Gamow-Teller response in the configuration space of DFT-rooted no-core configuration-interaction model

The atomic nucleus is a unique laboratory to study fundamental aspects of the electroweak interaction. This includes a question concerning in medium renormalization of the axial-vector current, which still lacks satisfactory explanation. Study of spin-isospin or Gamow-Teller (GT) response may provide valuable information on both the quenching of the axial-vector coupling constant as well as on nuclear structure and nuclear astrophysics. We have performed a seminal calculation of the GT response by using the no-core-configuration-interaction approach rooted on multi-reference density functional theory (DFT-NCCI). The model treats properly isospin and rotational symmetries and can be applied to calculate both the nuclear spectra and transition rates in atomic nuclei, irrespectively of their mass and particle-number parity. The method is applied to compute the GT strength distribution in selected $N\approx Z$ nuclei including the $p$-shell $^8$Li and $^8$Be nuclei and the $sd$-shell well-deformed nucleus $^{24}$Mg. In order to demonstrate a flexibility of the approach we present also a calculation of the superallowed GT beta decay in doubly-magic spherical $^{100}$Sn and the low-spin spectrum in $^{100}$In. It is demonstrated that the DFT-NCCI model is capable to capture the GT response satisfactorily well by using relatively small configuration space exhausting simultaneously the GT sum rule. The model, due to its flexibility and broad range of applicability, may either serve as a complement or even as an alternative to other theoretical approaches including the conventional nuclear shell model.


I. INTRODUCTION
Single Reference Density Functional Theory (SR-DFT) has proven to be extremely successful in accounting for the bulk nuclear properties like masses, radii, or quadrupole moments over the entire nuclear chart, see [1,2] and references quoted therein. The success of SR-DFT or, alternatively, self-consistent mean-field theory has its roots in the spontaneous symmetry breaking which allows to incorporate correlations into a single Slater determinant. Deformed wave function does not allow, however, for quantum-mechanically-rigorous treatment of neither the nuclear spectra nor the nuclear decay rates. So far, this domain was traditionally reserved for the Nuclear Shell Model (NSM), a configurationinteraction (CI) approach involving strict laboratoryframe treatment of symmetries, see [3] for a review.
An expanse of applicability of the mean-field or Single Reference Energy Density Functional (SR-EDF) based methods is ultimately related with symmetry restoration. Recently, strenuous effort was devoted to a development of symmetry-projected multi-reference DFT (MR-DFT) and to extend it towards No-Core Configuration-Interaction (NCCI) approach. Bally and coworkers proposed a DFT-NCCI framework involving Skyrme superfluid functional and applied it successfully to compute spectra and electromagnetic transition rates in 25 Mg [4]. Our group has developed a variant involving unpaired Skyrme functional and a unique combination of angularmomentum and isospin projections and applied to calculate the spectra and beta-decay rates in N ≈ Z nuclei from p-shell to medium mass nuclei around 62 Zn [5][6][7][8].
Recently, the DFT-NCCI method was applied to calculate spectra in neutron-rich 44 S and 64 Cr nuclei with Gogny force [9,10], within relativistic framework [11] in 54 Cr, or within pairing-plus-quadrupole model in magnesium chain [12].
The results obtained so far have been very promising. In particular, they indicate that relatively limited number of configurations is needed to obtain accurate description of low-energy, low-spin physics in complex nuclei. However, further tests of these methods are still required.
The DFT-NCCI method allows to address many important physics questions in a way which is complementary to the conventional NSM. The flagship example concerns physical origin of the quenching effect of the weak axial coupling constant (for free-neutron decay g A = −1.2701 (25)) being a subject of a vivid discussion since the first Gamow-Teller (GT) beta-decay calculations were performed. The DFT-NCCI calculations in T = 1/2 mirror nuclei [8] rather contradict with the statement that the quenching has its roots in a model space and therefore support the two-body current based explanation, put forward in Refs. [13,14], see also [15][16][17].
