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Large-scale shell-model calculations were carried out for the half-lives andbranching ratios of the $2\nu\beta\beta$ decay of $^{76}$Ge to the ground stateand the lowest three excited states $2_1^+$, $0_2^+$ and $2_2^+$ in $^{76}$Se.In total, the wave functions of more than 10,000 intermediate $1^+$ states in$^{76}$As were calculated in a three-step procedure allowing an efficient useof the available computer resources. In the first step, 250 lowest states,below some 5 MeV of excitation energy, were calculated without truncationswithin a full major shell $0f_{5/2}-1p-0g_{9/2}$ for both protons and neutrons.The wave functions of the rest of the states, up to some 30 MeV, were computedin two more steps by introducing two consecutive stages of truncation. Thecomputed magnitudes of the $2\nu\beta\beta$ nuclear matrix elements (includingthe value of the axial-vector coupling $g_{\rm A}$), $\vert M_{2\nu}\vertg_{\rm A}^2$, converged to the values 0.168$g_{\rm A}^2$,$1.2\times10^{-3}$$g_{\rm A}^2$, 0.121$g_{\rm A}^2$, and$3.1\times10^{-3}$$g_{\rm A}^2$ for the $0^+_{\rm g.s.}$, $2^+_1$, $0^+_2$, and$2^+_2$ states, respectively. Using up-to-date phase-space integrals, thecorresponding branching ratios were derived to be 99.926\%,4.4$\times10^{-5}$\%, 0.074\% and 2.5$\times10^{-7}$\%. The experimentalhalf-life $(1.926\pm0.094)\times10^{21}$ yr of the ground-state transition wasused to derive the value $g_{\rm A}=0.80\pm0.01$ for the axial-vector coupling,which is consistent with other shell-model calculations suggesting a quenchedvalue of $g_{\rm A}$. Using this value of $g_{\rm A}$, predictions for thetransition half-lives were derived.

The first large-scale shell-model calculation of the two-neutrino double beta decay of $^{76}$Ge to the excited states in $^{76}$Se