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Abstract

The antisymmetrized quasi-cluster model (AQCM) was proposed to describe |$\alpha$|-cluster and |$jj$|-coupling shell models on the same footing. In this model, the cluster–shell transition is characterized by two parameters, |$R$| representing the distance between |$\alpha$| clusters and |$\varLambda$| describing the breaking of |$\alpha$| clusters, and the contribution of the spin–orbit interaction, very important in the |$jj$|-coupling shell model, can be taken into account starting with the |$\alpha$|-cluster model wave function. Not only the closure configurations of the major shells but also the subclosure configurations of the |$jj$|-coupling shell model can be described starting with the |$\alpha$|-cluster model wave functions; however, the particle–hole excitations of single particles have not been fully established yet. In this study we show that the framework of AQCM can be extended even to the states with the character of single-particle excitations. For |$^{12}\mathrm{C}$|⁠, two-particle–two-hole (2p2h) excitations from the subclosure configuration of |$0p_{3/2}$| corresponding to a BCS-like pairing are described, and these shell model states are coupled with the three |$\alpha$|-cluster model wave functions. The correlation energy from the optimal configuration can be estimated not only in the cluster part but also in the shell model part. We try to pave the way to establish a generalized description of the nuclear structure.

1. Introduction

One of the important goals of nuclear structure physics is the description of both shell and cluster aspects on the same footing. Real nuclear systems have characteristics of both aspects, and the mixing or competition between these two is an important subject for the physics of quantum many-body systems [1-3]. Our strategy is to establish a framework, which starts with the cluster model side, in contrast with standard approaches, and includes shell correlations.

As it is well known, when we take zero limit for the relative distances between clusters, the model space coincides with that of the lowest shell model configuration. This is called the SU(3) limit [4,5], and for |$N=Z$| nuclei with magic numbers of a 3D harmonic oscillator (⁠|$N=Z=2,8,20,\ldots$|⁠), the cluster model wave functions agree with the doubly closed shell configurations. Here both spin–orbit favored (⁠|$j$|-upper) and unfavored (⁠|$j$|-lower) single-particle orbits are filled and we can forget about the spin–orbit contribution. However, the spin–orbit effect exists in other cases, and in most of the conventional cluster models, this effect cannot be taken into account; the spin–orbit contribution cancels because of the assumption of the |$\alpha$| cluster that four nucleons have the same form of the spatial wave function.

To overcome this difficulty with the cluster model, we proposed an antisymmetrized quasi-cluster model (AQCM) [2,3,6–11], which enables us to describe the |$jj$|-coupling shell model states with the spin–orbit contribution starting with the cluster model wave function. In AQCM, the transition from the cluster- to the shell-model structure can be described by two parameters; |$R$| representing the distance between |$\alpha$| clusters, and |$\varLambda$|⁠, which characterizes the transition of |$\alpha$| cluster(s) to quasi-cluster(s) and quantifies the role of the spin–orbit interaction. In Ref. [2], the AQCM wave function was shown to correspond to the |$(0s_{1/2})^4 (0p_{3/2})^8$| closed shell configuration of |$^{12}\mathrm{C}$|⁠, and the strong contribution of the spin–orbit interaction was taken into account. The optimal ground state of |$^{12}\mathrm{C}$| was shown to have an intermediate character between the three |$\alpha$| clusters and shell model states. In a similar way, the subclosure configuration of |$0d_{5/2}$| was described in |$^{28}\mathrm{Si}$|⁠, and characteristic magic numbers of the |$jj$|-coupling shell model, |$28$| and |$50$|⁠, were successfully described in |$^{56}\mathrm{Ni}$| and |$^{100}\mathrm{Sn}$| [10].

However, the particle–hole excitations of single particles are not yet fully established from a cluster model point of view. The purpose of the present study is to show that the framework of AQCM can be extended even to the states with the character of single-particle excitations. The first example is |$^{12}\mathrm{C}$|⁠. In this paper, we prepare two different models and perform a variational calculation. One is the cluster model; we prepare three |$\alpha$|-cluster states, which give almost converged energy for the ground state and the second |$0^+$| state, but this is only within the isosceles triangular configurations. The other one is the |$jj$|-coupling shell model, which is prepared by using AQCM. In addition, we can prepare the states corresponding to the 2p2h excitation of the |$jj$|-coupling shell model using AQCM. The |$jj$|-coupling shell model is in principle a complete set, and we have to limit the model space. Here we focus on the time reversal excitation of two particles, which corresponds to the BCS-like parings of proton–proton, neutron–neutron, and proton–neutron. We discuss the competition between normal pairing and proton–neutron pairing effects.

So far the features of |$^{12}\mathrm{C}$| have been investigated using many different models; various cluster models [12-14], shell models including modern ab initio ones [15,16], and so on. The |$0_2^+$| state, which is known as the Hoyle state, is nicely described by the three |$\alpha$|-cluster models; however, they cannot describe detailed properties related to the |$\alpha$|-cluster breaking effect, especially in the ground state rotational band. On the other hand, in principle the shell model provides a complete set, but the cluster states are in practice difficult to describe within finite model space. Takigawa et al. have introduced a hybrid model to mix the |$\alpha$|-cluster model and |$p$| shell SU(3) basis states [17]. Our concept is based on this idea; however, we transform the cluster model wave functions directly into those of the |$jj$|-coupling shell model and try to pave the way to establish a generalized description of the nuclear structure. Also, antisymmetrized molecular dynamics (AMD) and fermionic molecular dynamics (FMD) have been successfully introduced to describe both characters of shell and cluster models [18-22]. In these models, the central positions of all the nucleons are optimized under some constraints. On the other hand, in our approach, we introduce much fewer and more controllable parameters, which allow the description of excited configurations.

Concerning the universal description of the nuclear structure, the philosophy of the present study has some similarity to that of Wildermuth and Tang [23]. There, harmonic oscillator shell model states (SU(3) states) are generated from the |$\alpha$|-cluster model. There are some technical similarities between this work and our AQCM; an imaginary part is introduced to express the boost of the nucleons, and a stretched configuration of the spherical harmonics is used to simplify the discussion. However, we do not only study harmonic oscillator shell model states; we further describe |$jj$|-coupling shell model states starting with the |$\alpha$|-cluster model, and the breaking effect of the clusters can be discussed.

In AQCM, we transform a Brink-type |$\alpha$|-cluster model wave function [24] into the |$jj$|-coupling shell model wave function by giving an imaginary part for the Gaussian center parameters. This procedure has some similarity to the idea of Fock–Bargmann space developed by Filippov et al. [25]. In Ref. [25], they discussed |$^6\mathrm{He}$|⁠, introduced hyperspherical harmonics basis states for the description of two valence neutrons outside of the |$\alpha$| core, and extracted the matrix elements of the Hamiltonian from the expectation value obtained by using a Gaussian wave packet. We also use Gaussian wave packets; however, in our study, we directly transform the wave function into the |$jj$|-coupling shell model and the breaking effect of the |$\alpha$|-cluster part can be discussed.

In |$^{12}$|C, the presence of |$D_{3h}$| symmetry, which reflects the equilateral triangular symmetry of three |$\alpha$| clusters, was recently confirmed [26]. When a system has this symmetry, both |$K^\pi = 0^+$| and |$3^-$| rotational bands are possible, and recently these bands were experimentally confirmed. All the observed states in the low-lying region of |$^{12}$|C are well described by the algebraic cluster model [27–29]. In addition, tetrahedral symmetry of four |$\alpha$| clusters is discussed in |$^{16}$|O based on this model [30]. Although the model space and the symmetry of the wave function are slightly different, our approach is expected to be a complementary one to the algebraic cluster model.

This paper is organized as follows. We describe our formulation in this work including the review of AQCM in Sect. 2. The results and discussion are given in Sect. 3. Finally, we present the conclusion and outlook in Sect. 4.

2. Formulation

2.1. AQCM wave function

As in many conventional models, the single-particle wave function of AQCM (⁠|$\phi_i$|⁠) consists of spatial (⁠|$\psi_i$|⁠), spin (⁠|$\chi_i$|⁠), and isospin (⁠|$\tau_i$|⁠) parts:

\begin{align} \phi_i=\psi_i\chi_i\tau_i. \end{align}
(1)

The spatial part of the single-particle wave function has a Gaussian shape [24],

\begin{align} \psi_i=\left(\frac{2\nu}{\pi}\right)^{\frac{3}{4}}\exp[-\nu({\boldsymbol r}-{\boldsymbol \zeta}_i)^2], \end{align}
(2)
where |$\nu$| is the width parameter. From these single-particle wave functions, the Slater determinant of an |$A$| nucleon system |$\varPsi=\mathcal{A}[\phi_1,\ldots,\phi_A]$| is constructed, where |$\mathcal{A}$| is the antisymmetrizer for all nucleons. If we give the same value for the Gaussian center parameter |${\boldsymbol \zeta}_i$| of four nucleons (spin-up proton, spin-down proton, spin-up neutron, and spin-down neutron) as in the so-called Brink model, they form an |$\alpha$| cluster, and the contribution of the spin–orbit interaction vanishes because of the antisymmetrization effect.
In Ref. [6], the AQCM wave functions were shown to describe subclosure configurations of the |$jj$|-coupling shell model. The Gaussian center parameters |$\{ {\boldsymbol \zeta}_i \}$| are complex vectors, and the imaginary parts are introduced as
\begin{align} \mathrm{Im}{\boldsymbol \zeta}_i=\varLambda{\boldsymbol e}_i^{(\mathrm{spin})}\times\mathrm{Re}{\boldsymbol \zeta}_i, \end{align}
(3)
where |${\boldsymbol e}_i^{(\mathrm{spin})}$| is a unit vector for the intrinsic-spin orientation of the |$i$|th nucleon, and |$\varLambda$| is an order parameter for the dissolution of the cluster. Note that |${\boldsymbol e}_i^{(\mathrm{spin})}$| and |$\varLambda$| are chosen as real values. By introducing |$\varLambda$|⁠, |$\alpha$| clusters are transformed to quasi-clusters with the spin–orbit contribution.

