The THSR (Tohsaki-Horiuchi-Schuck-Röpke) wave function was first introduced in order to investigate the Bose-Einstein-condensation-like character of the 3-\(\alpha \) cluster state of \(^{12}\)C which is the \(0^+_2\) state known as the Hoyle state [1]. The THSR wave function for a \(n\alpha \)-condensate-like structure is given as

$$\begin{aligned} \Psi _{n\alpha }(b,B) & =\left( \frac{4}{B^2\pi }\right) ^{\frac{3n}{4}} {{\mathcal {A}}} \left\{ \prod _{i=1}^n {\textrm{e}}^{-2 {\varvec{X}}_i^2/{B^2}} \prod _{j=1}^n \phi (\alpha _j) \right\} \nonumber \\ & = \left( \frac{4R_0^2b}{B^3}\right) ^{\frac{3n}{2}}\int \prod _{i=1}^n d{\varvec{D}}_i \exp \left( -\sum _{j=1}^n\frac{{\varvec{D}}_j^2}{R_0^2}\right) \nonumber \\ & \quad \times {{\mathcal {A}}} \left\{ \psi (\alpha _1,{\varvec{D}}_1)\cdots \psi (\alpha _n,{\varvec{D}}_n)\right\} , \end{aligned}$$
(1)

where \(B^2 = b^2 + 2R_0^2 \). In Eq. (1), \({{\mathcal {A}}}\) stands for the antisymmetrization operator of nucleons. \({\varvec{X}}_j\) is the center of mass coordinate of the j-th alpha cluster. The relation of the first and second expressions of \(\Psi _{n\alpha }(b,B)\) is easily obtained by noting the relation \(\psi (\alpha _j,{\varvec{D}}_j)=g({\varvec{X}}_j-{\varvec{D}}_j,4\nu )\phi (\alpha )\). Here \(\psi (\alpha ,{\varvec{D}})\) stands for the \((0s)^4\) shell-model wave function of an \(\alpha \)-cluster located around the spatial position \({\varvec{D}}\) with the single-nucleon oscillator parameter \(\nu = 1/(2b^2)\). The 0 s wave function has the form proportional to \(\exp (-\nu r^2)\), and \(g({\varvec{X}}, \mu ) = (2\mu /\pi )^{3/4} \exp (-\mu {\varvec{X}}^2)\). \(\phi (\alpha )\) stands for the normalized internal wave function of an \(\alpha \)-cluster. It depends on the three Jacobi coordinates of four nucleons composing the \(\alpha \) cluster. The second expression of \(\Psi _{n\alpha }(b,B)\) means that \(\Psi _{n\alpha }(b,B)\) is the superposition of the Brink wave function

\({{\mathcal {A}}} \left\{ \psi (\alpha _1,{\varvec{D}}_1)\cdots \psi (\alpha _n,{\varvec{D}}_n)\right\} \) with respect to the \(\alpha \)-cluster position parameters \({\varvec{D}}_j\) with the weight \(\exp [-({\varvec{D}}_j^2/R_0^2)]\) for \(j=1 \dots n\). \(\Psi _{n\alpha }(b,B)\) has spin and parity \(J^\pi =0^+\).

The THSR wave function expresses that n \(\alpha \)-clusters have the same center of mass motion given by the 0 s-wave function with the size parameter B. When B is much larger than b, the THSR wave function expresses that the n \(\alpha \)-clusters have a gas-like structure and all the n \(\alpha \)-clusters occupy the same 0 s-orbits with the size parameter B. This means that the THSR wave function \(\Psi _{n\alpha }(b,B)\) stands for the Bose-Einstein-condensation-like state of the n \(\alpha \)-clusters. In \(^{12}\)C, the 3\(\alpha \) THSR wave function with a large value of B proved to give almost the same results as the 3\(\alpha \) cluster-model calculations in 1970 s, see [2] and references given therein, which succeeded to reproduce the observed data well.