The goal of this work is to compute spin-isospin response by using, for the first time, the DFT-NCCI approach. The spin-isospin, or GT response, provides valuable information on both the electroweak beta decay and nuclear structure. Since DFT-NCCI originates from very intuitive and powerful concept of spontaneous symmetry breaking, it gives a unique opportunity to discuss complex patterns that emerge in the response function in terms of simple deformed single-particle Nilsson levels which are the primary building blocks of the formalism. In this sense the DFT-NCCI can be considered again as complementary method with respect to the NSM [3,18,19], coupled cluster [14], or Quasiparticle Random Phase Approximation (QRPA) [20][21][22][23][24][25] which was, until now, the only possible mean-field-based alternative to the NSM concerning global studies of GT strength distribution. Last but not least, beta decay in pf -shell nuclei is studied in variation-after-projection Excited Vampir approach with G-matrix-driven realistic effective interaction [26]. Although the method is based on a mean-field concept, its model space and treatment of correlations are entirely different from the DFT-NCCI model. This paper is organized as follows. In Sec. II we discuss the foundations of the DFT-NCCI model paying special attention to the concept of configuration and model spaces. In Sec. III we present the results for the structure and GT strength distribution in A = 8 nuclei. In Sec. IV we discusses the spin-isospin response in the sdmidshell nucleus 24 Mg. Eventually, in Sec. V, we focus on the 100 Sn → 100 In superallowed GT beta decay and the low-spin spectrum of 100 In. Summary and conclusions are presented in Sec. VI. All calculations presented in this work were done using developing version of the HFODD solver [27] equipped with the NCCI module.

II. THE DFT-ROOTED NO-CORE-CONFIGURATION-INTERACTION MODEL
The DFT-NCCI models are post Hartree-Fock(-Bogliubov) approaches which mix non-orthogonal manybody states projected from symmetry breaking meanfield solutions. Their sole ingredients are therefore independent-particle (quasi)particle-(quasi)hole configurations and projection techniques that are used to restore spontaneously broken symmetries. In practical applications, the projections are handled by using the gener-alized Wick's theorem (GWT) which leads from a SR to multi-reference (MR) formulation of the DFT (MR-DFT).
The GWT allows to handle theory numerically, however it leads to singular kernels once modern densitydependent Skyrme or Gogny forces are used for the beyond-mean-field part of the calculation. The intensive work to overcome this problem of projection-induced singularities is currently under way. The attempts to regularize the kernels [28,29] have not provided a satisfactory solution so far. Hence, at present, the theory can be safely carried on only for true interactions like the SV T [30], used in the present work, or the SLyM0 [31], which both are density-independent Skyrme pseudopotentials. In addition to these, recently developed regularized finite-range pseudopotential [32] aims also for beyond-mean-field calculations. It is worth mentioning that these pseudopotentials are characterized by anomalously low effective mass which affects the single-particle (s.p.) level density and, in turn, influences spectroscopic properties of the calculated nuclei.
The MR-DFT approach developed by our group is unique in the sense that it restores angular momentum and treats rigorously the isospin symmetry i.e. is retaining only physical sources of its breaking. It provides wave functions which are isospin (T ) and K (projection of angular momentum onto intrinsic z-axis) mixed as whereP I andP T are projection operators of SU(2) group generated by angular-momentum and isospin, respectively, and N (i) ϕ;IM ;Tz is a normalization constant. Index i enumerates different solutions of a given spin I. The Slater determinant, ϕ, is calculated self-consistently by using Hartree-Fock (HF) method with the SV T Skyrme and Coulomb forces.
The MR-DFT wave functions (1) can be successfully used to compute, for example, beta-decay transition rates between the ground states as shown in Refs. [8,33]. In order to account for beta-decay strength distribution, the MR-DFT concept needs to be extended by including the states (1) projected from many Slater determinants ϕ j corresponding to different (multi)particle-(multi)hole excitations. The projected states, which are generally nonorthogonal to each other, are mixed by solving the Hill-Wheeler-Griffin equation with, typically, the same Hamiltonian that was used to generate them at the HF stage [34]. In effect, one obtains a set of linearly independent DFT-NCCI eigenstates of the form of |ψ k;IM ;Tz together with the corresponding energy spectrum. More details concerning our method can be found in Ref. [7].
Contrary to the standard NSM, the model space of our DFT-NCCI approach is not fixed. It is built step by step, by adding physically relevant low-lying particlehole (p-h) mean field configurations which correspond to self-consistent HF solutions conserving parity and signature symmetries. The basic idea is to explore all relevant single-particle Nilsson levels. Hence, in even-even nuclei, we include in the first place the ground-state configuration and low-lying aligned (|h ⊗|p or |h ⊗|p ) and antialigned (|h ⊗ |p or |h ⊗ |p ) 1p-1h configurations where |p and |p (|h and |h ) label single-particle (single-hole) states of opposite signature. In an odd-A nuclei we explore first configurations built by exciting the unpaired nucleon within a fixed signature block. In the second step we test stability of the predictions with respect to low-lying broken-pair configurations. Similar strategy is used in odd-odd nuclei. In this case, however, one has to consider both aligned and anti-aligned configurations.
