Efficacy of the symmetry-adapted basis for ab initio nucleon-nucleus interactions
for light- and intermediate-mass nuclei

Ab initio descriptions of spherical and deformed nuclei up through the calcium region are now possible within a no-core shell-model framework, by utilizing emerging symplectic symmetry in nuclei. In particular, the symmetry-adapted no-core shell-model (SA-NCSM) Launey et al. (2016); Dytrych et al. (2020) uses a physically relevant symmetry-adapted (SA) basis that can achieve significantly reduced model spaces compared to the corresponding complete ultra-large model spaces, without compromising the accuracy of results for various observables Dytrych et al. (2015); Launey et al. (2016); Baker et al. (2020a). This enables the SA-NCSM to accommodate contributions from more shells and to describe heavier nuclei, such as ²⁰Ne Dytrych et al. (2020), ²¹Mg Ruotsalainen et al. (2019), ²²Mg Henderson et al. (2018), ²⁸Mg Williams et al. (2019), as well as ³²Ne and ⁴⁸Ti Launey et al. (2018, 2021). The access to higher-lying shells makes the SA basis suitable for describing nuclear reactions Launey et al. (2021), the processes that are typically studied in experiments and govern stellar evolution. Remarkable progress has been made in first-principle descriptions to scattering and nuclear reactions for light nuclei (for an overview, see Johnson et al. (2020); Quaglioni and Navrátil (2020)), including studies of elastic scattering Nollett et al. (2007); Hagen et al. (2007); Quaglioni and Navrátil (2008a); Elhatisari et al. (2015); Zhang et al. (2020); Mercenne et al. (2019); Burrows et al. (2020), photoabsorption Bacca et al. (2014), transfer Navrátil and Quaglioni (2012) and capture reactions Girlanda et al. (2010), $\alpha$ widths Kravvaris and Volya (2017); Dreyfuss et al. (2020) and resonant states Lazauskas et al. (2019), as well as thermonuclear fusion Hupin et al. (2019). In this paper, we show that expanding the reach of ab initio reactions to deformed and heavier targets is now feasible with the SA basis.

Microscopic approaches to nuclear reactions take into account nucleon degrees of freedom along with their correlations within and among the reaction fragments. Coupled with realistic inter-nucleon interactions, such as the ones derived in the framework of chiral effective field theory Bedaque and van Kolck (2002); Epelbaum et al. (2002); Entem and Machleidt (2003); Epelbaum (2006), these approaches provide ab initio calculations of reaction observables. One of the earliest and very successful microscopic approaches to nuclear reactions is the resonating-group method (RGM) Wildermuth and Tang (1977); Tang et al. (1978). In the RGM, nucleons are organized within different groups, or clusters, “resonating” through the inter-cluster exchange of nucleons. Most importantly, the cluster system is translationally invariant, and the Pauli exclusion principle is enforced by the antisymmetrization between the different clusters. All of these features make this method particularly suitable for unified descriptions of nuclear structure and reaction observables. Following the success of the Elliott model Elliott (1958a, b), showing that a leading (most deformed) $\mathrm{SU}(3)$ shell-model configuration describes reasonably well the ground-state rotational band in intermediate-mass nuclei, the RGM has been extensively used with an $\mathrm{SU}(3)$ basis and its no-core shell-model extension, the symplectic $\mathrm{Sp}(3,\mathbb{R})$ basis Hecht (1977); Hecht and Zahn (1978); Hecht and Suzuki (1982). Applications of the model with Gaussian interactions have successfully calculated $\alpha$ and ⁸Be cluster amplitudes, spectroscopic amplitudes for heavy-fragment clusters, and sub-Coulomb ¹²C+¹²C resonances Hecht et al. (1981); Hecht and Braunschweig (1978); Suzuki and Hecht (1982). The formalism has been extended by utilizing a mixed no-core shell-model $\mathrm{Sp}(3,\mathbb{R})$ plus RGM cluster basis Suzuki (1986); Suzuki and Hecht (1986, 1987), and applied to studies of the monopole and quadrupole strengths in light nuclei Suzuki (1987); Suzuki and Hara (1989), as well as the $\alpha$+¹²C cluster system Suzuki (1976a, b); Suzuki et al. (2003).