The surprising fact was soon discovered that the 3\(\alpha \) THSR wave functions have almost 100 % overlap with the traditional 3\(\alpha \) cluster-model wave functions for the same effective inter-nucleon forces [2]. Traditional cluster models use the RGM (resonating group method) wave functions or Brink-type cluster model wave functions. This fact was considered to be due to the special nature of the gas-like cluster structure. However, about ten years later, it was found that this fact is not due to the gas-like nature of the Hoyle state. Namely, the THSR-type wave functions for the non-gas-like cluster structure were found to have almost 100 % overlaps with the traditional cluster-model wave functions. At first, THSR-type wave functions for the \(^{16}\)O + \(\alpha \) cluster structures in \(^{20}\)Ne were found to be almost equivalent to the wave functions of traditional \(^{16}\)O + \(\alpha \) cluster model wave functions which succeeded to reproduce observed data well [3, 4]. Needless to say, the \(^{16}\)O + \(\alpha \) cluster structures in \(^{20}\)Ne are entirely different from the gas-like cluster structure. The THSR-type wave function for the \(^{16}\)O + \(\alpha \) cluster structure is given as

$$\begin{aligned} \Phi _{\textrm{Ne}}({\varvec{\beta }},{\varvec{S}}) & = \int d^3R ~ \exp \left[ -\frac{4}{5}\left( \frac{R^2_x}{\beta ^2_x} + \frac{R^2_y}{\beta ^2_y} +\frac{R^2_z}{\beta ^2_z}\right) \right] \nonumber \\ & \quad \times \Phi ^{{\textrm{Brink}}}_{\textrm{Ne}} \left( \frac{4}{5}({\varvec{R}}+{\varvec{S}}), \frac{1}{5}({\varvec{R}}+{\varvec{S}})\right) \nonumber \\ & \quad * \propto \exp \left( -\frac{10X_G^2}{b^2}\right) \, {{{\hat{\Phi }}}}_{\textrm{Ne}}({\varvec{\beta }},{\varvec{S}}), \end{aligned}$$
(2)
$$\begin{aligned} {{{\hat{\Phi }}}}_{\textrm{Ne}}({\varvec{\beta }},{\varvec{S}}) & = {{\mathcal {A}}} \left\{ \exp \left[ -\hspace{-3mm}\sum _{k=x,y,z} \frac{8(r_k - S_k)^2}{5B_k^2}\right] \phi ( ^{16}{\textrm{O}}) \phi (\alpha ) \right\} .\nonumber \\ \end{aligned}$$
(3)

Here \({\varvec{X}}_G\) is the total center-of-mass (c.m.) coordinate, the coordinate \({\varvec{r}}\) stands for the relative coordinate between the clusters \(^{16}{\textrm{O}}\) and \(\alpha \). \(\Phi ^{\textrm{Brink}}_{\textrm{Ne}}\) stands for the Brink-type cluster wave function expressing the \(^{16}\)O + \(\alpha \) cluster structure, and \(\phi ( ^{16}{\textrm{O}})\) stands for the internal wave function of an \( ^{16}{\textrm{O}}\) cluster. The vector parameter \({\varvec{\beta }}\) stands for \({\varvec{\beta }}=(\beta _x,\beta _y,\beta _z)\), and the parameters \(B_k\) are given by \(B^2_k=b^2+2\beta ^2_k\), \((k=x,y,z)\). This form of the THSR-type wave function shows that the relative wave function between \(^{16}\)O and \(\alpha \) clusters is expressed by a single Gaussian function around the vector \({\varvec{S}}=(S_x,S_y,S_z)\) whose width parameters are given by \(B_k\) \((k=x,y,z)\) which are larger than b, namely \(B_k > b\). If \(B_k=b \) \((k=x,y,z)\), the THSR-type wave function is the same as the traditional Brink wave function. A very striking fact is that the traditional cluster-model wave functions of \(^{20}\)Ne for both the ground-state band states and the first negative-parity band states obtained in the generator coordinate method (GCM) calculation by using the \(^{16}\)O + \(\alpha \) Brink wave functions are almost equivalent to single THSR-type wave functions having vanishingly small \({\varvec{S}}\) parameters. The squared overlap value between the Brink-GCM wave function and the THSR-type wave function with \(S_k \approx 0\) is 0.9929, 0.9879, 0.9775, 0.9998, and 0.9987, for \(J^\pi =0^+, 2^+, 4^+\), \(1^-,\) and \(3^-\), respectively [4]. This is striking because the ground-rotational-band states and the negative-parity (\(K = 0^-\)) rotational-band states have been considered to be the inversion-doublet rotational-band states which have an \(^{16}\)O - \(\alpha \) intrinsic cluster structure [5]. The inversion-doublet rotational bands mean that they have an intrinsic structure which has parity-violating symmetry. This parity-violating structure comes from the shape of the intrinsic density distribution where the \(^{16}\)O and \(\alpha \)-clusters have their own density distributions which are spatially apart from each other.

