  The present work discusses, from an ab-initio standpoint, the definition, the
meaning, and the usefulness of effective single-particle energies (ESPEs) in
doubly closed shell nuclei. We perform coupled-cluster calculations to quantify
to what extent selected closed-shell nuclei in the oxygen and calcium isotopic
chains can effectively be mapped onto an effective independent-particle
picture. To do so, we revisit in detail the notion of ESPEs in the context of
strongly correlated many-nucleon systems and illustrate the necessity to
extract ESPEs through the diagonalization of the centroid {\it matrix}, as
originally argued by Baranger. For the purpose of illustration, we analyse the
impact of correlations on observable one-nucleon separation energies and
non-observable ESPEs in selected closed-shell oxygen and calcium isotopes. We
then state and illustrate the non-observability of ESPEs. Similarly to
spectroscopic factors, ESPEs can indeed be modified by a redefinition of
inaccessible quantities while leaving actual observables unchanged. This leads
to the absolute necessity to employ consistent structure and reaction models
based on the same nuclear Hamiltonian to extract the shell structure in a
meaningful fashion from experimental data.