In most of the applications, isospin symmetry restoration allows to reduce the configuration space in N = Z nuclei by a factor of two due to similarity between the neutron and proton 1p-1h excitations. The effect is illustrated in Fig. 1 for a representative example of 24 Mg. In the present calculation, SR ground state (g.s.) and the lowest proton (πp-πh) and neutron (νp-νh) 1p-1h HF configurations were taken into account. Energies of excited states differ by 80 keV as shown in the left column of Fig. 1. By applying the angular-momentum and isospin projections with I = 4 + , one obtains the corresponding, symmetry restored, I = 4 + 1 from the ground state and four almost doubly-degenerated excited I = 4 + states as shown in the second column of Fig. 1. The third and fourth columns show the DFT-NCCI results. The third column depicts configuration mixing calculation involving two HF configurations, the g.s. and the lowest neutron 1p-1h excitation. The fourth column shows the results of three-configurations mixing, including, in addition to the previous case, the lowest proton 1p-1h excitation. Addition of the proton 1p-1h configuration almost does not influence neither the spectrum nor the GT matrix elements for | 24 Al; 4 + 1 → | 24 Mg; 4 + i decay.

III. LOW-ENERGY SPECTRA AND GAMOW-TELLER BETA DECAY FOR A=8 NUCLEI
In this section we will investigate the structure and beta-decay properties of very light nuclei 8 Be, 8 Li, and 8 He by using the DFT-NCCI framework. The p-shell nuclei offer an excellent playground to test, in particular, a configuration-space dependence of our scheme. One should bear in mind, however, that light nuclei are weakly bound. Hence, they may exhibit a variety of phenomena which either emerge or strongly depend on the coupling to continuum [35,36] which is beyond our approach. These effects include clustering, appearance of low-lying broad resonances or particle-decay channels  that may compete with beta decay and, in turn, significantly influence beta-decay strength distribution. We shall focus on GT strength distributions of 8 He,0 + g.s. and 8 Li,2 + g.s. beta-decays paying special attention to physical interpretation of particular peaks. For the first time these peaks can be interpreted in terms of deformed Nilsson states and deformed Nilsson configurations used in the mixing. We shall also investigate the saturation of GT sum rules for the lowest 1 + , 2 + , and 3 + states in 8 Li in order to verify the completeness of the model space.

A. Configuration space in A=8 nuclei
Let us start the discussion by recalling the strategy of building configuration space. As already discussed in Sect. II, we start by calculating self-consistently the HF g.s. configuration. The s.p. Nilsson levels of both signatures (the signature symmetry is superimposed on our HF solutions) in the g.s. are used next as a guide to construct excited configurations. In the first place we include all relevant 1p-1h configurations. If needed, we extend the configuration space by adding low-lying 2p-2h configurations etc.
The neutron and proton s.p. Nilsson levels calculated for the ground states of 8 Be, 8 Li, and 8 He are shown in   8 Li is due to breaking of the time-reversal symmetry. In the case of N = Z nucleus 8 Be, we built the space by taking into account the g.s. configuration. Next, we attempt to compute all four possible (aligned and anti-aligned) neutron 1p-1h excitations among the available |N =1n z Λ Ω ± Nilsson states, where ± refers to the signature quantum number r = ±i. It appears, however, that one of them, the anti-aligned excitation to the first Nilsson s.p orbital |101 3/2 , does not converge. Eventually, in an attempt to cover the missing correlations from the s.p orbital |101 3/2 , the configuration space consisting the g.s. and three νp − νh is extended by adding three lowest 2p-2h excitations.
In the semi-magic nucleus 8 He we include in the model space the g.s. and four 1p-1h neutron excitations.
In odd-odd 8 Li, we compute first the aligned and anti-aligned g.s. configurations. Next, keeping the two neutrons paired in the lowest available signature reversed Nilsson states, we calculate several possible excited |ν ⊗ |π configurations by distributing the unpaired proton and unpaired neutron over the available s.p. states. Eventually, we break the neutron pair and attempt to compute fully unpaired configurations. These configurations are highly excited and difficult to converge. We were able to converge two such low-K axial configurations. As it will be shown below, in Sect. III, they do not influence the low-energy part of the spectrum but have quite significant impact on the GT resonance.