The above-mentioned contradiction between the localized traditional cluster-model wave functions and the nonlocalized THSR-type wave functions can be explained [3, 4] by considering the fermionic nature of nucleons which is expressed by the use of the antisymmetrization operator \({{\mathcal {A}}}\). Even though the relative wave function between \(^{16}\)O and \(\alpha \) clusters is expressed by a single Gaussian function, the antisymmetrization operation by \({{\mathcal {A}}}\) prohibits two clusters to approach too close spatially with each other. It was known already before 1960 from the analysis of the resonating group method (RGM)-type cluster-model wave function that the inter-cluster relative wave function expressed by the harmonic oscillator (HO) wave function can not have the oscillator quanta N smaller than some number \(N_A\) [6, 7]. In the case of the \(\alpha + \alpha \) two-cluster system, \(N_A\) = 4 and for \(N < N_A\) the \(\alpha + \alpha \) cluster wave function \(\mathcal{A} \{R_{n,\ell }(r)Y_{\ell ,m}(\theta ,\varphi )\} \phi (\alpha )\phi (\alpha )\}\) vanishes, where \(R_{n,\ell }(r)\) stands for the H.O. radial wave function with oscillator quanta N = \(2n + \ell \). In the case of the \( ^{16}{\textrm{O}} + \alpha \) two-cluster system, \(N_A\) = 8 and for \(N < N_A\) the \( ^{16}{\textrm{O}} + \alpha \) cluster wave function vanishes; \({{\mathcal {A}}} \{R_{n,\ell }(r)Y_{\ell ,m}(\theta ,\varphi )\} \phi ( ^{16}{\textrm{O}})\phi (\alpha )\}\) = 0. The inter-cluster relative wave functions \(R_{n,\ell }(r)Y_{\ell ,m}(\theta ,\varphi )\) with \(N = 2n + \ell < N_A\) are called Pauli-forbidden relative states. The existence of the Pauli-forbidden relative states means that two clusters can not approach too close spatially to each other.

The \({\textrm{C}}_1 + {\textrm{C}}_2\) cluster state \({{\mathcal {A}}} \{R_{n,\ell }(r)Y_{\ell ,m}(\theta ,\varphi )\} \phi ( C_1)\phi ({\textrm{C}}_2)\}\) which has the lowest HO quanta \(N_A\) (\(N=2n +\ell = N_A\)) for its relative wave function \(R_{n,\ell }(r)Y_{\ell ,m}(\theta ,\varphi )\) has a very important property which is known as the duality of cluster structure and shell-model structure. In the case of the \(\alpha + \alpha \) two-cluster system which has \(N_A\) = 4, there holds [8, 9]

$$\begin{aligned} & {{\mathcal {A}}} \{R_{n,\ell }(r)Y_{\ell ,m}(\theta ,\varphi )\} \phi (\alpha )\phi (\alpha )\} \exp \{-8\nu X_G^2\} \nonumber \\ & \quad \propto \Psi _{\mathrm{shell-model}}\{(0s)^4(0p)^4,[44]\} \end{aligned}$$
(4)

for \(N=2n+\ell =N_A\).

In the case of \( ^{16}{\textrm{O}} + \alpha \) two-cluster system which has \(N_A\) = 8, there holds [8, 9]

$$\begin{aligned} & {{\mathcal {A}}} \{R_{n,\ell }(r)Y_{\ell ,m}(\theta ,\varphi )\} \phi ( ^{16}{\textrm{O}})\phi (\alpha )\} \exp \{-20\nu X_G^2\} \nonumber \\ & \quad \propto \Psi _{\mathrm{shell-model}}\{(0s)^4(0p)^8(1s0d)^4,\nonumber \\ & \quad [44444] \, (\lambda ,\mu )=(8,0)\}\nonumber \\ \end{aligned}$$
(5)

for \(N=2n+\ell =N_A\).

The cluster state with \(N=2n+\ell =N_A\) is the most spatially compact cluster state and it is equivalent to a shell-model wave function. This duality property comes from the fermionic nature of nucleons. It means that it is impossible for clusters to have overlapping spatial position of cluster c.m. coordinates because of \(N=2n+\ell =N_A\ne 0\).

