  In recent years impressive progress has been made in the development of
highly accurate energy density functionals, which allow to treat medium-heavy
nuclei. In this approach one tries to describe not only the ground state but
also the first relevant excited states. In general, higher accuracy requires a
larger set of parameters, which must be carefully chosen to avoid redundancy.
Following this line of development, it is unavoidable that the connection of
the functional with the bare nucleon-nucleon interaction becomes more and more
elusive. In principle, the construction of a density functional from a density
matrix expansion based on the effective nucleon-nucleon interaction is
possible, and indeed the approach has been followed by few authors. However, to
what extent a density functional based on such a microscopic approach can reach
the accuracy of the fully phenomenological ones remains an open question. A
related question is to establish which part of a functional can be actually
derived by a microscopic approach and which part, on the contrary, must be left
as purely phenomenological. In this paper we discuss the main problems that are
encountered when the microscopic approach is followed. To this purpose we will
use the method we have recently introduced to illustrate the different aspects
of these problems. In particular we will discuss the possible connection of the
density functional with the nuclear matter Equation of State and the distinct
features of finite size effects proper of nuclei.