All configurations included in the configuration spaces of 8 Be, 8 Li, and 8 He are listed in Table I. The configurations are labeled by means of the Nilsson and signature quantum numbers, |N n z Λ Ω ± , pertaining to the unpaired valence particles. Table also includes quadrupole   TABLE I. Mean-field self-consistent configurations in 8 He, 8 Be, and 8 Li. Configurations are ordered according to their excitation energies (index i) and labeled by the asymptotic Nilsson quantum numbers and the signature of unpaired valence particles and holes. Last four columns list their properties including HF energy in MeV, quadrupole deformation parameters β2 and γ, and the total alignment j and its orientation in the intrinsic frame, respectively.
We use the Nilsson quantum numbers to label not only deformed but also near-spherical configurations. This is partly justified since some of these configurations, in particular those in 8 Li, exhibit very peculiar isovector shape effects. For example, the near-spherical configuration 6 in 8 Li is a superposition of prolate (oblate) density distribution of neutrons (protons), respectively, while the configurations 5 and 8, are superpositions of oblate (prolate) density distributions of neutrons (protons), respec-tively. The near-spherical configurations 7, 9 and 10, on the other hand, are built of near-spherical density distribution of protons 7 (neutrons 9 and 10) and deformed density distribution of neutrons (protons), respectively. Note, that these isovector shape effects may lead to different Ω-ordering of the neutron and proton s.p. levels. Beta decay from the 0 + g.s. of 8 He populates four 1 + states in 8 Li within the experimental Q β energy window. Except for the lowest 1 + state, the remaining 1 + states may decay through different particle emission channels. It makes therefore both the energy and B GT of decayingstates extremely difficult to determine experimentally. In fact, the so called experimental determination of β-decay properties of 8 He, is based on multi-parameter R-matrix formalism. The initial values of the R-matrix parameters are taken from shell-model calculations. These parameters are varied next to best fit the available data on half-life, branching ratios and energy spectra of betadelayed particles [37,40,41]. The inclusion of the particle emission channels reduces the experimental B GT to the resonant 1 + states in 8 Li and shifts their energies (centroids) as compared to the initial shell-model values with the largest impact on the GT resonance -the fourth 1 + state -which can decay through both the neutron and triton emission, as shown in Fig. 3.
In Fig. 3 we present GT strength function for the decay of 0 + g.s. of 8 He, smoothed with the Lorentzian distribution of half-width of Γ = 0.5 MeV. The peaks in the dis-tribution reflect excitation energies of the GT-populated 1 + states in 8 Li. Distribution is normalized with respect to the first 1 + state which is bound.
The DFT-NCCI model predicts a 2 + g.s. in 8 Li at the energy of −41.9 MeV, which is only ∼0.6 MeV below the experimental value. The resonant peak in the DFT-NCCI spectrum is shifted by ∼1 MeV towards higher energies as compared to experimental data, whereas the second and third peaks are roughly 2 MeV higher than the experiment. The height of the peaks is overestimated, in particular, for the GT resonance. Naturally, such a big difference cannot be explained solely by the quenching factor, which is a fortiori expected to be close to unity in light nuclei. The discrepancy is mostly due to lack of the coupling to particle-emission channels in the DFT-NCCI. In this respect, our results should be considered as an input to the R-matrix and compared directly to the shell-model input to the R-matrix. Such a comparison shows, see Fig. 3, that the results on the GT strength distribution are very similar. This is benefiting for us since our calculations are free from any adjustable parameter at variance to the shell-model results of Ref. [39]. Part of the discrepancy may be also due to the three-nucleon forces which, in ab initio NCCI calculation may become prominent, as was found in the beta decay of 14 C [42].
The DFT-NCCI approach allows for rather unique analysis of the GT strength distribution in terms of HF configurations which are the primary building blocks of the model. This is particularly useful in deformed nuclei where HF configurations, corresponding to certain p-h excitations, can be conveniently and rather intuitively labeled by Nilsson quantum numbers. The content of the n-th HF configuration in the k-th DFT-NCCI state of a given I and T z , see Eq. 2, that corresponds to the k-th peak in the spectrum is given by the following formula: Fig. 4 shows a decomposition of the wave functions of first and fourth 1 + states in 8 Li in terms of the included HF configurations. These HF configurations are the same as listed in Table I. As shown in the figure, the first peak is a mixture of the very well deformed aligned ground state, | 8 Li; ϕ 1 , with two very weakly deformed proton excitations i = 5 and 6. The lowest proton-excited configuration corresponds to oblate shape. The second is proton-excited configuration corresponding to prolate shape.
