Fully self-consistent solution of the Dyson equation using a plane-wave basis set

Self-consistent solutions of the Dyson equation are obtained using a plane-wave basis set for seven small molecules. Such self-consistent solutions can help to unify the different $GW$ self-consistent schemes, reduce the scatter of results in current $GW$ calculations, and shed light on the true effects of $GW$ self-consistency. Unlike other works of self-consistent $GW$ calculations, in the present work the Green's function is expressed as a matrix under the plane-wave basis set. The algorithmic details which enable such calculations are presented. The ability to solve the full Green's function using a plane-wave basis set may open the door for future beyond-$GW$ many-body perturbation theory calculations.

Figure 1

The $G_0(q_1,q_2,i\omega )$ for an typical off diagonal $(q_1,q_2)$ element (a), and (b) its fitting error within each interval $[{\omega }_k,{\omega }_{k+1}]$ using the analytical formula with coefficients obtained from Eqs. (8)–(10). The horizontal line with ticks in (a) indicates the ${\omega }_k$ positions.

Figure 2

The numerically integrated $G_0(q_1,q_2,i\tau )$ from the $G_0(q_1,q_2,i\omega )$ in Fig. 1; the error of the numerically integrated $G_0(q_1,q_2,i\tau )$ when compared with the analytical expression of Eq. (16) (b). The ticks on the horizontal axis in (a) indicate the ${\tau }_k$ positions.

Figure 3

(a) The self-consistent field (SCF) convergence of the GW calculations. The convergence is measured by the convergence of the eigenvalues of $H_0+\Sigma (\omega =0)$. See main text for more details. (b) The convergence using the difference of the Green's function between adjacent steps. (c) The convergence between the LDA initial wave functions and eigenenergies and the HF initial wave functions and eigenenergies.

Figure 4

The expectation value $\overline{\Sigma }\left(i\omega \right)=\langle {\psi }_i^{\prime }\left(0\right)\left|\Sigma \left(i\omega \right)\right|{\psi }_i^{\prime }\left(0\right)\rangle$ for $i=6$ on the imaginary axis (a). The crosses are the directly calculated values, and the continued lines are the analytical fitting curve; the analytically extended $\overline{\Sigma }\left(\omega \right)$ on real $\omega$ axis (b).

Figure 5

The spread of the eigenvectors along the imaginary $i\omega$ axis. The $d_i\left(i\omega \right)$ is defined in the text. The numbers are the state index $i$'s.

Figure 6

The spectral functions as calculated from Eq. (19). The dotted lines are for $G_0{W}_0$ results starting with LDA input wave functions and eigenenergies, and the solid lines are for sc-GW results. The Fermi energy $\mu$ for ${\mathrm{Al}}_2$, CHF, and HNO are around $-3$, $-5$, and $-5$ eV, respectively.