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 sd-shell nuclei ²⁰Ne, ²⁴Mg, and ³⁶Ar, starting from the Hartree-Fock plus Bardeen-Cooper-Schrieffer (BCS) approximation of the universal 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.