2.2. Description of subclosure configuration |$(^{12}\mathrm{C}$| case|$)$|

Before extending AQCM to describe single-particle excitations, here we review the description of the subclosure configuration for the |$^{12}\mathrm{C}$| case [2]. This part is the mathematical interpretation of AQCM and not needed in the actual calculation; however, we have to recall the important parts for further extension of the model. Since the neutron part is introduced in exactly the same way, here we concentrate on the proton part. The protons |$i=1$| and |$2$| are in a common quasi-cluster with spin-up and spin-down. Based on the original idea of Eq. (3), the Gaussian center parameters are introduced as
\begin{align} {\boldsymbol \zeta}_{i=1}=R({\boldsymbol e}_x+i\varLambda{\boldsymbol e}_y) \end{align}
(4)
and
\begin{align} {\boldsymbol \zeta}_{i=2}=R({\boldsymbol e}_x-i\varLambda{\boldsymbol e}_y), \end{align}
(5)
where |${\boldsymbol e}_x$| and |${\boldsymbol e}_y$| are unit vectors on the |$x$| and |$y$| axes, respectively. These are located on the |$x$| axis at a position with the distance |$R$| from the origin, and imaginary parts are given in the |$y$| and |$-y$| directions, since their intrinsic spins are quantized along the |$z$| axis (⁠|$z$| and |$-z$| directions). They are introduced as time reversal partners.
After setting the Gaussian center parameters, basically we can calculate the matrix elements of the Hamiltonian. Therefore, the following discussion is just the mathematical explanation that the AQCM model can describe the lowest configuration of the |$jj$|-coupling shell model including the subclosure configuration. For protons |$i=1$| and 2, the squares in the powers of the single-particle wave functions can be expanded as
\begin{align} \phi_{i=1}&=\left(\frac{2\nu}{\pi}\right)^{\frac{3}{4}}\exp[-\nu({\boldsymbol r}^2+{\boldsymbol \zeta}_1^2)+2\nu{\boldsymbol r}\cdot{\boldsymbol \zeta}_1]\chi_\uparrow\tau_1, \end{align}
(6)
\begin{align} \phi_{i=2}&=\left(\frac{2\nu}{\pi}\right)^{\frac{3}{4}}\exp[-\nu({\boldsymbol r}^2+{\boldsymbol \zeta}_2^2)+2\nu{\boldsymbol r}\cdot{\boldsymbol \zeta}_2]\chi_\downarrow\tau_2, \end{align}
(7)
where |$\chi_\uparrow$| and |$\chi_\downarrow$| stand for spin-up and down, respectively, and |$\tau_1$| and |$\tau_2$| are isospin wave functions of the protons. Here, both |$\tau_1$| and |$\tau_2$| are protons, and “1” and “2” are labels for each proton. In Eq. (6), the cross-term part in the power of the exponential can be Taylor expanded, and by substituting Eq. (4), this factor is described as
\begin{align} \exp[2\nu{\boldsymbol r}\cdot{\boldsymbol \zeta}_1]=\sum_{l=0}^\infty\frac{1}{l!}(2\nu{}Rr)^l\left(\frac{x+i\varLambda{}y}{r}\right)^l. \end{align}
(8)
For |$\varLambda=1$|⁠, by using the spherical harmonics |$Y_{lm}(\varOmega)$| and introducing the radial part of the spatial wave function |$R_{0l}(r)$|⁠, the single-particle wave function of the proton |$i=1$| can be expressed as
\begin{align} \phi_{i=1}=\left(\frac{2\nu}{\pi}\right)^{\frac{3}{4}}\sum_{l=0}^\infty\frac{(2\nu{}R)^l}{l!s_lt_l}R_{0l}(r)Y_{ll}(\varOmega)\chi_\uparrow\tau_1, \end{align}
(9)
where
\begin{align} \left(\frac{x+iy}{r}\right)^l=\frac{1}{s_l}Y_{ll}(\varOmega) \end{align}
(10)
and
\begin{align} r^l\exp[-\nu{\boldsymbol r}^2] \equiv \frac{1}{t_l}R_{0l}(r), \end{align}
(11)
and |$s_l$| and |$t_l$| are the normalization factors of |$Y_{ll}(\varOmega)$| and |$R_{0l}(r)$|⁠, respectively. The proton |$i=1$| has spin-up, and the spherical harmonics |$Y_{ll}(\varOmega)$| with spin-up has |$j_z=l+1/2$|⁠, which only couples to |$j=l+1/2$| (stretched configuration), and the spin–orbit interaction works attractively. Thus the proton |$i=1$| is described as a linear combination of |$j$|-upper orbits with |$j=l+1/2$| and |$j_z = j$|⁠,
\begin{align} \phi_{i=1}=\sum_{j=1/2}^\infty{}a_jR^{j-\frac{1}{2}} \langle {\boldsymbol r} |j,j\rangle\tau_1, \end{align}
(12)
where |$a_j$| is a coefficient for the |$\langle {\boldsymbol r} |j,j\rangle$| orbit with a separated factor of |$R^{j-\frac{1}{2}}$|⁠. Note that when |$R$| is small, small |$j$| orbits are more important. The proton |$i=2$| with spin-down is the time reversal partner of |$i=1$|⁠:
\begin{align} \phi_{i=2}=\sum_{j=1/2}^\infty{}a_{-j}R^{j-\frac{1}{2}} \langle {\boldsymbol r} |j,-j\rangle\tau_2. \end{align}
(13)
Next we have to define the Gaussian center parameters of the other four protons, |${\boldsymbol \zeta}_{i=3}$|⁠, |${\boldsymbol \zeta}_{i=4}$|⁠, |${\boldsymbol \zeta}_{i=5}$|⁠, and |${\boldsymbol \zeta}_{i=6}$|⁠. We have already shown that by introducing |${\boldsymbol \zeta}_{i=1}$| and |${\boldsymbol \zeta}_{i=2}$| as in Eqs. (4) and (5), the resultant single-particle wave functions are linear combinations of |$j$|-upper orbits with stretched configurations (the |$z$| component of the angular momentum is |$j$| or |$-j$|⁠). To describe the subclosure configuration of the |$jj$|-coupling shell model, we have to define other Gaussian center parameters of other protons so as to describe other |$j$|-upper orbits, which do not have the stretched configuration (the |$z$| components of the angular momentum are not |$j$| or |$-j$|⁠). Since the rotation does not change the total angular momentum |$j$|⁠, if we rotate both the spatial and spin parts of |$\phi_{1}$| and |$\phi_{2}$|⁠, we can prepare these orbits. The protons |$i=3$| and |$4$| are introduced as in the same quasi-cluster, Their wave functions are introduced by rotating both the spatial and spin parts of the protons |$i=1,2$| about the |$y$| axis as
\begin{align} \phi_{3}=\hat{R}(\alpha=0, \beta=\theta_1, \gamma=0)\phi_1, \end{align}
(14)
\begin{align} \phi_{4}=\hat{R}(\alpha=0, \beta=\theta_1, \gamma=0)\phi_2. \end{align}
(15)
Here, |$\alpha$|⁠, |$\beta$|⁠, |$\gamma$| are the Euler angles, and |$\hat{R}$|(⁠|$\alpha$|⁠, |$\beta$|⁠, |$\gamma$|⁠) is the rotation operator. The parameter |$\theta_1$| is a rotational angle, and |$\theta_1=2\pi/3$|⁠. Here the rotation of the spatial part of the single-particle wave functions is nothing but the rotation of Gaussian center parameters. The protons |$i=5$| and |$6$| also belong to the same quasi-cluster, but this is different from the one for |$i=3$| and |$i=4$|⁠. They are also introduced by rotating both the spatial and spin parts of the protons |$i=1,2$| about the |$y$| axis as
\begin{align} \phi_{5}=\hat{R}(\alpha=0, \beta=\theta_2, \gamma=0)\phi_1, \end{align}
(16)
\begin{align} \phi_{6}=\hat{R}(\alpha=0, \beta=\theta_2, \gamma=0)\phi_2; \end{align}
(17)
the rotation angle is |$\theta_2=4\pi/3$|⁠, and six protons give an equilateral triangular shape when |$\varLambda$| is equal to zero. These six protons are described as linear combinations of |$j$|-upper orbits, and their |$z$| components of the angular momentum are different. As a result, at the small-|$R$| limit and |$\varLambda = 1$|⁠, we can describe the lowest configuration of the |$jj$|-coupling shell model (two protons occupy |$(0s_{1/2})^2$| and four protons occupy |$(0p_{3/2})^4$|⁠) owing to the antisymmetrization effect [2,10], since smaller |$j$| orbits are more important when |$R$| is small, as shown in Eq. (9).

2.3. Extension of AQCM

Here we explain our new model, which is the extension of AQCM.

2.3.1. Total wave function

The total wave function is expressed as a linear combination of different Slater determinants based on the generator coordinate method (GCM) as
\begin{align} \varPhi^n&=\sum_kc_k^n\varPsi_k, \end{align}
(18)
\begin{align} \varPsi_k&=\hat{P}^J_{MK}\hat{P}^\pi\mathcal{A}[\phi_1\cdots\phi_{12}]_k, \end{align}
(19)
where |$\hat{P}^J_{MK}$| and |$\hat{P}^\pi$| are the angular momentum and parity projection operators. Here |$k=1,2\ldots$| is a label for different basis states. The coefficients |$\{ c_k^n \}$| are determined by solving the Hill–Wheeler equation, and |$n=1,2,\ldots$| denotes the |$n$|th excited state obtained after the diagonalization of the Hamiltonian. In this paper, we particularly pay attention to the |$0^+$| states; thus |$J=M=K=0$| and |$\pi=+$|⁠.

For the basis states, we prepare both the shell and cluster model ones. For the shell model part, we use AQCM, and in addition to the subclosure configuration of |$0p_{3/2}$|⁠, we introduce five different two-particle–two-hole (2p2h) configurations. For the cluster model space, we introduce thirty different three-|$\alpha$| configurations. In total, we superpose |$6+30=36$| basis states and diagonalize the Hamiltonian. For the width parameter |$\nu$||$(=1/2b^2)$| in Eq. (2), we take |$b=1.4\,\mathrm{fm}$|⁠.

For the shell model basis states, as shown in the previous subsection, we can transform the |$\alpha$|-cluster model wave function to the |$0p_{3/2}$| subclosure configuration of the |$jj$|-coupling shell model using AQCM. This is |$(0s_{1/2})^2(0p_{3/2})^4$| for the proton and neutron parts, and here we call it the zero-particle–zero-hole (0p0h) state. To generate this state, we take a small enough |$R$| value of |$R=0.1\,\mathrm{fm}$| in Eqs. (4) and (5). In addition, we introduce five 2p2h configurations. Four of them correspond to the normal BCS-like pairing effect of protons or neutrons; |$(0s_{1/2})^2(0p_{3/2})^2(0p_{1/2})^2$| and |$(0s_{1/2})^2(0p_{3/2})^2(0d_{5/2})^2$| are introduced for the proton part or neutron part. We further take into account the proton–neutron pairing effect. For this purpose, we prepare a basis state, where one proton and one neutron are excited from |$0p_{3/2}$| to |$0p_{1/2}$|⁠; |$(0s_{1/2})^2(0p_{3/2})^3(0p_{1/2})^1$| for both proton and neutron parts. This is also 2p2h, but each isospin (proton or neutron) part is one-particle–one-hole (1p1h). In total we introduce six configurations for the |$0^+$| states of |$^{12}\mathrm{C}$|⁠; 0p0h for both proton and neutron parts (⁠|$pn$|–0p0h), 2p2h excitation to |$0p_{1/2}$| for the proton part (⁠|$pp$|–|$p_{1/2}$|–2p2h), that for the neutron part (⁠|$nn$|–|$p_{1/2}$|–2p2h), 2p2h excitation to |$0d_{5/2}$| for the proton part (⁠|$pp$|–|$d_{5/2}$|–2p2h), that for the neutron part (⁠|$nn$|–|$d_{5/2}$|–2p2h), and 1p1h to |$0p_{1/2}$| for both proton and neutron parts (⁠|$pn$|–|$p_{1/2}$|–2p2h). The 1p1h configuration is explained in Sect. 3.3, and the 2p2h configurations are explained in Sects. 3.2 and 3.4.

For the cluster model basis states, as schematically shown in Fig. 1, the configurations are introduced with isosceles triangular shapes. The parameters |$d$| and |$h$| are the base and height of the isosceles triangle, respectively, and they are taken as |$d=1,2,\ldots,5\,\mathrm{fm}$| and |$h=1,2,\ldots,6\,\mathrm{fm}$|⁠. There are |$5\times6=30$| basis states for the cluster model side.

Fig. 1.
Schematic figure for the cluster model configurations. The red spheres show the $\alpha$ clusters.
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Schematic figure for the cluster model configurations. The red spheres show the |$\alpha$| clusters.

2.3.2. Hamiltonian

The Hamiltonian used in the present calculation is
\begin{align} \hat{H}=\hat{T}-\hat{T}_\mathrm{G}+\hat{V}_\mathrm{C}+\hat{V}_\mathrm{LS}+\hat{V}_\mathrm{Coulomb}, \end{align}
(20)
where |$\hat{T}$| is the total kinetic energy operator and |$\hat{T}_\mathrm{G}$| is the kinetic energy operator of the center of mass motion. For the central force |$\hat{V}_\mathrm{C}$|⁠, we use the Volkov No.2 force [31] given by
\begin{align} \hat{V}_\mathrm{C}=\sum_{i<j}^A\left[V_a\exp\left(-\frac{\hat{{\boldsymbol r}}_{ij}^2}{\alpha^2}\right)+V_r\exp\left(-\frac{\hat{{\boldsymbol r}}_{ij}^2}{\rho^2}\right)\right]\left[W+B\hat{P}^\sigma_{ij}-H\hat{P}^\tau_{ij}-M\hat{P}^\sigma_{ij}\hat{P}^\tau_{ij}\right]\!, \end{align}
(21)
where |$V_a=-60.65\,\mathrm{MeV}$|⁠, |$V_r=61.14\,\mathrm{MeV}$|⁠, |$\alpha=1.80\,\mathrm{fm}$|⁠, and |$\rho=1.01\,\mathrm{fm}$| are the original values. We take |$M=1-W=0.6$|⁠. Here, |$B$| and |$H$| denote the Bartlett and Heisenberg terms, which are added to remove the bound state of two neutrons. We take |$B=H=0.125$|⁠. For the spin–orbit force |$\hat{V}_\mathrm{LS}$|⁠, we use the spin–orbit part of the G3RS force [32] given by
\begin{align} \hat{V}_\mathrm{LS}=\sum_{i<j}^A\left[V_\mathrm{LS1}\exp\left(-\frac{\hat{{\boldsymbol r}}_{ij}^2}{\eta_1^2}\right)+V_\mathrm{LS2}\exp\left(-\frac{\hat{{\boldsymbol r}}_{ij}^2}{\eta_2^2}\right)\right]\hat{P}_{ij}(^3O)\hat{{\boldsymbol L}}_{ij}\cdot\hat{{\boldsymbol S}}_{ij}, \end{align}
(22)
where |$\eta_1=0.447\,\mathrm{fm}$| and |$\eta_2=0.6\,\mathrm{fm}$| are the original values. The coefficients |$V_\mathrm{LS1}=-V_\mathrm{LS2}=1600\,\mathrm{MeV}$| are determined to give a reasonable energy for the ground state in |$^{12}\mathrm{C}$|⁠. Also, the validity of |$V_\mathrm{LS1}=-V_\mathrm{LS2}=1600\,\mathrm{MeV}$| is checked in Ref. [2]. The operator |$\hat{V}_\mathrm{Coulomb}$| is the Coulomb potential for protons.

