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In the framework of nuclear energy density functional (EDF) methods, manynuclear phenomena can be related to the deformation of intrinsic states. Theiraccurate modeling relies on the correct description of the change of nuclearbinding energy with deformation. The two most important contributions to thedeformation energy have their origin in shell effects and the surface energycoefficient of nuclear matter.  In a first step, we build nine series of parametrizations with asystematically varied surface-energy coefficient a_surf for threefrequently-used options for the CM correction (none, one-body term only, fullone-body and two-body contributions) combined with three values for theisoscalar effective mass m^*_0/m (0.7, 0.8, 0.85) and analyse how well each ofthese parametrizations can be adjusted to the properties of spherical nucleiand infinite nuclear matter. In a second step, we performed additional fitswithout the constraint on surface energy, adding one ``best-fit"parametrization to each of the nine series. We then benchmark theseparametrizations to the deformation properties of heavy nuclei by means ofthree-dimensional Hartree-Fock-Bogoliubov calculations that allow for non-axialand/or non-reflection symmetric configurations.  We perform a detailed correlation analysis between surface and volumeproperties of nuclear matter using the nine series of parametrizations. Thebest fits out of each series are then benchmarked on the fission barriers ofPu240 and Hg180, as well as on the properties of deformed states at normal andsuperdeformation for actinides and nuclei in the neutron-deficient Hg region.  (see paper for full abstract)

On the impact of the scheme for center-of-mass correction on the surface energy of Skyrme Energy Density Functionals