The resonance is centered around weakly deformed oblate configuration, | 8 Li; ϕ 8 , corresponding to 1p-1h aligned excitation from |ν101 3/2 + Nilsson level to its spin-orbit partner |ν101 1/2 − with drastic shape-change of neutron density. An admixture of broken-neutron-pair configuration 9, to the resonance is of the order of 25%. And finally, 20% of a resonant peak comes from the lowest proton excitation to the |π101 3/2 + Nilsson level.  Table I.
C. GT strength distribution for 8 Li to 8 Be decay The 8 Be nucleus is a cluster composed of two α particles. Its molecular structure is characterized by very elongated distribution of nuclear matter which is well accounted for by our mean-field calculation which predicts a sudden increase of deformation in 8 Be to β 2 = 0.68 as compared to its neighbours. It appears, however, that neither the HF nor the DFT-NCCI can account for all correlations associated with the clustering. The g.s. energy calculated using the DFT-NCCI equals −52.8 MeV, underestimating the experimental value by 3.7 MeV. This should be compared to the g.s. energy of 8 Li which was overestimated only by 0.6 MeV.
The low-spin positive-parity levels in 8 Be are shown in Fig. 5. Apart of experimental data and the results of DFT-NCCI, the figure includes also, for a sake of comparison, the results of ab initio NCCI calculations of Ref. [43] with JISP16 interaction. This calculation predicts the g.s. at −57.5 MeV i.e. roughly 1 MeV below the experiment. A similar ab initio NCCI calculation in Ref. [44] with NNLO chiral potential results to under bounded g.s. energy.
As shown in the figure, our calculations reproduce relatively well odd-spin states. The level of agreement is comparable, if not better, to the ab initio NCCI results. The calculated isospin doublet of 1 + states around 24 MeV may represent a doublet seen experimentally at 23 MeV. Spins for this doublet has not yet been assigned. Even-spin states, on the other hand, are systematically overbound. The lowest 2 + and 4 + states are interpreted as a members of a rotational band built atop of the 0 + g.s. Their empirical excitation energy ratio, , equals 3.75 and thus belongs to the largest over the entire nuclear chart. Our model captures quite well the ratio giving R 4/2 = 3.77. This means The panels show, counting from the left, the groups of levels having spins I π = 0 + , 1 + , 2 + , 3 + , and 4 + , respectively. Each panel shows experimental (left), DFT-NCCI (center) and ab initio NCCI (right) spectra, each normalized with respect to its g.s. energy.
that our DFT-rooted calculation reproduces change of the moment of inertia along the band well but strongly overestimates its magnitude. Too strong a quadrupole collectivity (2 + 1 and 4 + 1 are to low in energy) and missing correlations in the calculated g.s. are well seen in the GT transition strengths of ( 8 Li, 2 + g.s. ) to ( 8 Be, 2 + i ) decays. The DFT-NCCI results and experimental data are compared in Table II. The DFT-NCCI results are calculated within 1p-1h configuration space. Inclusion of 2p-2h configurations has only a marginal impact on the results. The transition strength to the 2 + 1 state is clearly overestimated by our model, in contrary to the transition strength to the 2 + 2 resonance, which seems to be underestimated. One should bear in mind, however, that the empirical strength to the resonance is uncertain and can be affected by the close-lying 2 + 3 , T = 1 state. The Gamow-Teller sum rule (GTSR) is commonly considered as a convenient indicator of the completeness of a model space. Under the assumption of completeness the GTSR reads as follows where the sum extends over all final states I f = I i + k with k = 0, ±1. The strength is defined as where M ± GT stands for reduced matrix element for the Gamow-Teller one-body operator.
In this section we shall discuss the GTSR in 8 Li calculated within the DFT-NCCI with a particular emphasis on its dependence on the configuration and model spaces. The configurations (HF solutions) are numbered and labeled as in Table I. The model space, on the other hand, is spanned by so-called natural states. These are linearly independent linear combinations of projected states Eq.( 2) having eigenvalues of the norm matrix, n i , larger than a certain externally provided cut-off parameter ε.
