Effective shell model Hamiltonians from density functional theory: Quadrupolar and pairing correlations

We describe a procedure for mapping a self-consistent mean-field theory (also known as density functional theory) onto a shell-model Hamiltonian that includes quadrupole-quadrupole and monopole pairing interactions in a truncated space. We test our method in the deformed $N=Z\hspace{0.5em}\mathit{sd}$-shell nuclei ${}^{20}\mathrm{Ne}$, ${}^{24}\mathrm{Mg}$, and ${}^{36}\mathrm{Ar}$, starting from the Hartree-Fock plus Bardeen-Cooper-Schrieffer (BCS) approximation of the universal $\mathit{sd}$ shell-model interaction. A similar method is then followed using the SLy4 Skyrme energy density functional in the particle-hole channel plus a zero-range density-dependent force in the pairing channel. Based on the ground-state solution of this density functional theory at the Hartree-Fock plus BCS level, an effective shell-model Hamiltonian is constructed. We apply this mapped Hamiltonian to extract quadrupolar and pairing correlation energies beyond the mean-field approximation. The rescaling of the mass quadrupole operator in the truncated shell-model space is found to be almost independent of the coupling strength used in the pairing channel of the underlying mean-field theory.

Figure 1

Correlation energies $E_{\mathrm{corr}}$ versus mass number $A$ for the deformed $N=Z\hspace{0.5em}\mathit{sd}$-shell nuclei ${}^{20}\mathrm{Ne}$, ${}^{24}\mathrm{Mg}$, and ${}^{36}\mathrm{Ar}$. Results for the mapped CISM Hamiltonian (20) (solid squares) are compared with the quadrupolar correlation energies calculated in Ref. [18] (open squares) and with the full correlation energies of the USD Hamiltonian (open circles).

Figure 2

HF+BCS energy curves for the nuclei ${}^{20}\mathrm{Ne}$ (left column), ${}^{24}\mathrm{Mg}$ (middle column), and ${}^{36}\mathrm{Ar}$ (right column) as functions of the axially symmetric mass quadrupole moment $\langle \hat{Q}\rangle$. The corresponding surfaces are shown for DFT pairing strengths of $g^{\mathrm{DFT}}=900$ and 1000 MeV fm${}^3$. The SLy4 parametrization of the Skyrme interaction is used in the particle-hole channel, whereas a zero-range density-dependent interaction is included in the pairing channel. All energies are measured with respect to the global minimum of the corresponding surface.

Figure 3

The occupation probabilities $\langle {\hat{n}}_j\rangle /(2j+1)$ of the valence spherical orbitals as a function of the single-particle energy ${\epsilon }_j^{(0)}$ for the nuclei ${}^{20}\mathrm{Ne}$, ${}^{24}\mathrm{Mg}$, and ${}^{36}\mathrm{Ar}$ (see Table ). The DFT results (diamonds) are compared with the results in the HF+BCS ground-state solution of the CISM Hamiltonian (20) (triangles). Also shown are the occupations calculated in a shell-model approach for the effective CISM Hamiltonian (squares). Open (solid) symbols are for protons (neutrons).

Figure 4

Neutron BCS occupations $v_k^2$ of the deformed ground-state solution of the CISM Hamiltonian (20) (solid squares) are compared with the BCS occupations in the deformed basis of the DFT ground-state solution (open squares). These BCS occupations are plotted versus the deformed single-particle HF energies ${\epsilon }_k$. Results are shown for ${}^{20}\mathrm{Ne}$ using the SLy4 parametrization of the Skyrme force and a zero-range density-dependent force with $g^{\mathrm{DFT}}=900$ MeV fm${}^3$.

Figure 5

Correlation energies $E_{\mathrm{corr}}$ [see Eq. (24)] versus mass number A for the $N=Z\hspace{0.5em}\mathit{sd}$-shell nuclei ${}^{20}\mathrm{Ne}$, ${}^{24}\mathrm{Mg}$, and ${}^{36}\mathrm{Ar}$, calculated using the mapped CISM Hamiltonian (20) (solid squares). The SCMF theory used to construct the map is the SLy4 Skyrme force plus a zero-range density-dependent force in the pairing channel of strength $g^{\mathrm{DFT}}=900$ MeV fm${}^3$. The open squares are the correlation energies found from a mapped CISM Hamiltonian that includes quadrupolar correlations alone [18].