$\mathrm{GW}\mathrm{\Gamma }+$ Bethe-Salpeter equation approach for photoabsorption spectra: Importance of self-consistent $\mathrm{GW}\mathrm{\Gamma }$ calculations in small atomic systems

The self-consistent $\mathrm{GW}\mathrm{\Gamma }$ method satisfies the Ward-Takahashi identity (i.e., the gauge invariance or the local charge continuity) for arbitrary energy ($\omega$) and momentum ($\mathit{q}$) transfers. Its self-consistent first-principles treatment of the vertex $\mathrm{\Gamma }={\mathrm{\Gamma }}_v$ or ${\mathrm{\Gamma }}_W$ is possible to first order in the bare ($v$) or dynamically screened ($W$) Coulomb interaction. It is developed within a linearized scheme and combined with the Bethe-Salpeter equation (BSE) to accurately calculate photoabsorption spectra (PAS) and photoemission (or inverse photoemission) spectra (PES) simultaneously. The method greatly improves the PAS of Na, ${\mathrm{Na}}_3, {\mathrm{B}}_2$, and ${\mathrm{C}}_2{\mathrm{H}}_2$ calculated using the standard one-shot $G_0{W}_0+\mathrm{BSE}$ method that results in significantly redshifted PAS by 0.8–3.1 eV, although the PES are well reproduced already in $G_0{W}_0$.

Figure 1

Photoabsorption spectra of ${\mathrm{Na}}_3$ calculated using $G_0{W}_0$, GW, LGW, $\mathrm{LGW}{\mathrm{\Gamma }}_v$, and $\mathrm{LGW}{\mathrm{\Gamma }}_W$. Experimental data are taken from Ref. [12].

Figure 2

Skeleton diagrams of the first-order vertex part (a) ${\mathrm{\Gamma }}_{v\sigma }$ and (b) ${\mathrm{\Gamma }}_{W\sigma }$; (c), (d), and (e) are the polarization part $P$; (f), (g), and (h) are the exchange-correlation part of the self-energy ${\mathrm{\Sigma }}_{\sigma }^{\mathrm{xc}}$; (i), (j), and (k) are the interaction kernel ${\tilde{I}}^{{\sigma }_1{\sigma }_1^{\prime }}$ of the BSE. (c), (f), and (i) are usual diagrams without vertex correction; (a), (d), (g), (j), and (k) involve the first-order vertex in $v$ (dotted line); (b), (e), and (h) involve the first-order vertex in $W$ (wavy line).

Figure 3

Flow chart of the $\mathrm{SC} \mathrm{LGW}{\mathrm{\Gamma }}_W$ method. $W$ and ${\mathrm{\Gamma }}_W$ are replaced by $v$ and ${\mathrm{\Gamma }}_v$ in the $\mathrm{SC} \mathrm{LGW}{\mathrm{\Gamma }}_v$ method.

Figure 4

QP (or KS) energy spectra of ${\mathrm{Na}}_3$ calculated using the LDA, $G_0{W}_0$, and $\mathrm{LGW}{\mathrm{\Gamma }}_W$. Unphysically bound KS wave functions at the 21st spin-down level and the 23rd spin-up level are also depicted. Red balls are Na atoms, while yellow and blue indicate the positive and negative regions of the wave function, respectively.