In Fig. 6 we show the saturation of GTSRs for the 8 Li 1 + 1 , 2 + 1 and 3 + 1 initial states versus a number of configurations used in 8 Be final state. In the calculations, the B + GT was kept fixed at a value calculated using the entire 1p-1h configuration space in 8 He and 8 Li, see Table I. It is benefiting to observe that already with five configurations in 8 Be, the calculated GTSR reaches a level of 90%. The remaining 2p-2h provide circa 5% of the strength. It is interesting to note also that the unconverged 1p-1h configuration involving |101 3/2 Nilsson orbit can be effectively replaced by 2p-2h excitations to this orbital.   In many cases the natural states, corresponding to small eigenvalues of the norm matrix, lead to instabilities in DFT-NCCI calculation. The instabilities can be controlled to some extent by applying the appropriate cut-off parameter ε. The choice of the cut-off parameter is, however, not unique. Typically, its value is correlated with discontinuities (or jumps) seen in the eigenvalues of the norm matrix plotted in ascending (or descending) order. In 8 Li, see Fig. 8, the most natural choice is ε ≈ 0.01. This choice, as shown in Fig. 7, has almost no impact on the GTSR. With increasing ε, more physical states are being removed, which, in turn, gives a rise to large variations of the GTSR versus number of configurations.

IV. GAMOW-TELLER STRENGTH DISTRIBUTION IN THE SD-MIDSHELL NUCLEUS 24 MG
In this section we present the DFT-NCCI results for the Gamow-Teller strength distribution (GTSD) in 24 Mg following the g.s. beta decay of 24 Al (I π g.s. = 4 + ). For similar analysis of the GTSD in the neighbouring nucleus 20 Ne we refer reader to our conference publication [46].
Within the conventional spherical shell-model terminology, 24 Mg is a sd-shell nucleus having eight valence particles. Mean-field calculations, on the other hand, predict 24 Mg to be well deformed system. Hence, the DFT-NCCI configuration space is built by promoting particles among the deformed s.p. Nilsson levels as shown in Fig. 9.
In order to facilitate the discussion below, let us recall that the Nilsson levels |220 1/2 , |211 3/2 , and |202 5/2 originate from the spherical d 5/2 sub-shell, the level |200 1/2 comes from the spherical s 1/2 sub-shell, and the levels |211 1/2 and |202 3/2 originate from the spherical d 3/2 sub-shell. Moreover, the levels |200 1/2 and |211 1/2 are predicted to mix through the quadrupole field when intrinsic deformation is β 2 ∼ 0.1 − 0.3, see for example the Nilsson diagram in Ref. [34].   24 Mg, labeled by the index i and asymptotic Nilsson quantum numbers of excited p-h states. Listed are also the HF binding energy EHF in MeV, excitation energy ∆E in MeV, quadrupole deformation parameters β2, and the total alignment K together with its orientation in the intrinsic frame. The configuration spaces for 24 Mg is built by following the general rules sketched in Sect. II. We include the ground state and all possible 1p-1h excitations among active N = 2 Nilsson levels shown in Fig. 9. In selfconjugated nuclei the isospin projection allows to reduce the space by considering p − h excitations of a single, say neutron, charge. Indeed, the proton s.p. levels are almost identically spaced and just pushed higher in energy due to the Coulomb interaction. Simple counting shows that there is 16 different νp − νh excitations. In addition, we include in the configuration spaces two lowest 2p-2h configurations. These configurations are added in order to test stability of the GTSD with respect to higher order excitations. All HF states included in the configuration space of 24 Mg are listed in Table III. They are prolate deformed axially symmetric configurations.
A systematic study of the GT matrix elements (GTMEs) in T = 1/2 mirror nuclei [8] allowed us to conclude that these g.s. to g.s. I π → I π matrix elements are fairly insensitive on the configuration mixing. Similar property appears to hold here as demonstrated in Fig. 10.
The figure shows calculated 4 + → 4 + GTMEs between the g.s. of 24 Al and the 4 + states in 24 Mg calculated by using the DFT-NCCI model with 17 configurations involving the g.s. and all 1p-1h excitations. Each of the panel differs by the treatment of the g.s. DFT-NCCI wave function of the parent nucleus. We start with the wave function projected from the so called aligned SR g.s. configuration (a) and enrich it by admixing first the anti-aligned g.s. (b) and, eventually, the lowest 1p-1h excitation (c). As clearly visible, the calculated GTMEs are almost insensitive to the wave function in the parent nucleus. Hence, in all calculations shown below, the correlated g.s. DFT-NCCI wave function in 24 Al includes three SR Slater determinants: the aligned g.s., the antialigned g.s., and the lowest 1p-1h excitation.

