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The determination of the electron neutrino mass by electron capture in$^{163}$Ho relies on a precise understanding of the deexcitation of a core holeafter an electron capture event. We here present an \textit{ab intio}calculation of the electron capture spectrum in $^{163}$Ho, including allintra-atomic decay channels into bound-states. We use theoretical methodsdeveloped for the calculation of core level spectroscopy on correlated electroncompounds. Our comparison critically tests the reality of these theories. Wefind that relativistic interactions beyond the Dirac equation, i.e.quantum-electro dynamics, only lead to minor shifts of the spectral peaks. Theelectronic relaxation after an electron capture event due to the changednuclear potential leads to a mixing of different edges, but due to conservationof angular momentum of each scattered electron, no additional structuresemerge. Many-body Coulomb interactions lead to the formation of multiplets andto additional peaks with multiple core-holes due to Auger decay. Multipletscrucially change the appearance of the resonances on a Rydberg energy scale.The additional structures due to Auger decay are, although clearly visible,relatively weak compared to the one core hole states and accidentally far awayfrom the end-point region of the spectrum. As the end-point of the spectrum iseffected most by the neutrino mass these additional states do not influence thestatistics for determining the neutrino mass directly. The multiplet broadeningand Auger shake-up of the main core-level edges do change the apparentline-width and accompanying lifetime of these edges, thereby invalidatingexperimentally obtained lifetimes at the resonance for regions far away fromthe resonance.

$\textit{Ab initio}$ calculation of the calorimetric electron capture spectrum of $^{163}$Holmium: Intra-atomic decay into bound-states