Optimized Dirac Woods-Saxon basis for covariant density functional theory

The Woods-Saxon basis has achieved great success in both nonrelativistic and covariant density functional theories in recent years. Due to its nonanalytical nature, however, applications of the Woods-Saxon basis are numerically complicated and computationally time consuming. In this paper, based on the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc), we check in detail the convergence with respect to the basis space in the Dirac sea. An optimized Dirac Woods-Saxon basis is proposed, whose corresponding potential is close to the nuclear mean field. It is shown that the basis space of the optimized Dirac Woods-Saxon basis required for convergence is substantially reduced compared with the original one. In particular, it does not need to contain the bases from continuum in the Dirac sea. The application of the optimized Woods-Saxon basis would greatly reduce computing resource for large-scale density functional calculations.

Figure 1

The total energy is shown as a function of the energy cutoff in the Dirac sea $E_{\mathrm{cut}}^D$ for ${}^{20}\mathrm{Ne}$ (a), ${}^{112}\mathrm{Mo}$ (b), and ${}^{300}\mathrm{Th}$ (c). The horizontal dashed lines show their total energies, calculated with the numerical condition that the number of DWS bases in the Dirac sea is the same as that in the Fermi sea, and are regarded as the converged results. In (d), the relative difference of the total energy from the converged one, $(E_{\mathrm{Tot}}-E_{\mathrm{Con}})/E_{\mathrm{Con}}$, is plotted as a function of $E_{\mathrm{cut}}^D$ for ${}^{20}\mathrm{Ne}$, ${}^{112}\mathrm{Mo}$, and ${}^{300}\mathrm{Th}$. The energy cutoff in the Fermi sea $E_{\mathrm{cut}}^F$ is taken as 300 MeV.

Figure 2

The number of DWS bases in the Fermi sea, $N_F$, and the ratio of the number of DWS bases in the Dirac sea to it, $N_D/N_F$, are shown as functions of the absolute value of the quantum number $\kappa$ in the calculation of ${}^{20}\mathrm{Ne}$, ${}^{112}\mathrm{Mo}$, and ${}^{300}\mathrm{Th}$. For clearness, the ratios for positive parity ($+$) and negative parity ($-$) are shown respectively in (b) and (c), where the values for protons ($P$) and neutrons ($N$) are distinguished by symbols. The energy cutoff in the Fermi sea $E_{\mathrm{cut}}^F$ and that in the Dirac sea $E_{\mathrm{cut}}^D$ are taken as 300 and 50 MeV, respectively.

Figure 3

The Woods-Saxon potentials $V+S$ (a) and $V-S$ (b) used to generate the DWS basis and those corresponding to the converged mean field are plotted as functions of the radial coordinate $r$ for ${}^{300}\mathrm{Th}$. Here the angular dependence of the deformed mean-field potentials has been averaged. The energy cutoff in the Fermi sea $E_{\mathrm{cut}}^F$ and that in the Dirac sea $E_{\mathrm{cut}}^D$ are taken as 300 and 50 MeV, respectively.

Figure 4

The difference of the total energy from the converged result as a function of $E_{\mathrm{cut}}^F$ in (a) and of $E_{\mathrm{cut}}^D$ in (b) for ${}^{300}\mathrm{Th}$. The results from $E_{\mathrm{cut}}^D=0,10,\cdots ,50$ MeV in (a) and $E_{\mathrm{cut}}^F=200,220,\cdots ,300$ MeV in (b) are given by different symbols. Empty and filled symbols represent the results with the DWS and ODWS basis, respectively.

Figure 5

Single-neutron levels of ${}^{300}\mathrm{Th}$ near the continuum threshold calculated using DWS and ODWS basis with $E_{\mathrm{cut}}^F=200$ MeV and $E_{\mathrm{cut}}^D=0$ MeV as well as those from the converged result. The quantum numbers $m^{\pi }$ are given on the right, and the solid and dashed lines represent occupied and unoccupied levels, respectively.