  We compute inclusive electron-nucleus cross sections using ab initio spectral
functions of $^4$He and $^{16}$O obtained within the Self Consistent Green's
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 Green's 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 Green's 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.
