Φ long for G = 0 (16) where the new terms that add up to Φ(G) = 4π/G 2 come from the difference between the Fourier transform Φ long (G) and the Fourier series components Φ long (G) of the long distance part, that acts as a screen of the interactions between the isolated system and its infinite periodic images. Note that the singularity of Φ(0) would be exactly canceled by that of Φ long (0) and both were ignored in Eq.16, but the non-singular difference lim G→0 Φ(G) -Φ long (G) = π α 2 must be included. It has been shown in practice that for αL ∼ 7, where L is the smallest size of the parallelepiped box, Φ long (G) can be efficiently evaluated via a FFT that converges rapidly with respect to the Cartesian grid. In the framework of PW/pseudopotential DFT, the evaluation of Coulomb-like integrals with the MT Poisson solver provides accurate energies, provided that the integration domain spans about twice the size of the electron density. Thus, as noted by MT, increasing the size of the supercell be-comes analogous to converging the energy according to the largest width diffuse function included in a Gaussian basis set. BAP (Eq.13) and MT (Eq.16) schemes look very similar for G = 0, where the sum over G = 0 vectors actually corresponds to the electrostatic energy of a Gaussian charge distribution interacting with a compensating uniform background in a periodic setup . The first term in Eq.15 accounts for the electrostatic self-energy of an isolated Gaussian charge in the supercell so that, intuitively, χ corrects the singularity using the difference between the electrostatic energy of an isolated probe charge and its periodically repeated analogues in a compensating background . For MT instead, the Gaussian charge distribution is used to construct the screening function Φ long (G) -Φ long (G) of the long-range electrostatic interactions after an Ewald-type splitting of the Coulomb potential . Very few analyses of the BAP or the MT treatment have been performed on HF calculations with full exact exchange and, to the best of our knowledge, none has been done on the MP2 correlation energy. It is therefore crucial to understand how these act on such energy contributions to ensure the convergence and accuracy of PW results in what follows.