The development of a modern and more realistic nuclear energy density functional

(EDF) for accurate predictions of properties of nuclei and nuclear matter (NM) is

the subject of enhanced activity, since it is very important for the study of the

properties of rare nuclei and the equation of state of NM, the ingredient needed in

the study of the structure and evolution of compact astrophysical objects.

      We will first describe a method for determining a modern EDF, based on the

effective nucleon-nucleon Skyrme type interaction. The parameters of the new and

improved EDF (named KDE0 and KDE0v1) will be presented. We will also present

and discuss the prediction of these EDFs for the nuclear fission barrier and for the

relation between the mass and radius on neutron stars, which were not included in

the parameter fit for the EDF. Next we present results of Hartree-Fock-based

random-phase-approximation calculations of properties rare nuclei and of the

strength distributions and centroid energies of multipole isoscalar and isovector giant

resonances and their sensitivities to NM properties, such as the incompressibility

coefficient, symmetry energy density and effective mass, and thereby deduce limits

for their values which are needed for determining the next generation EDF.