More recently, a successful first-principle description of scattering and reactions has been realized by implementing the RGM using ab initio NCSM Navrátil et al. (2009); Barrett et al. (2013) wave functions for the clusters in a formalism known as NCSM/RGM Quaglioni and Navrátil (2008b, 2009); Hupin et al. (2013) and, later, by fully combining the two approaches into the generalized ab initio cluster expansion of the no-core shell model with continuum (NCSMC) Baroni et al. (2013a, b). These methods, which have enabled predictions of nucleon Hupin et al. (2014); Calci et al. (2016), deuteron Hupin et al. (2015) and alpha Kravvaris et al. (2020) scattering off light targets, as well as polarized deuterium-tritium fusion Hupin et al. (2019) from chiral nucleon-nucleon (NN) and three—nucleon (3N) forces, are reviewed in Refs. Navrátil et al. (2016); Quaglioni and Navrátil (2020).

In addition, the Gamow shell model coupled-channel approach combines the RGM with a continuum core-valence shell approach and allows for descriptions of nuclear reactions of heavier systems Jaganathen et al. (2014); Fossez et al. (2015); Mercenne et al. (2019).

The goal of this paper is to show the efficacy of a new approach that can extend the study of ab initio reactions to medium mass nuclei by using the SA-NCSM approach Launey et al. (2016); Dytrych et al. (2020). The SA framework takes advantage of symmetries inherent to nuclei and of group theoretical algorithms, and reorganizes the model space into a physically relevant basis. This allows us to account for the relevant correlations within only a few dominant components and, hence, achieve manageable Hamiltonian matrix sizes. In this paper, we present a new formalism of the RGM, one that admits the use of the SA basis, and we demonstrate the capability and potential of the approach for light and intermediate-mass nuclei. The formalism of the SA-RGM framework is presented in Sec. II, where we discuss RGM kernels computed using the SA basis. The sensitivity of the kernels on different selected model spaces and model space sizes is discussed in Sec. III.1 for a ⁴He target and in Sec. III.2 for intermediate-mass ¹⁶O and ²⁰Ne targets. Section III.3 presents an analysis of the basis dimension and its scaling with model space sizes and particle numbers. Finally, Sec. IV outlines the conclusions.

Traditionally, the RGM adopts microscopic cluster wave functions as basis functions to describe the motion of a system of two or more clusters (see, e.g., Refs. Quaglioni and Navrátil (2009); Quaglioni et al. (2013)). We consider two nuclear fragments, or binary-cluster nuclear reactions. For two clusters $A$ and $a$, the cluster states for a channel $c$ are defined as:

where the cluster system is defined for a channel $c=\{\mathfrak{a}_{1},I_{1}^{\pi_{1}},\mathfrak{a}_{2},I_{2}^{\pi_{2}},I,\ell\}$, which is labeled by the angular momentum (spin) and parity of each of the clusters and the total spin of the clusters $I$ (the labels $\mathfrak{a}_{1}$ and $\mathfrak{a}_{2}$ denote all other quantum numbers needed to fully characterize their respective states), and the orbital momentum $l$. For particle laboratory coordinates $\vec{r}_{i}$ (used in this study), the separation distance between the center-of-mass of the two clusters is determined from ${\vec{r}}_{A,a}={1\over A}\sum_{i=1}^{A}\vec{r}_{i}-{1\over a}\sum_{i=A+1}^{A+a}\vec{r}_{i}$. The distance $r$ between the clusters defines the cluster states and the RGM kernels, as shown below, and as an integration variable facilitates the treatment of the inter-cluster antisymmetrization. Namely, the $A+a$ nuclear wave function is expressed in terms of the cluster states as

with unknown amplitudes ${{g}_{c}^{J^{\pi}}(r)}$ that are determined by solving the integral Hill-Wheeler equations for a given total energy $E$ in the $A+a$ center-of-mass frame:

