  Recently, a microscopically motivated nuclear energy density functional was
derived by applying the density matrix expansion to the Hartree-Fock (HF)
energy obtained from long-range chiral effective field theory two- and
three-nucleon interactions. However, the HF approach cannot account for all
many-body correlations. One class of correlations is included by
Brueckner-Hartree-Fock (BHF) theory, which gives an improved definition of the
one-body HF potential by replacing the interaction by a reaction matrix $G$. In
this paper, we find that the difference between the $G$-matrix and the
nucleon-nucleon potential $V_{\mathrm{NN}}$ can be well accounted for by a
truncated series of contact terms. This is consistent with renormalization
group decoupling generating a series of counterterms as short-distance physics
is integrated out. The coefficients $C_{n}$ of the power series expansion $\sum
C_{n}q^{n}$ for the counterterms are examined for two potentials at different
renormalization group resolutions and at a range of densities. The success of
this expansion for $G-V_{\mathrm{NN}}$ means we can apply the density matrix
expansion at the HF level with low-momentum interactions and density-dependent
zero-range interactions to model BHF correlations.
