In this project we will evaluate the performance of new many-electron approximations based on the Green’s function, starting from the so-called random phase approximation (RPA). The RPA has shown to combine high accuracy with reasonable computational cost, making it a good starting point for further refinements. We will start by looking at the hydrogen dimer, a simple system where the degree of correlation varies with the interatomic separation. Approximations will be studied self-consistently and compared to exact results. The same techniques will then be applied to study realistic antiferromagnetic insulators, such as NiO, that exhibit a Mott insulator-metal transition at finite pressure.