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This paper opens a series of publications, devoted to review of new results of the original nuclear-geophysical researches at the OSENU mathematics department and its laboratories (computational and quantum mathematics and mechanics, environmental radioactivity, nuclear geophysics). Primarily the first material will focus on fundamental issues that constitute the theoretical basis of the further applied nuclear-geophysical research. It is shortly presented a modern approach to computing the key parameters of the beta decay processes. There are discussed the cooperative electron β-nuclear processes in atoms and molecules, including the excitation, ionization, electronic rearrangement, induced by the nuclear reactions and β-decay. A few factors are taken into account: changing the integration limits in the Fermi function integral, energy corrections for different chemical substances, and the possibility of the bound β-decay or other exotic decay channels.