  The excitation of electron shell of a daughter atom in a neutrinoless double
$\beta$-decay causes change in shape of the total energy peak of $\beta$
electrons at the end of the energy spectrum. The main parameters of the
modified distribution are the average energy and variance of the excitation
energy of the electron shell. We derive an expression for the variance taking
into account exchange effects and make numerical estimates of the average
excitation energy and variance based on the non-relativistic
Roothaan-Hartree-Fock method and the relativistic Dirac-Hartree-Fock method
implemented in the General Relativistic Atomic Structure Package (Grasp2018).
Estimates are made for eleven isotopes, two-neutrino double-$\beta$ decay of
which is observed experimentally. The results are determined by the first two
negative moments of the electron radii in the parent atom. The values obtained
for change in the peak shape can be used to parameterize the energy
distribution of $\beta$ electrons, taking into account the excitation of the
electron shell of the daughter atom.
