Theory

The isospin-projected DFT technique [11,12,13]
utilizes the ability of the
self-consistent MF method to properly describe the balance between
the long-range Coulomb force and the short-range nuclear interaction,
represented in this work by the Skyrme-type energy density functional (EDF). To
remove the spurious isospin-symmetry-breaking effects, we use the
standard one-dimensional isospin projection after variation, which
allows us to decompose the Slater determinant into
good isospin states :

where denotes the Euler angle associated with the rotation operator about the -axis in the isospace, is the Wigner function [17], and is the third component of the total isospin . The normalization factors , or interchangeably the expansion coefficients that encode the isospin content of , read:

where is the so-called overlap kernel. For technical aspects concerning the calculation of the overlap and Hamiltonian kernels, we refer the reader to Ref. [13]. The isospin-projected DFT technique utilizes the ability of the HF solver HFODD [18] to produce fully symmetry-unrestricted Slater determinants .

The isospin projection determines the set of good isospin states
(called the *basis* in the following), which in the next step is used to rediagonalize
the entire nuclear Hamiltonian, consisting of the kinetic energy, Skyrme EDF,
and the isospin-breaking Coulomb force. The rediagonalization leads to the
eigenstates:

where stands for the squared norm of the dominant amplitude in the wave function . It is worth stressing that the isospin projection, unlike particle-number or angular-momentum projections, is essentially non-singular; hence, it can be safely used with the local EDFs. The rigorous analytical proof of this useful property can be found in Ref. [13].

The combined isospin and angular-momentum projection
leads to the set of states,

which are labeled by the index and by the conserved quantum numbers , , and [cf. Eq. (4)].