isosurface values $\pm 2/\sqrt{V}$, showing $\text{Ti}{d}_{z^2}$ character with a small $\text{O}{p}_z$ antibonding contribution, for LDA and for an “enhanced” GW function which exaggerates the difference as explained in the text.

Figure 3

(Color online) ${\text{SrTiO}}_3$ LDA band structure for the $\text{O}2p$ upper valence bands and the low-lying conduction bands. The gray dashed lines are full LDA calculations, and the solid red lines are the Wannier interpolation. The dash-dotted OW and FW lines indicate the range of the outer and frozen energy windows used in the conduction-band MLWF construction.

Figure 4

(Color online) ${\text{SrTiO}}_3$ band structure comparing Wannier-interpolated LDA (solid red) and QSGW (dashed blue) upper valence and lower conduction bands. The open circles at the symmetry points denote the exact QSGW results on $\mathbf{k}$-mesh points.

Figure 5

Second conduction band of solid Ar: squared modulus of the Bloch function along the direction (110) at $\mathbf{k}=\left(-1/8,-3/8,1/4\right)$. White circles represent the location of the argon atoms.

Figure 6

(Color online) Full and interpolated solid Ar LDA conduction bands following the conventions of Fig. 3. The lower limits of the energy windows can be anywhere in the gap.

Figure 7

(Color online) Isosurface plots of QSGW MLWFs for solid Ar conduction bands at isosurface values $\pm 0.75/\sqrt{V}$. (a) One of six $s\text{−}p\text{−}d$ hybridlike functions pointing along the positive and negative Cartesian axes; (b) one of three $t_{2g}$ $d$-like functions.

Figure 8

(Color online) Comparison of Wannier-interpolated LDA and QSGW conduction bands for solid Ar, following the conventions of Fig. 4.