We present a theoretical study of the two-neutrino $0^{+}\to 0^{+}$ double beta decay of ${}^{76}\mathrm{Ge}$ within the no-core configuration-interaction framework based on the Skyrme SV density functional. We analyze three allowed decay scenarios distinguished by the $[n,m]\equiv [{\left(\nu g_{9/2}\right)}^n,{\left(\pi g_{9/2}\right)}^m]$ occupancy of the $0g_{9/2}$ intruder orbital, which remains conserved to high precision, as well as by the triaxiality of the daughter nucleus. The resulting $2\nu \beta \beta$ nuclear matrix element is found to depend strongly on the scenario. For the energetically favored [4,2] occupancy, we obtain $|{\mathcal{M}}^{2\nu }|=0.069\left(7\right) {\mathrm{MeV}}^{-1}$. For the [6,0] occupancy, the matrix element further depends on the triaxiality parameter $\gamma$ of the two coexisting, closely lying minima in ${}^{76}\mathrm{Se}$, yielding $|{\mathcal{M}}^{2\nu }|=0.040\left(4\right) {\mathrm{MeV}}^{-1}$ at $\gamma =17.7^{\circ }$ and $|{\mathcal{M}}^{2\nu }|=0.22\left(2\right) {\mathrm{MeV}}^{-1}$ at $\gamma =41.9^{\circ }$. The latter result is consistent with the empirical value reported by Barabash, $|{\mathcal{M}}^{2\nu }|=0.204\left(14\right) {\mathrm{MeV}}^{-1}$ [Barabash, Universe 6, 159 (2020)], while the two former results are comparable to existing calculations based on energy-density-functional frameworks. Our calculations reveal challenges in the precise determination of $|{\mathcal{M}}^{2\nu }|$ for the ${}^{76}\mathrm{Ge}$ decay. The structural complexity, triaxiality, and shape coexistence identified in the analyzed nuclei imply a strong sensitivity to fine details of the interaction and configuration mixing. This, in turn, explains the difficulties in theoretical modeling of the $|{\mathcal{M}}^{2\nu }|$ matrix elements for the ${}^{76}\mathrm{Ge}$ decay, which vary by almost an order of magnitude in the available literature.

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• Double beta decay
• Nuclear forces
• 59 ≤ A ≤ 89
• Nuclear density functional theory