Existing parameterizations

Although one might disagree with the rationale for neglecting the
terms, it is not easy to adjust the coupling
constants *C*_{t}^{T} to spectral data. Large values for
*C*_{t}^{T} can be ruled out because they spoil the previously
obtained agreement for single-particle spectra, but there are
broad regions of values where they influence the usual time-even
observables too weakly to be uniquely determined [31].
Only once in the published literature has there been an attempt
to do so [32].

All first-generation Skyrme interactions, e.g., SI, SII [33],
and SIII [30], used a three-body delta force instead of a
density-dependent two-body delta-force to obtain reasonable
nuclear-matter properties. The three-body interactions led
to
for
in Eq. (9),
but a different density dependence of the *C*_{t}^{s}.
is too large to get the
incompressibility
right, and causes a spin instability
in infinite nuclear matter [34] and finite nuclei
[35] (again only within a microscopic potential framework).
Both problems are cured with smaller values of
(between 1/6
and 1/3 [23]) but the second-generation interactions that did
so still had problems in the time-odd channels, giving a poor
description of spin and spin-isospin excitations and prompting several
attempts to describe finite nuclei with extended Skyrme interactions.
Krewald *et al.* [36], Waroquier *et al.* [37], and Liu
*et al.* [27], for example, introduced additional three-body
momentum-dependent forces. Waroquier *et al.* added an admixture of the
density-dependent two-body delta force and a three-body delta force,
while Liu *et al.* considered a tensor force. But none of these
interactions has been used subsequently.

Van Giai and Sagawa [38] developed the more durable parameterization SGII, which gave a reasonable description of GT resonance data known at the time and is still used today. The fit to ground state properties was made without the terms, however, even though they were used in the QRPA. Consequently, in such an approach, the QRPA does not correspond to the small-amplitude limit of time-dependent HFB.

All these attempts to improve the description of the time-odd channels impose severe restrictions on the coupling by linking them to the HF expectation value of a Skyrme force, leading to one difficulty or another. The authors of Refs. [18,39] proceed differently, treating the Skyrme energy functional as the result of a local-density approximation. The interpretation of the Skyrme interaction as an energy-density functional, besides relaxing the restrictions on the time-odd couplings, endows the spin-orbit interaction with a more flexible isospin structure [40,41,42] than can be obtained from the standard Skyrme force [43]. Some of the parameterizations used here will take advantage of that freedom. But the authors of Ref. [39] include only time-odd terms that are determined by gauge invariance; the other couplings are tentatively set to zero ( ). Such a procedure is reasonable when describing natural parity excitations within the (Q)RPA, but the neglected spin-spin terms are crucial for the unnatural parity states that we discuss.

In this study, we use the energy-functional approach (12) with fully independent time-even and time-odd coupling constants. Our hope is that this more general formulation will improve the description of the GT properties while leaving the good description of ground-state properties in even nuclei untouched.