Introduction

Density functional theory (DFT) is a method of choice in large-scale calculations of nuclear properties. In spite of certain difficulties related to rigorous formulation of the DFT for self-bound systems like atomic nuclei, the method is potentially exact, which is guaranteed by the Hohenberg-Kohn-Scham (HKS) theorems[1,2], see recent discussion in Refs.[3,4,5].

Due to the complexity of nuclear many-body problem in general, and of the effective nucleon-nucleon interaction in particular, there exist no universal rules for constructing nuclear energy density functional (EDF). In this respect, our intuition is almost solely based on symmetry properties and practical knowledge, accumulated over the years for density-dependent effective interactions of Skyrme[6] or Gogny[7] type being applied within the mean-field (MF) approximation. Free parameters of these interactions or functionals are fitted to empirical data. Hence, the quality and performance of these methods strongly depends on adopted fitting strategies and datasets[8].

Conventional fitting methods use datasets that are dominated by bulk nuclear matter data and by nuclear binding energies of selected double-magic nuclei, with essentially no data on the single-particle (s.p.) energies. Such strategies have quite dramatic consequences concerning mostly spin-orbit (SO) and tensor parts of the EDF. In particular, they lead to an artificial isoscalar-effective-mass scaling of the SO strengths[9], contradicting scaling in selected s.p. splittings[10], and perpetual problems in reproducing evolution of the proton (neutron) s.p. energy splittings versus the neutron (proton) shell filling along the isotopic or isotonic chains of nuclei.

The most prominent examples of such chains include: neutron-rich oxygen[11], neon[12], sodium[13,14], magnesium[15], titanium[16,17], or, in medium-mass region, antimony[18] isotopes. In fact, a non-conventional shell evolution found in these neutron-rich nuclei directly motivated the shell-model theorists to introduce the so-called monopole shifts, to account for empirical trends. The physical origin of these shifts was, in turn, attributed to the shell-model tensor interaction[19,20,21]. Connection between the monopole-shifts and the tensor interaction was later on confirmed within the self-consistent MF models using either finite-range Gogny force[22] or contact Skyrme interaction[23,24,25,26,27,28,29] augmented by a strong tensor interaction.

Single-particle spectra provide for a clear evidence of strong tensor interaction and call for a new strategy of fitting the nuclear EDF in general, and of the SO and tensor terms in particular, directly to the s.p. data[28,30]. The use of the s.p. levels was usually contested, because of the isoscalar-effective-mass scaling ($m^*$) of s.p. levels. Several authors[31,32,33] argued that the physical density of s.p. levels around the Fermi energy can be reinstated only after the inclusion of particle-vibration coupling, that is, by going beyond MF. In our opinion, effective EDF theories should warrant a proper value of the effective mass through the fit to empirical data and readjust other coupling constants to this particular value of $m^*$, leading to fairly $m^*$ independent predictions. Hence, fitting strategies can include information on s.p. levels provided that the s.p. levels are understood through the binding-energy differences between doubly-magic cores and the lowest s.p. states in odd-$A$ single-particle/hole neighbors[34,28]. Spherical s.p. energies or, more precisely, the Kohn-Sham s.p. energies computed in even-even double-magic core should serve only as auxiliary quantities.

In our recent study[28], we have proposed a novel fitting strategy of the SO and tensor terms in the nuclear EDF. It is based on a direct fit to the $f_{7/2} - f_{5/2}$ SO splittings in spin-saturated isoscalar nucleus $^{40}$Ca, spin-unsaturated isoscalar nucleus $^{56}$Ni, and spin-unsaturated isovector nucleus $^{48}$Ca. The procedure allows for fixing three out of four coupling constants in this sector, namely, the isoscalar strengths of the SO and tensor interactions and the ratio of the isovector coupling constants. The procedure indicates a clear need for a major reduction of the SO strength and for strong attractive tensor fields. The aim of the present work is to address further consequences of strong attractive tensor and weak SO fields on binding energies, two-neutron separation energies, and nuclear deformability.