Improved density matrices for accurate molecular ionization potentials

The ionization potential and the other quasiparticle energies of electrons in molecules or solids are often evaluated within the $GW$ approximation. Though this approximation can be considered as a fantastic tradeoff between accuracy and simplicity, there exist ionization potentials of simple molecules that are unacceptably wrong. We derive here a working approximation to the density matrix that we name the $GW$ density matrix. This relatively light approximation improves several physical properties and, in particular, it yields excellent electronic dipoles for the examined molecules. As a direct consequence, the ionization potentials are affected through the change in the electrostatic potential and in the exchange operator. Benchmarking on a set of 34 molecules, we demonstrate that most of the error in the $GW$ ionization potentials is indeed eliminated thanks to this simple addition. This contribution is not a vertex correction, but nevertheless it is crucial when going beyond the standard $GW$.