The relativistic density functional with minimal density dependent nucleon-meson couplings for nuclei and nuclear matter is extended to include tensor couplings of the nucleons to the vector mesons. The dependence of the minimal couplings on either vector or scalar densities is explored. New parametrisations are obtained by a fit to nuclear observables with uncertainties that are determined self-consistently. The corresponding nuclear matter parameters at saturation are determined including their uncertainties. An improvement in the description of nuclear observables, in particular for binding energies and diffraction radii, is found when tensor couplings are considered, accompanied by an increase of the Dirac effective mass. The equations of state for symmetric nuclear matter and pure neutron matter are studied for all models. The density dependence of the nuclear symmetry energy, the Dirac effective masses and scalar densities is explored. Problems at high densities for parametrisations using a scalar density dependence of the couplings are identified due to the rearrangement contributions in the scalar self-energies that lead to vanishing Dirac effective masses.
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