by introducing a correlated Gaussian [@boys60; @vs95; @book], provided each part of $\psi_{L_a L_b (L_{ab}) \ell L}^{(\rm space)}$ is given in terms of (a combination of) Gaussian functions as in the present case. In what follows we will demonstrate how it is performed. Because the formulation with the correlated Gaussian is not restricted to four nucleons but can be applied to a system including more particles, the number of nucleons is assumed to be $N$ in this and next sections as well as in Appendices B and C unless otherwise mentioned.

The relative and center of mass coordinates of the $N$ nucleons, ${{\mbox{\boldmath $x$}}}_i\, (i=1,\ldots,N)$, and the single-particle coordinates, ${{\mbox{\boldmath $r$}}}_i\, (i=1,\ldots,N)$, are mutually related by a linear transformation matrix $U$ and its inverse $U^{-1}$ as follows: $${{\mbox{\boldmath $x$}}}_i=\sum_{j=1}^{N}U_{ij}{{\mbox{\boldmath $r$}}}_j,\ \ \ \ \ 
{{\mbox{\boldmath $r$}}}_i=\sum_{j=1}^{N}(U^{-1})_{ij}{{\mbox{\boldmath $x$}}}_j.
\label{def.matU}$$ We use a matrix notation as much as possible in order to simplify formulas and expressions. Let ${{\mbox{\boldmath $x$}}}$ denote an $(N\!-\!1)$-dimensional column vector comprising all ${{\mbox{\boldmath $x$}}}_i$ but the center of mass coordinate ${{\mbox{\boldmath $x$}}}_N$. Its transpose is a row vector and it is expressed as $$\widetilde{{{\mbox{\boldmath $x$}}}}=({{\mbox{\boldmath $x$}}}_1,{{\mbox{\boldmath $x$}}}_2,...,{{\mbox{\boldmath $x$}}}_{N-1}).$$ The choice for ${{\mbox{\boldmath $x$}}}$ is not unique but a 