3. Results and discussion

As already seen, the closure configurations of the major shells can be described by conventional |$\alpha$|-cluster models, and subclosure configurations of the |$jj$|-coupling shell model can be described by AQCM. Here we extend AQCM. First we discuss the AQCM wave functions with general |$\varLambda$| values and next show how to describe particle–hole excitations. For |$^{12}\mathrm{C}$|⁠, single-particle excitations from |$0p_{3/2}$| to |$0p_{1/2}$| and |$0d_{5/2}$| are introduced for the proton part, neutron part, and proton–neutron part, and the effect of BCS-like pairing is incorporated. Finally these shell-model-like wave functions are coupled with the three |$\alpha$|-cluster wave functions.

3.1. AQCM wave functions with general |$\varLambda$| values

We have already discussed that |$\varLambda = 0$| corresponds to |$\alpha$|-cluster states and |$\varLambda=1$| with small |$R$| corresponds to the |$jj$|-coupling shell model states. However, the discussion for general |$\varLambda$| values (⁠|$\varLambda\neq 0,1$|⁠) is insufficient. In this subsection, we investigate the feature of the AQCM wave functions with general |$\varLambda$| values in |$^{12}\mathrm{C}$|⁠.

Using the relations for the spherical harmonics
\begin{align} \frac{x+iy}{r}&=\frac{1}{s_1}Y_{11}(\varOmega), \end{align}
(23)
\begin{align} \frac{x-iy}{r}&=\frac{1}{s_{-1}}Y_{1-1}(\varOmega)=-\frac{1}{s_1}Y_{1-1}(\varOmega), \end{align}
(24)
and Eqs. (8) and (11), the single-particle wave function of the proton |$i=1$| (Eq. (6)) with general |$\varLambda$| becomes
\begin{align} \phi_{i=1}&=\left(\frac{2\nu}{\pi}\right)^{\frac{3}{4}}e^{-\nu{}R^2(1-\varLambda^2)} \nonumber \\ &\quad\times\left[\frac{1}{s_0t_0}R_{00}(r)Y_{00}(\varOmega)+\frac{2\nu{}R}{s_1t_1}R_{01}(r)\left(\frac{1+\varLambda}{2}Y_{11}(\varOmega)-\frac{1-\varLambda}{2}Y_{1-1}(\varOmega)\right)+\mathcal{O}(R^2)\right]\chi_\uparrow\tau_1. \end{align}
(25)
We introduce |$jj$|-coupling bases using the Clebsch–Gordan coefficients:
\begin{align} \left|j=\frac{3}{2},j_z=\frac{1}{2}\right\rangle&=R_{01}(r)\left(\frac{1}{\sqrt{3}}Y_{11}(\varOmega)\chi_\downarrow+\sqrt{\frac{2}{3}}Y_{10}(\varOmega)\chi_\uparrow\right)\!, \end{align}
(26)
\begin{align} \left|j=\frac{3}{2},j_z=-\frac{1}{2}\right\rangle&=R_{01}(r)\left(\sqrt{\frac{2}{3}}Y_{10}(\varOmega)\chi_\downarrow+\frac{1}{\sqrt{3}}Y_{1-1}(\varOmega)\chi_\uparrow\right)\!, \end{align}
(27)
\begin{align} \left|j=\frac{1}{2},j_z=\frac{1}{2}\right\rangle&=R_{01}(r)\left(\sqrt{\frac{2}{3}}Y_{11}(\varOmega)\chi_\downarrow-\frac{1}{\sqrt{3}}Y_{10}(\varOmega)\chi_\uparrow\right)\!, \end{align}
(28)
\begin{align} \left|j=\frac{1}{2},j_z=-\frac{1}{2}\right\rangle&=R_{01}(r)\left(\frac{1}{\sqrt{3}}Y_{10}(\varOmega)\chi_\downarrow-\sqrt{\frac{2}{3}}Y_{1-1}(\varOmega)\chi_\uparrow\right)\!. \end{align}
(29)
Thus the single-particle wave function of proton |$i=1$| with general |$\varLambda$| becomes
\begin{align} \phi_{i=1}&=e^{-\nu{}R^2(1-\varLambda^2)} \nonumber \\ &\quad\times\left[\vphantom{\sqrt{\frac{2}{3}}}a_{1/2}\left|s\frac{1}{2}\frac{1}{2}\right\rangle\right. \nonumber \\ &\qquad\quad \left.+a_{3/2}R\left(\frac{1+\varLambda}{2}\left|p\frac{3}{2}\frac{3}{2}\right\rangle-\frac{1-\varLambda}{2}\left(\frac{1}{\sqrt{3}}\left|p\frac{3}{2}\frac{-1}{2}\right\rangle-\sqrt{\frac{2}{3}}\left|p\frac{1}{2}\frac{-1}{2}\right\rangle\right)\right)+\mathcal{O}(R^2)\right]\tau_1, \end{align}
(30)
where |$s$| and |$p$| are indexes to distinguish the |$s$| and |$p$| orbits. Similarly, the single-particle wave function of the proton |$i=2$|⁠, which is the time reversal of |$i=1$|⁠, becomes
\begin{align} \phi_{i=2}&=e^{-\nu{}R^2(1-\varLambda^2)} \nonumber \\ &\quad\times\left[\sqrt{\frac{2}{3}}a_{-1/2}\left|s\frac{1}{2}\frac{-1}{2}\right\rangle\right. \nonumber \\ &\qquad\quad \left.+a_{-3/2}R\left(-\frac{1-\varLambda}{2}\left(\frac{1}{\sqrt{3}}\left|p\frac{3}{2}\frac{1}{2}\right\rangle+\sqrt{\frac{2}{3}}\left|p\frac{1}{2}\frac{1}{2}\right\rangle\right)+\frac{1+\varLambda}{2}\left|p\frac{3}{2}\frac{-3}{2}\right\rangle\right)+\mathcal{O}(R^2)\right]\tau_2. \end{align}
(31)

The single-particle wave functions of protons |$i=3$|–6 are generated by multiplying the rotational operators |$\hat{R}(0,2\pi/3,0)$| or |$\hat{R}(0,4\pi/3,0)$| for the single-particle wave functions of protons |$i=1,2$| as in the original AQCM [2]:

\begin{align} \phi_{i=3}&=e^{-\nu{}R^2(1-\varLambda^2)} \nonumber \\ &\quad\times\left[a_{1/2}\left(\frac{1}{2}\left|s\frac{1}{2}\frac{1}{2}\right\rangle+\frac{\sqrt{3}}{2}\left|s\frac{1}{2}\frac{-1}{2}\right\rangle\right)\right. \nonumber \\ &\qquad\quad+a_{3/2}R\left(-\frac{1-2\varLambda}{8}\left|p\frac{3}{2}\frac{3}{2}\right\rangle+\frac{1+2\varLambda}{8}\left|p\frac{3}{2}\frac{1}{2}\right\rangle-\sqrt{2}\frac{1-\varLambda}{4}\left|p\frac{1}{2}\frac{1}{2}\right\rangle\right. \nonumber \\ &\qquad\quad\left.\left.+\frac{1}{\sqrt{3}}\frac{7+2\varLambda}{8}\left|p\frac{3}{2}\frac{-1}{2}\right\rangle+\sqrt{\frac{2}{3}}\frac{1-\varLambda}{4}\left|p\frac{1}{2}\frac{-1}{2}\right\rangle+\sqrt{3}\frac{1+2\varLambda}{8}\left|p\frac{3}{2}\frac{-3}{2}\right\rangle\right)+\mathcal{O}(R^2)\right]\tau_3, \end{align}
(32)
\begin{align} \phi_{i=4}&=e^{-\nu{}R^2(1-\varLambda^2)} \nonumber \\ &\quad\times\left[a_{-1/2}\left(-\frac{\sqrt{3}}{2}\left|s\frac{1}{2}\frac{1}{2}\right\rangle+\frac{1}{2}\left|s\frac{1}{2}\frac{-1}{2}\right\rangle\right)\right. \nonumber \\ &\qquad\quad+a_{-3/2}R\left(-\sqrt{3}\frac{1+2\varLambda}{8}\left|p\frac{3}{2}\frac{3}{2}\right\rangle+\frac{1}{\sqrt{3}}\frac{7+2\varLambda}{8}\left|p\frac{3}{2}\frac{1}{2}\right\rangle-\sqrt{\frac{2}{3}}\frac{1-\varLambda}{4}\left|p\frac{1}{2}\frac{1}{2}\right\rangle\right. \nonumber \\ &\qquad\quad\left.\left.-\frac{1+2\varLambda}{8}\left|p\frac{3}{2}\frac{-1}{2}\right\rangle-\sqrt{2}\frac{1-\varLambda}{4}\left|p\frac{1}{2}\frac{-1}{2}\right\rangle-\frac{1-2\varLambda}{8}\left|p\frac{3}{2}\frac{-3}{2}\right\rangle\vphantom{\sqrt{\frac{2}{3}}}\right)+\mathcal{O}(R^2)\right]\tau_4, \end{align}
(33)
\begin{align} \phi_{i=5}&=e^{-\nu{}R^2(1-\varLambda^2)} \nonumber \\ &\quad\times\left[a_{1/2}\left(-\frac{1}{2}\left|s\frac{1}{2}\frac{1}{2}\right\rangle+\frac{\sqrt{3}}{2}\left|s\frac{1}{2}\frac{-1}{2}\right\rangle\right)\right. \nonumber \\ &\qquad\quad+a_{3/2}R\left(\frac{1-2\varLambda}{8}\left|p\frac{3}{2}\frac{3}{2}\right\rangle+\frac{1+2\varLambda}{8}\left|p\frac{3}{2}\frac{1}{2}\right\rangle-\sqrt{2}\frac{1-\varLambda}{4}\left|p\frac{1}{2}\frac{1}{2}\right\rangle\right. \nonumber \\ &\qquad\quad\left.\left.-\frac{1}{\sqrt{3}}\frac{7+2\varLambda}{8}\left|p\frac{3}{2}\frac{-1}{2}\right\rangle-\sqrt{\frac{2}{3}}\frac{1-\varLambda}{4}\left|p\frac{1}{2}\frac{-1}{2}\right\rangle+\sqrt{3}\frac{1+2\varLambda}{8}\left|p\frac{3}{2}\frac{-3}{2}\right\rangle\right)+\mathcal{O}(R^2)\right]\tau_5, \end{align}
(34)
\begin{align} \phi_{i=6}&=e^{-\nu{}R^2(1-\varLambda^2)} \nonumber \\ &\quad\times\left[a_{-1/2}\left(-\frac{\sqrt{3}}{2}\left|s\frac{1}{2}\frac{1}{2}\right\rangle-\frac{1}{2}\left|s\frac{1}{2}\frac{-1}{2}\right\rangle\right)\right. \nonumber \\ &\qquad\quad+a_{-3/2}R\left(-\sqrt{3}\frac{1+2\varLambda}{8}\left|p\frac{3}{2}\frac{3}{2}\right\rangle-\frac{1}{\sqrt{3}}\frac{7+2\varLambda}{8}\left|p\frac{3}{2}\frac{1}{2}\right\rangle+\sqrt{\frac{2}{3}}\frac{1-\varLambda}{4}\left|p\frac{1}{2}\frac{1}{2}\right\rangle\right. \nonumber \\ &\qquad\quad\left.\left.-\frac{1+2\varLambda}{8}\left|p\frac{3}{2}\frac{-1}{2}\right\rangle-\sqrt{2}\frac{1-\varLambda}{4}\left|p\frac{1}{2}\frac{-1}{2}\right\rangle+\frac{1-2\varLambda}{8}\left|p\frac{3}{2}\frac{-3}{2}\right\rangle\vphantom{\sqrt{\frac{2}{3}}}\right)+\mathcal{O}(R^2)\right]\tau_6. \end{align}
(35)
Thus the proton part of the wave function becomes
\begin{align} \varPsi_p&=\mathcal{A}[\phi_1,\ldots,\phi_6] \nonumber \\ &=-\frac{9}{16}e^{-6\nu{}R^2(1-\varLambda^2)}(a_{1/2}a_{-3/2}-a_{-1/2}a_{3/2})^2a_{3/2}a_{-3/2}R^4 \nonumber \\ &\quad\times\mathcal{A}\left[\left|s\frac{1}{2}\frac{1}{2}\right\rangle,\left|s\frac{1}{2}\frac{-1}{2}\right\rangle, -\frac{1}{2}\left|p\frac{3}{2}\frac{3}{2}\right\rangle+\frac{\sqrt{3}}{2}\left|p\frac{3}{2}\frac{-1}{2}\right\rangle, \frac{\sqrt{3}}{2}\left|p\frac{3}{2}\frac{1}{2}\right\rangle-\frac{1}{2}\left|p\frac{3}{2}\frac{-3}{2}\right\rangle,\right. \nonumber \\ &\qquad\qquad \left.-\frac{1-\varLambda}{\sqrt{2}}\left|p\frac{1}{2}\frac{1}{2}\right\rangle+\frac{1+2\varLambda}{\sqrt{3}}\left|p\frac{3}{2}\frac{-3}{2}\right\rangle, -\frac{1+2\varLambda}{\sqrt{3}}\left|p\frac{3}{2}\frac{3}{2}\right\rangle-\frac{1-\varLambda}{\sqrt{2}}\left|p\frac{1}{2}\frac{-1}{2}\right\rangle\vphantom{\frac{\sqrt{3}}{2}}\right] \nonumber \\ &\quad+\mathcal{O}(R^5), \end{align}
(36)
where we omit the isospin part of the wave function. The neutron part is introduced in exactly the same way. In Eq. (36), two protons occupy |$0s$| orbits and the others are described by the superposition of four different |$0p$| orbits. We can easily check that |$\varLambda=1$| gives the |$(0s_{1/2})^2(0p_{3/2})^4$| configuration with a small enough |$R$| value as in the original AQCM [2].

