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We perform a systematic study of the impact of the J 2 tensor term in the Skyrme energy functional on properties of spherical nuclei. In the Skyrme energy functional, the tensor terms originate from both zero-range central and tensor forces. We build a set of 36 parametrizations, covering a wide range of the parameter space of the isoscalar and isovector tensor term coupling constants with a fit protocol very similar to that of the successful SLy parametrizations. We analyze the impact of the tensor terms on a large variety of observables in spherical mean-field calculations, such as the spin-orbit splittings and single-particle spectra of doubly-magic nuclei, the evolution of spin-orbit splittings along chains of semi-magic nuclei, mass residuals of spherical nuclei, and known anomalies of radii. The major findings of our study are as follows: (i) Tensor terms should not be added perturbatively to existing parametrizations; a complete refit of the entire parameter set is imperative. (ii) The free variation of the tensor terms does not lower the χ 2 within a standard Skyrme energy functional. (iii) For certain regions of the parameter space of their coupling constants, the tensor terms lead to instabilities of the spherical shell structure, or even to the coexistence of two configurations with different spherical shell structures. (iv) The standard spin-orbit interaction does not scale properly with the principal quantum number, such that single-particle states with one or several nodes have too large spin-orbit splittings, whereas those of nodeless intruder levels are tentatively too small. Tensor terms with realistic coupling constants cannot cure this problem.(v) Positive values of the coupling constants of proton-neutron and like-particle tensor terms allow for a qualitative description of the evolution of spin-orbit splittings in chains of Ca, Ni, and Sn isotopes. (vi) For the same values of the tensor term coupling constants, however, the overall agreement of the single-particle spectra in doubly-magic nuclei is deteriorated, which can be traced back to features of the single-particle spectra that are not related to the tensor terms. We conclude that the currently used central and spin-orbit parts of the Skyrme energy density functional are not flexible enough to allow for the presence of large tensor terms.well-defined transformation properties under rotations: (7) where δ µν is the Kronecker symbol and µνκ is the Levi-Civita tensor. The pseudoscalar, vector, and pseudotensor parts expressed in terms of the Cartesian tensor are given byν=x κµν J µν (r), 014312-6 TENSOR PART OF THE SKYRME ENERGY DENSITY . . . PHYSICAL REVIEW C 76, 014312 (2007) some authors [59]: z µ,ν=x J t,µν J t,µν = 1 3 J (0) t 2 + 1 2 J 2 t + z µ,ν=x J (2) t,µν J (2) t,µν , (24) 1 2 z µ=x J t,µµ 2 + z µ,ν=x J t,µν J t,νµ = 2 3 J (0) t 2 − 1 4 J 2 t + 1 2 z µ,ν=x J (2) t,µν J (2) t,µν . (25)