Binding energies

Parameters of the Skyrme functional have been fitted to reproduce several physical quantities, with emphasis on the masses of magic nuclei. Therefore it is not surprising that dramatic modifications of the SO and tensor terms of the functional, described in Sec. 3, while improving the agreement between the calculated and measured single-particle properties, can destroy the quality of the mass fit. Hence, it is interesting to know whether this disagreement is significant and whether it can be healed by refitting the remaining parameters of the functional.

Table 5 shows differences between calculated and experimental (Ref. [54]) ground-state energies, $E_{\mathrm{calc}}-E_{\mathrm{exp}}$ , (in MeV) for a set of spherical nuclei. Negative values mean that nuclei are overbound. Results given in the second column, denoted as SLy4, correspond to the standard SLy4 [19] parametrization. The third column, denoted as SLy4 $_{T}$ , illustrates significant deterioration of the quality of fit when parameters $C^{\nabla J}_0$ ,

and

are modified (see Sec. 3). Values presented in the last column, SLy4 $_{T\mathrm{min}}$ , were obtained by minimizing the rms of relative discrepancies between the calculated and measured masses Values of $C^{\nabla J}_0$ ,

and

were kept fixed at their SLy4 $_{T}$ values, while minimization was performed by varying the remaining parameters of the functional

and

. Note that for the standard SLy4 functional, the tensor coupling constants are set equal to zero independently of the values of

and

. For the minimization, we have used the same methodology, namely, the influence of parameters

and

on tensor coupling constants was disregarded.

It should be emphasized that no attempt has been made to find the global minimum -- the minimization was purely local, in the vicinity of the standard SLy4 values of the parameters

and

. One can see that even this very limited procedure can lead to significant reduction of discrepancies, down to quite reasonable values (with an exception of $^{56}$ Ni nucleus).

Table 5: Differences between calculated and experimental ground-state energies, $E_{\mathrm{calc}}-E_{\mathrm{exp}}$ , (in MeV) for a set of spherical nuclei. Column denoted as SLy4 $_{T\mathrm{min}}$ shows results obtained after (local) minimization with respect to parameters $t_i, x_i,\, i = 0,1,2,3$ . See text for details.

Nucleus	SLy4	SLy4 $_{T}$	SLy4 $_{T\mathrm{min}}$
$^{40}$ Ca	2.197	1.830	5.775
$^{48}$ Ca	1.912	5.039	0.279
$^{56}$ Ni	0.625	15.138	9.127
$^{90}$ Zr	1.845	7.492	3.032
$^{132}$ Sn	0.660	19.898	2.222
$^{208}$ Pb	0.822	24.910	3.048

It is worth noting that the resulting modifications of the

and

parameters turned out to be very small. Table 6 shows the values of the

and

parameters in the standard SLy4 parametrization (second column) and those obtained as the result of the minimization procedure (third column). The last column shows relative changes of parameters (in percent). As one can see, they are at most of the order of one percent. Nevertheless, even such small changes were sufficient to improve significantly the agreement between calculated and experimental masses.

We stress again that the refitting procedure is used here only for illustration purposes and the global fit to masses must probably include extended functionals and improved methodology. For example, the Wigner energy correction [55] was not included in the fit, as it was neither included in the fit of the SLy4 parametrization. This correction alone may change the balance of discrepancies obtained for the

and $N\neq Z$ nuclei, and strongly impact the results. Systematic studies of these effects will be performed in the near future.

Table 6: Skyrme force parameters $t_i, x_i,\, i = 0,1,2,3$ of the standard SLy4 parametrization (second column) compared with those obtained from the minimization procedure described in the text (third column). The last column shows relative change of parameters (in percent).

param.	SLy4	SLy4 $_{T\mathrm{min}}$	change (%)
	2488.913	2490.00300	0.04
	486.818	486.78460	0.01
	546.395	545.35849	0.19
	13777.000	13767.77776	0.07
	0.834	0.83257	0.17
	0.344	0.34227	0.50
	1.000	0.99798	0.20
	1.354	1.36128	0.54