3.2. Description of 2p2h |$(p_{1/2}-2p2h$)$|

We show that the AQCM wave functions describe the 2p2h excitations of protons from the subclosure configuration to |$0p_{1/2}$| [|$(0s_{1/2})^4(0p_{3/2})^6(0p_{1/2})^2$|] by using a different |$\varLambda$| parameter for each quasi-|$\alpha$| cluster (exactly the same procedure can also be applied to the neutron part). We introduce two |$\varLambda$| parameters, |$\varLambda_a$| for protons |$i=$|1–2 and |$\varLambda_b$| for protons |$i=3$|–6. After the antisymmetrization, the proton part of the wave function becomes
\begin{align} \varPsi_p&=\mathcal{A}[\phi_1,\ldots,\phi_6] \nonumber \\ &=e^{-2\nu{}R^2(3-\varLambda_a^2-2\varLambda_b^2)}a_{1/2}a_{-1/2}a_{3/2}^2a_{-3/2}^2R^4 \nonumber \\ &\quad\times\mathcal{A}\left[\left|s\frac{1}{2}\frac{1}{2}\right\rangle,\left|s\frac{1}{2}\frac{-1}{2}\right\rangle, \right. \nonumber \\ &\qquad -\frac{3+2(\varLambda_a-\varLambda_b)}{8}\left|p\frac{3}{2}\frac{3}{2}\right\rangle+\frac{1}{\sqrt{3}}\frac{9-2(\varLambda_a-\varLambda_b)}{8}\left|p\frac{3}{2}\frac{-1}{2}\right\rangle+\sqrt{\frac{2}{3}}\frac{\varLambda_a-\varLambda_b}{4}\left|p\frac{1}{2}\frac{-1}{2}\right\rangle, \nonumber \\ &\qquad\quad \frac{1}{\sqrt{3}}\frac{9-2(\varLambda_a-\varLambda_b)}{8}\left|p\frac{3}{2}\frac{1}{2}\right\rangle-\sqrt{\frac{2}{3}}\frac{\varLambda_a-\varLambda_b}{4}\left|p\frac{1}{2}\frac{1}{2}\right\rangle-\frac{3+2(\varLambda_a-\varLambda_b)}{8}\left|p\frac{3}{2}\frac{-3}{2}\right\rangle, \nonumber \\ &\qquad -\frac{1-2(\varLambda_a+\varLambda_b)}{4}\left|p\frac{3}{2}\frac{1}{2}\right\rangle-\frac{2-(\varLambda_a+\varLambda_b)}{\sqrt{2}}\left|p\frac{1}{2}\frac{1}{2}\right\rangle+\sqrt{3}\frac{3+2(\varLambda_a+\varLambda_b)}{4}\left|p\frac{3}{2}\frac{-3}{2}\right\rangle, \nonumber \\ &\qquad \left.-\sqrt{3}\frac{3+2(\varLambda_a+\varLambda_b)}{4}\left|p\frac{3}{2}\frac{3}{2}\right\rangle+\frac{1-2(\varLambda_a+\varLambda_b)}{4}\left|p\frac{3}{2}\frac{-1}{2}\right\rangle-\frac{2-(\varLambda_a+\varLambda_b)}{\sqrt{2}}\left|p\frac{1}{2}\frac{-1}{2}\right\rangle\right] \nonumber \\ &\quad+\mathcal{O}(R^5). \end{align}
(37)
Here, we have used the time reversal relations, |$a_{-1/2}=a_{1/2}$| and |$a_{-3/2}=-a_{3/2}$|⁠, and omitted the isospin part. As is easily recognized, the conditions |$3+2(\varLambda_a-\varLambda_b)=0$| and |$3+2(\varLambda_a+\varLambda_b)=0$|⁠, namely |$(\varLambda_a,\varLambda_b)=(-3/2,0)$|⁠, allow us to remove the components of |$|p,3/2,3/2\rangle$| and |$|p,3/2,-3/2\rangle$|⁠. In this case, the proton part of the wave function (Eq. (37)) becomes
\begin{align} \varPsi_p&=6e^{-\frac{3}{2}\nu{}R^2}a_{1/2}a_{-1/2}a_{3/2}^2a_{-3/2}^2R^4 \nonumber \\ &\quad\times\mathcal{A}\left[\left|s\frac{1}{2}\frac{1}{2}\right\rangle,\left|s\frac{1}{2}\frac{-1}{2}\right\rangle,\left|p\frac{3}{2}\frac{1}{2}\right\rangle,\left|p\frac{3}{2}\frac{-1}{2}\right\rangle,\left|p\frac{1}{2}\frac{1}{2}\right\rangle,\left|p\frac{1}{2}\frac{-1}{2}\right\rangle\right]+\mathcal{O}(R^5), \end{align}
(38)
and it coincides with the |$(0s_{1/2})^2(0p_{3/2})^2(0p_{1/2})^2$| configuration (⁠|$p_{1/2}$|–2p2h) at the limit of |$R\to0$|⁠. By imposing other conditions, we can change the hole configuration. The conditions |$9-2(\varLambda_a-\varLambda_b)=0$| and |$1-2(\varLambda_a+\varLambda_b)=0$|⁠, namely |$(\varLambda_a,\varLambda_b)=(5/2,-2)$|⁠, allow us to remove the components of |$|p,3/2,1/2\rangle$| and |$|p,3/2,-1/2\rangle$|⁠. Thus the proton part of the wave function (Eq. (37)) becomes
\begin{align} \varPsi_p&=\frac{27}{8}e^{\frac{25}{2}\nu{}R^2}a_{1/2}a_{-1/2}a_{3/2}^2a_{-3/2}^2R^4 \nonumber \\ &\quad\times\mathcal{A}\left[\left|s\frac{1}{2}\frac{1}{2}\right\rangle,\left|s\frac{1}{2}\frac{-1}{2}\right\rangle,\left|p\frac{3}{2}\frac{3}{2}\right\rangle,\left|p\frac{3}{2}\frac{-3}{2}\right\rangle,\left|p\frac{1}{2}\frac{1}{2}\right\rangle,\left|p\frac{1}{2}\frac{-1}{2}\right\rangle\right]+\mathcal{O}(R^5), \end{align}
(39)
and it coincides with the |$(0s_{1/2})^2(0p_{3/2})^2(0p_{1/2})^2$| configuration (⁠|$p_{1/2}$|–2p2h) at the limit of |$R\to0$|⁠. This is also the 2p2h excitation from |$0p_{3/2}$| to |$0p_{1/2}$|⁠, but the configuration is slightly different from the previous case.

3.3. Description of 1p1h

Next we further improve the AQCM wave function to describe the 1p1h excitations from the subclosure configuration. For this purpose, we generalize the Gaussian center parameter in Eq. (2) as
\begin{align} {\boldsymbol \zeta}_i=R(a_i{\boldsymbol e}_x+ib_i{\boldsymbol e}_y+c_i{\boldsymbol e}_z), \end{align}
(40)
where |$R$| is a real number with a dimension of length, and |$a_i$|⁠, |$b_i$|⁠, and |$c_i$| are dimensionless real numbers. Here |${\boldsymbol e}_x$|⁠, |${\boldsymbol e}_y$|⁠, and |${\boldsymbol e}_z$| are unit vectors for the |$x$|⁠, |$y$|⁠, and |$z$| axes, respectively. The spin orientation is no longer fixed along the |$z$| axis, and the spin wave function |$\chi_i$| is more generalized as
\begin{align} \chi_i=\cos\frac{\beta_i}{2}\chi_\uparrow+\sin\frac{\beta_i}{2}\chi_\downarrow, \end{align}
(41)
where |$\beta_i$| is taken as a real parameter for simplicity. If |$(a_i,b_i,c_i)=(1,\varLambda,0)$| and |$\beta_i=0$| are satisfied, Eq. (40) coincides with the original AQCM in Eq. (4).
The single-particle wave function is expanded with the |$jj$|-coupling shell model bases, |$\{|j,j_z\rangle\}$|⁠, as in the previous subsections. Using the relation
\begin{align} \frac{a_ix+ib_iy+c_iz}{r}&=A_i\frac{x+iy}{r}+B_i\frac{x-iy}{r}+C_i\frac{\sqrt{2}z}{r} \nonumber \\ &=\frac{1}{s_1}\left(A_iY_{11}(\varOmega)-B_iY_{1-1}(\varOmega)-C_iY_{10}(\varOmega)\right)\!, \end{align}
(42)
and Eqs. (11), (26)–(29), this generalized single-particle wave function becomes
\begin{align} \phi_i&=e^{-\nu{}R^2(a_i^2-b_i^2+c_i^2)} \nonumber \\ &\quad\times\left[a_{1/2}\left(\cos\frac{\beta_i}{2}\left|s\frac{1}{2}\frac{1}{2}\right\rangle+\sin\frac{\beta_i}{2}\left|s\frac{1}{2}\frac{-1}{2}\right\rangle\right)\right. \nonumber \\ &\qquad\quad+a_{3/2}R\left(A_i\cos\frac{\beta_i}{2}\left|p\frac{3}{2}\frac{3}{2}\right\rangle+\frac{1}{\sqrt{3}}\left(A_i\sin\frac{\beta_i}{2}-\sqrt{2}C_i\cos\frac{\beta_i}{2}\right)\left|p\frac{3}{2}\frac{1}{2}\right\rangle\right. \nonumber \\ &\qquad\quad+\frac{1}{\sqrt{3}}\left(\sqrt{2}A_i\sin\frac{\beta_i}{2}+C_i\cos\frac{\beta_i}{2}\right)\left|p\frac{1}{2}\frac{1}{2}\right\rangle-\frac{1}{\sqrt{3}}\left(\sqrt{2}C_i\sin\frac{\beta_i}{2}+B_i\cos\frac{\beta_i}{2}\right)\left|p\frac{3}{2}\frac{-1}{2}\right\rangle \nonumber \\ &\qquad\quad\left.\left.-\frac{1}{\sqrt{3}}\left(C_i\sin\frac{\beta_i}{2}-\sqrt{2}B_i\cos\frac{\beta_i}{2}\right)\left|p\frac{1}{2}\frac{-1}{2}\right\rangle-B_i\sin\frac{\beta_i}{2}\left|p\frac{3}{2}\frac{-3}{2}\right\rangle\right)+\mathcal{O}(R^2)\right]\tau_i, \end{align}
(43)
where |$A_i=(a_i+b_i)/2$|⁠, |$B_i=(a_i-b_i)/2$|⁠, and |$C_i=c_i/\sqrt{2}$|⁠, respectively. The single-particle wave function has all four components of |$0p_{3/2}$| orbits and both components of |$0p_{1/2}$| orbits with different coefficients.

