Two neutrino double-$β$ decay of $94\leq A\leq 150$ nuclei for the 0$^{+}\rightarrow $2$^{+}$ transition

Within the PHFB approach, the $0^{+}\rightarrow 2^{+}$ transition of two neutrino double-$β$ decay of $^{94,96}$Zr, $^{100}$Mo, $^{104}$Ru, $^{110}$Pd, $^{128,130}$Te and $^{150}$Nd isotopes is studied employing wave functions generated with four different parametrizations of the pairing plus multipole type of two-nucleon interaction and the summation method. In comparison to the $0^{+}\rightarrow 0^{+}$ transition, the nuclear transition matrix elements $M_{2ν}(2^{+})$ are quite sensitive to the deformation of the yrast 2$^{+}$ state. Consideration of the available theoretical and experimental results suggest that the observation of the 0$% ^{+}\rightarrow $2$^{+}$ transition of $2νβ^{-}β^{-}$ decay may be possible in $^{96}$Zr, $^{100}$Mo, $^{130}$Te and $^{150}$Nd isotopes. The effect of deformation on the $M_{2ν}(2^{+})$ is also studied.