Microscopically based energy density functionals for nuclei using the density matrix expansion. II. Full optimization and validation

Background: Energy density functional methods provide a generic framework to compute properties of atomic nuclei starting from models of nuclear potentials and the rules of quantum mechanics. Until now, the overwhelming majority of functionals have been constructed either from empirical nuclear potentials such as the Skyrme or Gogny forces, or from systematic gradient-like expansions in the spirit of the density functional theory for atoms.

Purpose: We seek to obtain a usable form of the nuclear energy density functional that is rooted in the modern theory of nuclear forces. We thus consider a functional obtained from the density matrix expansion of local nuclear potentials from chiral effective field theory. We propose a parametrization of this functional carefully calibrated and validated on selected ground-state properties that is suitable for large-scale calculations of nuclear properties.

Methods: Our energy functional comprises two main components. The first component is a non-local functional of the density and corresponds to the direct part (Hartree term) of the expectation value of local chiral potentials on a Slater determinant. Contributions to the mean field and the energy of this term are computed by expanding the spatial, finite-range components of the chiral potential onto Gaussian functions. The second component is a local functional of the density and is obtained by applying the density matrix expansion to the exchange part (Fock term) of the expectation value of the local chiral potential. We apply the UNEDF2 optimization protocol to determine the coupling constants of this energy functional.

Results: We obtain a set of microscopically constrained functionals for local chiral potentials from leading order up to next-to-next-to-leading order with and without three-body forces and contributions from $\mathrm{\Delta }$ excitations. These functionals are validated on the calculation of nuclear and neutron matter, nuclear mass tables, single-particle shell structure in closed-shell nuclei, and the fission barrier of ${}^{\text{240}}\mathrm{Pu}$. Quantitatively, they perform noticeably better than the more phenomenological Skyrme functionals.

Conclusions: The inclusion of higher-order terms in the chiral perturbation expansion seems to produce a systematic improvement in predicting nuclear binding energies while the impact on other observables is not really significant. This result is especially promising since all the fits have been performed at the single-reference level of the energy density functional approach, where important collective correlations such as center-of-mass correction, rotational correction, or zero-point vibrational energies have not been taken into account yet.

Figure 1

Difference between the potentials $V_C\left(\mathit{r}\right)$ and $W_C\left(\mathit{r}\right)$ up to a certain chiral order and their corresponding approximations by a sum of five Gaussian functions shown in Eqs. (32) and (33).

Figure 2

Difference between the two-body density dependent couplings $g_t^{\rho \rho }$ and their corresponding interpolating function ${\tilde{g}}_t^{\rho \rho }$ given by Eq. (47).

Figure 3

Difference between the three-body density-dependent couplings $g^{{\rho }_0^3}$ and $g^{{\rho }_0{\rho }_1^2}$ and their corresponding interpolating functions ${\tilde{g}}^{{\rho }_0^3}$ and ${\tilde{g}}^{{\rho }_0{\rho }_1^2}$ given by Eq. (47). The irregularities in the curves show that the interpolation has an accuracy similar to the numerical multidimensional integral.

Figure 4

Symmetric nuclear matter (top) and pure neutron matter (bottom) for the new DME functionals. For comparison we mark with red pentagons the calculation by Logoteta, Bombaci, and Kievsky (LBK) for a local chiral potential with delta isobar, labeled as $\mathrm{N}3\mathrm{LO}\mathrm{\Delta }+\mathrm{N}2\mathrm{LO}\mathrm{\Delta }1$ in Ref. [58].

Figure 5

Deviations between theoretical and experimental binding energies for even-even nuclei. Experimental binding energies are extracted from the 2016 mass evaluation [60, 61] and only actual measured values are used. Left, EDFs without $\mathrm{\Delta }$ excitations; right, EDFs with $\mathrm{\Delta }$ excitations.

Figure 6

Proton radii residuals for the UNEDF2 (top) and $\mathrm{NLO}\mathrm{\Delta }$+$3N$ (bottom) functionals. Experimental data are taken from Ref. [62].

Figure 7

Neutron s.p. levels in ${}^{\text{208}}\mathrm{Pb}$ extracted from blocking calculations. In the top panel, EDFs derived from $NN$ and $3N$ forces without $\mathrm{\Delta }$ excitations are included; in the bottom panel, EDFs derived from $NN$+$3N$ forces with $\mathrm{\Delta }$ excitations are shown.

Figure 8

Deformation potential energy surface in ${}^{\text{240}}\mathrm{Pu}$ as a function of the axial quadrupole moment. Top: Energy functionals at LO, NLO, and N2LO, with and without $3N$ forces (when applicable). Bottom: Same for EDF based on potentials including the $\mathrm{\Delta }$ contribution. For comparison, each panel also shows the results for the UNEDF0, UNEDF1, and UNEDF2 functionals of Refs. [42, 43, 44].