Now a proton has all six components of |$0p$| orbits, and we remove some of them by imposing conditions. If |$A_i\sin(\beta_i/2)-\sqrt{2}C_i\cos(\beta_i/2)=0$| is satisfied, we can eliminate the component of |$|p,3/2,1/2\rangle$|⁠. Similarly, if |$C_i\sin(\beta_i/2)-\sqrt{2}B_i\cos(\beta_i/2)=0$| is satisfied, the component of |$|p,1/2,-1/2\rangle$| vanishes. Thus, if |$A_i\sin(\beta_i/2)-\sqrt{2}C_i\cos(\beta_i/2)=0$| and |$C_i\sin(\beta_i/2)-\sqrt{2}B_i\cos(\beta_i/2)=0$| are simultaneously satisfied, the single-particle wave function does not have the |$|p,3/2,1/2\rangle$| and |$|p,1/2,-1/2\rangle$| components. This was for one proton; however, if all the protons satisfy the same conditions, the proton part of the wave function also does not have the components of |$|p,3/2,1/2\rangle$| and |$|p,1/2,-1/2\rangle$|⁠. This is nothing but 1p1h excitation to |$0p_{1/2}$|⁠.

In the following part, we simplify the conditions to describe 1p1h. The conditions |$A_i\sin(\beta_i/2)-\sqrt{2}C_i\cos(\beta_i/2)=0$| and |$C_i\sin(\beta_i/2)-\sqrt{2}B_i\cos(\beta_i/2)=0$| are equivalent to |$\tan(\beta_i/2)=\sqrt{2}C_i/A_i=\sqrt{2}B_i/C_i$|⁠. Substituting |$A_i=(a_i+b_i)/2$|⁠, |$B_i=(a_i-b_i)/2$|⁠, and |$C_i=c_i/\sqrt{2}$|⁠, the conditions become |$a_i^2-b_i^2=2c_i^2$|⁠. As |$a_i$|⁠, |$b_i$|⁠, and |$c_i$| are real numbers, another condition of |$|a_i|\ge|b_i|$| is required. As a result, the conditions for the spin part of the wave function become |$\tan(\beta_i/2)=\mathrm{sign}(a_ic_i)\sqrt{2(a_i-b_i)/(a_i+b_i)}$|⁠, where |$\mathrm{sign}(\xi)=\xi/|\xi|$|⁠. As an example that realizes the conditions |$a_i^2-b_i^2=2c_i^2$| and |$\tan(\beta_i/2)=\mathrm{sign}(a_ic_i)\sqrt{2(a_i-b_i)/(a_i+b_i)}$|⁠, we show a set for the six protons in Table 1. The center of mass of the system is set to the origin. The parameters for protons |$i=1$| and |$2$| are equivalent to the ones for the original AQCM wave function [2]. We can confirm that the presence of a real parameter |$\varLambda$| avoids the risk that some of the single-particle orbits are not linearly independent. For the range of |$\varLambda$|⁠, only |$0<\varLambda<1$| is allowable. Using these parameters, the wave function of the proton part |$\varPsi_p=\mathcal{A}[\phi_1,\ldots,\phi_6]$| describes the |$(0s_{1/2})^2(0p_{3/2})^3(0p_{1/2})^1$| configuration at the limit of |$R\to0$|⁠.

Table 1.

Example of |$\{(a_i,ib_i,c_i)\}$| and coefficients for the spin wave functions for the six protons (⁠|$i=$|1–6), which describes the 1p1h configuration. The required conditions are |$a_i^2-b_i^2=2c_i^2$| and |$\tan(\beta_i/2)=\mathrm{sign}(a_ic_i)\sqrt{2(a_i-b_i)/(a_i+b_i)}$|⁠. The center of mass of the system is set to the origin. The parameters |$a_i$|⁠, |$b_i$|⁠, and |$c_i$| are introduced in Eq. (40), and the parameter |$\beta_i$| is introduced in Eq. (41).

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|$a_i$||$1$||$1$||$-\frac{1}{2}$||$-\frac{1}{2}$||$-\frac{1}{2}$||$-\frac{1}{2}$|
|$ib_i$||$i$||$-i$||$\frac{1}{2}i\varLambda$||$-\frac{1}{2}i\varLambda$||$\frac{1}{2}i\varLambda$||$-\frac{1}{2}i\varLambda$|
|$c_i$||$0$||$0$||$\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$-\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$-\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$|
coefficient for |$\chi_\uparrow$| (⁠|$\cos\frac{\beta_i}{2}$|⁠)|$1$||$0$||$\sqrt{\frac{1-\varLambda}{3+\varLambda}}$||$\sqrt{\frac{1+\varLambda}{3-\varLambda}}$||$\sqrt{\frac{1-\varLambda}{3+\varLambda}}$||$\sqrt{\frac{1+\varLambda}{3-\varLambda}}$|
coefficient for |$\chi_\downarrow$| (⁠|$\sin\frac{\beta_i}{2}$|⁠)|$0$||$1$||$-\sqrt{2}\sqrt{\frac{1+\varLambda}{3+\varLambda}}$||$-\sqrt{2}\sqrt{\frac{1-\varLambda}{3-\varLambda}}$||$\sqrt{2}\sqrt{\frac{1+\varLambda}{3+\varLambda}}$||$\sqrt{2}\sqrt{\frac{1-\varLambda}{3-\varLambda}}$|
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|$a_i$||$1$||$1$||$-\frac{1}{2}$||$-\frac{1}{2}$||$-\frac{1}{2}$||$-\frac{1}{2}$|
|$ib_i$||$i$||$-i$||$\frac{1}{2}i\varLambda$||$-\frac{1}{2}i\varLambda$||$\frac{1}{2}i\varLambda$||$-\frac{1}{2}i\varLambda$|
|$c_i$||$0$||$0$||$\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$-\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$-\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$|
coefficient for |$\chi_\uparrow$| (⁠|$\cos\frac{\beta_i}{2}$|⁠)|$1$||$0$||$\sqrt{\frac{1-\varLambda}{3+\varLambda}}$||$\sqrt{\frac{1+\varLambda}{3-\varLambda}}$||$\sqrt{\frac{1-\varLambda}{3+\varLambda}}$||$\sqrt{\frac{1+\varLambda}{3-\varLambda}}$|
coefficient for |$\chi_\downarrow$| (⁠|$\sin\frac{\beta_i}{2}$|⁠)|$0$||$1$||$-\sqrt{2}\sqrt{\frac{1+\varLambda}{3+\varLambda}}$||$-\sqrt{2}\sqrt{\frac{1-\varLambda}{3-\varLambda}}$||$\sqrt{2}\sqrt{\frac{1+\varLambda}{3+\varLambda}}$||$\sqrt{2}\sqrt{\frac{1-\varLambda}{3-\varLambda}}$|
Open in new tab
Table 1.

Example of |$\{(a_i,ib_i,c_i)\}$| and coefficients for the spin wave functions for the six protons (⁠|$i=$|1–6), which describes the 1p1h configuration. The required conditions are |$a_i^2-b_i^2=2c_i^2$| and |$\tan(\beta_i/2)=\mathrm{sign}(a_ic_i)\sqrt{2(a_i-b_i)/(a_i+b_i)}$|⁠. The center of mass of the system is set to the origin. The parameters |$a_i$|⁠, |$b_i$|⁠, and |$c_i$| are introduced in Eq. (40), and the parameter |$\beta_i$| is introduced in Eq. (41).

i123456
|$a_i$||$1$||$1$||$-\frac{1}{2}$||$-\frac{1}{2}$||$-\frac{1}{2}$||$-\frac{1}{2}$|
|$ib_i$||$i$||$-i$||$\frac{1}{2}i\varLambda$||$-\frac{1}{2}i\varLambda$||$\frac{1}{2}i\varLambda$||$-\frac{1}{2}i\varLambda$|
|$c_i$||$0$||$0$||$\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$-\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$-\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$|
coefficient for |$\chi_\uparrow$| (⁠|$\cos\frac{\beta_i}{2}$|⁠)|$1$||$0$||$\sqrt{\frac{1-\varLambda}{3+\varLambda}}$||$\sqrt{\frac{1+\varLambda}{3-\varLambda}}$||$\sqrt{\frac{1-\varLambda}{3+\varLambda}}$||$\sqrt{\frac{1+\varLambda}{3-\varLambda}}$|
coefficient for |$\chi_\downarrow$| (⁠|$\sin\frac{\beta_i}{2}$|⁠)|$0$||$1$||$-\sqrt{2}\sqrt{\frac{1+\varLambda}{3+\varLambda}}$||$-\sqrt{2}\sqrt{\frac{1-\varLambda}{3-\varLambda}}$||$\sqrt{2}\sqrt{\frac{1+\varLambda}{3+\varLambda}}$||$\sqrt{2}\sqrt{\frac{1-\varLambda}{3-\varLambda}}$|
i123456
|$a_i$||$1$||$1$||$-\frac{1}{2}$||$-\frac{1}{2}$||$-\frac{1}{2}$||$-\frac{1}{2}$|
|$ib_i$||$i$||$-i$||$\frac{1}{2}i\varLambda$||$-\frac{1}{2}i\varLambda$||$\frac{1}{2}i\varLambda$||$-\frac{1}{2}i\varLambda$|
|$c_i$||$0$||$0$||$\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$-\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$-\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$|
coefficient for |$\chi_\uparrow$| (⁠|$\cos\frac{\beta_i}{2}$|⁠)|$1$||$0$||$\sqrt{\frac{1-\varLambda}{3+\varLambda}}$||$\sqrt{\frac{1+\varLambda}{3-\varLambda}}$||$\sqrt{\frac{1-\varLambda}{3+\varLambda}}$||$\sqrt{\frac{1+\varLambda}{3-\varLambda}}$|
coefficient for |$\chi_\downarrow$| (⁠|$\sin\frac{\beta_i}{2}$|⁠)|$0$||$1$||$-\sqrt{2}\sqrt{\frac{1+\varLambda}{3+\varLambda}}$||$-\sqrt{2}\sqrt{\frac{1-\varLambda}{3-\varLambda}}$||$\sqrt{2}\sqrt{\frac{1+\varLambda}{3+\varLambda}}$||$\sqrt{2}\sqrt{\frac{1-\varLambda}{3-\varLambda}}$|
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We need to couple all the nucleons to |$0^+$|⁠, and 1p1h for the proton part and that for the neutron part must be introduced as time reversal partners. For the protons, we choose the parameters in Table 1, and the wave function becomes
\begin{align} \mathcal{A}\left[\left|s\frac{1}{2}\frac{1}{2}\right\rangle,\left|s\frac{1}{2}\frac{-1}{2}\right\rangle,\left|p\frac{3}{2}\frac{3}{2}\right\rangle,\left|p\frac{1}{2}\frac{1}{2}\right\rangle,\left|p\frac{3}{2}\frac{-1}{2}\right\rangle,\left|p\frac{3}{2}\frac{-3}{2}\right\rangle\right]\!. \end{align}
(44)
Thus the neutron part must be introduced as
\begin{align} \mathcal{A}\left[\left|s\frac{1}{2}\frac{1}{2}\right\rangle,\left|s\frac{1}{2}\frac{-1}{2}\right\rangle,\left|p\frac{3}{2}\frac{3}{2}\right\rangle,\left|p\frac{3}{2}\frac{1}{2}\right\rangle,\left|p\frac{1}{2}\frac{-1}{2}\right\rangle,\left|p\frac{3}{2}\frac{-3}{2}\right\rangle\right]\!. \end{align}
(45)

