Effective pseudopotential for energy density functionals with higher-order derivatives

We derive a zero-range pseudopotential that includes all possible terms up to sixth order in derivatives. Within the Hartree-Fock approximation, it gives the average energy that corresponds to a quasilocal nuclear energy density functional (EDF) built of derivatives of the one-body density matrix up to sixth order. The direct reference of the EDF to the pseudopotential acts as a constraint that divides the number of independent coupling constants of the EDF by two. This allows, e.g., for expressing the isovector part of the functional in terms of the isoscalar part, or vice versa. We also derive the analogous set of constraints for the coupling constants of the EDF that is restricted by spherical, space-inversion, and time-reversal symmetries.

Figure 1

Number of terms of the pseudopotential (2), plotted as a function of the order in derivatives.

Figure 2

Number of terms of the spherical EDF that is related to a pseudopotential (solid lines). Full squares and circles show results for the Galilean and gauge invariance, respectively. For reference, dashed lines with open squares and circles show the corresponding results for the general spherical EDF studied in Ref. [2].