Conclusions

Applicability of functional-based self-consistent mean-field or energy-density-functional methods to nuclear structure is hampered by their unsatisfactory s.p. properties. This fact seems to be a mere consequence of strategies used to select datasets that were applied in the process of adjusting free parameters of these effective theories. In spite of the fact that the s.p. energies are at the heart of these methods, the datasets are heavily oriented towards reproducing bulk nuclear properties in large-$N$ limit, with only a marginal influence of the s.p. levels or level splittings in finite nuclei.

In this work we suggest a necessity of shifting attention from bulk to s.p. properties and to look for spectroscopic-quality EDF, even at the expense of deteriorating its quality in reproducing binding energies. Such a strategy requires well-defined empirical input related to the s.p. energies, to be used directly in the fitting process. We argue that odd-even mass differences around magic nuclei not only provide unambiguous direct information about nuclear s.p. energies but are also well anchored within the spirit of the EDF formalism. Indeed, the theorems due to Hohenberg and Kohn [56] and Levy [57], see also Refs. [58], imply existence of universal EDF capable, at least in principle, treating ground-states energies of nuclei exactly. One can, therefore, argue that this implies essentially exact treatment of at least the lowest s.p. levels forming ground states in one-particle (one-hole) odd-$A$ nuclei with respect to even-even cores or, alternatively, almost exact description of core-polarization phenomena caused by odd single-particle (single-hole).

An attempt to refit the EDF is preceded by a systematic analysis of the s.p. energies and self-consistent core-polarization effects within the state-of-the-art Skyrme-force-inspired EDF. Three major sources of core-polarization, including mass, shape and spin (time-odd) effects, are identified and discussed. The analysis is performed for even-even doubly-magic cores and the lowest s.p. states in odd-$A$ one-particle(hole) nuclei. The discussion is supplemented by analysis of the s.p. SO splittings.

New strategy in fitting the EDF is applied to the SO and tensor parts of the nuclear EDF. Instead of large-scale fit to binding energies we propose simple and intuitive three-step procedure that can be used to fit the isoscalar strength of the SO interaction as well as the isoscalar and isovector strengths of the tensor interaction. The entire idea is based on the observation that the $f_{7/2} - f_{5/2}$ SO splittings in spin-saturated isoscalar $^{40}$Ca, spin-unsaturated isoscalar $^{56}$Ni, and spin-unsaturated isovector $^{48}$Ca form distinct pattern that can neither be understood nor reproduced based solely on the conventional SO interaction. The procedure indicates a clear need for major reduction (from $\sim$20% till $\sim$35% depending on the parameterization) of the SO strength and for strong tensor fields. It is verified that the suggested changes lead to systematic improvements of the functional performance concerning such s.p. properties like the SO splittings or magic gaps. It is also shown that destructive impact of these changes on the binding energies can be improved, to a large extent, by relatively small refinements of the remaining coupling constants.

This work was supported in part by the Polish Ministry of Science and by the Academy of Finland and University of Jyväskylä within the FIDIPRO programme.