The conditions that eliminate the components of |$|p,3/2,-1/2\rangle$| and |$|p,1/2,1/2\rangle$| in Eq. (43) are |$\sqrt{2}C_i\sin(\beta_i/2)+B_i\cos(\beta_i/2)=0$| and |$\sqrt{2}A_i\sin(\beta_i/2)+C_i\cos(\beta_i/2)=0$|⁠. These conditions are equivalent to |$a_i^2-b_i^2=2c_i^2$| and |$\tan(\beta_i/2)=-\mathrm{sign}(a_ic_i)\sqrt{(a_i-b_i)/[2(a_i+b_i)]}$|⁠. As an example that satisfies these conditions, we show a set of |$\{(a_i,ib_i,c_i)\}$| and coefficients for the spin wave functions for the six neutrons in Table 2. The center of mass of the system is set to the origin. We choose these parameters in Table 2 for the neutron part.

Table 2.

Example of |$\{(a_i,ib_i,c_i)\}$| and coefficients for the spin wave functions for the six neutrons (⁠|$i=7$|–12), which describes the 1p1h configuration. The required conditions are |$a_i^2-b_i^2=2c_i^2$| and |$\tan(\beta_i/2)=-\mathrm{sign}(a_ic_i)\sqrt{(a_i-b_i)/[2(a_i+b_i)]}$|⁠. The center of mass of the system is set to the origin. The parameters |$a_i$|⁠, |$b_i$|⁠, and |$c_i$| are introduced in Eq. (40), and the parameter |$\beta_i$| is introduced in Eq. (41).

i789101112
|$a_i$||$1$||$1$||$-\frac{1}{2}$||$-\frac{1}{2}$||$-\frac{1}{2}$||$-\frac{1}{2}$|
|$ib_i$||$i$||$-i$||$\frac{1}{2}i\varLambda$||$-\frac{1}{2}i\varLambda$||$\frac{1}{2}i\varLambda$||$-\frac{1}{2}i\varLambda$|
|$c_i$||$0$||$0$||$\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$-\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$-\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$|
coefficient for |$\chi_\uparrow$| (⁠|$\cos\frac{\beta_i}{2}$|⁠)|$1$||$0$||$\sqrt{2}\sqrt{\frac{1-\varLambda}{3-\varLambda}}$||$\sqrt{2}\sqrt{\frac{1+\varLambda}{3+\varLambda}}$||$\sqrt{2}\sqrt{\frac{1-\varLambda}{3-\varLambda}}$||$\sqrt{2}\sqrt{\frac{1+\varLambda}{3+\varLambda}}$|
coefficient for |$\chi_\downarrow$| (⁠|$\sin\frac{\beta_i}{2}$|⁠)|$0$||$1$||$\sqrt{\frac{1+\varLambda}{3-\varLambda}}$||$\sqrt{\frac{1-\varLambda}{3+\varLambda}}$||$-\sqrt{\frac{1+\varLambda}{3-\varLambda}}$||$-\sqrt{\frac{1-\varLambda}{3+\varLambda}}$|
i789101112
|$a_i$||$1$||$1$||$-\frac{1}{2}$||$-\frac{1}{2}$||$-\frac{1}{2}$||$-\frac{1}{2}$|
|$ib_i$||$i$||$-i$||$\frac{1}{2}i\varLambda$||$-\frac{1}{2}i\varLambda$||$\frac{1}{2}i\varLambda$||$-\frac{1}{2}i\varLambda$|
|$c_i$||$0$||$0$||$\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$-\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$-\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$|
coefficient for |$\chi_\uparrow$| (⁠|$\cos\frac{\beta_i}{2}$|⁠)|$1$||$0$||$\sqrt{2}\sqrt{\frac{1-\varLambda}{3-\varLambda}}$||$\sqrt{2}\sqrt{\frac{1+\varLambda}{3+\varLambda}}$||$\sqrt{2}\sqrt{\frac{1-\varLambda}{3-\varLambda}}$||$\sqrt{2}\sqrt{\frac{1+\varLambda}{3+\varLambda}}$|
coefficient for |$\chi_\downarrow$| (⁠|$\sin\frac{\beta_i}{2}$|⁠)|$0$||$1$||$\sqrt{\frac{1+\varLambda}{3-\varLambda}}$||$\sqrt{\frac{1-\varLambda}{3+\varLambda}}$||$-\sqrt{\frac{1+\varLambda}{3-\varLambda}}$||$-\sqrt{\frac{1-\varLambda}{3+\varLambda}}$|
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Table 2.

Example of |$\{(a_i,ib_i,c_i)\}$| and coefficients for the spin wave functions for the six neutrons (⁠|$i=7$|–12), which describes the 1p1h configuration. The required conditions are |$a_i^2-b_i^2=2c_i^2$| and |$\tan(\beta_i/2)=-\mathrm{sign}(a_ic_i)\sqrt{(a_i-b_i)/[2(a_i+b_i)]}$|⁠. The center of mass of the system is set to the origin. The parameters |$a_i$|⁠, |$b_i$|⁠, and |$c_i$| are introduced in Eq. (40), and the parameter |$\beta_i$| is introduced in Eq. (41).

i789101112
|$a_i$||$1$||$1$||$-\frac{1}{2}$||$-\frac{1}{2}$||$-\frac{1}{2}$||$-\frac{1}{2}$|
|$ib_i$||$i$||$-i$||$\frac{1}{2}i\varLambda$||$-\frac{1}{2}i\varLambda$||$\frac{1}{2}i\varLambda$||$-\frac{1}{2}i\varLambda$|
|$c_i$||$0$||$0$||$\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$-\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$-\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$|
coefficient for |$\chi_\uparrow$| (⁠|$\cos\frac{\beta_i}{2}$|⁠)|$1$||$0$||$\sqrt{2}\sqrt{\frac{1-\varLambda}{3-\varLambda}}$||$\sqrt{2}\sqrt{\frac{1+\varLambda}{3+\varLambda}}$||$\sqrt{2}\sqrt{\frac{1-\varLambda}{3-\varLambda}}$||$\sqrt{2}\sqrt{\frac{1+\varLambda}{3+\varLambda}}$|
coefficient for |$\chi_\downarrow$| (⁠|$\sin\frac{\beta_i}{2}$|⁠)|$0$||$1$||$\sqrt{\frac{1+\varLambda}{3-\varLambda}}$||$\sqrt{\frac{1-\varLambda}{3+\varLambda}}$||$-\sqrt{\frac{1+\varLambda}{3-\varLambda}}$||$-\sqrt{\frac{1-\varLambda}{3+\varLambda}}$|
i789101112
|$a_i$||$1$||$1$||$-\frac{1}{2}$||$-\frac{1}{2}$||$-\frac{1}{2}$||$-\frac{1}{2}$|
|$ib_i$||$i$||$-i$||$\frac{1}{2}i\varLambda$||$-\frac{1}{2}i\varLambda$||$\frac{1}{2}i\varLambda$||$-\frac{1}{2}i\varLambda$|
|$c_i$||$0$||$0$||$\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$-\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$||$-\frac{\sqrt{1-\varLambda^2}}{2\sqrt{2}}$|
coefficient for |$\chi_\uparrow$| (⁠|$\cos\frac{\beta_i}{2}$|⁠)|$1$||$0$||$\sqrt{2}\sqrt{\frac{1-\varLambda}{3-\varLambda}}$||$\sqrt{2}\sqrt{\frac{1+\varLambda}{3+\varLambda}}$||$\sqrt{2}\sqrt{\frac{1-\varLambda}{3-\varLambda}}$||$\sqrt{2}\sqrt{\frac{1+\varLambda}{3+\varLambda}}$|
coefficient for |$\chi_\downarrow$| (⁠|$\sin\frac{\beta_i}{2}$|⁠)|$0$||$1$||$\sqrt{\frac{1+\varLambda}{3-\varLambda}}$||$\sqrt{\frac{1-\varLambda}{3+\varLambda}}$||$-\sqrt{\frac{1+\varLambda}{3-\varLambda}}$||$-\sqrt{\frac{1-\varLambda}{3+\varLambda}}$|
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3.4. Description of |$(0s_{1/2})^2(0p_{3/2})^2(0d_{5/2})^2$| configuration |$(d_{5/2}$|–2p2h|$)$|

In this subsection, we describe the 2p2h excitations to a higher major shell. In the conventional Brink model, the excitation was described by changing the spatial configuration of |$\alpha$| clusters. The excitation to the |$sd$| shell, |$(0s)^2(0p)^2(0d)^2$| for the protons, was described by assuming a configuration in which the three |$\alpha$| clusters are on a straight line. Because of the antisymmetrization effect, two protons are excited from |$0s$| to |$0p$| orbits, and two protons are further excited to |$0d$| orbits. At that time there was no spin–orbit effect, but now we have to transform these |$0d$| orbits to |$0d_{5/2}$| orbits of the |$jj$|-coupling shell model. Here we describe the |$(0s_{1/2})^2(0p_{3/2})^2(0d_{5/2})^2$| configuration (⁠|$d_{5/2}$|–2p2h) of |$^{12}\mathrm{C}$| in two ways. If we take up to the second order of |$R$|⁠, the single-particle wave functions in Eqs. (12) and (13) become
\begin{align} \phi_{i=1}&=\left[a_{1/2}\left|\frac{1}{2}\frac{1}{2}\right\rangle+a_{3/2}R\left|\frac{3}{2}\frac{3}{2}\right\rangle+a_{5/2}R^2\left|\frac{5}{2}\frac{5}{2}\right\rangle+\mathcal{O}(R^3)\right]\tau_1, \end{align}
(46)
\begin{align} \phi_{i=2}&=\left[a_{-1/2}\left|\frac{1}{2}\frac{-1}{2}\right\rangle+a_{-3/2}R\left|\frac{3}{2}\frac{-3}{2}\right\rangle+a_{-5/2}R^2\left|\frac{5}{2}\frac{-5}{2}\right\rangle+\mathcal{O}(R^3)\right]\tau_2, \end{align}
(47)
respectively. These two single-particle wave functions are used in both methods.

