  In the full quantum theory, the energy of a many-body quantum system with a
given one-body density is described by the Levy-Lieb functional. It is exact,
but very complicated to compute. For practical computations, it is useful to
introduce the Local Density Approximation which is based on the local energy of
constant densities. The aim of this paper is to make a rigorous connection
between the Levy-Lieb functional theory and the Local Density Approximation.
Our justification is valid for fermionic systems with a general class of smooth
short range interaction potentials, in the regime of slowly varying densities.
We follow a general approach developed by Lewin, Lieb and Seiringer for Coulomb
potential, but avoid using any special properties of the potential including
the scaling property and screening effects for the localization of the energy.
