  We use the Negele-Vautherin density matrix expansion to derive a quasi-local
density functional for the description of systems of fermions interacting with
short-ranged interactions composed of arbitrary finite-range central,
spin-orbit, and tensor components. Terms that are absent in the original
Negele-Vautherin approach owing to the angle averaging of the density matrix
are fixed by employing a gauge invariance condition. We obtain the Kohn-Sham
interaction energies in all spin-isospin channels, including the exchange
terms, expressed as functions of the local densities and their derivatives up
to second (next to leading) order. We illustrate the method by determining the
coupling constants of the Skyrme functional or Skyrme force that correspond to
the finite-range Gogny central force. The resulting self-consistent solutions
reproduce the Gogny-force binding energies and radii within the precision of
1-2%.
