  A Density Matrix Functional theory is constructed semi-empirically for the
two-level Lipkin model. This theory, based on natural orbitals and occupation
numbers, is shown to provide a good description for the ground state energy of
the system as the two-body interaction and particle number vary. The
application of Density Matrix Functional theory to the Lipkin model illustrates
that it could be a valuable tool for systems presenting a shape
phase-transition such as nuclei. The improvement of one-body observables
description as well as the interest for Energy Density Functional theory are
discussed.