Here, $H_{c^{\prime}c}(r^{\prime},r)={\bra{{\Phi}_{c^{\prime}r^{\prime}}^{J^{\pi}}}{\mathcal{A}}_{c^{\prime}}\hat{H}{\mathcal{A}}_{c}\ket{{\Phi}_{cr}^{J^{\pi}}}}$ is the Hamiltonian kernel and $N_{c^{\prime}c}(r^{\prime},r)={\bra{{\Phi}_{c^{\prime}r^{\prime}}^{J^{\pi}}}{\mathcal{A}}_{c^{\prime}}{\mathcal{A}}_{c}\ket{{\Phi}_{cr}^{J^{\pi}}}}$ is the norm kernel, where ${{\mathcal{A}}}$ is the inter-cluster antisymmetrizer. The kernels are computed by using the microscopic wave functions of the clusters that can be obtained in the ab initio NCSM and SA-NCSM. Once the kernels are computed, Eq. (3) can then be solved using the microscopic ${R}$-matrix approach Descouvemont and Baye (2010); Descouvemont (2016).

In the SA-RGM, the target nucleus of $A$ particles is described by SA-NCSM many-body wave functions. In the SA-NCSM, the many-body basis is labeled by irreducible representations (irreps) according to the group chain Elliott (1958a, b):

The ${(\lambda\,\mu)}$ quantum numbers label an $\mathrm{SU}(3)$ irrep and can be related to the average deformation through the established link with the well-known parameters, deformation ${\beta}$ and triaxiality ${\gamma}$ Castaños et al. (1988); Mustonen et al. (2018). The label ${\kappa}$ distinguishes multiple occurrences of the same orbital momentum ${L}$ in the parent irrep ${(\lambda\,\mu)}$, and ${M}_{L}$ is the projection. These quantum numbers define the spatial degrees of freedom, which can then be coupled to the intrinsic spin ($S$) to yield a good total angular momentum.

Specifically, a target state with total angular momentum and parity ${I_{1}}^{\pi_{1}}$ (and projection ${M_{1}}$) is constructed in terms of the SA basis:

where the labels are defined as $\mathfrak{b}\equiv\left\{\dots\omega_{\rm p}{\omega}_{\rm n}\rho N;{S}_{\rm p}{S}_{\rm n}\right\}$ and deformation $\omega\equiv(\lambda\,\mu)$ (it is understood that the coefficients $C$ are for given ${\pi_{1}}$, which is omitted from labeling). Protons and neutrons are labeled by p and n, respectively, and $S$ labels the intrinsic spin (“$\dots$” denotes all additional quantum numbers including $\mathfrak{a}_{1}$). The $\mathrm{SU}(3)$ outer multiplicity ${\rho}$ Draayer et al. (1989) results from the coupling of the proton deformation with that of neutrons to total deformation $\omega_{1}$. ${N}$ labels the total HO excitations above the valence-shell configuration and is truncated at a maximum value ($N\leq N_{\rm max}$), which determines the model space size.

For a single-particle projectile, the SA-RGM channel basis states can thus be defined for a channel $\{{\nu_{1};\nu}\}=\{{\omega}_{1}{\kappa}_{1}(L_{1}{S}_{1});\omega\kappa(LS)\}$ [related to channel $c$ in Eq. (19)] as:

where the $\mathrm{SU}(3)$ basis states for the target are coupled to the HO single-particle states of the projectile with $(\eta\,0)$ SU(3) quantum numbers and spin ${1\over 2}$ (we will omit the parity $\pi$ from the notation throughout the paper for simplicity). We note that the SU(3) outer multiplicity associated with the coupling of $\omega_{1}$ and $(\eta\,0)$ is 1, and hence, omitted from the notations. Remarkably, there is no dependence on the orbital momentum of the projectile, only on the shell number it occupies, $\eta$. Furthermore, the summation over ${{\mathfrak{b}}_{1}}$ implies that the SA-RGM basis requires only a part of the information present in the SA basis.

