To obtain a \( V \)-matrix in a h.o. basis, we need the transformation $$ \langle nNlL{\cal J}ST\vert\hat{V}\vert n'N'l'L'{\cal J}S'T\rangle, $$ with \( n \) and \( N \) the principal quantum numbers of the relative and center-of-mass motion, respectively. $$ \vert nlNL{\cal J}ST\rangle= \int k^{2}K^{2}dkdKR_{nl}(\sqrt{2}\alpha k) R_{NL}(\sqrt{1/2}\alpha K) \vert klKL{\cal J}ST\rangle. $$ The parameter \( \alpha \) is the chosen oscillator length.