<i>Ab initio</i>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 ${\text{SnO}}_{2}$.