Conclusions

In the present study, we derived a set of compact expressions giving the local energy density corresponding to the Kohn-Sham potential energy for an arbitrary local finite-range central, spin-orbit, and tensor interactions. The method is based on the Negele-Vautherin density matrix expansion augmented by the odd-power gradient terms fulfilling the gauge-invariance condition. The coupling constants of the local energy density depend on a set of moments of the interaction conforming to the ideas of the effective theory. The expansion is based on the separation of scales between the range of the force and space characteristics of the one-body density matrix. It leads to a representation of dynamical properties of the system in terms of a set of numbers, whereby complicated short-range characteristics of effective interactions remain unresolved.

We pointed out the fact that to correctly describe the exchange properties of the functional, proper treatment of the density matrix in the nonlocal direction is essential. This immediately leads to the local energy density that does not correspond to an averaged zero-range pseudopotential. Therefore, the Negele-Vautherin expansion performed up to NLO leads to the Skyrme functional and not to the Skyrme force. Within this formalism, the only way to define the Skyrme force is to match it to the time-even properties of the non-local functional, disregarding those pertaining to the time-odd channel.

We applied the general NLO expressions to the case of the central finite-range part of the Gogny interaction. It turns out that the obtained coupling constants of the local Skyrme functional quite strongly depend on the Fermi momentum or on the density. Nevertheless, the equation of state obtained for fixed coupling constants, calculated at the saturation point of $k_F=1.35$fm$^{-1}$, fairly well reproduces the exact Gogny-force result. On the other hand, partial density dependence, inferred from the explicit dependence of the coupling constants on $k_F$, gives very unsatisfactory results.

By solving the self-consistent equations with the Skyrme-force parameters derived from the Gogny force, one obtains an excellent agreement (up to 1-2%) of binding energies and radii with those corresponding to the true Gogny force. This shows that the ideas of the effective theory, whereby the finite-range nuclear forces are sufficiently short-range to be replaced by contact quasi-potentials, are applicable to low-energy nuclear observables.

We also discussed properties of the one-body density matrix for the spin and isospin polarized infinite nuclear matter. In this case, one obtains the nonlocal densities with mixed spin-isospin channels. As a result, the local energy density is not invariant but covariant with respect to rotational and isospin symmetries; that is, it does not have the form of the standard Skyrme functional. It means that the standard Negele-Vautherin expansion can only be performed for unpolarized densities.

Interesting comments by Thomas Duguet are gratefully acknowledged. This work was supported by the Polish Ministry of Science and Higher Education under Contract No. N N 202 328234, by the Academy of Finland and University of Jyväskylä within the FIDIPRO program, and by the U.S. Department of Energy under Contract Nos. DE-FC02-09ER41583 (UNEDF SciDAC Collaboration) and DE-FG02-96ER40963 (University of Tennessee).