The SA-RGM basis is used to calculate the RGM kernels, which is the main computational task in RGM Quaglioni and Navrátil (2009). These include the norm kernel, which is the overlap between antisymmetrized non-orthogonal RGM basis states. It consists of a direct part (a Dirac delta function), which dominates at large relative distances, and an exchange part that takes into account the Pauli principle at short distances. The exchange norm kernel is related to the permutation operator ${{{P}}_{A,A+1}}$ that exchanges the nucleon projectile with another nucleon within the target, thereby ensuring antisymmetrization (cf. Quaglioni and Navrátil (2009)):

The exchange norm kernel in the SA-RGM basis is thus reduced to evaluating the following (similarly, for the Hamiltonian kernels):

where ${U\left[\dots\right]}$ is the $\mathrm{SU}(3)$ 6-$(\lambda\,\mu)$ recoupling coefficient Draayer and Akiyama (1973), analogous to the SU(2) 6-$j$ symbol, $\dim\,(\lambda\,\mu)={1\over 2}(\lambda+1)(\mu+1)(\lambda+\mu+2)$, and the $\mathrm{SU}(3)$ one-body density matrix elements are defined as:

where $a^{\dagger}_{(\eta\,0)lm_{l}\frac{1}{2}m_{s}}\equiv a^{\dagger}_{\eta lm_{l}\frac{1}{2}m_{s}}$ and $a_{\eta lm_{l}\frac{1}{2}m_{s}}$ creates and annihilates, respectively, a particle of spin 1/2 in the $\eta$-th HO shell, and $\tilde{a}_{(0\,\eta)l-m_{l}\frac{1}{2}-m_{s}}\equiv(-1)^{\eta+l-m_{l}+s-m_{s}}a_{\eta lm_{l}\frac{1}{2}m_{s}}$ is the annihilation SU(3) tensor operator. The matrix elements of the $\rho$ density can be quickly computed on the fly in the SA basis. Their computation can utilize an efficacious algorithm that exploits organization of SA basis states in terms of subspaces of SU(3) irreps and the factorization of spatial SU(3) and SU(2) spin degrees of freedom Dytrych et al. (2007); Launey et al. (2016), and this can be done prior to the computation of the kernels. It is notable that, as a result of the Kronecker delta function $\delta_{\nu\nu^{\prime}}$ in Eq. (11), the exchange part of the norm kernel turns out to be block-diagonal in the SA-RGM basis. The reason is that the operator ${{{P}}}$ is an SU(3) scalar and spin scalar, and therefore preserves deformation and spin of the composite $A+1$ system (note that it may change the $\omega_{1}$ deformation of the target itself).

Eq. (11) allows the kernels to be calculated, for each ${J^{\pi}M}$, through the SA-RGM channel basis of Eq. (6) that only depends on the deformation, rotation, and spin of the target $\nu_{1}$ (that is, ${\omega}_{1}{\kappa}_{1}L_{1}{S}_{1}$), and the deformation, rotation, and spin of the target-projectile system $\nu$ (that is, $\omega\kappa LS$). From this, it is clear that the SA offers two main advantages: first, calculations utilize group-theoretical algorithms that use a reduced subset of quantum numbers $\nu$ and $\nu_{1}$, and second, the number of $\mathrm{SU}(3)$ configurations in the target wave function, we find, is a manageable number when compared to the complete model-space size. This results in a manageable number of configurations for the target-projectile system based on $\mathrm{SU}(3)$ and SU(2) selection rules, namely, $\omega=\omega_{1}\times(\eta\,0)$ and $S=S_{1}\times{1\over 2}$ (for further details on scalability, see Sec. III.3).

