Within the MF approximation, the isospin symmetry breaking has two
sources. The spontaneous breaking of isospin associated with the MF
approximation itself [36,26,29]
is, in this theory, intertwined with the explicit symmetry breaking
due to the Coulomb interaction. The essence of our
method [28,29] is to retain
the explicit isospin mixing. It is achieved by rediagonalizing the
total effective nuclear Hamiltonian
in good-isospin
basis. The basis is generated by using the
projection-after-variation technique, that is, by acting with
the standard one-dimensional isospin-projection operator on
the MF product state
:

where denotes the Euler angle associated with the rotation operator about the -axis in the isospace, is the Wigner function [37], and is the third component of the total isospin . The wave function is obtained self-consistently by solving the Skyrme-Hartree-Fock (SHF) equations. In the HF limit, the -component is strictly conserved. The overlap , or interchangeably the normalization factor , are given by

where

stands for the overlap kernel. The good-isospin basis is spanned by the states that have tangible contributions to the MF state

In practice, we retain states having , which sets the limit of .

The total nuclear Hamiltonian consists of the kinetic
energy , the Skyrme interaction
and the
Coulomb interaction
that breaks isospin:

where

are, respectively, isoscalar, covariant rank-1 (isovector), and covariant rank-2 axial (isotensor) spherical tensor components of the Coulomb interaction. They are constructed by coupling spherical components of the one-body isospin operator:

(9) |

where denote Pauli matrices and symbol stands for the scalar product of isovectors.

The Coulomb interaction is the only source of
the isospin symmetry violation in our model. Charge symmetry breaking
components of the strong interaction and the isovector kinetic
energy (which is quenched as compared to its isoscalar counterpart
by a factor
due to very small
mass difference between the neutron and the proton relative to
mean nucleonic mass ) are not taken into account.
Hence, the isoscalar part of the total
Hamiltonian reads
. Its matrix elements can be cast into
one-dimensional integrals:

involving the Hamiltonian kernel

Calculation of matrix elements of the isovector (7) and isotensor (8) components of the Coulomb interaction is slightly more complicated. It appears, however, that these matrix elements can also be reduced to one-dimensional integrals over the Euler angle :

where

and symbols stand for the Clebsch-Gordan coefficients [37]. The non-axial (0) isovector and isotensor components of the Coulomb interaction appearing in Eq. (13) are:

(14) | |||

(15) | |||

(16) |

Equations (12)-(13) can be derived by using the standard transformation rules for a covariant spherical-tensor operator of rank- under the three-dimensional rotation by the Euler angles , that is,

where the SO(3) projection operator commutes with spherical tensors as [38]

(18) |

The cornerstone of the isospin projection scheme described above is a calculation of the Hamiltonian and norm kernels and subsequent one-dimensional integration over the Euler angle . The integrals are calculated numerically using the Gauss-Legandre quadrature, which is very well suited for this problem provided that the calculated kernels are non-singular.