Residual interactions are often summarized by
the Landau parameters that appear in Eq. (13).
The parameters can be derived as the corresponding coupling constants
when Eq. (14) is evaluated for infinite spin-saturated
symmetric nuclear matter (see Appendix 10).
In the literature, the infinite nuclear matter (INM)
properties of the Skyrme interactions are
usually calculated from Eqs. (34). For the generalized
energy functional (12) discussed here, the
time-even INM properties such as the saturation density, energy per
particle, effective mass, incompressibility, symmetry coefficient,
and the time-even Landau parameters fi, fi'
are unchanged, but properties of polarized INM and expressions for the
time-odd Landau parameters gi and gi' are different.
We derive them in Appendix 10. Here we are most
concerned with the Landau parameters in the spin and spin-isospin
Table 1 also gives values for the Landau parameters calculated for the Gogny forces D1  and D1s  from the expressions provided in Appendix 11. In the spirit of the Gogny force as a two-body potential, one has no freedom to choose the time-odd terms independently from the time-even ones. (Note that the Gogny force, however, employs the same local-density approximation for the density-dependence as the Skyrme energy functional that contributes to the Landau parameters.) The higher-order Landau parameters are uniquely fixed by the finite-range part of the Gogny force.
Figures 1 and 2 show the summed GT strength B(GT) in 208Pb and 124Sn, calculated with all the selected Skyrme forces. The ground-state energies are calculated as described in Ref. , and all strengths are divided by 1.262, following common practice, to account for GT quenching. Although the GT resonance in 208Pb comes out at about the right energy for SGII, SLy4, SkO, and SkO', it is too low for SkP and SLy5. These latter two interactions also leave too much GT strength at small excitation energies. It is tempting to interpret these findings in terms of the Landau parameters for these interactions. Schematic models suggest  that an increase of g0' results in an increased resonance energy and more GT strength in the resonance. The nucleus 208Pb indeed behaves in this way, as can be seen in Fig. 1. The forces SkP and SLy5, with small values of g0', yield more low-lying strength and a lower resonance energy than the remaining forces which correspond to .
In 124Sn, however, this simple picture does not hold, as Fig. 2 shows. The resonance energies are similar (and close to the experimental value) for SkP, SLy5, SkO, and SkO' forces with very different values of g0', while SGII and SLy4 push the resonance energy too high. Only the amount of the low-lying strength seems to scale with g0'. It is interesting, though, that the related forces SLy4 and SLy5 (which predict very similar single-particle spectra, but have quite different GT residual interactions) agree with the schematic model in that SLy4, with larger g0', puts the GT resonance at a higher excitation energy.
It is clear that the scaling predicted by the schematic model is too simple, and Fig. 3 demonstrates this clearly. There we show the calculated strengths in the GT resonances relative to the sum-rule value , and the calculated GT resonance energies relative to the experimental values . [For 90Zr, 112Sn, 124Sn, and 208Pb we used MeV, 8.9MeV, 13.7MeV, and 15.5MeV, respectively . Note that the calculated resonance energy depends on a prescription (see ) not strictly dictated by the QRPA.] The scatter near , in both the resonance energy and in the amount of low-lying strength, shows that other combinations of parameters in the residual interaction besides g0' affect the GT distribution. This is not entirely surprising given the complexity of finite nuclei and of the interaction (14). In Sect. 4.3 we quantify these other important combinations and discuss their effects.
But another factor, this one determined by the time-even part of the Skyrme functional, affects the GT distribution: the underlying single-particle spectrum. Since GT transitions are especially sensitive to proton spin-orbit splittings, small changes in the time-even part of the force can, in principle, move the GT resonance considerably. Sensitivity to the spin-orbit splitting is particularly obvious in 90Zr, where detailed information has been obtained from a recent experiment by Wakasa et al. . Unlike in 124Sn and 208Pb, which respond to a GT excitation in a collective way, the 90Zr GT spectrum is dominated by two single-particle transitions, from the neutron 1g9/2 state to the proton 1g9/2 and 1g7/2 states. The difference between the locations of the two peaks in the GT spectrum is the sum of the proton 1g spin-orbit splitting and a contribution from the residual interaction (which can be expected to increase the difference). As Fig. 4 shows, all interactions, whatever their value for g0', overestimate this difference; the resonance energy is always too large, even when the residual interaction is switched off completely.
Most Skyrme interactions give spin-orbit splittings in heavy nuclei that are too large . We can therefore expect errors in their predicted GT strength distributions [45,47]. Figure 5 shows errors in the predicted spin-orbit energies for the same forces as in Fig. 3. Interactions such as SkI3, SkI4, or SLy4 that overestimate the proton spin-orbit splittings give the largest resonance energies (and tend to overestimate them). The best interaction, in view of the combined information from Figs. 3 and 5, appears to be SkO'. Therefore, below, we use its time-even energy functional for further exploration of the time-odd terms.
We have included some new forces in Fig. 5; in a recent paper , Sagawa et al. attempt to improve the spin-orbit interaction for the standard Skyrme forces SIII, SkM*, and SGII, aiming at better GT-response predictions. They generalize the spin-orbit interaction through the condition and include the term with a coupling given by Eq. (34). Although the modified forces SkM*-u, and SGII-u give slightly better descriptions of GT resonances than the original interactions, they generate unacceptable errors in total binding energies and do not substantially improve the overall description of single-particle spectra in 208Pb.
A few remarks are in order before proceeding: (i) The spin-orbit splittings shown in Fig. 5 are calculated from intrinsic single-particle energies. Since experimental data are obtained from binding-energy differences between even-even and adjacent odd-mass nuclei, core polarization induced by the unpaired nucleon, which depends partly on time-odd channels of the interaction [7,8], alters single-particle energies. The effect is largest in small nuclei (of the order of 20 % in 16O), decreasing rapidly with mass number . (ii) GT distributions are also affected by the particle-particle channel of the effective interaction, but mainly at low energies. The GT resonance is not materially altered , so we can safely neglect the particle-particle interaction here.