Discussion of the energy density at N 3 LO with conserved spherical, space-inversion, and time-reversal symmetries (Sec. VI in Ref. [1]) was left unfinished in the sense that properties of secondary densities were not taken into account. By this omission, within these symmetry conditions, five extra independent terms appeared in the energy density functional (EDF).The problem was related to the fact that the spherical symmetry imposes specific relations between derivatives of vector or tensor fields. In particular, following Eq. (44), an arbitrary vector ( J) or tensor ( ↔ R) field must have the form dictated by the generalized Cayley-Hamilton theorem [2]:where J (r) and R(r) are scalar functions, that is, functions of r = |r|. Then it is a matter of simple algebra to show thatwhere primes denote derivatives with respect to r. The first two equalities constitute relations between pairs of secondary densities, which are imposed by the spherical symmetry. Therefore, the corresponding terms in the energy densities at fourth and sixth orders (Eqs. (61) and (62) in Ref.[1]) are pairwise identical. Specifically, this concerns pairs of terms represented by the coupling constants C 1 21 and D 1 21 in fourth order andConsequently, the numbers of terms given in Table XX in Ref.[1] at fourth and sixth orders are smaller by one and four, respectively, and are given in the second column in Table I here.In addition, we correct here two misprints found in Ref. [1]. First, the description of rules that were used to select terms in the functional, which was given before Eq. (31), should read as follows: To avoid double-counting one takes only terms with

No abstract

No abstract

No abstract

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 quasi-local 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.