The present work constitutes a step towards the search for a spectroscopic-quality energy density functional, which would properly and precisely account for the single-particle structure of atomic nuclei. We analyzed dependence of single-particle energies on coupling constants of the Skyrme functional and showed that this dependence is almost linear. Such an observation allowed us to focus on linear regression coefficients, which characterize variations of individual single-particle states with changing coupling constants.
We discussed the regression coefficients obtained for single-particle
states in 8 doubly magic spherical nuclei. The results turn out to be
rather generic, with only a weak dependence on the particular variant
of the Skyrme-functional parametrization. We showed that for nuclei
near stability, isovector coupling constants have significantly
smaller impact on single-particle energies than isoscalar ones. We
also showed that the effective-mass coupling constant
cannot be considered as merely changing the overall density of
states, which was up to now a commonly accepted view. This coupling
constants also significantly influences relative positions of
single-particle levels, including the splitting of spin-orbit
partners. We explained these effects by a detailed analysis of
densities and mean fields related to this coupling constant.
By applying the singular value decomposition to the matrix of the regression coefficients, we performed fits of calculated single-particle energies to experimental data. We showed that even by fitting all the 12 coupling constants of the standard Skyrme functional to experiment, one is unable to obtain the rms deviations of single-particle energies below about 1.1MeV. This discouraging result points out to a necessity of extending the form of the Skyrme functional beyond the standard parametrization.
This work was supported in part by the Academy of Finland and University of Jyväskylä within the FIDIPRO programme and by the Polish Ministry of Science.