“Sloppy” nuclear energy density functionals. II. Finite nuclei

A study of parameter sensitivity of nuclear energy density functionals, initiated in the first part of this work [Nikšić and Vretenar, Phys. Rev. C 94, 024333 (2016)], is extended by the inclusion of data on ground-state properties of finite nuclei in the application of the manifold boundary approximation method (MBAM). Density functionals used in self-consistent mean-field calculations, and nuclear structure models based on them, are generally “sloppy” and exhibit an exponential range of sensitivity to parameter variations. Concepts of information geometry are used to identify the presence of effective functionals of lower dimension in parameter space associated with parameter combinations that can be tightly constrained by data. The MBAM is used in an iterative procedure that systematically reduces the complexity and dimension of parameter space of a sloppy functional, with properties of nuclear matter and data on finite nuclei determining not only the values of model parameters but also the optimal functional form of the density dependence.

Figure 1

Eigenvectors and eigenvalues of the $10\times 10$ Hessian matrix of second derivatives $\mathcal{M}$ of ${\chi }^2\left(\mathbf{p}\right)$ at the best-fit point for the functional defined by the couplings of Eq. (1), plus the strength parameter of the derivative term ${\delta }_s$. The empty and filled bars indicate that the corresponding amplitudes contribute with opposite signs.

Figure 2

The parameters of the isoscalar part of the ten-parameter functional set 1, as functions of the affine parametrization, along the geodesic path determined by the eigenvector of the Hessian matrix that corresponds to the smallest eigenvalue (cf. Fig. 1). Panel (a) displays the four parameters of the scalar channel, while the evolution of the three parameters that determine the vector channel is shown in panel (b).

Figure 3

Same as in the caption to Fig. 2 but for the two parameters of the isovector part of the functional defined in Eq. (1) (a) and the strength parameter of the derivative term (b).

Figure 4

The initial (best-fit point) and final (at the boundary of the model manifold) eigenspectrum of the FIM for the ten-parameter functional SET 1 (a). The eigenvectors that correspond to the initial and final smallest eigenvalues are shown in panels (b) and (c).

Figure 5

Same as in the caption to Fig. 1 but for the functional with nine parameters (set 2) obtained by applying the MBAM to the ten-parameter functional set 1.

Figure 6

Same as in the caption to Fig. 2 but for the nine-parameter functional (set 2).

Figure 7

Same as in the caption to Fig. 3 but for the nine-parameter functional (set 2).

Figure 8

Same as in the caption to Fig. 4 but for the nine-parameter functional (set 2).

Figure 9

Equation of state of symmetric nuclear matter (a) and neutron matter (b). Energy as a function of nuclear (neutron) matter calculated with the functional DD-PC1, and the three functionals adjusted in this study, sets 1, 2, and 3, are shown in comparison to the microscopic equations of state [13]. The points to which the parameters of the functionals have been fitted are shown with the adopted uncertainty $10\%$.

Figure 10

Differences between the theoretical and experimental values for the charge radii (a) and binding energies (b) of the eight nuclei used to adjust the parameters of the functionals sets 1, 2, and 3.