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For the first time, the calculation of the nuclear matrix element of thedouble-$\beta$ decay of $^{100}$Mo, with and without the emission of twoneutrinos, is performed in the framework of the nuclear shell model. This taskis accomplished starting from a realistic nucleon-nucleon potential, then theeffective shell-model Hamiltonian and decay operators are derived within themany-body perturbation theory. The exotic features which characterize thestructure of Mo isotopes -- such as shape coexistence and triaxiality softness-- push the shell-model computational problem beyond its present limits, makingit necessary to truncate the model space. This has been done with the goal topreserve as much as possible the role of the rejected degrees of freedom in aneffective approach that has been introduced and tested in previous studies.This procedure is grounded on the analysis of the effective single-particleenergies of a large-scale shell-model Hamiltonian, that leads to a truncationof the number of the orbitals belonging to the model space. Then, the originalHamiltonian generates a new one by way of a unitary transformation onto thereduced model space, to retain effectively the role of the excludedsingle-particle orbitals. The predictivity of our calculation of the nuclearmatrix element for the neutrinoless double-$\beta$ decay of $^{100}$Mo issupported by the comparison with experiment of the calculated spectra,electromagnetic transition strengths, Gamow-Teller transition strengths and thetwo-neutrino double-beta nuclear matrix elements.

Shell-model calculation of $^{100}$Mo double-$\beta$ decay