B. Gamow-Teller strength distribution
In order to pin down a specific role played by different Nilsson levels we studied diagonal I π → I π GTMEs as a function of the configuration space size in the daughter nucleus. The results are illustrated in Fig. 11. The bottom panel (a) shows the GTME distribution calculated using HF configurations involving the g.s. and all 1p-1h excitations among the Nilsson levels originating from the spherical d 5/2 sub-shell. Panel (b) contains additionally all 1p-1h configurations involving the |211 1/2 Nilsson state originating from the spherical d 3/2 sub-shell. This level plays critical role in shaping up the GTSD in 24 Mg around the excitation energy of ∼8 MeV. This ex- influence predominantly the high-energy part of GTME distribution, above the experimental Q β window. The result from the DFT-NCCI calculation, including transitions from the 4 + g.s. of 24 Al to all 3 + , 4 + , and 5 + states in 24 Mg, is shown in Fig. 12. The calculated GTSD is compared to USDb shell-model calculation and experiment [47]. Both the shell model and DFT-NCCI results are in a perfect agreement with experiment concerning position of a centroid describing 4 + → 4 + transition but the theoretical peaks are roughly two times higher compared to experiment. Moreover, in the DFT-NCCI calculations the first resonant peak, called hereafter the first GT resonance (GTR1), splits into two close-lying peaks. The second GT resonance (GTR2) seen in the DFT-NCCI calculation at high excitations energies is well above the experimental Q β -energy.
In order to reveal the nature of resonant transitions, their wave functions has been decomposed in terms of HF configurations, shown in Fig. 13. The 4 + g.s. of 24 Al is dominated by the aligned HF configuration having an unpaired proton on |202 5/2 s.p. level. This level has a large GT s.p. matrix elements with the |202 5/2 level and its spin-orbit partner, |202 3/2 , in 24 Mg. Hence, the  Table III GTR1 is due to transition to the aligned p − h excitation involving neutron particle in |202 5/2 . Its structure, however, is strongly affected by the aligned p − h excitation involving neutron particle in |211 1/2 orbit due to proximity of the |202 5/2 and |211 1/2 levels in the potential well, see Fig. 9. The near-degeneracy causes mixing between the states projected from these HF configurations since the K quantum number is not conserved.
By slightly increasing the spacing between the |202 5/2 and |211 1/2 s.p. levels, the mixing becomes reduced, which increases purity of the GTR1 wave function and further improves agreement with the data. The spacing can be increased, for example, by slight increase of the spin-orbit strength. The result of such a test study is shown in Fig. 14. The figure shows a series of DFT-NCCI calculations using the SV T Skyrme force with the spin-orbit strength increased by 10%, 20%, 30%, and 40% with respect to the original value. The configuration space in 24 Mg used in this test study was constrained to the SR g.s. and two aligned p − h excitations to |202 5/2 and |211 1/2 levels. The calculation shows that the centroid of the main peak and its height weakly depends on the spin-orbit strength at variance to the secondary peak associated with the |211 1/2 Nilsson level. Indeed, with increasing the spin-orbit strength the secondary peak moves toward higher energies and its magnitude decreases. The study suggest that the optimal spin-orbit strength should be around 25% larger than the original value. At contrast to the GTR1, the structure of GTR2 is predicted to be very pure with 80% of its wave function content coming from the aligned 1p-1h excitation involving neutron particle in the |202 3/2 Nilsson level.

V. SUPERALLOWED GAMOW-TELLER BETA DECAY OF 100 SN
In this section we have computed the superallowed Gamow-Teller beta decay of the heaviest N = Z nucleus 100 Sn and the low-spin structure, I ≤ 8, in the daughter nucleus 100 In. The transition, which proceeds from the 0 + g.s. of 100 Sn to the first 1 + 1 state in 100 In, is the fastest GT decay observed so far, see Ref. [48]. The GTME is well reproduced by the dedicated Large-Scale-Shell-Model calculations under the assumption that the axial coupling constant is quenched by 40% [48]. The aim is to test the universality of DFT-NCCI approach which, at least in principle, can be applied to calculate both the nuclear spectra and transition rates in any atomic nucleus, irrespectively of its mass and particle-number parity. Hence, in the calculation we have used exactly the same formalism as in the preceding sections. The HF configurations were calculated by using the SV SO variant of the Skyrme SV T force within the space consisting of 12 spherical harmonic oscillator shells.
The SV SO has a 20% stronger spin-orbit interaction strength compared to SV T [8]. As discussed in Ref. [8], the use of SV SO variant considerably improves calculated masses in N ∼ Z nuclei as compared to the DFT-NCCI calculations based on the SV T force.
In case of the doubly-magic 100 Sn we considered only a single mean field configuration representing its g.s. The calculated binding energy 827.7 MeV of 100 Sn is in a fair agreement with the experimental value 825.3±0.3 MeV overestimating it by 0.3%.