3.4.1. Description of |$(0s_{1/2})^2(0p_{3/2})^2(0d_{5/2})^2$| configuration by a linear structure

Here we assume a linear shape, and the single-particle wave functions of protons |$i=3,4$| are generated by multiplying the rotational operator |$\hat{R}(\alpha=0,\beta=\pi,\gamma=0)$| for the protons |$i=1,2$|⁠, respectively, and
\begin{align} \phi_{i=3}&=\hat{R}(\alpha=0,\beta=\pi,\gamma=0)\phi_{i=1} \nonumber \\ &=\left[a_{1/2}\left|\frac{1}{2}\frac{-1}{2}\right\rangle+a_{3/2}R\left|\frac{3}{2}\frac{-3}{2}\right\rangle+a_{5/2}R^2\left|\frac{5}{2}\frac{-5}{2}\right\rangle+\mathcal{O}(R^3)\right]\tau_3, \end{align}
(48)
\begin{align} \phi_{i=4}&=\hat{R}(\alpha=0,\beta=\pi,\gamma=0)\phi_{i=2} \nonumber \\ &=\left[-a_{-1/2}\left|\frac{1}{2}\frac{1}{2}\right\rangle-a_{-3/2}R\left|\frac{3}{2}\frac{3}{2}\right\rangle-a_{-5/2}R^2\left|\frac{5}{2}\frac{5}{2}\right\rangle+\mathcal{O}(R^3)\right]\tau_4. \end{align}
(49)
Note that the rotation angle is |$\pi$| so as to generate a linear structure. The Gaussian center parameters of protons |$i=5,6$| are set to the origin, and |${\boldsymbol \zeta}_{i=5}={\boldsymbol \zeta}_{i=6}={\boldsymbol 0}$|⁠, where |$i=5$| and |$6$| are spin-up and spin-down protons, respectively:
\begin{align} \phi_{i=5}&=a_{1/2}\left|\frac{1}{2}\frac{1}{2}\right\rangle\tau_5, \end{align}
(50)
\begin{align} \phi_{i=6}&=a_{-1/2}\left|\frac{1}{2}\frac{-1}{2}\right\rangle\tau_6. \end{align}
(51)
These six protons are arranged on a straight line, which creates additional nodes owing to the antisymmetrization effect. Thus the proton part of the wave function becomes
\begin{align} \varPsi_p&=\mathcal{A}[\phi_1,\ldots,\phi_6] \nonumber \\ &=-\,a_{1/2}a_{-1/2}(a_{3/2}a_{-5/2}-a_{-3/2}a_{5/2})^2R^6 \nonumber \\ &\quad\times\mathcal{A}\left[\left|\frac{1}{2}\frac{1}{2}\right\rangle,\left|\frac{1}{2}\frac{-1}{2}\right\rangle,\left|\frac{3}{2}\frac{3}{2}\right\rangle,\left|\frac{3}{2}\frac{-3}{2}\right\rangle,\left|\frac{5}{2}\frac{5}{2}\right\rangle,\left|\frac{5}{2}\frac{-5}{2}\right\rangle\right]+\mathcal{O}(R^7), \end{align}
(52)
where the isospin part is omitted. This wave function coincides with the |$(0s_{1/2})^2(0p_{3/2})^2(0d_{5/2})^2$| configuration (⁠|$d_{5/2}$|–2p2h) at the limit of |$R\to0$|⁠. The same procedure can be applied to the neutron part in exactly the same way.

3.4.2. Description of |$(0s_{1/2})^2(0p_{3/2})^2(0d_{5/2})^2$| configuration by a regular triangle structure

Here we describe the |$(0s_{1/2})^2(0p_{3/2})^2(0d_{5/2})^2$| configuration (⁠|$d_{5/2}$|–2p2h) in a different way; we do not assume a linear shape and the configuration remains with a regular triangular shape. However, in return, the protons |$i=3$|–6 are rotated not about the |$y$| axis but about the |$z$| axis. Then, not only the total angular momentum |$j$| but also the |$z$| component |$j_z$| is unchanged after the rotation. The single-particle wave functions of protons |$i=3,4$| are generated by multiplying the rotational operator |$\hat{R}(\alpha=2\pi/3,\beta=0,\gamma=0)$| by the protons |$i=1,2$|⁠, respectively, and
\begin{align} \phi_{i=3}&=\hat{R}(\alpha=2\pi/3,\beta=0,\gamma=0)\phi_{i=1} \nonumber \\ &=\left[a_{1/2}e^{-i\frac{\pi}{3}}\left|\frac{1}{2}\frac{1}{2}\right\rangle+a_{3/2}e^{-i\pi}R\left|\frac{3}{2}\frac{3}{2}\right\rangle+a_{5/2}e^{-i\frac{5\pi}{3}}R^2\left|\frac{5}{2}\frac{5}{2}\right\rangle+\mathcal{O}(R^3)\right]\tau_3, \end{align}
(53)
\begin{align} \phi_{i=4}&=\hat{R}(\alpha=2\pi/3,\beta=0,\gamma=0)\phi_{i=2} \nonumber \\ &=\left[a_{-1/2}e^{i\frac{\pi}{3}}\left|\frac{1}{2}\frac{-1}{2}\right\rangle+a_{-3/2}e^{i\pi}R\left|\frac{3}{2}\frac{-3}{2}\right\rangle+a_{-5/2}e^{i\frac{5\pi}{3}}R^2\left|\frac{5}{2}\frac{-5}{2}\right\rangle+\mathcal{O}(R^3)\right]\tau_4. \end{align}
(54)
The single-particle wave functions of protons |$i=5,6$| are generated by multiplying the rotational operator |$\hat{R}(\alpha=4\pi/3,\beta=0,\gamma=0)$| by the single-particle wave functions of protons |$i=1,2$|⁠, respectively, and
\begin{align} \phi_{i=5}&=\hat{R}(\alpha=4\pi/3,\beta=0,\gamma=0)\phi_{i=1} \nonumber \\ &=\left[a_{1/2}e^{-i\frac{2\pi}{3}}\left|\frac{1}{2}\frac{1}{2}\right\rangle+a_{3/2}e^{-2i\pi}R\left|\frac{3}{2}\frac{3}{2}\right\rangle+a_{5/2}e^{-i\frac{10\pi}{3}}R^2\left|\frac{5}{2}\frac{5}{2}\right\rangle+\mathcal{O}(R^3)\right]\tau_5, \end{align}
(55)
\begin{align} \phi_{i=6}&=\hat{R}(\alpha=4\pi/3,\beta=0,\gamma=0)\phi_{i=2} \nonumber \\ &=\left[a_{-1/2}e^{i\frac{2\pi}{3}}\left|\frac{1}{2}\frac{-1}{2}\right\rangle+a_{-3/2}e^{2i\pi}R\left|\frac{3}{2}\frac{-3}{2}\right\rangle+a_{-5/2}e^{i\frac{10\pi}{3}}R^2\left|\frac{5}{2}\frac{-5}{2}\right\rangle+\mathcal{O}(R^3)\right]\tau_6. \end{align}
(56)
These six protons are arranged in a regular triangular shape and the proton part of the wave function becomes
\begin{align} \varPsi_p&=\mathcal{A}[\phi_1,\ldots,\phi_6] \nonumber \\ &=[(e^{-i\pi}-e^{-i\frac{\pi}{3}})(e^{-i\frac{10\pi}{3}}-e^{-i\frac{2\pi}{3}})-(e^{-2i\pi}-e^{-i\frac{2\pi}{3}})(e^{-i\frac{5\pi}{3}}-e^{-i\frac{\pi}{3}})] \nonumber \\ &\quad\times[(e^{i\pi}-e^{i\frac{\pi}{3}})(e^{i\frac{10\pi}{3}}-e^{i\frac{2\pi}{3}})-(e^{2i\pi}-e^{i\frac{2\pi}{3}})(e^{i\frac{5\pi}{3}}-e^{i\frac{\pi}{3}})]a_{1/2}a_{-1/2}a_{3/2}a_{-3/2}a_{5/2}a_{-5/2}R^6 \nonumber \\ &\quad\times\mathcal{A}\left[\left|\frac{1}{2}\frac{1}{2}\right\rangle,\left|\frac{1}{2}\frac{-1}{2}\right\rangle,\left|\frac{3}{2}\frac{3}{2}\right\rangle,\left|\frac{3}{2}\frac{-3}{2}\right\rangle,\left|\frac{5}{2}\frac{5}{2}\right\rangle,\left|\frac{5}{2}\frac{-5}{2}\right\rangle\right]+\mathcal{O}(R^7), \end{align}
(57)
where the isospin part is omitted. This wave function coincides with the |$(0s_{1/2})^2(0p_{3/2})^2(0d_{5/2})^2$| configuration (⁠|$d_{5/2}$|–2p2h) at the limit of |$R\to0$|⁠. This is another method to create the |$d_{5/2}$|–2p2h configuration. The same procedure can be applied to the neutron part.

3.5. Energy levels and principal quantum numbers

We couple all of the 2p2h configurations to the subclosure configuration of |$0p_{3/2}$|⁠, and finally the three |$\alpha$|-cluster wave functions are mixed. Concerning the |$R$| and |$\varLambda $| values, for the 1p1h configuration in Sect. 3.3, |$R=0.1\,\mathrm{fm}$| and |$\varLambda=0.1$| are employed, and for the 2p2h excitation to |$0p_{1/2}$| in Sect. 3.2, we take |$R=0.1\,\mathrm{fm}$| and |$(\varLambda_a,\varLambda_b)=(-3/2,0)$|⁠. For the 2p2h excitation to |$0d_{5/2}$| in Sect. 3.4, we take |$R=0.1\,\mathrm{fm}$| in Eqs. (4) and (5). We discuss the obtained |$0^+$| energy levels, principal quantum numbers, and |$E0$| transition matrix elements.

3.5.1. Energy levels with the shell model basis states

We start with the shell model configurations introduced in Sect. 2.3. Figure 2(a) shows the |$0^+$| energy of |$^{12}\mathrm{C}$| with the subclosure configuration of |$0p_{3/2}$| (⁠|$pn$|–0p0h), which is |$-84.5\,\mathrm{MeV}$| (the experimental value is |$-92.2\,\mathrm{MeV}$| [33]). In Fig. 2(b), the |$0^+$| levels obtained after coupling with the 2p2h configurations are shown. Here, we mixed five different 2p2h configurations; two nucleons are excited from |$0p_{3/2}$| to |$0p_{1/2}$| (⁠|$pp$|–|$p_{1/2}$|–2p2h, |$nn$|–|$p_{1/2}$|–2p2h), or they are excited to |$0d_{5/2}$| (⁠|$pp$|–|$d_{5/2}$|–2p2h, |$nn$|–|$d_{5/2}$|–2p2h). In addition, we couple a configuration that one proton and one neutron are excited from |$0p_{3/2}$| to |$0p_{1/2}$| (⁠|$pn$|–|$p_{1/2}$|–2p2h). In this way, we include the effects of BCS-like pairing for the proton part, neutron part, and proton–neutron part. The energy of the ground state becomes |$-86.9\,\mathrm{MeV}$|⁠, and this is lower than that of the subclosure configuration by |$2.4\,\mathrm{MeV}$|⁠. The reduction is caused by the coherent effects of the three BCS-like pairings. The squared overlap between the ground state of the shell-model basis states and subclosure configuration of |$0p_{3/2}$| (⁠|$pn$|–0p0h) is |$0.91$|⁠.

Fig. 2.
$0^+$ energy levels of $^{12}\mathrm{C}$ calculated with shell model-like basis states. (a) $0^+$ energy of the subclosure configuration of $0p_{3/2}$ ($pn$–0p0h), (b) $0^+$ levels obtained after coupling with the 2p2h states.
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|$0^+$| energy levels of |$^{12}\mathrm{C}$| calculated with shell model-like basis states. (a) |$0^+$| energy of the subclosure configuration of |$0p_{3/2}$| (⁠|$pn$|–0p0h), (b) |$0^+$| levels obtained after coupling with the 2p2h states.

3.5.2. Competition of shell and cluster structures

Finally we couple the shell and cluster basis states. In Table 3, the |$0^+$| energies (⁠|$E\,(\mathrm{MeV})$|⁠) and principle quantum numbers (⁠|$N$|⁠) of |$^{12}\mathrm{C}$| are shown. The present interaction gives a slightly lower ground state energy for the cluster basis states (⁠|$-89.1\,\mathrm{MeV}$|⁠) compared with the one for the shell model basis states (⁠|$-86.9\,\mathrm{MeV}$|⁠), but this is related to the fine tuning of the interaction parameters. The ground state energy is reduced by |$3.5\,\mathrm{MeV}$| from the one for the cluster model basis states by mixing both the shell and cluster model basis states (⁠|$-92.6\,\mathrm{MeV}$|⁠), since the spin–orbit interaction was not taken into account within the cluster model basis states. If we calculate without the 2p2h basis states, namely only within the subclosure configuration of |$0p_{3/2}$| and cluster model basis states, the energy is |$-91.8\,\mathrm{MeV}$|⁠. This is higher by |$0.8\,\mathrm{MeV}$| than the final result, and the mixing of the 2p2h configurations is found to have a certain effect. The principal quantum number for the ground state obtained with the shell model basis states is close to |$8$|⁠, which is the lowest possible value, even though 2p2h excitations to |$0d_{5/2}$| are allowed. On the other hand, the cluster model gives a rather large value of |$11.22$|⁠, and this is reduced to |$9.15$| after coupling with the shell model basis states. The three-|$\alpha$| configuration shrinks after coupling with the |$jj$|-coupling shell model states, as discussed in many preceding works including ours [3,18,19,21].

