  We describe a finite-field approach to compute density response functions,
which allows for efficient $G_0W_0$ and $G_0W_0\Gamma_0$ calculations beyond
the random phase approximation. The method is easily applicable to density
functional calculations performed with hybrid functionals. We present results
for the electronic properties of molecules and solids and we discuss a general
scheme to overcome slow convergence of quasiparticle energies obtained from
$G_0W_0\Gamma_0$ calculations, as a function of the basis set used to represent
the dielectric matrix.
