  We introduce a finite-range pseudopotential built as an expansion in
derivatives up to next-to-next-to-next-to-leading order (N$^3$LO) and we
calculate the corresponding nonlocal energy density functional (EDF). The
coupling constants of the nonlocal EDF, for both finite nuclei and infinite
nuclear matter, are expressed through the parameters of the pseudopotential.
All central, spin-orbit, and tensor terms of the pseudopotential are derived
both in the spherical-tensor and Cartesian representation. At next-to-leading
order (NLO), we also derive relations between the nonlocal EDF expressed in the
spherical-tensor and Cartesian formalism. Finally, a simplified version of the
finite-range pseudopotential is considered, which generates the EDF identical
to that generated by a local potential.