The structure of 100 In was computed by using nine axially-deformed mean field configurations. Counting with respect to the 100 Sn core, eight of them correspond to p-h configurations with the neutron particle occupying different s.p. states originating from the d 5/2 and g 7/2 spherical sub-shells and the proton hole being in the s.p. orbital originating from g 9/2 spherical subshell. In spherical language these are νd 5/2 ⊗ πg −1 9/2 and νf 5/2 ⊗ πg −1 9/2 configurations. The ninth configuration, involving the lowest πp-πh excitation through the Z=50 shell gap, was added to test stability against the crossshell excitations. The calculation shows that it does not affect neither the low-lying spectrum nor the GTME.
The calculated spectrum, which includes the first 1 + ≤ I π ≤ 8 + states in 100 In, is depicted in Fig. 15 and compared to the LSSM results for the first 1 + ≤ I π ≤ 6 + , taken from Ref. [48]. We refrain from showing the experimental spectrum since neither the spins nor the excitation energies are firmly assigned [48]. Theoretical spectra were normalized to the g.s. energy which is predicted to have I = 6 + by both the models. The predicted DFT-NCCI binding energy for this state is in perfect agreement with the experimental binding energy of 100 In underestimating it only by 9 keV. Concerning excited states, the DFT-NCCI model predicts the following values: 0.618 MeV (5 + 1 ), 0.637 MeV (7 + 1 ), 0.927 MeV (8 + 1 ), 1.176 MeV (4 + 1 ), 1.912 MeV (3 + 1 ), 2.194 MeV (2 + 1 ), and 4.475 MeV (1 + 1 ). The excitation energy of 8 + state may be somewhat uncertain due to too small number of knots used in the integration over the angles in the angularmomentum projection procedure. The level ordering agrees relatively well with the LSSM calculations but the excitation energies are systematically larger. Note, that the DFT-NCCI model predicts the low-lying doublet composed of 5 + 1 and 7 + 1 states at variance to the LSSM which predicts near-degeneracy of the first 5 + 1 and 4 + 1 states.
The calculated B (NCCI) GT ≈ 10.2 after using the effective axial-vector strength, g (eff) A = qg A , quenched by 40% [48] with respect to the free-neutron value of g A = −1.2701. The quenching factor q=0.6 is typical for A ≈ 100 mass region [49]. The quenched B

VI. CONCLUSIONS
In the present work we have presented pioneering calculations of the Gamow-Teller transitions by using the no-core-configuration-interaction approach based on multi-reference density functional theory treating properly the isospin and rotational symmetries. The DFT-NCCI formalism was applied to compute the GTSD in the p-shell 8 Li and 8 Be nuclei. Although the model lacks the coupling to continuum essential to describe broaden resonances and in turn beta-decay properties, we have shown that it can provide an input to theories exploring open-channel physics such as R-matrix. Shellmodel calculation applied to such approach supported experimental-data analysis. It may be of particular interest to follow the path with entirely different DFT-rooted theory. Moreover, we have demonstrated that the model is capable to capture the GTSD satisfactorily well using relatively small configuration space in the sd-shell 24 Mg as well. It was also shown that the model allows for interpretation of the GTSD peaks in terms of specific Nilsson orbits of deformed mean field, i.e. in a way that is complementary to the traditional nuclear shell model calculations.
The DFT-NCCI model can be, at least in principle, applied to calculate both the nuclear spectra and transition rates in atomic nuclei irrespectively of their mass and particle-number parity. In order to demonstrate its flexibility, the model was also applied to compute the superallowed GT beta decay in 100 Sn and the low-spin spectrum in 100 In. It is shown that, after applying the standard quenching factor of q ≈ 0.6, the calculated matrix element agrees well with the experimental value. The low-spin spectrum agrees quite well with the large-spaceshell-model calculation of Hinke [48]. Eventually, let us stress that all the results presented above were obtained without any readjustment of the model parameters to experimental data.
In conclusion, we have demonstrated that DFT-NCCI formalism can be successfully used to study nuclear beta decay in diverse set of nuclei, thus offering a complementary method to ab initio and shell model approaches. This study paves a way for more systematic studies of nuclear beta decay rates, for exploring forbidden betadecays, and for tackling the double-beta-decay process within the DFT-NCCI framework. With forbidden betadecays, the spectrum-shape method may offer valuable hints for the g A quenching puzzle [50]. Although the correspondence to experimental results was generally found to be rather good, the underlying effective interaction used to construct the EDF has its limitations. A work towards developing novel EDFs applicable for beyondmean-field calculations is under way.