To this end, we first find the self-consistent MF state and then build a normalized angular-momentum- and isospin-conserving basis by using the projection method:

where and stand for the standard isospin and angular-momentum projection operators:

(5) | |||

(6) |

where, is the rotation operator about the -axis in the isospace, is the Wigner function, and is the third component of the total isospin . As usual, is the three-dimensional rotation operator in space, are the Euler angles, is the Wigner function, and and denote the angular-momentum components along the laboratory and intrinsic -axis, respectively [22,25]. Note that unpaired MF states conserve the third isospin component ; hence, the one-dimensional isospin projection suffices.

The set of states (4) is, in general, overcomplete because
the quantum number is not conserved. This difficulty is overcome
by selecting first the subset of linearly independent states
known as *collective space* [22], which
is spanned, for each and , by the so-called *natural states*
[26,27].
The entire Hamiltonian - including
the ISB terms - is rediagonalized in the collective space, and the resulting
eigenfunctions are:

where the index labels the eigenstates in ascending order according to their energies. The amplitudes define the degree of isospin mixing through the so-called isospin-mixing coefficients (or isospin impurities), determined for a given th eigenstate as:

where the sum of norms corresponds to the isospin dominating in the wave function .

One of the advantages of the projected DFT as compared to the shell-model-based approaches [28,3] is that it allows for a rigorous quantum-mechanical evaluation of the Fermi matrix element using the bare isospin operators:

(9) |

where denotes the rank-one covariant one-body spherical-tensor operators in the isospace, see the discussion in Ref. [29,30]. Indeed, noting that each th eigenstate (7) can be uniquely decomposed in terms of the original basis states (4),

with microscopically determined mixing coefficients , the expression for the Fermi matrix element between the parent state and daughter state can be written as:

where tilde indicates the Slater determinant rotated in space: . The matrix element appearing on the right-hand side of Eq. (11) can be expressed through the transition densities that are basic building blocks of the multi-reference DFT [31,32,33,24]. Indeed, with the aid of the identity

(12) |

which results from the general transformation rule for spherical tensors under rotations or isorotations,

the matrix element entering Eq. (11) can be expressed as:

For unpaired Slater determinants considered here, the double integral over the isospace Euler angles in Eq. (11) can be further reduced to a one-dimensional integral over the angle using the identity

(15) |

which is the one-dimensional version of the transformation rule (13) valid for rotations around the axis in the isospace. The final expression for the matrix element in Eq. (14) reads:

where is the isovector transition density, and the double-tilde sign indicates that the right Slater determinant used to calculate this density is rotated both in space as well as in isospace: . The symbol denotes the overlap kernel.

Since the natural states have good isospin,
the states (7) are free from spurious
isospin mixing. Moreover, since the isospin projection is applied to self-consistent
MF solutions, our model accounts for a subtle balance
between the long-range Coulomb polarization,
which tends to make proton and neutron wave functions different,
and the short-range nuclear attraction, which
acts in an opposite way. The long-range polarization
affects globally all s.p. wave functions.
Direct inclusion of this effect in open-shell heavy nuclei is possible essentially only within
the DFT, which is the only *no-core* microscopic
framework that can be used there.

Recent experimental data on the isospin impurity deduced in Zr from the giant dipole resonance -decay studies [34] agree well with the impurities calculated using isospin-projected DFT based on modern Skyrme-force parametrizations [23,17]. This further demonstrates that the isospin-projected DFT is capable of capturing the essential piece of physics associated with the isospin mixing.