Introduction

Recent experimental studies of nuclei in the nobelium region provided rich spectroscopic data [1,2], which, in principle, can be used as a benchmark information for extrapolations into the region of superheavy nuclei. Numerous theoretical studies are aimed at modelling of these spectroscopic data [3,4,5,6,7,8,9,10,11,,13,14,15,16,17,18,19,20,21] so as to make such extrapolations as reliable as possible.

The estimation of theoretical uncertainties is one of the most essential aspects of extrapolating nuclear models into exotic nuclei [22]. One, fairly easy part of it, is the evaluation of statistical uncertainties of observables that are related to the uncertainties of model parameters adjusted, in one way or another, to experimental data. Another one, very difficult, pertains to those systematic uncertainties related to the definition and contents of the different terms that make up the models themselves. An obvious strategy, which, anyhow, gives us only a limited glimpse on possible systematic uncertainties, is to study a set of variants of a given model, and to analyze differences obtained for calculated observables.

In the present study, we aim at such an analysis of results obtained within three fairly different energy-density-functional (EDF) approaches. Namely, we employ the covariant EDFs [9], with one classic (NL1 [23]) and one recent (NL3* [24]) parameter set, Skyrme EDFs [25], with one classic (SLy4 [26]) and one recent (UNEDF2 [27]) parameter set, as well as Gogny EDFs [28], again with one classic (D1S [29]) and one recent (D1M [30]) parameter set.

Our goal is thus to determine, present, and compare results obtained within these six models for a common set of calculated observables. We aim at performing these analyses within the most similar and/or equivalent conditions, so as to meaningfully discuss general qualitative similarities and differences. In all cases, pairing correlations are treated on the Bogoliubov level. In the literature, it is customary to label such calculations as Hartree-Fock-Bogoliubov (HFB) in the non-relativistic cases and as relativistic Hartree-Bogoliubov (RHB) for the specific variant of relativistic mean-field model used here, thereby emphasizing that HFB-like equations are solved instead of simpler HF+BCS equations. Only in the calculations with a Gogny force, however, the same effective interaction is used to determine direct, exchange, and pairing matrix elements. By contrast, in case of Skyrme EDF and the relativistic approach particle-hole and pairing matrix elements relate to different effective interactions, which can be used to simplify their form and phenomenological adjustment. Also, in the RHB approach all exchange terms are neglected, whereas in the case of many Skyrme parameterizations some are modified in order to improve the description of data [25]. None of these formal differences is relevant for our discussion, and we will use whenever possible the generic notion of an EDF method for all three approaches. Indeed, in all cases the total energy can be cast into the form of a functional of normal and anomalous one-body density matrices from which the equations-of-motion are then derived by variation. We do not attribute too much of an importance to quantitative similarities and differences between the obtained results, especially when the models are compared to experimental data. Indeed, a detailed agreement with the data may crucially depend on specific model-parameter adjustments, or on various corrections taken into account or disregarded. The phenomenological EDF used include a limited set of parameters (typically between seven and twenty) and aim at a global description of a wide variety of nuclear properties and therefore a perfect agreement with experimental data is out of reach at present. Certainly, in the future all EDF approaches will be improved; here we only look into generic properties obtained for selected current global parameterisations thereof.

In the present analysis, we systematically calculated the ground states of even-even and odd-mass nuclei from uranium ($Z=92$) to rutherfordium ($Z=104$) and for neutron numbers between $N=144$ and 156. The selection of this region of heaviest actinides/lightest superheavy nuclei is guided by the need for reliable experimental data on spectroscopic properties (in particular, on the single-particle energies of deformed one-quasiparticle states) based on which the extrapolability of a given theory/functional towards region of superheavy nuclei may be judged. In addition, we determined low-lying quasiparticle spectra of odd-mass nuclei and low-spin moments of inertia of even-even nuclei. The main thrust of the analysis was on the attempt to identify single-particle and shell-structure properties of these nuclei by looking at many-body observables such as masses, odd-even and two-particle mass staggering, and excitation energies.

The paper is organised as follows. Selected theoretical aspects of our calculations are presented in section 2, with four subsections discussing the methods related to obtaining results for the Skyrme EDF SLy4 (2.1), Skyrme EDF UNEDF2 (2.2), Gogny EDFs (2.3), and covariant EDFs (2.4). The results of the calculations are given in section 3, with subsections devoted to the Nilsson diagrams (3.1), quasiparticle spectra (3.2 and 3.3), odd-even and two-particle mass staggering (3.4), and moments of inertia (3.5). Conclusions are presented in section 4.