Another advantage of the SA scheme is that the dependence on the orbital momentum $\ell$ is recovered in the very last step:

This wave function is then used in a microscopic ${R}$-matrix approach Descouvemont and Baye (2010) to calculate phase shifts and cross sections.

To study the efficacy of the SA scheme, we focus on the norm and potential kernels. For the potential kernel, we consider only the part that involves the projectile and a single nucleon in the target (similarly to Ref. Burrows et al. (2020)), that is, the potential kernel of particle-rank one, denoted here as ${V}_{c^{\prime}c}^{(1)}(r^{\prime},r)$ (cf. Quaglioni and Navrátil (2009)):

Note that the exchange of two nucleons that interact with each other is part of this kernel. We do not consider the particle-rank two potential kernel that accounts for the projectile exchanging with one nucleon in the target and interacting with another nucleon (called exchange potential kernel in Quaglioni and Navrátil (2009)). Since the goal of this study is to validate the use of the SA scheme against the use of complete model spaces, we expect that the particle-rank two potential kernel will benefit from advantages similar to those shown in the next section. The reason is that the main advantage stems from the reductions of the number of basis states needed to describe the target wavefunctions. Such a reduction ensures that one-body densities, along with the two-body densities that will be needed for the particle-rank two potential kernels, are computed for wavefunctions that span only a fraction of the complete model space (as discussed in Sec. III.3).

The derivation of the potential kernel in the SA-RGM basis follows a procedure similar to that for the norm kernel:

where $\omega_{\rm p}$, $\omega_{0}$, and $\omega_{0}^{\prime}$ denote the SU(3) rank of the operator that transforms the initial state to the final state of the projectile, target, and the $A+1$ system, respectively.

It is important to validate the use of the SA basis in the SA-RGM, by comparing selected model spaces with the corresponding complete $N_{\rm max}$, to ensure that the selection does not remove configurations relevant for the reaction processes under consideration. For this, we study single-projectile scattering off the spherical ⁴He and ¹⁶O nuclei, as well as for the deformed ²⁰Ne nucleus. We present kernels that use target ground state (g.s.) wavefunctions computed with the SA basis in a complete $N_{\rm max}$ model space (equivalent to NCSM/RGM calculations Quaglioni and Navrátil (2009)) and we compare these to the results that use wavefunctions calculated in a selected SA model space. In general, SA selections are denoted as $\langle N_{\max}^{\rm C}\rangle N_{\max}$. For example, the ${{N}_{\text{max}}=\langle 6\rangle 14}$ model space includes the complete set of excitations up to ${6\hbar\Omega}$ and selected excitations in the $8\hbar\Omega$ - $14\hbar\Omega$ subspaces, following a prescription detailed in Ref. Launey et al. (2020). This allows the mixing of all possible shapes within the complete subspaces, whereas the higher selected subspaces accommodate spatially expanded collective modes Launey et al. (2021).

In this paper, we have studied the efficacy of the new ab initio SA-RGM approach that combines the SA-NCSM and RGM frameworks. We have discussed nucleon-nucleus interactions and the use of the SA framework for ⁴He and ¹⁶O targets, as well as the intermediate-mass ²⁰Ne and ²³Mg targets feasible in the SA-NCSM. We have shown that the SU(3) selection of the model space has almost negligible effect on the SA-RGM norm and particle-rank one potential kernels that provide the input to calculations of phase shifts and cross sections. The results demonstrate that the nonnegligible components that are included in the calculations account for the correlations needed to describe the single nucleon scattering process in this mass region.

In addition, we have studied the scalability of the SA-RGM approach, showing its computational advantages that stem from the largely reduced number of SU(3) basis states needed to describe the target, as well as the manageable number of the SA-RGM basis states for the target+N system that scale polynomially with the increase in the model space size. The demonstrated efficacy of the SA basis and its scalability with particle numbers and model space dimensions opens the way to ab initio calculations up through the medium-mass region of nucleon-nucleus interactions that enter nucleon scattering and nucleon capture reactions.