Self-consistent $GW$: All-electron implementation with localized basis functions

This paper describes an all-electron implementation of the self-consistent $GW$ (sc-$GW$) approach—i.e., based on the solution of the Dyson equation—in an all-electron numeric atom-centered orbital basis set. We cast Hedin's equations into a matrix form that is suitable for numerical calculations by means of (i) the resolution-of-identity technique to handle four-center integrals and (ii) a basis representation for the imaginary-frequency dependence of dynamical operators. In contrast to perturbative $G_0{W}_0$, sc-$GW$ provides a consistent framework for ground- and excited-state properties and facilitates an unbiased assessment of the $GW$ approximation. For excited states, we benchmark sc-$GW$ for five molecules relevant for organic photovoltaic applications: thiophene, benzothiazole, 1,2,5-thiadiazole, naphthalene, and tetrathiafulvalene. At self-consistency, the quasiparticle energies are found to be in good agreement with experiment and, on average, more accurate than $G_0{W}_0$ based on Hartree-Fock or density-functional theory with the Perdew-Burke-Ernzerhof exchange-correlation functional. Based on the Galitskii-Migdal total energy, structural properties are investigated for a set of diatomic molecules. For binding energies, bond lengths, and vibrational frequencies sc-$GW$ and $G_0{W}_0$ achieve a comparable performance, which is, however, not as good as that of exact-exchange plus correlation in the random-phase approximation and its advancement to renormalized second-order perturbation theory. Finally, the improved description of dipole moments for a small set of diatomic molecules demonstrates the quality of the sc-$GW$ ground-state density.

Figure 1

(a) Values of $\Delta$—defined in Eq. (24)—as a function of the number of iterations of the sc-$GW$ loop for H${}_2$, H${}_2$O, and C${}_6$H${}_6$ in their equilibrium geometry in a Tier 2 basis set. A linear mixing parameter $\alpha =0.2$ was used for all molecules. ${\Delta }_{\mathrm{th}}$ indicates the default value of the convergence threshold. (b) Values of $\Delta$ for H${}_2$O as a function of the number of iterations for different values of $\alpha$.

Figure 2

Total time (in seconds, on a single CPU) per iteration of the sc-$GW$ loop for linear hydrogen chains of different lengths. The total time required for the evaluation of the self-energy in $G_0{W}_0$ is included for comparison.

Figure 3

sc-$GW$ total energy of H${}_2$O as a function of the convergence parameters ${\varepsilon }_{\mathrm{orth}}$ (left) and ${\varepsilon }_{\mathrm{SVD}}$ (right), evaluated with a Tier 2 basis set. The number of product basis functions corresponding to each value of ${\varepsilon }_{\mathrm{orth}}$ is also reported.

Figure 4

Comparison of the analytic structure of the real (in blue, top) and imaginary parts (in orange, bottom) of $f_n\left(i\omega \right)$ for different values of $b_n$ and a matrix element of the Green's function (black).

Figure 5

Mean absolute error (MAE) introduced by Fourier transforming the Green's function of N${}_2$ for different numbers of poles and frequency points.

Figure 6

Spectral function calculated from Eq. (27) using a sc-$GW$ Green's function for H${}_2$O, NH${}_3$, and N${}_2$ with Tier 1 (black line), Tier 2 (orange line), and Tier 3 (blue dashed line) NAO basis sets. Panels (a), (b), and (c) show the peaks corresponding to the first valence states; peaks relative to conduction states are reported in panels (d), (e), and (f). Finally, panels (g), (h), and (i) report the convergence of the highest occupied molecular orbital (HOMO) level as a function of the number of numerical orbitals used in the basis set.

Figure 7

Comparison between theoretical and experimental vertical ionization energies of thiophene, benzothiazole, $1,2,5-$thiadiazole, naphthalene, and tetrathiafulvalene. Experimental photoemission data are from Refs. 47, 48, 49, 50, 51. The molecular geometries were optimized with PBE in a Tier 2 basis set and are reported on the right. $G_0{W}_0$ ionization energies are obtained with a Tier 4 basis set, the sc-$GW$ ones with a Tier 2 basis.

Figure 8

Comparison between sc-$GW$ and $G_0{W}_0$@PBE0 spectral functions for thiophene and $1,2,5$-thiadiazole evaluated with a Tier 2 basis set. Vertical dashed lines indicates experimental photoemission data from Refs. 47 and 48, respectively. $G_0{W}_0$@PBE0 spectra are obtained with a Tier 4 basis set, and the sc-$GW$ spectra are obtained with a Tier 2 basis.

Figure 9

Spectral function of benzene evaluated from sc-$GW$ and $G_0{W}_0$ based on PBE, PBE0 with different mixture of exact exchange (EX) and HF. Vertical dashed lines (in green) indicate the energy threshold $\Delta$ for the formation of an electron-hole pair, which depends explicitly on the HOMO and LUMO levels of the underlying DFT/HF calculation. The quasiparticle peaks acquire a finite broadening only for states below $\Delta$. For peaks above $\Delta$, the residual broadening stems from the parameter $\eta ={10}^{-4}$ discussed in the text.

Figure 10

Spectral function of the CO dimer evaluated for the first five iterations of the Dyson equation from the HF, PBE, and hybrid PBE0 starting points. The first iteration corresponds to the $G_0{W}_0$ approximation.

Figure 11

BSSE corrected binding energy of CO evaluated with Tiers 1, 2, 3, and 4. Each curve is aligned at their respective Tier 4 value. All data are in eV.

Figure 12

Mean absolute error (MAE) of bond lengths (top panel), binding energies (middle panel), and vibrational frequencies (bottom panel) of LiH, LiF, HF, CO, H${}_2$, and N${}_2$ evaluated at different levels of theory. The estimated zero-point motion correction has been subtracted from the experimental binding energies (reported from Ref. 69). Calculations were done with a Tier 3 basis set (the largest basis set available) for H${}_2$, LiH, and HF, whereas a Tier 4 basis was used for N${}_2$ and CO. Numerical values are reported in Tables , , and in Appendix .

Figure 13

Difference between the sc-$GW$ and HF densities for the hydrogen fluoride dimer at its experimental equilibrium geometry ($d=0.917$ Å). At self-consistency, electron density is shifted from negative (dark) to positive regions (light). Units are Å${}^{-3}$ and the calculation were performed using a Tier 3 basis set.