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The electron shell of the daughter atoms often appears excited in thedouble-$\beta$ decays, which causes a change in the energy taken away by$\beta$-electrons. The average value and variance of the excitation energy ofthe electron shell of the daughter atom are calculated for the double-$\beta$decay of germanium $_{32}^{76}\mathrm{Ge} \rightarrow_{34}^{76}\mathrm{Se}^*+2\beta^-(+~2\bar{\nu_e})$ in both the Thomas--Fermimodel and the relativistic Dirac--Hartree--Fock theory. Using the resultsobtained, a two-parameter model of the energy spectrum of $\beta$-electrons inthe neutrinoless mode is constructed, taking into account reaction energyredistribution in the decay channels. The shift in total energy of$\beta$-electrons is found to be under 50 eV at a confidence level of 90%. Theaverage excitation energy, on the other hand, is an order of magnitude higherand equal to $\sim 400$ eV, while the square root of the variance is equal to$\sim 2900$ eV, which is presumably explained by the contribution of the coreelectrons to the energy characteristics of the process. The probability isnearly saturated with excitations with a small amount of released energy, whichis common for the outermost electrons. The distortion of the peak shape of theneutrinoless double-$\beta$ decay should be taken into consideration whenanalyzing data from detectors with a resolution of $\sim 100$ eV or higher.

Impact of electron shell excitations on the energy spectrum of $\beta$-electrons in neutrinoless double-$\beta$ decay