Introduction

Self-consistent mean-field (MF) approach is practically the only formalism allowing for large-scale no-core computations in heavy open-shell nuclei with many valence particles. Inherent to the MF approach is the mechanism of spontaneous symmetry breaking, which is essentially the only way to allow for incorporating a significant part of many-body correlations into a single intrinsic (symmetry-breaking) Slater determinant. However, unlike in the cases of rotational or translational symmetry-breaking schemes, violation of the isobaric symmetry has two distinctively different sources. The unwanted or unphysical source pertains directly to the MF approximation[1,2,3,4]. It manifests itself very clearly in the ground-state wave functions calculated by using isospin-invariant interactions, like Skyrme or Gogny forces, with Coulomb force neglected. Indeed, such calculations manifestly break the isospin symmetry in all but the $N=Z$ systems, simply because the self-consistent proton and neutron wave functions are then different. The second source of the isospin-symmetry violation is of strictly physical nature and is caused mostly by the Coulomb field and, to a much lesser extent, by strong-force isospin-non-invariant components.

Hereby, we report on the development of a new theoretical tool that allows for isospin projection (after variation) of Slater determinants. It has been implemented within the Hartree-Fock code HFODD[5,6]. First, the Slater determinants are determined in a standard way by minimizing the Skyrme functional plus the Coulomb energy. Both direct and exchange terms of the Coulomb energy are calculated exactly. We allow for arbitrary spatial deformations of these intrinsic states. Second, the isospin-projected components are determined, and third, they are mixed so as to rediagonalize the total Skyrme-plus-Coulomb Hamiltonian.

Such a three-step procedure allows, respectively, for (i) taking into account the competition between the nuclear and Coulomb interactions in building up the single Slater determinant, which is becoming 'deformed' in the isospace, (ii) restoring the isospin symmetry, and (iii) letting the nuclear and Coulomb interactions pick the correct mixtures of symmetry-restored eigenstates of the isospin.

In the present study, we briefly overview the main theoretical building blocks of the formalism (Sect. 2) and discuss preliminary applications. In particular, we give results for the isospin-mixing parameters ($\alpha _C$) calculated in the ground states of $N=Z$ nuclei (Sect. 3). The ultimate goal will be to perform simultaneous isospin and angular momentum[7,8] projections and to systematically calculate the isospin-symmetry breaking corrections to the Fermi matrix element ($\delta_C$) for the set of nuclei undergoing the superallowed $0^+ \rightarrow 0^+$ Fermi beta decay[9,10].