4      $-$2.10        1.63          0      $-$2.20        1.95           0
         MN         $t(\frac{1}{2}^+)$     15     $-$8.38        1.70          0      $-$8.38        1.71           0
                    $h(\frac{1}{2}^+)$     15     $-$7.70        1.72          0      $-$7.71        1.74           0
                      $^4$He$(0^+)$      (1140)   $-$29.94       1.41          0      $-$29.94       1.41           0
  ---------------- -------------------- -------- ---------- --------------- -------- ---------- --------------- ---------

  : Energies $E$, rms radii $R^{\rm rms}$ and $D$-state probabilities $P_D$ of the clusters that appear in four-nucleon scattering and $^4$He with the AV8$^{\prime}$ (with TNF) and MN potentials. $N_k$ is the number of basis functions used to approximate the wave function of cluster $k$. The values in the last three columns for three- and four-body systems are taken from Ref. [@hiyama04] for AV8$^{\prime}$ and Ref. [@DGVR] for MN.

\[sub1\]

Correlated Gaussian function with triple global vectors {#sect.3}
=======================================================

As explained in the previous section, the calculation of the $S$-matrix reduces to that of the Hamiltonian and overlap matrix elements with the functions defined by (\[wf-ch1\]) and (\[wf.cluster\]), and it is conveniently performed by transforming that wave function into an $LS$-coupled form, $$\begin{aligned}
&&{\cal A}
\left[\left[\left[\psi^{(\rm space)}_{L_a}\psi^{(\rm space)}_{L_b}\right]_{L_{ab}}
\chi_\alpha({{\mbox{\boldmath $\rho$}}}_\alpha)\right]_L
\left[\psi^{(\rm spin)}_{S_a} \psi^{(\rm spin)}_{S_b}\right]_{S}
\right]_{JM}.\end{aligned}$$ The transformation can be done as $$\begin{aligned}
& &{\hspace{-10mm}}
{\cal A}\left[ \left[
[\psi_{L_a}^{(\rm{space})}\psi_{S_a}^{(\rm{spin})}]_{I_a}
[\psi_{L_b}^{(\rm{space})}\psi_{S_b}^{(\rm{spin})}]_{I_b}
\right]_I \chi_\alpha({{\mbox{\boldmath $\rho$}}}_\alpha)\right]_{JM}
\psi^{(\rm{isospin})}_{T_a M_{T_a}}\psi^{(\rm{isospin})}_{T_b M_{T_b}}
\nonumber \\
&=&\sum_{L_{ab}LS}
\left[
