Ab initio calculations of electronic excitations: Collapsing spectral sums

We present a method for the evaluation of electronic excitations of advanced materials by reformulating spectral sum-over-states expressions such that only occupied states appear. All empty states are accounted for by one effective energy. Thus we keep the simplicity and precision of the sum-over-states approach while speeding up calculations by more than an order of magnitude. We demonstrate its power by applying it to the $GW$ method, where a huge summation over empty states appears twice (screening and self-energy). The precision is shown for bulk Si and solid and atomic Ar. We then use it to determine the band gap of the technologically important oxide ${\text{SnO}}_2$.

Figure 1

(Color online) The real part of ${\Sigma }_c\left(\omega \right)$ for the highest occupied band at $\Gamma$ for Si around the LDA orbital energy (set to 0 eV). Solid line (black): $G^0{W}^0$ (SOS); dotted line (green): $G^0{W}^0$ (EET: ${\delta }^{\prime \left(2\right)}$ and ${\delta }^{\left(0\right)}$); dashed line (red): $G^0{W}^0$ (EET: ${\delta }^{\prime \left(2\right)}$ and ${\delta }^{\left(2\right)}$); dotted-dashed line (blue): $G^0{W}^0$ (EET: ${\delta }^{\prime \left(4\right)}$ and ${\delta }^{\left(4\right)}$); double-dotted-dashed line (violet): COHSEX.

Figure 2

(Color online) Band structure of solid argon. Solid line (black): $G^0{W}^0$ (SOS); dashed line (red): $G^0{W}^0$ (EET); dotted line (blue): LDA.