Nuclear energy density functionals from empirical ground-state densities

A model is developed, based on the density functional perturbation theory and the inverse Kohn-Sham method, that can be used to improve relativistic nuclear energy density functionals towards an exact but unknown Kohn-Sham Hartree-exchange-correlation functional. The improved functional is determined by empirical exact ground-state densities of finite systems. A test of the model and an illustrative calculation are performed, starting from two different approximate functionals, to reproduce the parameters and density dependence of a target functional, using exact ground-state densities of symmetric $N=Z$ systems.

Figure 1

(a) The sum of neutron vector and scalar densities in the $N=Z=8$ system as a function of the radial coordinate. The target density obtained using the DD-PC1 functional (dot-dashed green curve) is compared to the the density calculated in the initial step of the inversion method (dashed red) with a Woods-Saxon potential, and to the final IKS density (solid black). (b) Same as in (a) but for the difference between the neutron vector and scalar densities. (c) Same as in (a) but for the neutron vector density. (d) Same as in (a) but for the neutron scalar density.

Figure 2

(a) The sum of neutron vector and scalar potentials in the $N=Z=8$ system as a function of the radial coordinate. The target DD-PC1 Kohn-Sham potential (dot-dashed green curve) is compared to the initial Woods-Saxon potential (dashed red), and to the final IKS potential (solid black). (b) Same as in (a) but for the difference between the neutron vector and scalar potentials. (c) Same as in (a) but for the neutron vector potential. (d) Same as in (a) but for the neutron scalar potential.

Figure 3

Values of the constants ${\left(b_{\text{s}}^{\left(1\right)}\right)}_i$, ${\left(c_{\text{s}}^{\left(1\right)}\right)}_i$, and ${\left(b_{\text{v}}^{\left(1\right)}\right)}_i$ at different iteration steps. The dashed lines denote the target values that correspond to the functional DD-PC1.

Figure 4

Values of the constants ${\left(b_{\text{s}}^{\left(1\right)}\right)}_i$, ${\left(c_{\text{s}}^{\left(1\right)}\right)}_i$ [(a)], ${\left(b_{\text{v}}^{\left(1\right)}\right)}_i$ and ${\left(c_{\text{v}}^{\left(1\right)}\right)}_i$ [(b)] at different iteration steps of the $\text{IKS}+\text{DFPT}$ calculation. The dashed lines denote the corresponding parameters of the linear and quadratic term in the Taylor expansion of the DD-PC1 couplings.

Figure 5

(a) Scalar and (b) vector $\text{IKS}+\text{DFPT}$ couplings as functions of vector density, compared to the corresponding DD-PC1 target coupling functions.

Figure 6

The vector densities of the four symmetric systems: $N=Z=8$, $N=Z=20$, $N=Z=28$, and $N=Z=50$. The dashed red curves are the densities that correspond to the unperturbed initial functional $E_{\text{int}}^{\left(0\right)}$ shown in Eq. (41). The dot-dashed green and solid black curves denote the densities obtained with the target functional DD-PC1 and the final results of the $\text{IKS}+\text{DFPT}$ calculation, respectively.