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The density matrix expansion (DME) of Negele and Vautherin is a convenienttool to map finite-range physics associated with vacuum two- and three-nucleoninteractions into the form of a Skyme-like energy density functional (EDF) withdensity-dependent couplings. In this work, we apply the improved formulation ofthe DME proposed recently in arXiv:0910.4979 by Gebremariam {\it et al.} to thenon-local Fock energy obtained from chiral effective field theory (EFT)two-nucleon (NN) interactions at next-to-next-to-leading-order (N$^2$LO). Thestructure of the chiral interactions is such that each coupling in the DME Fockfunctional can be decomposed into a cutoff-dependent coupling {\it constant}arising from zero-range contact interactions and a cutoff-independent coupling{\it function} of the density arising from the universal long-range pionexchanges. This motivates a new microscopically-guided Skyrme phenomenologywhere the density-dependent couplings associated with the underlyingpion-exchange interactions are added to standard empirical Skyrme functionals,and the density-independent Skyrme parameters subsequently refit to data. AMathematica notebook containing the novel density-dependent couplings isprovided.

Microscopically-constrained Fock energy density functionals from chiral effective field theory. I. Two-nucleon interactions