To properly account for the isospin mixing effects, following Refs. [28,29], the total Hamiltonian (5) (strong interaction plus the Coulomb interaction) is rediagonalized in the space spanned by the good-isospin wave functions (1), and the resulting eigenstates are denoted by

and numbered by index . The amplitudes define the degree of isospin mixing. In particular, the isospin-mixing parameter for the lowest energy solution is defined as .

Precise determination of the Coulomb mixing constitutes a notoriously difficult problem, but is strongly motivated by its impact on fundamental physics tested through the super-allowed decays [50,51,52]. Of importance here is to capture the proper balance between the short-range strong interaction and the long-range Coulomb polarization. This balance is naturally taken into account by the DFT approach, but is not accessible within a perturbative-analysis theory [53] or hydrodynamical model [54]. This is illustrated in Fig. 3 that shows the excitation energies of the doorway =1 states in e-e = nuclei, i.e., the energies of the states in Eq. (59) relative to the HF g.s. energies . The estimates given by the perturbative/hydrodynamical approaches are clearly at variance with the self-consistent results.

Figure 3 shows that not only the values but also the -dependence of the doorway excitation energies differ substantially from the self-consistent results, pointing very clearly to a non-perturbative origin of the Coulomb mixing. One can also notice that the mean excitation energies of the doorway states (directly impacting the Coulomb mixing) strongly depend on the EDF parameterization. The currently used Skyrme functionals are not sufficiently constrained in the isospin sector to provide reliable estimates of . At this point is also not at all obvious what parts of EDF have to be refined in order to improve on this situation, see Ref. [29] for further details.