  Empirical energy density functionals (EDFs) are generally successful in
describing nuclear properties across the table of nuclides. But their
limitations motivate using the density-matrix expansion (DME) to embed
long-range pion interactions into a Skyrme functional. Recent results on the
impact of the pion were both encouraging and puzzling, necessitating a careful
re-examination of the DME implementation. Here we take the first steps,
focusing on two-body scalar terms in the DME. Exchange energies with long-range
one-pion contributions are well approximated by all DME implementations
considered, with preference for variants that do not truncate at two
derivatives in every EDF term. The use of the DME for chiral pion contributions
is therefore supported by this investigation. For scalar-isovector energies it
is important to treat neutrons and protons separately. The results are found to
apply under broad conditions, although self-consistency is not yet tested.
