The puzzle of direct dark matter searches can find solution in the model of OHe dark atom, which consists of a stable O^-- lepton core and nuclear interacting (alpha particle) shell of primordial helium nuclei. In this model positive results of DAMA group can be explained by annual modulation of radiative capture of OHe atoms to low-energy bound states with sodium nuclei, which doesn’t take place under the conditions of other underground experiments. The existence of such a low-energy bound state is the key problem of the OHe model of composite dark matter. The complexity of this problem, which has not found a correct solution during last 15 years, requires a consistent approach to its solution. Within the framework of the proposed approach to such modeling, in order to reveal the essence of the processes of interaction of OHe with the nuclei of baryonic matter, a classical model is used, to which the effects of quantum physics and final size of nuclei are successively added. The numerical model of the interaction of the “dark” OHe atom with the nuclei is developed by successive addition of realistic features of quantum-mechanical description to the initial classical problem of three point-like bodies ( O^-- particle, the He nucleus and the target nucleus). The developed approach leads to a numerical model describing the OHe-nucleus system with self-consistent accounting for nuclear attraction and electromagnetic interaction of dark atom with nuclei. The model can prove the interpretation of the results of the direct underground experimental dark matter search in the terms of dark atom hypothesis.