No-core configuration-interaction model

The static variant of our model is based on the double projection, on isospin and angular momentum, of a single Slater determinant. In an even-even nucleus, the Slater determinant representing the ground-state is uniquely defined. In an odd-odd nucleus, the conventional MF theory that gives Slater determinants separably for neutrons and protons faces problems. First, there is no single Slater determinant representing the $I=0^+, T=1$ state, see [4,9]. In our approach, this obstacle is removed by projecting from the so-called anti-aligned Slater determinant. This configuration, by construction, has no net alignment and manifestly breaks the isospin symmetry, being an almost fifty-fifty mixture of the $T=0$ and $T=1$ states. In this way, the needed $T=1$ component can be recovered. The problem is, however, that the anti-aligned states are not uniquely defined. In the general case of a triaxial nucleus, there exist three linearly-dependent Slater determinants, built of valence neutron and proton single-particle states that carry angular momenta aligned along the X ( $\vert\varphi^{\rm (X)}\rangle$), Y ( $\vert\varphi^{\rm (Y)}\rangle$), or Z ( $\vert\varphi^{\rm (Z)}\rangle$) axes of the intrinsic frame of reference or, respectively, along the long, intermediate, and short axes of the core. In our calculations, no tilted-axis anti-aligned solution was found so far.

In the static approach, the only way to cope with this ambiguity is to calculate three independent $\beta$-decay matrix elements and to take the average of the resulting $\delta_{\rm C}$ values. Such a solution is not only somewhat artificial, but also increases the theoretical uncertainty of the calculated ISB corrections. This deficiency motivated our development of the dynamic model, which allowed for mixing states projected from the three reference states $\vert \varphi^{\rm (k)}\rangle$ for k=X, Y, and Z, with the mixing matrix elements derived from the same Hamiltonian that was used to calculate them. The dynamic model further evolved towards a full no-core configuration-interaction (NCCI) model, in which we allow for mixing states projected from different low-lying (multi)particle-(multi)hole Slater determinants $\vert\varphi_i\rangle$. This final variant has all features of the no core shell model, with two-body effective interaction (including the Coulomb force) and a basis-truncation scheme dictated by the self-consistent deformed HF solutions. The computational scheme proceeds in four major steps:

• First, a set of relevant low-lying (multi)particle-(multi)hole HF states $\{ \varphi_i \}$ is calculated along with their HF energies $e^{{\rm (HF)}}_i$, which form a subspace of reference states for subsequent projections.

• Second, the projection techniques are applied to the set of states $\{ \varphi_i \}$, so as to determine the family of states $\{ \Psi_{TIK}^{(i)}\}$ having good isospin $T$, angular momentum $I$, and angular-momentum projection on the intrinsic axis $K$.

• Third, states $\{ \Psi_{TIK}^{(i)}\}$ are mixed, so as to properly take into account the $K$ mixing and Coulomb isospin mixing - this gives the set of good angular-momentum states $\{\Psi_I^{(i)}\}$ of the static model [3,4].

• Finally, the mixing of, in general, non-orthogonal states $\{\Psi_I^{(i)}\}$ for all configurations $i$ is performed by solving the Hill-Wheeler equation in the collective space spanned by the natural states corresponding to non-zero eigenvalues of their norm matrix, that is, by applying the same technique, which was used to handle the $K-$mixing alone [10].

The numerical stability of the method is affected by necessary truncations of the model space, namely, numerically unstable solutions are removed by truncating either the high-energy states $\{\Psi_I^{(i)}\}$ or the natural states corresponding to small eigenvalues of the norm matrix, or by applying both truncations simultaneously. Although such truncation procedure gives reliable values of the energy, the results shown below must still be considered as preliminary.