Gallagher-Moszkowski splitting in deformed odd-odd nuclei within a microscopic approach

Background: Low-lying bandhead states in axially prolate deformed odd-odd nuclei have long been described essentially within the $\text{rotor}+\text{two}$-quasiparticle picture. This approach allows one to explain the appearance of so-called Gallagher-Moszkowski doublets of bandheads with $K={\mathrm{\Omega }}_n\pm {\mathrm{\Omega }}_p$, sum and difference of neutron and proton angular momentum projections on the symmetry axis. According to an empirical rule stated by Gallagher and Moszkowski the spin-aligned configuration lies lower in energy than the spin-antialigned configuration. A recent study by Robledo, Bernard, and Bertsch [Phys. Rev. C 89, 021303(R) (2014)] within the Gogny energy-density functional with self-consistent blocking of the unpaired nucleons showed that calculations fail to reproduce this rule in about half of the cases, and points to the density-dependent term of the functional as responsible of this failure.