Ab initio self-energy corrections in systems with metallic screening

The calculation of self-energy corrections to the electron bands of a metal requires the evaluation of the intraband contribution to the polarizability in the small-$\mathbf{q}$ limit. When neglected, as in standard $GW$ codes for semiconductors and insulators, a spurious gap opens at the Fermi energy. Systematic methods to include intraband contributions to the polarizability exist, but require a computationally intensive Fermi-surface integration. We propose a numerically cheap and stable method based on a fit of the power expansion of the polarizability in the small-$\mathbf{q}$ region. We test it on the homogeneous electron gas and on real metals such as sodium and aluminum.

Figure 1

(Color online) $G_0{W}_0$ band structure of Na (110 direction), showing the appearance of an unphysical gap, and its dependence on different numerical convergence parameters. (a) shows the dependence with respect to the number of empty states in Eq. (13); (b) with respect to the smearing temperature; and (c) with respect to the $\mathbf{k}$-point mesh. (d) shows the dependence of the unphysical gap on the inverse number of $\mathbf{k}$ points in each direction; the dashed line is a fitted $a_1{N}_{\mathrm{kpt}}^{-1∕3}+a_2{N}_{\mathrm{kpt}}^{-2∕3}$ curve.

Figure 2

(Color online) Band structure for the HEG $(r_s=3.5a_0)$ computed with the standard implementation of the $G_0{W}_0$ method (dashed line). The spurious gap, caused by the lack of the intraband term in the screening, is removed when the computed polarizability is replaced by the Lindhard one (dot-dashed line). The KS band is also displayed for reference (dotted line).

Figure 3

(Color online) Numerically computed HEG screening function ${ϵ}^{-1}(\mathbf{q},\omega )$ $(r_s=3.5a_0)$, for $\omega =0$ and $\omega =i{\omega }_P$, compared to the Lindhard function. For $\mathbf{q}\to \mathbf{0}$, the incorrect discontinuous points—pointed at by arrows—appear due to the lack of the intraband term. The differences at large $\mathbf{q}$ are due to the finite number of empty states included in the sums of Eq. (13).

Figure 4

(Color online) The parabolic polarizability ${\chi }_0^{\text{fit}}$ [Eq. (17)] fitted to the computed ${\chi }_{0\mathbf{0},\mathbf{0}}(\mathbf{q},\omega )$ of the HEG (dots) restricted to $\mathbf{q}$ points within a sphere of radius $q_c$ centered at $\mathbf{q}=\mathbf{0}$, and compared to the computed polarizability itself and to the exact (Lindhard) function, for (a) $\omega =0$ and (b) $\omega =i{\omega }_P$. The computation involves a cutoff energy of $3\text{hartree}$, $N_{\mathrm{kpt}}=16\times 16\times 16$, and a smearing temperature $T_{\mathrm{smear}}=0.005\text{hartree}$. [(c)–(f)] Convergence of the fitted values ${\chi }_0^{\text{fit}}({\mathbf{q}}_s,\omega )$ [where the tiny ${\mathbf{q}}_s=(7,14,21){10}^{-6}{a}_0^{-1}$] as a function of the cutoff radius $q_c$, for different $\mathbf{k}$-point sampling, and with (c) $\omega =0$ and (d) $\omega =i{\omega }_P$, and for different smearing temperatures, and with (e) $\omega =0$ and (f) $\omega =i{\omega }_P$. Horizontal lines, the exact (Lindhard) values.

Figure 5

(Color online) Comparison of the band energy obtained via the naive $G_0{W}_0$ calculation (dashed) to those obtained with the $\mathbf{q}\simeq 0$ corrected polarizability (solid). (a) For the HEG, the figure also shows $G_0{W}_0$ results obtained with the analytic (Lindhard) polarizability. For (b) sodium and (c) aluminum, the bands are plotted along the (110) and $\left(11\overline{1}\right)$ directions, respectively.