GW method with the self-consistent Sternheimer equation

We propose an approach to quasiparticle GW calculations which does not require the computation of unoccupied electronic states. In our approach the screened Coulomb interaction is evaluated by solving self-consistent linear-response Sternheimer equations and the noninteracting Green's function is evaluated by solving inhomogeneous linear systems. The frequency dependence of the screened Coulomb interaction is explicitly taken into account. In order to avoid the singularities of the screened Coulomb interaction the calculations are performed along the imaginary axis, and the results are analytically continued to the real axis through Pad\'e approximants. As a proof of concept we implemented the proposed methodology within the empirical pseudopotential formalism and we validated our implementation using silicon as a test case. We examine the advantages and limitations of our method and describe promising future directions.