  Large-scale shell-model calculations have been performed for the study of two
neutrino double-beta ($2\nu\beta\beta$) decay in $^{82}$Se, $^{94}$Zr,
$^{108}$Cd, $^{124}$Sn, $^{128}$Te, $^{130}$Te, $^{136}$Xe, and $^{150}$Nd. We
have employed JUN45 interaction to calculate the nuclear matrix element (NME)
for $2\nu\beta\beta$ decay in $^{82}$Se. In the case of $^{94}$Zr, the glekpn
effective interaction is used. For $^{108}$Cd, we have used a realistic
effective interaction derived through the G-matrix approach. In the case of
$^{124}$Sn, $^{128,130}$Te and $^{136}$Xe, the sn100pn effective interaction is
employed. For $^{150}$Nd, we have used KHHE effective interaction based on
holes in a $^{208}$Pb core. We have extracted the half-lives of these nuclei
for the $2\nu\beta\beta$ decay with the help of calculated NME. Our results are
consistent with the available experimental half-lives. The variation of
cumulative $2\nu\beta\beta$ NME with respect to the excitation energy of the
intermediate $1^+$ states is also shown, and in all cases, it is ensured that
their values are almost saturated. In the present work we have calculated more
intermediate $1^+$ states as much as possible in comparison to results
available in the literature.
