Microscopically based energy density functionals for nuclei using the density matrix expansion: Implementation and pre-optimization

In a recent series of articles, Gebremariam, Bogner, and Duguet derived a microscopically based nuclear energy density functional by applying the density matrix expansion (DME) to the Hartree-Fock energy obtained from chiral effective field theory two- and three-nucleon interactions. Owing to the structure of the chiral interactions, each coupling in the DME functional is given as the sum of a coupling constant arising from zero-range contact interactions and a coupling function of the density arising from the finite-range pion exchanges. Because the contact contributions have essentially the same structure as those entering empirical Skyrme functionals, a microscopically guided Skyrme phenomenology has been suggested in which the contact terms in the DME functional are released for optimization to finite-density observables to capture short-range correlation energy contributions from beyond Hartree-Fock. The present article is the first attempt to assess the ability of the newly suggested DME functional, which has a much richer set of density dependencies than traditional Skyrme functionals, to generate sensible and stable results for nuclear applications. The results of the first proof-of-principle calculations are given, and numerous practical issues related to the implementation of the new functional in existing Skyrme codes are discussed. Using a restricted singular value decomposition optimization procedure, it is found that the new DME functional gives numerically stable results and exhibits a small but systematic reduction of our test ${\chi }^2$ function compared to standard Skyrme functionals, thus justifying its suitability for future global optimizations and large-scale calculations.

Figure 1

INM saturation curves for symmetric (a) and neutron (b) nuclear matter calculated with the standard SLy4 Skyrme functional (dotted lines), and the DME functional at LO (dashed lines), NLO (dot-dashed lines), ${\mathrm{N}}^2$LO without $\mathit{NNN}$ contributions (short-dashed lines), and ${\mathrm{N}}^2$LO (solid lines). Ab initio results [35] are plotted as reference points. All functionals are constrained to reproduce the reference INM saturation values from Eq. (45).

Figure 2

INM saturation curves for symmetric (a) and neutron (b) nuclear matter calculated with the modified SLy$4^{\prime }$ Skyrme functional (dotted lines) and the DME functional at LO (dashed lines), NLO (dot-dashed lines), and ${\mathrm{N}}^2$LO (solid lines) using the parameters from Table .

Figure 3

Comparison of ${}^{208}\mathrm{Pb}$ experimental single-particle energies (in MeV) for neutrons and protons with canonical single-particle energies, calculated with the standard SLy$4^{\prime }$ functional and the LO, NLO, and ${\mathrm{N}}^2$LO DME functionals.

Figure 4

Two-neutron separation energies (a), neutron rms radii (b), and average neutron pairing gaps (c) for the Ni isotopic chain of nuclei in the region of the neutron drip line, calculated with the standard SLy$4^{\prime }$ functional and the LO, NLO, and ${\mathrm{N}}^2$LO DME-based functionals.

Figure 5

Deformation energy for the nucleus ${}^{100}\mathrm{Zr}$ calculated with standard SLy$4^{\prime }$ functional (dotted line, squares) and the DME functional in LO (dashed line), NLO (long-dashed line), and ${\mathrm{N}}^2$LO (solid line, circles).

Figure 6

Comparison of the proton rms radii for nuclei along the Ca chain calculated with the standard SLy$4^{\prime }$ functional (long-dashed line with squares) and the DME functional in LO (dashed line with triangles), NLO (dashed line with diamonds), and ${\mathrm{N}}^2$LO (solid line with circles).