Table 3.

|$0^+$| energies (⁠|$E\,(\mathrm{MeV})$|⁠) and principle quantum numbers (⁠|$N$|⁠) of |$^{12}\mathrm{C}$| calculated using the shell (shell), cluster (cluster) model basis states. The values for the mixed model space, subclosure configuration of |$0p_{3/2}$|⁠, and cluster model basis states are shown in the column “|$pn$|–0p0h+cluster”. The values for the full model space, shell, and cluster basis states are shown in the column “shell+cluster”.

 shellcluster|$pn$|–0p0h+clustershell+cluster
 |$E$||$N$||$E$||$N$||$E$||$N$||$E$||$N$|
|$0_1^+$||$-86.9$||$8.00$||$-89.1$||$11.22$||$-91.8$||$\ \, 9.40$||$-92.6$||$\ \, 9.15$|
|$0_2^+$||$-58.9$||$8.01$||$-79.1$||$20.01$||$-83.2$||$13.82$||$-83.4$||$14.00$|
 shellcluster|$pn$|–0p0h+clustershell+cluster
 |$E$||$N$||$E$||$N$||$E$||$N$||$E$||$N$|
|$0_1^+$||$-86.9$||$8.00$||$-89.1$||$11.22$||$-91.8$||$\ \, 9.40$||$-92.6$||$\ \, 9.15$|
|$0_2^+$||$-58.9$||$8.01$||$-79.1$||$20.01$||$-83.2$||$13.82$||$-83.4$||$14.00$|
Open in new tab
Table 3.

|$0^+$| energies (⁠|$E\,(\mathrm{MeV})$|⁠) and principle quantum numbers (⁠|$N$|⁠) of |$^{12}\mathrm{C}$| calculated using the shell (shell), cluster (cluster) model basis states. The values for the mixed model space, subclosure configuration of |$0p_{3/2}$|⁠, and cluster model basis states are shown in the column “|$pn$|–0p0h+cluster”. The values for the full model space, shell, and cluster basis states are shown in the column “shell+cluster”.

 shellcluster|$pn$|–0p0h+clustershell+cluster
 |$E$||$N$||$E$||$N$||$E$||$N$||$E$||$N$|
|$0_1^+$||$-86.9$||$8.00$||$-89.1$||$11.22$||$-91.8$||$\ \, 9.40$||$-92.6$||$\ \, 9.15$|
|$0_2^+$||$-58.9$||$8.01$||$-79.1$||$20.01$||$-83.2$||$13.82$||$-83.4$||$14.00$|
 shellcluster|$pn$|–0p0h+clustershell+cluster
 |$E$||$N$||$E$||$N$||$E$||$N$||$E$||$N$|
|$0_1^+$||$-86.9$||$8.00$||$-89.1$||$11.22$||$-91.8$||$\ \, 9.40$||$-92.6$||$\ \, 9.15$|
|$0_2^+$||$-58.9$||$8.01$||$-79.1$||$20.01$||$-83.2$||$13.82$||$-83.4$||$14.00$|
Open in new tab

The |$0_2^+$| state is the famous Hoyle state, which has the character of three |$\alpha$| clusters. Experimentally the state appears at |$E_x = 7.65\,\mathrm{MeV}$|⁠, and our final result gives |$9.2\,\mathrm{MeV}$|⁠. Within the cluster model basis states only, the principal quantum number is |$20.01$|⁠, and this is reduced to |$14.00$| after coupling with the shell model basis states. Since the ground state wave function is drastically changed after mixing the shell model basis states with the cluster configurations, the |$0_2^+$| state is also influenced because of the orthogonal condition [34]. The matrix element of the |$E0$| transition between the |$0_1^+$| and |$0_2^+$| states is |$7.36\,e\,\mathrm{fm}^2$|⁠, which is |$9.22\,e\,\mathrm{fm}^2$| within the cluster model basis states only (the experimental value is |$5.52\,e\,\mathrm{fm}^2$|⁠).

In Table 4, we show the squared overlaps between the |$0_{1,2}^+$| states obtained with the full model space and the six shell model basis states introduced in the calculation. The squared overlap between the |$0_1^+$| state and the |$pn$|–0p0h is |$0.42$|⁠, which is reduced from the one within the shell model basis states only (⁠|$0.91$|⁠). This is because, using the present interaction parameters, the cluster model basis states give a lower ground state energy compared with the shell model basis states; however, this tendency may change when we use a slightly different parameter set. Concerning the 2p2h configurations, the squared overlaps between the |$0_1^+$| state and |$pn$|–, |$pp$|–, and |$nn$|–|$p_{1/2}$|–2p2h are about 0.04–0.07, which are not negligible. However, the squared overlap between the |$0_1^+$| state and |$pp$|–|$d_{5/2}$|–2p2h or |$nn$|–|$d_{5/2}$|–2p2h is more than an order of magnitude smaller. For the |$0_2^+$| state, it has a squared overlap with |$pn$|–0p0h of |$0.33$|⁠, but the squared overlaps with the 2p2h configurations are quite small. Note that for each |$0^+$| state, the sum these squared overlaps does not become unity. This is because the solution of our model contains both cluster model space and |$jj$|-coupling shell model space. The cluster model and |$jj$|-coupling shell model have some overlap, but they are different. The values in this table are just the projection to the |$jj$|-coupling shell model space, and the remaining part is the contribution of the cluster states.

Table 4.

Squared overlaps between the |$0_{1,2}^+$| states and the six shell model basis states.

 |$pn$|–0p0h|$pn$|–|$p_{1/2}$|–2p2h|$pp$|–|$p_{1/2}$|–2p2h|$nn$|–|$p_{1/2}$|–2p2h|$pp$|–|$d_{5/2}$|–2p2h|$nn$|–|$d_{5/2}$|–2p2h
|$0_1^+$||$4.21\times10^{-1}$||$3.96\times10^{-2}$||$6.78\times10^{-2}$||$6.86\times10^{-2}$||$9.26\times10^{-4}$||$9.64\times10^{-4}$|
|$0_2^+$||$3.28\times10^{-1}$||$1.10\times10^{-4}$||$4.30\times10^{-4}$||$5.27\times10^{-4}$||$8.41\times10^{-4}$||$7.92\times10^{-4}$|
 |$pn$|–0p0h|$pn$|–|$p_{1/2}$|–2p2h|$pp$|–|$p_{1/2}$|–2p2h|$nn$|–|$p_{1/2}$|–2p2h|$pp$|–|$d_{5/2}$|–2p2h|$nn$|–|$d_{5/2}$|–2p2h
|$0_1^+$||$4.21\times10^{-1}$||$3.96\times10^{-2}$||$6.78\times10^{-2}$||$6.86\times10^{-2}$||$9.26\times10^{-4}$||$9.64\times10^{-4}$|
|$0_2^+$||$3.28\times10^{-1}$||$1.10\times10^{-4}$||$4.30\times10^{-4}$||$5.27\times10^{-4}$||$8.41\times10^{-4}$||$7.92\times10^{-4}$|
Open in new tab
Table 4.

Squared overlaps between the |$0_{1,2}^+$| states and the six shell model basis states.

 |$pn$|–0p0h|$pn$|–|$p_{1/2}$|–2p2h|$pp$|–|$p_{1/2}$|–2p2h|$nn$|–|$p_{1/2}$|–2p2h|$pp$|–|$d_{5/2}$|–2p2h|$nn$|–|$d_{5/2}$|–2p2h
|$0_1^+$||$4.21\times10^{-1}$||$3.96\times10^{-2}$||$6.78\times10^{-2}$||$6.86\times10^{-2}$||$9.26\times10^{-4}$||$9.64\times10^{-4}$|
|$0_2^+$||$3.28\times10^{-1}$||$1.10\times10^{-4}$||$4.30\times10^{-4}$||$5.27\times10^{-4}$||$8.41\times10^{-4}$||$7.92\times10^{-4}$|
 |$pn$|–0p0h|$pn$|–|$p_{1/2}$|–2p2h|$pp$|–|$p_{1/2}$|–2p2h|$nn$|–|$p_{1/2}$|–2p2h|$pp$|–|$d_{5/2}$|–2p2h|$nn$|–|$d_{5/2}$|–2p2h
|$0_1^+$||$4.21\times10^{-1}$||$3.96\times10^{-2}$||$6.78\times10^{-2}$||$6.86\times10^{-2}$||$9.26\times10^{-4}$||$9.64\times10^{-4}$|
|$0_2^+$||$3.28\times10^{-1}$||$1.10\times10^{-4}$||$4.30\times10^{-4}$||$5.27\times10^{-4}$||$8.41\times10^{-4}$||$7.92\times10^{-4}$|
Open in new tab

4. Conclusion

We have developed the framework of AQCM to describe not only the subclosure configuration of the |$jj$|-coupling shell model but also the 2p2h configurations, in addition to the cluster model wave functions. In |$^{12}\mathrm{C}$|⁠, it was shown that the 2p2h excitations from |$0p_{3/2}$| to |$0p_{1/2}$| and that to |$0d_{5/2}$| were successfully described, which enables us to include the effects of BCS-like pairing for the proton part, neutron part, and proton–neutron part. The correlation energy from the optimal configuration can be estimated not only in the cluster part but also in the shell model part.

For the ground |$0^+$| state of |$^{12}\mathrm{C}$|⁠, the interaction of the present calculation gives a slightly lower energy for the cluster model basis states (⁠|$-89.1\,\mathrm{MeV}$|⁠) compared with the one for the shell model basis states (⁠|$-86.9\,\mathrm{MeV}$|⁠), and the ground state energy is reduced by |$3.5\,\mathrm{MeV}$| by mixing both the shell and cluster model basis states (⁠|$-92.6\,\mathrm{MeV}$|⁠). This is because the spin–orbit interaction is not taken into account within the cluster model basis states. If we calculate without the 2p2h basis states, the energy becomes |$-91.8\,\mathrm{MeV}$|⁠, about |$0.8\,\mathrm{MeV}$| higher, and the mixing of the 2p2h configurations is found to have a certain effect. Within the cluster model basis states only, the principal quantum number is rather large, |$11.22$|⁠, and this is reduced to |$9.15$| after coupling with the shell model basis states. The three-|$\alpha$| configuration shrinks after coupling with the |$jj$|-coupling shell model states. The squared overlap between the ground |$0^+$| state and the 0p0h configuration of the |$jj$|-coupling shell model is |$0.42$|⁠, and the overlaps with some of the 2p2h configurations are |$0.04$|–|$0.07$|⁠, which are not negligible.

The |$0_2^+$| state is the famous Hoyle state, and the present model gives |$E_x = 9.2\,\mathrm{MeV}$|⁠. The cluster model basis states give a principal quantum number of |$20.01$|⁠, and this is reduced to |$14.00$| after coupling with the shell model basis states. Since the ground state wave function is drastically changed after mixing the shell model basis with the cluster configurations, the |$0_2^+$| state is also influenced because of the orthogonal condition.

This method of describing particle–hole excitations is considered to be applicable to other light or even heavier nuclei. As an example, description of four-particle–four-hole configurations such as |$(0s_{1/2})^4(0p_{3/2})^8(0d_{5/2})^4$| and coupling with the cluster model wave functions (⁠|$^{12}\mathrm{C}$|+|$\alpha$|⁠, four |$\alpha$|⁠) are being investigated for |$^{16}\mathrm{O}$|⁠; the understanding of “the mysterious |$0^+$| state” is a longstanding problem [35]. The systematic description of competition between particle–hole excitations and cluster states is a challenging subject to be investigated in the near future.

Acknowledgements

The authors are grateful for the computer facility at Yukawa Institute for Theoretical Physics, Kyoto University. This work was supported by JSPS KAKENHI Grant Number 17K05440.

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© The Author(s) 2017. Published by Oxford University Press on behalf of the Physical Society of Japan.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
Subject
D11 Models of nuclear structure
Issue Section:
Nuclear Physics
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