The existence of the Pauli-forbidden states is intimately related to the inter-cluster potential. The inter-cluster potential has a repulsive force in its short-range region which prevents two clusters to come too close to each other. When the inter-cluster potential is complex and has an imaginary part, the existence of short-range repulsion is not easy to pin down phenomenologically, but it surely exists theoretically because there surely exist the Pauli-forbidden states for the inter-cluster motion. The inter-cluster wave function should be orthogonal to the Pauli-forbidden states and this causes the position of the innermost nodal point of the inter-cluster wave function to be rather stable with respect to the change of the energy of the inter-cluster relative motion. As is explained in Ref. [10] this position of the innermost nodal point of the inter-cluster relative wave function is very close to the radius of the repulsive core of the inter-cluster phenomenological potential. The effect of the Pauli-forbidden states to the inter-cluster potential is nicely formulated by the OCM orthogonality condition model (OCM) [11]. The inter-cluster relative wave function is constrained to be orthogonal to the Pauli-forbidden states and the inter-cluster potential is adopted to be similar to the doubly-folding potential which has no repulsive component.

The duality of the cluster structure and the shell-model structure for the cluster state with \(N=2n+\ell =N_A\) was the starting point of Karl Wildermuth for his research of cluster structure in nuclei [6, 7]. Because of the duality nature of the shell-model wave function of the ground state, the ground state can be expressed also by the cluster-model wave function. Wildermuth considered that the excitation of the shell-model feature of the ground state yields the excited states of the shell-model type while the excitation of the cluster-model feature of the ground state yields excited states with cluster-model type. Recently, the intimate relation between the excited states with cluster nature and the ground state was shown to be reflected in the strong monopole-transition rates between these states both experimentally [12] and theoretically [13].

The spatial localization of clusters of the state described by THSR-type wave function can be seen explicitly by calculating the nucleon density distribution of the THSR-type wave function. In Ref. [4], the nucleon density distribution of the THSR-type wave function for the \(^{16}\)O + \(\alpha \) cluster structure is displayed. It is the nucleon density distribution of the THSR-type intrinsic wave function with \(S_z\) = 0.6 fm and \((\beta _x,\beta _y,\beta _z)\) = (0.9, 0.9, 2.5 fm). We observe in this figure that, in spite of the small value of \(S_z\)= 0.6 fm, the inter-cluster distance between \(^{16}\)O and \(\alpha \) is about 3.6 fm. Namely, the large inter-cluster distance of about 3.6 fm between \(^{16}\)O and \(\alpha \) is not attributable to the parameter \(S_z\) but attributable to the effective spatial localization of \(^{16}\)O and \(\alpha \) clusters in the prolate THSR-type wave function. Another good example which shows that the THSR-type wave function can represent the localized clusters is the linear-chain-like \(n\alpha \) cluster system [4]. For example, the \(3\alpha \) THSR wave function with \((\beta _x,\beta _y,\beta _z)\) = (0.01, 0.01, 5.1 fm) which may be called a one-dimensional \(\alpha \)-condensate is demonstrated to have a nucleon-density distribution where \(3\alpha \) clusters are spatially localized in a line.

We can say that the spatial localization of clusters need not to be expressed by a wave function which has explicit spatial localization of clusters. The above-mentioned argument given in Refs. [3, 4] shows that the spatial localization of clusters comes from the Fermi-Dirac statistics of nucleons described by the antisymmetrizing operator \({{\mathcal {A}}}\) through the duality of shell-model structure and the cluster-model structure of spatially compact ground-state configuration. It is the reason why the THSR-type wave functions can describe spatial localization of clusters which has been traditionally described by cluster-model wave functions having explicit spatial localization of clusters. The model which describes the cluster structure by the use of THSR-type wave functions is called container model since the constituent clusters move in a container-like space showing no explicit localized density distribution of clusters.

Investigations of cluster dynamics by the use of THSR-type wave functions are now developing in many directions. As representative examples of investigation we here choose the following two ones. The first one is the investigation by Yasuro Funaki who discusses the formation of various cluster structures in nuclei following the increase of excitation energy [14, 15]. He adopts the container model to describe the formation of a cluster structure as the increase of excitation energy which he calls the container evolution of the cluster structure. The second one is the investigation [16] by Bo Zhou, Yasuro Funaki, and their coauthors who discuss, by the use of THSR wave function, the formation of the 5\(\alpha \) condensate states in \(^{20}\)Ne theoretically which were reported by Kawabata et al. experimentally in Ref. [17]. It is sad that we now can not discuss these developments with Peter Schuck. I cherish the memory of my first discussion with Peter about the possibility of the existence of the dilute gas-like cluster structure of \(\alpha \) clusters in finite nuclei which was made on the excursion boat on the Adriatic Sea at the case of international cluster conference in Croatia in 1999. This discussion was the starting point of the collaboration which produced the first paper on the \(\alpha \) condensation in finite nuclei [1].