We discuss two conditions needed for correct computation of $2 u betabeta$ nuclear matrix-elements within the realistic shell-model framework. An algorithm in which intermediate states are treated based on Whiteheads moment method is inspected, by taking examples of the double GT$^+$ transitions $mbox{$^{36}$Ar}rightarrowmbox{$^{36}$S}$, $mbox{$^{54}$Fe}rightarrowmbox{$^{54}$Cr}$ and $mbox{$^{58}$Ni} rightarrowmbox{$^{58}$Fe}$. This algorithm yields rapid convergence on the $2 ubetabeta$ matrix-elements, even when neither relevant GT$^+$ nor GT$^-$ strength distribution is convergent. A significant role of the shell structure is pointed out, which makes the $2 ubeta beta$ matrix-elements highly dominated by the low-lying intermediate states. Experimental information of the low-lying GT$^pm$ strengths is strongly desired. Half-lives of $T^{2 u}_{1/2}({rm EC}/{rm EC}; mbox{$^{36}$Ar}rightarrowmbox{$^{36}$S})=1.7times 10^{29}mbox{yr}$, $T^{2 u}_{1/2}({rm EC}/{rm EC};mbox{$^{54}$Fe}rightarrow mbox{$^{54}$Cr})=1.5times 10^{27}mbox{yr}$,$T^{2 u}_{1/2}({rm EC} /{rm EC};mbox{$^{58}$Ni}rightarrowmbox{$^{58}$Fe})=6.1times 10^{24}mbox{yr}$and $T^{2 u}_{1/2}(beta^+/{rm EC};mbox{$^{58}$Ni} rightarrowmbox{$^{58}$Fe})=8.6times 10^{25}mbox{yr}$ are obtained from the present realistic shell-model calculation of the nuclear matrix-elements.
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