We compute inclusive electron-nucleus cross sections using ab initio spectral functions of $^4$He and $^{16}$O obtained within the Self Consistent Greens Function approach. The formalism adopted is based on the factorization of the spectral function and the nuclear transition matrix elements. This allows to provide an accurate description of nuclear dynamics and to account for relativistic effects in the interaction vertex. Our calculations use a saturating chiral Hamiltonian in order reproduce the correct nuclear sizes. When final state interactions for the struck particle are accounted for, we find nice agreement between the data and the theory for the inclusive electron-$^{16}$O cross section. The results lay the foundations for future applications of the Self Consistent Greens Function method, in both closed and open shell nuclei, to neutrino data analysis. This work also presents results for the point-proton, charge and single-nucleon momentum distribution of the same two nuclei. The center of mass can affect these quantities for light nuclei and cannot be separated cleanly in most ab initio post-Hartree-Fock methods. In order to address this, we developed a Metropolis Monte Carlo calculation in which the center of mass coordinate can be subtracted exactly from the trial wave function and the expectation values. We gauged this effect for $^4$He by removing the center of mass effect from the Optimal Reference State wave function that is generated during the Self Consistent Greens Function calculations. Our findings clearly indicate that the residual center of mass contribution strongly modifies calculated matter distributions with respect to those obtained in the intrinsic frame. Hence, its subtraction is crucial for a correct description of light nuclei.
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